line::qsys Namespace Reference

Namespaces
namespace	mgck_sync_detail
namespace	mmapmap1detail

Classes
struct	BmapM1Result
	Everything qsys_bmapm1 returns, mirroring the MATLAB result struct. More...
struct	BmapPhNnRetrialOptions
	Options of qsys_bmapphnn_retrial (the name-value pairs of the .m file). More...
struct	BmapPhNnRetrialResult
	Result of qsys_bmapphnn_retrial. More...
struct	DmcResult
struct	Gig1ExtremalResult
	The bounds, all as TIMES IN QUEUE: add 1/mu for a response time. More...
struct	Gig1RqResult
struct	GigkRqtResult
struct	LdpsWorkloadResult
	Return value of qsys_ldps_workload, mirroring the three MATLAB outputs. More...
struct	LindleyResult
	Mirrors the struct MATLAB returns from qsys_mm1_lindley. More...
struct	MapD1Result
	Return value of qsys_mapd1, mirroring the MATLAB result struct. More...
struct	MapDcResult
	Return value of qsys_mapdc, mirroring the MATLAB result struct. More...
struct	MapG1kPerflowResult
	Return value of qsys_mapg1k_perflow, mirroring the MATLAB result struct. More...
struct	MapG1kResult
	Return value of qsys_mapg1k, mirroring the MATLAB result struct. More...
struct	MapG1Result
	Result of qsys_mapg1. More...
struct	MapMap1Result
	Return value of the MAP/MAP/1 family (qsys_mapmap1, qsys_mapph1, qsys_phph1), carrying the same quantities as the MATLAB result struct. More...
struct	MapMcResult
	Return value of qsys_mapmc and qsys_mapm1, mirroring the MATLAB struct. More...
struct	MapPhcResult
	Return value of qsys_mapphc, mirroring the MATLAB struct. More...
struct	MaximaResult
	Two-moment description of a maximum. More...
struct	MDcCrommelinResult
struct	Mg1DisciplineResult
struct	Mg1kLossMgsResult
struct	Mg1kLossResult
struct	Mg1PrioResult
struct	Mg1PsOptions
	Options of qsys_mg1_ps, mirroring the reference's name-value pairs. More...
struct	Mg1PsResult
	Everything qsys_mg1_ps returns. More...
struct	MgckSyncResult
	Return value of qsys_mgck_sync, mirroring the MATLAB result struct. More...
struct	MginfResult
	Return value of qsys_mginf, mirroring MATLAB's [L,Lq,W,Wq,p0,pk]. More...
struct	MgisrgiOptions
	Options of qsys_mgisrgi_whitt, all with the MATLAB defaults. More...
struct	Mm1DpsResult
struct	Mm1kLossResult
struct	Mm1PsResult
	Mirrors MATLAB's [W, W2, alpha] return list. More...
struct	Mm1TandemLindleyResult
	Mirrors the struct MATLAB returns from qsys_mm1_tandem_lindley. More...
struct	MmapG1kResult
	Return value of qsys_mmapg1k, mirroring the MATLAB result struct. More...
struct	MmapGk1Result
	Return value of qsys_mmapgk1, mirroring the MATLAB struct. More...
struct	MmapGk1Service
	One marked type's service law, in any of the three forms the reference accepts: a Distribution object, a (D0, D1) MAP pair, or a transform struct. More...
struct	MmapGk1Transform
	A service law given by its transform, the reference's struct form struct('lst', @(s) ..., 'moments', [m1 m2 ...]): the LST at a complex argument and its raw moments, of which at least numWMoms + 1 are needed. More...
struct	MmapMap1Result
	Return value of qsys_mmapmap1, mirroring the MATLAB struct. More...
struct	MmccRetrialFpResult
	Return value of qsys_mmcc_retrial_fp, mirroring the three MATLAB outputs. More...
struct	MmckResult
struct	MxM1Result
struct	NpfqnTvFluidResult
	Result of the network solve. More...
struct	Patience
	The patience (time-to-abandon) law, in the three forms the algorithm accepts. More...
struct	PhM1Result
struct	PhMcResult
struct	QsysAbandonResult
	Steady-state measures of a multiserver queue with customer abandonment. More...
struct	QsysFluidAbandonResult
	Steady state of the G/GI/s+GI fluid model. More...
struct	QsysGgnmResult
	Steady-state measures of the G/GI/n/m diffusion approximation. More...
struct	QsysMolResult
	MOL and PSA measures of a time-varying multiserver system. More...
struct	QsysMtginfResult
	Time-varying measures of the Mt/G/infinity queue. More...
struct	QsysQedResult
	QED measures of the M/M/s queue. More...
struct	QsysQedStaffingResult
	Outcome of the square-root staffing rule. More...
struct	QsysResult
	Return value of the qsys family, mirroring MATLAB's [W,rhohat] and the JAR's Ret.qsys. More...
struct	QsysTgaResult
	Steady-state measures of the G/GI/n+GI truncated Gaussian approximation. More...
struct	QsysTvFluidResult
	Trajectory of the Gt/Mt/st+GI fluid queue; every vector is on the time grid. More...
struct	ServiceLaw
	Service-time descriptor, the C++ form of the MATLAB svc struct. More...
struct	SvclawCoeffsResult
	Outputs [cn, Smean] of qsys_svclaw_coeffs.m, plus the truncation residual. More...
struct	TandemLindleyResult
	Mirrors the struct MATLAB returns from qsys_tandem_lindley. More...
struct	TandemUbResult
	Mirrors the struct MATLAB returns from qsys_tandem_ub_ciucu. More...
struct	TvFluidOptions
	Options of qsys_gtmtst_fluid, all with the MATLAB defaults. More...
struct	WorkloadServiceLaw
	The three things Cohen's formula needs from the required-service-time law: its mean, its squared coefficient of variation, and its CDF. More...

Typedefs
using	Cplx = std::complex<double>

Enumerations
enum class	RetrialPolicy { LINEAR , CONSTANT }
	Retrial admission policy (RetrialPolicy in MATLAB). More...
enum class	MapG1ServiceFit { Exponential , Erlang , Hyperexponential , Acyclic }
	Which branch of fitServiceToPH was taken, for the caller and for tests. More...
enum class	ServiceKind { Gamma , Deterministic , PhaseType , Density }
	Which family the service law belongs to. More...
enum class	PatienceForm { Exponential , Hazard , Ccdf }
	Which of the three accepted descriptions of the patience law is carried. More...
enum class	QedCriterion { Delay , MeanWait , ServiceLevel }
	Which target the staffing rule is asked to meet. More...

Functions
template<class T>
BmapM1Result< T >	qsys_bmapm1 (const std::vector< Matrix< T > > &D, const T &mu, const T &qParam, std::size_t maxLevelParam, unsigned maxIter, const T &tol, double tailTol)
	BMAP/M/1 by the matrix-analytic (M/G/1-type) method.
template<class T>
BmapM1Result< T >	qsys_bmapm1 (const std::vector< Matrix< T > > &D, const T &mu)
	The reference's defaults: adaptive truncation, 10000 iterations, tol 1e-12.
template<class T>
BmapPhNnRetrialResult< T >	qsys_bmapphnn_retrial (const std::vector< Matrix< T > > &D, const std::vector< T > &beta, const Matrix< T > &S, int N, const T &alpha, const T &gamma, const T &p, const std::vector< long > &R, const BmapPhNnRetrialOptions &opt)
	The BMAP/PH/N/N bufferless retrial queue.
template<class T>
BmapPhNnRetrialResult< T >	qsys_bmapphnn_retrial (const std::vector< Matrix< T > > &D, const std::vector< T > &beta, const Matrix< T > &S, int N, const T &alpha, const T &gamma, const T &p, long R)
	qsys_bmapphnn_retrial with the reference's automatic truncation level.
template<class T>
DmcResult< T >	qsys_dmc (const T &lambda, const T &mu, unsigned c, unsigned truncation, unsigned quadSteps)
	D/M/c: deterministic interarrival times, exponential service.
template<class T>
DmcResult< T >	qsys_dmc (const T &lambda, const T &mu, unsigned c)
	qsys_dmc with the MATLAB defaults, automatic truncation and 200 steps.
template<class T>
QsysResult< T >	qsys_gg1 (const T &lambda, const T &mu, const T &ca2, const T &cs2)
	G/G/1 dispatcher: exact where a two-moment description determines the answer, Allen-Cunneen otherwise.
template<class T>
QsysTgaResult< T >	qsys_ggingi_tga (const T &lambda, const T &mu, unsigned n, const T &ca, const T &cs, const std::function< T(const T &)> &patienceCcdf, const std::function< T(const T &)> &patiencePdf=std::function< T(const T &)>(), const std::function< T(const T &)> &serviceCcdf=std::function< T(const T &)>())
	Truncated Gaussian approximation (TGA-G) for the G/GI/n+GI queue.
template<class T>
QsysFluidAbandonResult< T >	qsys_ggisgi_fluid (const T &lambda, const T &mu, unsigned s, const std::function< T(const T &)> &patienceCcdf, const std::function< T(const T &)> &servingCcdf=std::function< T(const T &)>(), const std::vector< T > &agePoints=std::vector< T >(), double tol=1e-12, double maxTime=std::numeric_limits< double >::quiet_NaN())
	Steady state of the G/GI/s+GI fluid model.
template<class T>
QsysGgnmResult< T >	qsys_ggnm_diffusion (const T &lambda, const T &mu, unsigned n, double m, const T &ca, const T &cs, const std::function< T(const T &)> &serviceCcdf=std::function< T(const T &)>(), double tol=1e-12, std::size_t panels=4000)
	Diffusion approximation for the G/GI/n/m queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_allencunneen (const T &lambda, const T &mu, const T &ca, const T &cs)
	Allen-Cunneen approximation of the mean response time of a G/I/G/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_gelenbe (const T &lambda, const T &mu, const T &ca, const T &cs)
	Gelenbe diffusion approximation with instantaneous-return boundary.
template<class T>
QsysResult< T >	qsys_gig1_approx_heyman (const T &lambda, const T &mu, const T &ca, const T &cs)
	Heyman approximation of the mean response time of a G/I/G/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_kimura (const T &lambda, const T &mu, const T &ca, const T &cs)
	Kimura diffusion-interpolation approximation for the G/I/G/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_klb (const T &lambda, const T &mu, const T &ca, const T &cs)
	Kraemer and Langenbach-Belz approximation for the G/I/G/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_kobayashi (const T &lambda, const T &mu, const T &ca, const T &cs)
	Kobayashi diffusion approximation for the G/I/G/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_marchal (const T &lambda, const T &mu, const T &ca, const T &cs)
	Marchal approximation of the mean response time of a G/I/G/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_myskja (const T &lambda, const T &mu, const T &ca, const T &cs, const T &q0, const T &qa)
	Myskja's third-moment approximation of the mean response time of a G/I/G/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_myskja2 (const T &lambda, const T &mu, const T &ca, const T &cs, const T &q0, const T &qa)
	Myskja's enhanced third-moment approximation of the mean response time of a G/I/G/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_approx_whitt (const T &lambda, const T &mu, const T &ca, const T &cs)
	Whitt's approximation of the G/G/1 mean response time.
template<class T>
Gig1ExtremalResult< T >	qsys_gig1_bnds_extremal (const T &lambda, const T &mu, const T &ca, const T &cs, std::size_t K=4000, std::size_t N=2000, bool skipTight=false)
	Extremal two-moment bounds for the GI/GI/1 queue.
template<class T>
QsysResult< T >	qsys_gig1_lbnd (const T &lambda, const T &mu, const T &ca, const T &cs)
	Fundamental lower bound on the mean response time of a G/G/1 queue.
template<class T, class IaFun>
Gig1RqResult< T >	qsys_gig1_rq (const T &rho, const T &mu, const T &cs2, IaFun &&IaFun_)
	Robust Queueing (RQ) approximation of a G/GI/1 queue characterized by its arrival index of dispersion and the first two service moments.
template<class T>
GigkRqtResult< T >	qsys_gig1_rqt (const T &lambda, const T &mu, const T &Gamma_a, const T &Gamma_s, const T &alpha_a, const T &alpha_s)
	Robust Queueing Theory (RQT) worst-case system time of a G/G/1 FCFS queue, the single-server case of qsys_gigk_rqt (Theorem 2 and eq.
template<class T>
GigkRqtResult< T >	qsys_gig1_rqt (const T &lambda, const T &mu, const T &Gamma_a, const T &Gamma_s)
	Finite-variance case, alpha_a = alpha_s = 2.
template<class T>
QsysResult< T >	qsys_gig1_ubnd_kingman (const T &lambda, const T &mu, const T &ca, const T &cs)
	Kingman upper bound on the mean waiting time of a G/G/1 queue.
template<class T>
QsysResult< T >	qsys_gigk_approx (const T &lambda, const T &mu, const T &ca, const T &cs, unsigned k)
	Default G/I/G/k approximation of the mean response time.
template<class T>
QsysResult< T >	qsys_gigk_approx_cosmetatos (const T &lambda, const T &mu, const T &ca, const T &cs, unsigned k)
	Cosmetatos / Page interpolation approximation for the GI/G/k queue.
template<class T>
QsysResult< T >	qsys_gigk_approx_kingman (const T &lambda, const T &mu, const T &ca, const T &cs, unsigned k)
	Kingman (Lee-Longton) scaling of the exact M/M/k waiting time.
template<class T>
QsysResult< T >	qsys_gigk_approx_whitt (const T &lambda, const T &mu, const T &ca, const T &cs, unsigned k)
	Whitt (1993) approximation for the GI/G/k queue, eqs.
template<class T>
GigkRqtResult< T >	qsys_gigk_rqt (const T &lambda, const T &mu, const T &Gamma_a, const T &Gamma_s, std::size_t k, const T &alpha_a, const T &alpha_s)
	Robust Queueing Theory (RQT) worst-case system time of a G/G/k FCFS queue.
template<class T>
T	qsys_gigk_rqt_gamma (const T &rho, const T &mu, const T &Gamma_a, const T &sigma_s, std::size_t k, const T &alpha_a, const std::string &regime="independent")
	Service variability parameter of the Robust Queueing Theory (RQT) framework.
template<class T>
T	qsys_gm1 (const T &sigma, const T &mu)
	Exact mean response time of the G/M/1 queue.
template<class Tv>
QsysTvFluidResult< Tv >	qsys_gtmtst_fluid (const std::function< Tv(const Tv &)> &lambdaFun, const std::function< Tv(const Tv &)> &sFun, const std::function< Tv(const Tv &)> &muFun, const std::function< Tv(const Tv &)> &patienceCcdf, const Tv &T, const TvFluidOptions< Tv > &opts=TvFluidOptions< Tv >())
	The Gt/Mt/st+GI many-server fluid queue, and the network of them.
template<class Tv>
NpfqnTvFluidResult< Tv >	npfqn_gtmtst_fluid (const std::vector< std::function< Tv(const Tv &)> > &lambdaFuns, const std::vector< std::function< Tv(const Tv &)> > &sFuns, const std::vector< std::function< Tv(const Tv &)> > &muFuns, const std::vector< std::function< Tv(const Tv &)> > &patienceCcdfs, const std::function< std::vector< std::vector< Tv > >(const Tv &)> &Pfun, const Tv &T, const Tv &dt=num_traits< Tv >::from_int(0), const std::vector< Tv > &B0=std::vector< Tv >(), const std::vector< Tv > &w0=std::vector< Tv >(), double tol=1e-6, std::size_t maxIter=100)
	A time-varying open network of many-server fluid queues with abandonment.
template<class Tv>
NpfqnTvFluidResult< Tv >	npfqn_gtmtst_fluid (const std::vector< std::function< Tv(const Tv &)> > &lambdaFuns, const std::vector< std::function< Tv(const Tv &)> > &sFuns, const std::vector< std::function< Tv(const Tv &)> > &muFuns, const std::vector< std::function< Tv(const Tv &)> > &patienceCcdfs, const std::vector< std::vector< Tv > > &P, const Tv &T, const Tv &dt=num_traits< Tv >::from_int(0), const std::vector< Tv > &B0=std::vector< Tv >(), const std::vector< Tv > &w0=std::vector< Tv >(), double tol=1e-6, std::size_t maxIter=100)
	Constant routing matrix: MATLAB's matrix form of P, the same at every step.
template<class T>
LindleyResult< T >	qsys_hh1_lindley (const std::vector< T > &lambda, const std::vector< T > &pa, const std::vector< T > &mu, const std::vector< T > &ps, const std::vector< T > &Wn, unsigned mmax=2)
	Conditional waiting-time moments of the Hl/Hn/1 Lindley recursion.
template<class T>
LindleyResult< T >	qsys_hh1_lindley (const std::vector< T > &lambda, const std::vector< T > &pa, const std::vector< T > &mu, const std::vector< T > &ps, const T &Wn, unsigned mmax=2)
	Scalar-Wn overload.
template<class T>
LdpsWorkloadResult< T >	qsys_ldps_workload (const T &lambda, const WorkloadServiceLaw< T > &B, const std::vector< T > &alpha, std::size_t N, const std::vector< T > &t, std::size_t ngrid)
	Workload distribution of the load-dependent PS station with blocking.
template<class T>
LdpsWorkloadResult< T >	qsys_ldps_workload (const T &lambda, const WorkloadServiceLaw< T > &B, const std::vector< T > &alpha, std::size_t N)
	qsys_ldps_workload on the automatic grid with the reference default ngrid = 2001.
template<class T>
std::vector< T >	qsys_lindley_moment (const T &lambda, const T &mu, const std::vector< T > &Wn, unsigned m)
	One conditional Lindley moment for exponential primitives.
template<class T>
T	qsys_lindley_moment (const T &lambda, const T &mu, const T &Wn, unsigned m)
	Scalar overload of the same expression.
template<class T>
MapD1Result< T >	qsys_mapd1 (const mam::Map< T > &arrival, const T &s, std::size_t dist_size, unsigned max_arrivals, std::size_t max_levels, const T &tol, std::size_t num_steps)
	MAP/D/1 by the exact embedded M/G/1-type chain.
template<class T>
MapD1Result< T >	qsys_mapd1 (const mam::Map< T > &arrival, const T &s, std::size_t dist_size, unsigned max_arrivals, std::size_t max_levels, const T &tol)
	qsys_mapd1 with the reference's numSteps = 1.
template<class T>
MapD1Result< T >	qsys_mapd1 (const mam::Map< T > &arrival, const T &s)
	qsys_mapd1 with 100 materialized levels, an arrival-count cap of 4096, a level cap of 20000 and tolerance 1e-14.
template<class T>
MapDcResult< T >	qsys_mapdc (const mam::Map< T > &arrival, const T &s, unsigned c, std::size_t dist_size, unsigned max_arrivals, std::size_t max_levels, const T &tol, std::size_t num_steps)
	MAP/D/c by Crommelin's exact embedded lattice chain.
template<class T>
MapDcResult< T >	qsys_mapdc (const mam::Map< T > &arrival, const T &s, unsigned c, std::size_t dist_size, unsigned max_arrivals, std::size_t max_levels, const T &tol)
	qsys_mapdc with the reference's numSteps = 1.
template<class T>
MapDcResult< T >	qsys_mapdc (const mam::Map< T > &arrival, const T &s, unsigned c)
	qsys_mapdc with 100 materialized levels, an arrival-count cap of 4096, a super-level cap of 20000 and tolerance 1e-14, matching the qsys_mapd1 defaults.
template<class T>
mam::Map< T >	qsys_mapg1_service_fit (const std::vector< T > &moments, MapG1ServiceFit &kind)
	Fit a general service time to a PH, following qsys_mapg1.m's fitServiceToPH.
template<class T>
MapG1Result< T >	qsys_mapg1 (const mam::Map< T > &arrival, const std::vector< T > &moments, std::size_t dist_size, std::size_t numQLMoms, std::size_t numSTMoms)
	The MAP/G/1 FCFS queue.
template<class T>
MapG1Result< T >	qsys_mapg1 (const mam::Map< T > &arrival, const std::vector< T > &moments, std::size_t dist_size)
	qsys_mapg1 with the reference's defaults numQLMoms = 3, numSTMoms = 3.
template<class T>
MapG1Result< T >	qsys_mapg1 (const mam::Map< T > &arrival, const std::vector< T > &moments)
	qsys_mapg1 with the reference's default of 100 materialized levels.
template<class T>
SvclawCoeffsResult< T >	qsys_svclaw_coeffs (const ServiceLaw< T > &svc, const T &theta, const T &tol, std::size_t nmaxCap)
	qsys_svclaw_coeffs: uniformization coefficients of a service-time descriptor; see qsys_svclaw_coeffs.h.
template<class T>
MapG1kResult< T >	qsys_mapg1k (const mam::Map< T > &arrival, const ServiceLaw< T > &svc, std::size_t K, const T &tol, std::size_t nmaxCap)
	MAP/G/1/K with tail drop.
template<class T>
MapG1kResult< T >	qsys_mapg1k (const mam::Map< T > &arrival, const ServiceLaw< T > &svc, std::size_t K)
	qsys_mapg1k with the reference defaults tol = 1e-12, nmax = 200000.
template<class T>
MapG1kPerflowResult< T >	qsys_mapg1k_perflow (const std::vector< mam::Map< T > > &flows, const ServiceLaw< T > &svc, std::size_t K, const T &tol, std::size_t nmaxCap)
	Per-flow analysis of a MAP-fed tail-drop buffer.
template<class T>
MapG1kPerflowResult< T >	qsys_mapg1k_perflow (const std::vector< mam::Map< T > > &flows, const ServiceLaw< T > &svc, std::size_t K)
	qsys_mapg1k_perflow with the qsys_mapg1k defaults tol = 1e-12, nmax = 200000.
template<class T>
MapMcResult< T >	qsys_mapm1 (const mam::Map< T > &arrival, const T &mu, std::size_t dist_size)
	MAP/M/1 by the matrix-geometric solution, i.e.
template<class T>
MapMcResult< T >	qsys_mapm1 (const mam::Map< T > &arrival, const T &mu)
	qsys_mapm1 with 100 materialized levels.
template<class T>
MapMap1Result< T >	qsys_mapmap1 (const mam::Map< T > &arrival, const mam::Map< T > &service, std::size_t dist_size, std::size_t numQLMoms, std::size_t numSTMoms)
	MAP/MAP/1 as the reference solves it: the service MAP collapsed to its PH marginal (mapToPh), then the MAP/PH/1 queue by the exact QBD solution.
template<class T>
MapMap1Result< T >	qsys_mapmap1 (const mam::Map< T > &arrival, const mam::Map< T > &service, std::size_t dist_size)
	qsys_mapmap1 with the reference's defaults numQLMoms = 3, numSTMoms = 3.
template<class T>
MapMap1Result< T >	qsys_mapmap1 (const mam::Map< T > &arrival, const mam::Map< T > &service)
	qsys_mapmap1 with 100 materialized levels, the reference's numQLProbs.
template<class T>
MapPhcResult< T >	qsys_mapmapc (const mam::Map< T > &arrival, const mam::Map< T > &service, unsigned c, std::size_t dist_size=500, std::size_t num_w_moms=3)
	MAP/MAP/c FCFS with the service phase carried in start order, exactly.
template<class T>
MapMcResult< T >	qsys_mapmc (const mam::Map< T > &arrival, const T &mu, unsigned c, std::size_t dist_size)
	MAP/M/c by the matrix-geometric solution.
template<class T>
MapMcResult< T >	qsys_mapmc (const mam::Map< T > &arrival, const T &mu, unsigned c)
	qsys_mapmc with 500 materialized levels, the reference's maxNumComp default.
template<class T>
MapMap1Result< T >	qsys_mapph1 (const mam::Map< T > &arrival, const std::vector< T > &sigma, const Matrix< T > &S, std::size_t dist_size)
	MAP/PH/1 by the exact QBD solution of the equivalent MAP/MAP/1 queue.
template<class T>
MapMap1Result< T >	qsys_mapph1 (const mam::Map< T > &arrival, const std::vector< T > &sigma, const Matrix< T > &S)
	qsys_mapph1 with 100 materialized levels, the reference's numQLProbs.
template<class T>
MapPhcResult< T >	qsys_mapphc (const mam::Map< T > &arrival, const std::vector< T > &alpha, const Matrix< T > &S, unsigned c, std::size_t dist_size, std::size_t num_w_moms, const std::vector< T > &w_points)
	MAP/PH/c FCFS, exactly.
template<class T>
MapPhcResult< T >	qsys_mapphc (const mam::Map< T > &arrival, const std::vector< T > &alpha, const Matrix< T > &S, unsigned c)
template<class T>
MaximaResult< T >	qsys_maxima_twomoment (std::size_t n, const T &mean, const T &cs2, const T &q=num_traits< T >::from_int(0), bool exactFitted=true)
	Two-moment approximation for the maximum of n iid non-negative variables.
template<class T>
MDcCrommelinResult< T >	qsys_mdc_crommelin (const T &lambda_arr, const T &s, unsigned c, long truncation=-1)
	M/D/c by Crommelin's embedded chain.
template<class T>
QsysResult< T >	qsys_mg1 (const T &lambda, const T &mu, const T &cs)
	Exact mean response time of the M/G/1 queue (Pollaczek-Khinchine).
template<class T>
Mg1DisciplineResult< T >	qsys_mg1_fb (const std::vector< T > &lambda, const std::vector< T > &mu, const std::vector< T > &cs)
	M/G/1 under FB (feedback), also called LAS (least attained service).
template<class T>
Mg1DisciplineResult< T >	qsys_mg1_lrpt (const std::vector< T > &lambda, const std::vector< T > &mu, const std::vector< T > &cs)
	M/G/1 under LRPT (longest remaining processing time).
template<class T>
Mg1PrioResult< T >	qsys_mg1_prio (const std::vector< T > &lambda, const std::vector< T > &mu, const std::vector< T > &cs)
	M/G/1 with non-preemptive head-of-line priorities: per-class mean response times from the Cobham/Kleinrock formula.
Mg1PsResult	qsys_mg1_ps (double lambda, const std::vector< double > &alpha, const Matrix< double > &Tmat, const Mg1PsOptions &opt)
	Sojourn-time distribution of the M/G/1 processor-sharing queue.
Mg1PsResult	qsys_mg1_ps (double lambda, const std::function< Cplx(const Cplx &)> &bhat, double m1, const Mg1PsOptions &opt)
	Sojourn-time distribution of the M/G/1-PS queue for a service law given by its Laplace-Stieltjes transform, the reference's second form qsys_mg1_ps(lambda, bhat, m1, ...).
template<class T>
Mg1DisciplineResult< T >	qsys_mg1_psjf (const std::vector< T > &lambda, const std::vector< T > &mu, const std::vector< T > &cs)
	M/G/1 under PSJF (preemptive shortest job first).
template<class T>
Mg1DisciplineResult< T >	qsys_mg1_setf (const std::vector< T > &lambda, const std::vector< T > &mu, const std::vector< T > &cs)
	M/G/1 under SETF (shortest elapsed time first), the non-preemptive counterpart of FB/LAS.
template<class T>
Mg1DisciplineResult< T >	qsys_mg1_srpt (const std::vector< T > &lambda, const std::vector< T > &mu, const std::vector< T > &cs)
	M/G/1 under SRPT (shortest remaining processing time), by the Schrage-Miller formula.
template<class T, class Density>
Mg1kLossResult< T >	qsys_mg1k_loss (const T &lambda, Density &&density, unsigned K)
	Exact M/G/1/K loss probability, via the chain embedded at service-start epochs.
template<class T>
Mg1kLossMgsResult< T >	qsys_mg1k_loss_mgs (const T &lambda, const T &mu, const T &mu_scv, unsigned K)
	MacGregor Smith's closed-form approximation of the M/G/1/K loss probability.
template<class T>
MgckSyncResult< T >	qsys_mgck_sync (const T &lambda, std::size_t c, std::size_t K, const ServiceLaw< T > &svc, const std::string &variant, std::size_t nphases, const T &tol, std::size_t nmaxCap)
	Synchronous approximation of the M/G/c/K queue with dual bounds.
template<class T>
MgckSyncResult< T >	qsys_mgck_sync (const T &lambda, std::size_t c, std::size_t K, const ServiceLaw< T > &svc, const std::string &variant, std::size_t nphases=0)
	qsys_mgck_sync with the reference defaults tol = 1e-12, nmax = 200000.
template<class T>
MginfResult< T >	qsys_mginf (const T &lambda, const T &mu)
	Exact solution of the M/G/infinity queue.
template<class T>
MginfResult< T >	qsys_mginf (const T &lambda, const T &mu, unsigned k)
	Exact solution of the M/G/infinity queue.
template<class T>
QsysAbandonResult< T >	qsys_mgisrgi_whitt (const T &lambda, const T &mu, unsigned s, double r, const Patience< T > &patience, const MgisrgiOptions &opts=MgisrgiOptions())
	Engineering solution of the call-center model M/GI/s/r+GI.
template<class T>
QsysAbandonResult< T >	qsys_erlanga (const T &lambda, const T &mu, const T &theta, unsigned s, double r=std::numeric_limits< double >::infinity(), const MgisrgiOptions &opts=MgisrgiOptions())
	Exact analysis of the Erlang A model M/M/s/r+M.
template<class T>
QsysResult< T >	qsys_mm1 (const T &lambda, const T &mu)
	Exact mean response time of the M/M/1 queue.
template<class T>
Mm1DpsResult< T >	qsys_mm1_dps (const std::vector< T > &lambda, const std::vector< T > &mu, const std::vector< T > &w, const T &tol, unsigned maxCutoff)
	Multiclass M/M/1 under DPS (discriminatory processor sharing), solved numerically on the truncated population chain.
template<class T>
Mm1DpsResult< T >	qsys_mm1_dps (const std::vector< T > &lambda, const std::vector< T > &mu, const std::vector< T > &w)
	MATLAB defaults: tol = 1e-10, maxCutoff = 2048.
template<class T>
LindleyResult< T >	qsys_mm1_lindley (const T &lambda, const T &mu, const std::vector< T > &Wn, unsigned mmax=2)
	Conditional waiting-time moments of the M/M/1 Lindley recursion.
template<class T>
LindleyResult< T >	qsys_mm1_lindley (const T &lambda, const T &mu, const T &Wn, unsigned mmax=2)
	Scalar overload.
template<class T>
Mm1PsResult< T >	qsys_mm1_ps (const std::vector< T > &lambda, const std::vector< T > &mu)
	Exact sojourn-time moments of the multiclass M/M/1-PS queue.
template<class T>
Mm1TandemLindleyResult< T >	qsys_mm1_tandem_lindley (const T &lambda, const T &mu1, const T &mu2, const std::vector< T > &Wk, const std::vector< T > &Wk1)
	Conditional waiting time at the SECOND station of an M/M/1 -> /M/1 tandem.
template<class T>
Mm1TandemLindleyResult< T >	qsys_mm1_tandem_lindley (const T &lambda, const T &mu1, const T &mu2, const T &Wk, const T &Wk1)
	Scalar overload.
template<class T>
Mm1kLossResult< T >	qsys_mm1k_loss (const T &lambda, const T &mu, unsigned K)
	Blocking probability of the M/M/1/K queue.
template<class T>
MmapG1kResult< T >	qsys_mmapg1k (const Matrix< T > &D0, const std::vector< Matrix< T > > &D1c, const ServiceLaw< T > &svc, std::size_t K, const T &tol, std::size_t nmaxCap)
	MMAP[K]/G/1/K with tail drop.
template<class T>
MmapG1kResult< T >	qsys_mmapg1k (const Matrix< T > &D0, const std::vector< Matrix< T > > &D1c, const ServiceLaw< T > &svc, std::size_t K)
	qsys_mmapg1k with the qsys_mapg1k defaults tol = 1e-12, nmax = 200000.
template<class T>
MmapGk1Result< T >	qsys_mmapgk1 (const std::vector< Matrix< T > > &MMAP, const std::vector< MmapGk1Service< T > > &svc, const std::vector< T > &w_points, std::size_t num_w_moms, double tol, std::size_t iter_max, std::size_t quad_nodes)
	MMAP[K]/G[K]/1 FCFS, per type.
template<class T>
MmapGk1Result< T >	qsys_mmapgk1 (const std::vector< Matrix< T > > &MMAP, const std::vector< MmapGk1Service< T > > &svc, const std::vector< T > &w_points, std::size_t num_w_moms, double tol, std::size_t iter_max)
	qsys_mmapgk1 on service variants with the reference's quadNodes = 120.
template<class T>
MmapGk1Result< T >	qsys_mmapgk1 (const std::vector< Matrix< T > > &MMAP, const std::vector< lang::Distrib< T > > &svc, const std::vector< T > &w_points, std::size_t num_w_moms, double tol, std::size_t iter_max)
	qsys_mmapgk1 on Distribution objects.
template<class T>
MmapGk1Result< T >	qsys_mmapgk1 (const std::vector< Matrix< T > > &MMAP, const std::vector< lang::Distrib< T > > &svc)
	qsys_mmapgk1 on Distribution objects with the reference's defaults.
template<class T>
MmapGk1Result< T >	qsys_mmapgk1 (const std::vector< Matrix< T > > &MMAP, const std::vector< MmapGk1Service< T > > &svc)
	qsys_mmapgk1 on service variants with the reference's defaults.
template<class T>
MmapMap1Result< T >	qsys_mmapmap1 (const mam::Mmap< T > &arrival, const std::vector< mam::Map< T > > &svc, std::vector< bool > carry=std::vector< bool >())
	MMAP[K]/MAP[K]/1 FCFS with per-class carried service phases, exactly.
template<class T>
T	erlang_b_recursive (const T &a, unsigned c)
	Erlang's loss formula B(a, c) by the numerically stable rational recursion B_0 = 1, B_k = a B_{k-1}/(k + a B_{k-1}).
template<class T>
MmccRetrialFpResult< T >	qsys_mmcc_retrial_fp (const T &lambda, const T &mu, unsigned c, const T &tol, std::size_t maxiter)
	M/M/c/c with retrials by the Cohen fixed point.
template<class T>
MmccRetrialFpResult< T >	qsys_mmcc_retrial_fp (const T &lambda, const T &mu, unsigned c)
	qsys_mmcc_retrial_fp with the reference defaults tol = 1e-10, maxiter = 10000.
template<class T>
MmckResult< T >	qsys_mmck (const T &lambda, const T &mu, unsigned c, unsigned K)
	Exact analysis of the M/M/c/K queue (truncated Erlang form).
template<class T>
QsysResult< T >	qsys_mmk (const T &lambda, const T &mu, unsigned k)
	Exact mean response time of the M/M/k queue (Erlang-C).
template<class T>
T	qsys_mmk_qed_alpha (const T &beta)
	The Halfin-Whitt delay-probability function alpha(beta); 1 at beta <= 0.
template<class T>
T	qsys_mmk_qed_erlangc (unsigned s, const T &lambda, const T &mu)
	Erlang C by the Erlang B recursion B_j = a B_{j-1}/(j + a B_{j-1}), which never forms a^j/j!
template<class T>
QsysQedResult< T >	qsys_mmk_qed (const T &lambda, const T &mu, unsigned s)
	Halfin-Whitt QED approximation for the M/M/s queue, and the square-root staffing rule that inverts it.
template<class T>
QsysQedStaffingResult< T >	qsys_mmk_qed_staffing (const T &lambda, const T &mu, const T &target, QedCriterion crit=QedCriterion::Delay, const T &deadline=num_traits< T >::from_int(0), const T &level=num_traits< T >::from_int(0), bool exact=false)
	Square-root staffing of the M/M/s queue.
template<class T>
QsysMtginfResult< T >	qsys_mtginf (const std::function< T(const T &)> &lambdaFun, const std::function< T(const T &)> &serviceCcdf, const T &ES, const std::vector< T > &tvals, double startTime=-std::numeric_limits< double >::infinity(), double ES2=std::numeric_limits< double >::quiet_NaN(), const std::function< T(const T &)> &servicePdf=std::function< T(const T &)>(), double tol=1e-12, std::size_t panels=4000, double maxAge=1e12)
	Exact time-varying analysis of the Mt/G/infinity queue.
template<class T>
T	qsys_erlang_b (unsigned s, const T &a)
	Erlang B by the recursion B_j = a B_{j-1}/(j + a B_{j-1}), which never forms a^s/s!
template<class T>
T	qsys_erlang_c (unsigned s, const T &a)
	Erlang C from the same recursion; 1 when the load saturates the servers.
template<class T>
QsysMolResult< T >	qsys_mtgs0_mol (const std::function< T(const T &)> &lambdaFun, const std::function< T(const T &)> &serviceCcdf, const T &ES, unsigned s, const std::vector< T > &tvals, double startTime=-std::numeric_limits< double >::infinity(), bool delay=false)
	Modified-offered-load and pointwise-stationary approximations for a time-varying multiserver system.
template<class T>
MxM1Result< T >	qsys_mxm1 (const T &lambda_batch, const T &mu, const T &E_X, const T &E_X2)
	M^X/M/1: the batch-arrival queue with exponential service.
template<class T>
MxM1Result< T >	qsys_mxm1_variance (const T &lambda_batch, const T &mu, const T &E_X, const T &Var_X)
	Variance form: MATLAB's qsys_mxm1(..., Var_X, 'variance').
template<class T>
MxM1Result< T >	qsys_mxm1_pmf (const T &lambda_batch, const T &mu, const std::vector< T > &batch_sizes, const std::vector< T > &pmf)
	Support-and-pmf form: MATLAB's qsys_mxm1(..., batch_sizes, pmf).
template<class T>
PhM1Result< T >	qsys_phm1 (const std::vector< T > &alpha, const Matrix< T > &Tm, const T &mu, const T &tol)
	Exact PH/M/1, the GI/M/1 queue with phase-type interarrival times.
template<class T>
PhM1Result< T >	qsys_phm1 (const std::vector< T > &alpha, const Matrix< T > &Tm, const T &mu)
	qsys_phm1 with the fzero-equivalent default bracket tolerance.
template<class T>
PhMcResult< T >	qsys_phmc (const std::vector< T > &alpha, const Matrix< T > &Tm, const T &mu, unsigned c, unsigned maxIter, const T &tol)
	Exact PH/M/c by Neuts' matrix-geometric method.
template<class T>
PhMcResult< T >	qsys_phmc (const std::vector< T > &alpha, const Matrix< T > &Tm, const T &mu, unsigned c)
	qsys_phmc with the MATLAB defaults, 50000 iterations and tolerance 1e-14.
template<class T>
MapMap1Result< T >	qsys_phph1 (const std::vector< T > &alpha, const Matrix< T > &Tm, const std::vector< T > &beta, const Matrix< T > &S, std::size_t dist_size)
	PH/PH/1 by the exact QBD solution of the equivalent MAP/MAP/1 queue.
template<class T>
MapMap1Result< T >	qsys_phph1 (const std::vector< T > &alpha, const Matrix< T > &Tm, const std::vector< T > &beta, const Matrix< T > &S)
	qsys_phph1 with 100 materialized levels, the reference's numQLProbs.
template<class T>
TandemLindleyResult< T >	qsys_tandem_lindley (const std::vector< T > &A, const Matrix< T > &S, const std::vector< T > &W0=std::vector< T >())
	Sample-path Lindley recursion along a tandem of single-server FCFS queues.
template<class T>
TandemUbResult< T >	qsys_tandem_ub_ciucu (const std::vector< T > &x, const std::function< T(const T &)> &lst, const std::vector< T > &p, const std::vector< T > &mu, const std::function< T(const T &)> &dlst=std::function< T(const T &)>())
	Tail bounds for a GI/Hn/1 -> .

Typedef Documentation

Cplx

using line::qsys::Cplx = std::complex<double>

Definition at line 88 of file qsys_mg1_ps.h.

Enumeration Type Documentation

MapG1ServiceFit

enum class line::qsys::MapG1ServiceFit

strong

Which branch of fitServiceToPH was taken, for the caller and for tests.

Enumerator
Exponential	one moment, or cv2 exactly 1
Erlang	0 < cv2 < 1, or the cv2 <= 0 branch
Hyperexponential	cv2 > 1
Acyclic	three or more moments

Definition at line 108 of file qsys_mapg1.h.

PatienceForm

enum class line::qsys::PatienceForm

strong

Which of the three accepted descriptions of the patience law is carried.

Enumerator
Exponential
Hazard
Ccdf

Definition at line 74 of file qsys_mgisrgi_whitt.h.

QedCriterion

enum class line::qsys::QedCriterion

strong

Which target the staffing rule is asked to meet.

Enumerator
Delay
MeanWait
ServiceLevel

Definition at line 83 of file qsys_mmk_qed.h.

RetrialPolicy

enum class line::qsys::RetrialPolicy

strong

Retrial admission policy (RetrialPolicy in MATLAB).

Enumerator
LINEAR	each orbiting customer retries at rate alpha (orbit rate i*alpha)
CONSTANT	the orbit retries at rate alpha whenever it is non-empty

Definition at line 104 of file qsys_bmapphnn_retrial.h.

ServiceKind

enum class line::qsys::ServiceKind

strong

Which family the service law belongs to.

Enumerator
Gamma
Deterministic
PhaseType
Density

Definition at line 100 of file qsys_mapg1k.h.

Function Documentation

erlang_b_recursive()

template<class T>

T line::qsys::erlang_b_recursive	(	const T &	a,
		unsigned	c )

Erlang's loss formula B(a, c) by the numerically stable rational recursion B_0 = 1, B_k = a B_{k-1}/(k + a B_{k-1}).

Exact in any field: no logs, no factorials, no cancellation.

Definition at line 65 of file qsys_mmcc_retrial_fp.h.

References erlang_b_recursive().

Referenced by erlang_b_recursive(), and qsys_mmcc_retrial_fp().

npfqn_gtmtst_fluid() [1/2]

template<class Tv>

NpfqnTvFluidResult< Tv > line::qsys::npfqn_gtmtst_fluid	(	const std::vector< std::function< Tv(const Tv &)> > &	lambdaFuns,
		const std::vector< std::function< Tv(const Tv &)> > &	sFuns,
		const std::vector< std::function< Tv(const Tv &)> > &	muFuns,
		const std::vector< std::function< Tv(const Tv &)> > &	patienceCcdfs,
		const std::function< std::vector< std::vector< Tv > >(const Tv &)> &	Pfun,
		const Tv &	T,
		const Tv &	dt = num_traits<Tv>::from_int(0),
		const std::vector< Tv > &	B0 = std::vector<Tv>(),
		const std::vector< Tv > &	w0 = std::vector<Tv>(),
		double	tol = 1e-6,
		std::size_t	maxIter = 100 )

A time-varying open network of many-server fluid queues with abandonment.

Parameters

lambdaFuns	external arrival rate of each queue
sFuns	staffing of each queue
muFuns	service rate of each queue
patienceCcdfs	patience ccdf of each queue
Pfun	routing proportions P(t), substochastic at every grid time; the overload below takes a constant matrix
T	horizon
dt	grid step; non-positive takes T/2000
B0	initial fluid in service, empty for an empty network
w0	initial boundary waiting times, empty for an empty network
tol	sup-norm tolerance on the arrival-rate iteration
maxIter	cap on the iterations

Definition at line 333 of file qsys_gtmtst_fluid.h.

References line::qsys::NpfqnTvFluidResult< T >::arrivalRates, line::qsys::TvFluidOptions< T >::B0, line::qsys::TvFluidOptions< T >::dt, line::InputError::InputError(), line::qsys::NpfqnTvFluidResult< T >::iterations, npfqn_gtmtst_fluid(), qsys_gtmtst_fluid(), line::qsys::NpfqnTvFluidResult< T >::queues, line::qsys::NpfqnTvFluidResult< T >::residual, line::qsys::QsysTvFluidResult< T >::sigma, line::qsys::NpfqnTvFluidResult< T >::times, and line::qsys::TvFluidOptions< T >::w0.

Referenced by npfqn_gtmtst_fluid(), and npfqn_gtmtst_fluid().

npfqn_gtmtst_fluid() [2/2]

template<class Tv>

NpfqnTvFluidResult< Tv > line::qsys::npfqn_gtmtst_fluid	(	const std::vector< std::function< Tv(const Tv &)> > &	lambdaFuns,
		const std::vector< std::function< Tv(const Tv &)> > &	sFuns,
		const std::vector< std::function< Tv(const Tv &)> > &	muFuns,
		const std::vector< std::function< Tv(const Tv &)> > &	patienceCcdfs,
		const std::vector< std::vector< Tv > > &	P,
		const Tv &	T,
		const Tv &	dt = num_traits<Tv>::from_int(0),
		const std::vector< Tv > &	B0 = std::vector<Tv>(),
		const std::vector< Tv > &	w0 = std::vector<Tv>(),
		double	tol = 1e-6,
		std::size_t	maxIter = 100 )

Constant routing matrix: MATLAB's matrix form of P, the same at every step.

Definition at line 430 of file qsys_gtmtst_fluid.h.

References npfqn_gtmtst_fluid().

qsys_bmapm1() [1/2]

template<class T>

BmapM1Result< T > line::qsys::qsys_bmapm1	(	const std::vector< Matrix< T > > &	D,
		const T &	mu )

The reference's defaults: adaptive truncation, 10000 iterations, tol 1e-12.

Definition at line 364 of file qsys_bmapm1.h.

References qsys_bmapm1().

qsys_bmapm1() [2/2]

template<class T>

BmapM1Result< T > line::qsys::qsys_bmapm1	(	const std::vector< Matrix< T > > &	D,
		const T &	mu,
		const T &	qParam,
		std::size_t	maxLevelParam,
		unsigned	maxIter,
		const T &	tol,
		double	tailTol )

BMAP/M/1 by the matrix-analytic (M/G/1-type) method.

Parameters

D	the BMAP {D0, D1, ..., DK}; D0 carries hidden transitions, Dk the transitions releasing a batch of k
mu	exponential service rate
qParam	uniformization constant; <= 0 selects the reference's default
maxLevelParam	explicit level truncation; 0 selects the adaptive search
maxIter	iteration cap for the G matrix
tol	convergence tolerance for the G matrix
tailTol	relative truncation target for the level distribution

Definition at line 177 of file qsys_bmapm1.h.

References line::qsys::BmapM1Result< T >::A, line::qsys::BmapM1Result< T >::A0, line::qsys::BmapM1Result< T >::A1, line::qsys::BmapM1Result< T >::alpha, line::qsys::BmapM1Result< T >::B0, line::qsys::BmapM1Result< T >::Bk, line::mc::ctmc_solve(), line::qsys::BmapM1Result< T >::decayRate, line::qsys::BmapM1Result< T >::drift, line::mc::dtmc_solve(), line::qsys::BmapM1Result< T >::G, line::qsys::BmapM1Result< T >::gConverged, line::InputError::InputError(), line::qsys::BmapM1Result< T >::lambda, line::qsys::BmapM1Result< T >::levelProb, line::matmul(), line::Matrix< T >::Matrix(), line::qsys::BmapM1Result< T >::meanQueueLength, line::qsys::BmapM1Result< T >::pi0, line::qsys::BmapM1Result< T >::q, qsys_bmapm1(), line::qsys::BmapM1Result< T >::rho, line::Matrix< T >::size(), line::qsys::BmapM1Result< T >::theta, line::qsys::BmapM1Result< T >::throughput, line::qsys::BmapM1Result< T >::truncError, line::qsys::BmapM1Result< T >::truncLevel, line::qsys::BmapM1Result< T >::utilization, and line::vecmul().

Referenced by qsys_bmapm1(), qsys_bmapm1(), and line::mam::solver_mam_getmamresult().

qsys_bmapphnn_retrial() [1/2]

template<class T>

BmapPhNnRetrialResult< T > line::qsys::qsys_bmapphnn_retrial	(	const std::vector< Matrix< T > > &	D,
		const std::vector< T > &	beta,
		const Matrix< T > &	S,
		int	N,
		const T &	alpha,
		const T &	gamma,
		const T &	p,
		const std::vector< long > &	R,
		const BmapPhNnRetrialOptions &	opt )

The BMAP/PH/N/N bufferless retrial queue.

Parameters

D	the BMAP as {D0, D1, ..., DK}, each V x V
beta	PH service entry vector, length M
S	PH service sub-generator, M x M
N	number of servers, which is also the capacity
alpha	retrial rate per orbiting customer
gamma	abandonment rate per orbiting customer
p	probability that an overflowing batch is lost rather than joining the orbit
R	admission threshold per BMAP state; a single entry is broadcast
opt	truncation level

Definition at line 464 of file qsys_bmapphnn_retrial.h.

References line::qsys::BmapPhNnRetrialResult< T >::clipped, line::Matrix< T >::cols(), CONSTANT, line::qsys::BmapPhNnRetrialResult< T >::converged, line::lang::GlobalConstants::FineTol, line::InputError::InputError(), line::inverse(), line::qsys::BmapPhNnRetrialResult< T >::L_orbit, line::qsys::BmapPhNnRetrialResult< T >::L_system, line::mulvec(), line::qsys::BmapPhNnRetrialResult< T >::N_server, line::NumericError::NumericError(), line::ones(), line::qsys::BmapPhNnRetrialResult< T >::P_empty_orbit, line::qsys::BmapPhNnRetrialResult< T >::P_empty_system, line::qsys::BmapPhNnRetrialResult< T >::P_idle, line::qsys::BmapPhNnRetrialResult< T >::pi, qsys_bmapphnn_retrial(), line::Matrix< T >::rows(), line::solve(), line::qsys::BmapPhNnRetrialResult< T >::throughput, line::qsys::BmapPhNnRetrialResult< T >::topLevelMass, line::Matrix< T >::transpose(), line::qsys::BmapPhNnRetrialResult< T >::truncError, line::qsys::BmapPhNnRetrialResult< T >::truncLevel, and line::qsys::BmapPhNnRetrialResult< T >::utilization.

Referenced by qsys_bmapphnn_retrial(), qsys_bmapphnn_retrial(), and line::mam::solver_mam_retrial().

qsys_bmapphnn_retrial() [2/2]

template<class T>

BmapPhNnRetrialResult< T > line::qsys::qsys_bmapphnn_retrial	(	const std::vector< Matrix< T > > &	D,
		const std::vector< T > &	beta,
		const Matrix< T > &	S,
		int	N,
		const T &	alpha,
		const T &	gamma,
		const T &	p,
		long	R )

qsys_bmapphnn_retrial with the reference's automatic truncation level.

Definition at line 853 of file qsys_bmapphnn_retrial.h.

References qsys_bmapphnn_retrial().

qsys_dmc() [1/2]

template<class T>

DmcResult< T > line::qsys::qsys_dmc	(	const T &	lambda,
		const T &	mu,
		unsigned	c )

qsys_dmc with the MATLAB defaults, automatic truncation and 200 steps.

Definition at line 225 of file qsys_dmc.h.

References qsys_dmc().

qsys_dmc() [2/2]

template<class T>

DmcResult< T > line::qsys::qsys_dmc	(	const T &	lambda,
		const T &	mu,
		unsigned	c,
		unsigned	truncation,
		unsigned	quadSteps )

D/M/c: deterministic interarrival times, exponential service.

Parameters

lambda	deterministic arrival rate, interarrival time 1/lambda
mu	exponential service rate of one server
c	number of servers, c >= 1
truncation	state-space truncation; 0 selects the MATLAB default
quadSteps	trapezoid steps over one interarrival cycle (MATLAB 200)

Definition at line 122 of file qsys_dmc.h.

References line::eye(), line::InputError::InputError(), line::matmul(), line::qsys::DmcResult< T >::meanQueueLength, line::qsys::DmcResult< T >::meanSojournTime, line::qsys::DmcResult< T >::meanWaitingQueue, line::qsys::DmcResult< T >::meanWaitingTime, line::mulvec(), qsys_dmc(), line::solve(), and line::qsys::DmcResult< T >::utilization.

Referenced by qsys_dmc(), and qsys_dmc().

qsys_erlang_b()

template<class T>

T line::qsys::qsys_erlang_b	(	unsigned	s,
		const T &	a )

Erlang B by the recursion B_j = a B_{j-1}/(j + a B_{j-1}), which never forms a^s/s!

and so never overflows.

Parameters

s	number of servers
a	offered load in erlangs

Definition at line 76 of file qsys_mtgs0_mol.h.

References qsys_erlang_b().

Referenced by qsys_erlang_b(), qsys_erlang_c(), and qsys_mtgs0_mol().

qsys_erlang_c()

template<class T>

T line::qsys::qsys_erlang_c	(	unsigned	s,
		const T &	a )

Erlang C from the same recursion; 1 when the load saturates the servers.

Parameters

s	number of servers
a	offered load in erlangs

Definition at line 90 of file qsys_mtgs0_mol.h.

References qsys_erlang_b(), and qsys_erlang_c().

Referenced by qsys_erlang_c(), and qsys_mtgs0_mol().

qsys_erlanga()

template<class T>

QsysAbandonResult< T > line::qsys::qsys_erlanga	(	const T &	lambda,
		const T &	mu,
		const T &	theta,
		unsigned	s,
		double	r = std::numeric_limits<double>::infinity(),
		const MgisrgiOptions &	opts = MgisrgiOptions() )

Exact analysis of the Erlang A model M/M/s/r+M.

The number in system is the birth-and-death process with death rate min(k,s) mu + (k-s)^+ theta, so every measure is exact: this is the case in which the approximation above reproduces the model (eq. 7.12). theta = 0 recovers M/M/s/r, and then a finite r is required whenever lambda >= s mu.

Port of matlab/src/api/qsys/qsys_erlanga.m.

Parameters

lambda	arrival rate
mu	service rate of one server
theta	abandonment rate of a waiting customer
s	number of servers
r	extra waiting spaces; infinity for an unbounded queue
opts	cdf times, truncation controls and inversion settings

Definition at line 436 of file qsys_mgisrgi_whitt.h.

References line::qsys::Patience< T >::exponential(), line::InputError::InputError(), qsys_erlanga(), and qsys_mgisrgi_whitt().

Referenced by qsys_erlanga(), and line::mva::solver_mva_qsys_analyzer().

qsys_gg1()

template<class T>

QsysResult< T > line::qsys::qsys_gg1	(	const T &	lambda,
		const T &	mu,
		const T &	ca2,
		const T &	cs2 )

G/G/1 dispatcher: exact where a two-moment description determines the answer, Allen-Cunneen otherwise.

Parameters

lambda	arrival rate
mu	service rate
ca2	squared coefficient of variation of the interarrival time
cs2	squared coefficient of variation of the service time

Definition at line 118 of file qsys_gg1.h.

References line::num_abs(), qsys_gg1(), qsys_gig1_approx_allencunneen(), qsys_gm1(), qsys_mg1(), and qsys_mm1().

Referenced by qsys_gg1(), and line::mva::solver_mva_qsys_analyzer().

qsys_ggingi_tga()

template<class T>

QsysTgaResult< T > line::qsys::qsys_ggingi_tga	(	const T &	lambda,
		const T &	mu,
		unsigned	n,
		const T &	ca,
		const T &	cs,
		const std::function< T(const T &)> &	patienceCcdf,
		const std::function< T(const T &)> &	patiencePdf = std::function<T(const T&)>(),
		const std::function< T(const T &)> &	serviceCcdf = std::function<T(const T&)>() )

Truncated Gaussian approximation (TGA-G) for the G/GI/n+GI queue.

Parameters

lambda	arrival rate
mu	service rate of one server
n	number of servers
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
patienceCcdf	F^c(x) = P(patience > x)
patiencePdf	the patience density; empty differences the ccdf
serviceCcdf	G^c(x), used only in the underloaded branch

Definition at line 155 of file qsys_ggingi_tga.h.

References line::qsys::QsysTgaResult< T >::fluidQueueLength, line::qsys::QsysTgaResult< T >::fluidWait, line::InputError::InputError(), line::qsys::QsysTgaResult< T >::meanNumber, line::qsys::QsysTgaResult< T >::meanNumberInService, line::qsys::QsysTgaResult< T >::meanQueueLength, line::qsys::QsysTgaResult< T >::meanWait, line::qsys::QsysTgaResult< T >::probAbandon, line::qsys::QsysTgaResult< T >::probDelay, qsys_ggingi_tga(), line::qsys::QsysTgaResult< T >::regime, line::qsys::QsysTgaResult< T >::sigmaW, line::qsys::QsysTgaResult< T >::sigmaX, line::qsys::QsysTgaResult< T >::trafficIntensity, line::qsys::QsysTgaResult< T >::varQueueLength, and line::qsys::QsysTgaResult< T >::varWait.

Referenced by qsys_ggingi_tga(), and line::fluid::solver_fluid_qsys().

qsys_ggisgi_fluid()

template<class T>

QsysFluidAbandonResult< T > line::qsys::qsys_ggisgi_fluid	(	const T &	lambda,
		const T &	mu,
		unsigned	s,
		const std::function< T(const T &)> &	patienceCcdf,
		const std::function< T(const T &)> &	servingCcdf = std::function<T(const T&)>(),
		const std::vector< T > &	agePoints = std::vector<T>(),
		double	tol = 1e-12,
		double	maxTime = std::numeric_limits<double>::quiet_NaN() )

Steady state of the G/GI/s+GI fluid model.

Parameters

lambda	arrival rate
mu	service rate of one server
s	number of servers, s >= 1
patienceCcdf	F^c(t) = P(patience > t)
servingCcdf	G^c(x) = P(service > x), used only for the in-service age density; an empty callable takes the exponential of rate mu
agePoints	ages at which to return the densities
tol	bisection tolerance for w
maxTime	largest age searched for w; NaN grows the search

Definition at line 157 of file qsys_ggisgi_fluid.h.

References line::qsys::QsysFluidAbandonResult< T >::abandonRate, line::qsys::QsysFluidAbandonResult< T >::agePoints, line::InputError::InputError(), line::qsys::QsysFluidAbandonResult< T >::meanNumber, line::qsys::QsysFluidAbandonResult< T >::meanNumberInService, line::qsys::QsysFluidAbandonResult< T >::meanQueueLength, line::qsys::QsysFluidAbandonResult< T >::meanWait, line::qsys::QsysFluidAbandonResult< T >::meanWaitAbandon, line::qsys::QsysFluidAbandonResult< T >::meanWaitServed, line::qsys::QsysFluidAbandonResult< T >::offeredWait, line::qsys::QsysFluidAbandonResult< T >::probAbandon, qsys_ggisgi_fluid(), line::qsys::QsysFluidAbandonResult< T >::queueAgeDensity, line::qsys::QsysFluidAbandonResult< T >::regime, line::qsys::QsysFluidAbandonResult< T >::serviceAgeDensity, line::qsys::QsysFluidAbandonResult< T >::throughput, line::qsys::QsysFluidAbandonResult< T >::trafficIntensity, and line::qsys::QsysFluidAbandonResult< T >::utilization.

Referenced by qsys_ggisgi_fluid(), and line::fluid::solver_fluid_qsys().

qsys_ggnm_diffusion()

template<class T>

QsysGgnmResult< T > line::qsys::qsys_ggnm_diffusion	(	const T &	lambda,
		const T &	mu,
		unsigned	n,
		double	m,
		const T &	ca,
		const T &	cs,
		const std::function< T(const T &)> &	serviceCcdf = std::function<T(const T&)>(),
		double	tol = 1e-12,
		std::size_t	panels = 4000 )

Diffusion approximation for the G/GI/n/m queue.

Parameters

lambda	arrival rate
mu	service rate of one server
n	number of servers, n >= 1
m	extra waiting spaces; infinity for an unbounded queue
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
serviceCcdf	G^c(x) = P(S > x); empty takes the exponential of rate mu
tol	service-tail cut for the peakedness integral
panels	Simpson panels for it

Definition at line 144 of file qsys_ggnm_diffusion.h.

References line::qsys::QsysGgnmResult< T >::beta, line::qsys::QsysGgnmResult< T >::gamma, line::InputError::InputError(), line::qsys::QsysGgnmResult< T >::meanNumber, line::qsys::QsysGgnmResult< T >::meanQueueLength, line::qsys::QsysGgnmResult< T >::meanWait, line::num_abs(), line::qsys::QsysGgnmResult< T >::peakedness, line::qsys::QsysGgnmResult< T >::peakednessWeight, line::qsys::QsysGgnmResult< T >::probBlock, line::qsys::QsysGgnmResult< T >::probDelay, qsys_ggnm_diffusion(), line::qsys::QsysGgnmResult< T >::throughput, line::qsys::QsysGgnmResult< T >::trafficIntensity, line::qsys::QsysGgnmResult< T >::utilization, and line::qsys::QsysGgnmResult< T >::variability.

Referenced by qsys_ggnm_diffusion(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_approx_allencunneen()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_allencunneen	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Allen-Cunneen approximation of the mean response time of a G/I/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 40 of file qsys_gig1_approx_allencunneen.h.

References line::num_pow_int(), and qsys_gig1_approx_allencunneen().

Referenced by qsys_gg1(), qsys_gig1_approx_allencunneen(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_approx_gelenbe()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_gelenbe	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Gelenbe diffusion approximation with instantaneous-return boundary.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 45 of file qsys_gig1_approx_gelenbe.h.

References line::num_pow_int(), and qsys_gig1_approx_gelenbe().

Referenced by qsys_gig1_approx_gelenbe(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_approx_heyman()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_heyman	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Heyman approximation of the mean response time of a G/I/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 39 of file qsys_gig1_approx_heyman.h.

References line::num_pow_int(), and qsys_gig1_approx_heyman().

Referenced by qsys_gig1_approx_heyman(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_approx_kimura()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_kimura	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Kimura diffusion-interpolation approximation for the G/I/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 46 of file qsys_gig1_approx_kimura.h.

References line::num_pow_int(), and qsys_gig1_approx_kimura().

Referenced by qsys_gig1_approx_kimura(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_approx_klb()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_klb	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Kraemer and Langenbach-Belz approximation for the G/I/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 41 of file qsys_gig1_approx_klb.h.

References line::num_pow_int(), and qsys_gig1_approx_klb().

Referenced by qsys_gig1_approx_klb(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_approx_kobayashi()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_kobayashi	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Kobayashi diffusion approximation for the G/I/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 40 of file qsys_gig1_approx_kobayashi.h.

References line::num_pow_int(), and qsys_gig1_approx_kobayashi().

Referenced by qsys_gig1_approx_kobayashi(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_approx_marchal()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_marchal	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Marchal approximation of the mean response time of a G/I/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 40 of file qsys_gig1_approx_marchal.h.

References line::num_pow_int(), and qsys_gig1_approx_marchal().

Referenced by qsys_gig1_approx_marchal(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_approx_myskja()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_myskja	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs,
		const T &	q0,
		const T &	qa )

Myskja's third-moment approximation of the mean response time of a G/I/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
q0	smallest third relative moment for the given mean and SCV
qa	third relative moment E[A^3]/(6 E[A]^3) of the interarrival time

Definition at line 53 of file qsys_gig1_approx_myskja.h.

References qsys_gig1_approx_myskja().

Referenced by qsys_gig1_approx_myskja().

qsys_gig1_approx_myskja2()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_myskja2	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs,
		const T &	q0,
		const T &	qa )

Myskja's enhanced third-moment approximation of the mean response time of a G/I/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
q0	smallest third relative moment for the given mean and SCV
qa	third relative moment E[A^3]/(6 E[A]^3) of the interarrival time

Definition at line 56 of file qsys_gig1_approx_myskja2.h.

References line::num_abs(), line::num_pow_int(), qsys_gig1_approx_myskja2(), and qsys_mg1().

Referenced by qsys_gig1_approx_myskja2().

qsys_gig1_approx_whitt()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_approx_whitt	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Whitt's approximation of the G/G/1 mean response time.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 54 of file qsys_gig1_approx_whitt.h.

References qsys_gig1_approx_whitt().

Referenced by qsys_gig1_approx_whitt().

qsys_gig1_bnds_extremal()

template<class T>

Gig1ExtremalResult< T > line::qsys::qsys_gig1_bnds_extremal	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs,
		std::size_t	K = 4000,
		std::size_t	N = 2000,
		bool	skipTight = false )

Extremal two-moment bounds for the GI/GI/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
K	truncation of the negative binomial value
N	truncation of the random-walk length
skipTight	skip the O(K*N) tight bound and return the closed forms only

Definition at line 148 of file qsys_gig1_bnds_extremal.h.

References line::qsys::Gig1ExtremalResult< T >::delta, line::qsys::Gig1ExtremalResult< T >::heavyTraffic, line::InputError::InputError(), line::qsys::Gig1ExtremalResult< T >::lowerBound, qsys_gig1_bnds_extremal(), line::qsys::Gig1ExtremalResult< T >::relativeWidth, line::qsys::Gig1ExtremalResult< T >::tightComputed, line::qsys::Gig1ExtremalResult< T >::trafficIntensity, line::qsys::Gig1ExtremalResult< T >::upperBound, line::qsys::Gig1ExtremalResult< T >::upperBoundClosed, line::qsys::Gig1ExtremalResult< T >::upperBoundDaley, and line::qsys::Gig1ExtremalResult< T >::upperBoundKingman.

Referenced by qsys_gig1_bnds_extremal(), and line::mva::solver_mva_qsys_analyzer().

qsys_gig1_lbnd()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_lbnd	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Fundamental lower bound on the mean response time of a G/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time, unused
cs	coefficient of variation of the service time, unused

Definition at line 41 of file qsys_gig1_lbnd.h.

References qsys_gig1_lbnd().

Referenced by qsys_gig1_lbnd().

qsys_gig1_rq()

template<class T, class IaFun>

Gig1RqResult< T > line::qsys::qsys_gig1_rq	(	const T &	rho,
		const T &	mu,
		const T &	cs2,
		IaFun &&	IaFun_ )

Robust Queueing (RQ) approximation of a G/GI/1 queue characterized by its arrival index of dispersion and the first two service moments.

Parameters

rho	traffic intensity lambda/mu
mu	service rate
cs2	squared coefficient of variation of the service time
IaFun_	callable, IaFun_(x) -> the arrival IDC I_a(x) at x > 0

Definition at line 77 of file qsys_gig1_rq.h.

References line::InputError::InputError(), line::qsys::Gig1RqResult< T >::Q, qsys_gig1_rq(), line::qsys::Gig1RqResult< T >::W, line::qsys::Gig1RqResult< T >::X, and line::qsys::Gig1RqResult< T >::Z.

Referenced by qsys_gig1_rq(), line::mva::solver_mva_qsys_analyzer(), and line::mva::solver_rqna().

qsys_gig1_rqt() [1/2]

template<class T>

GigkRqtResult< T > line::qsys::qsys_gig1_rqt	(	const T &	lambda,
		const T &	mu,
		const T &	Gamma_a,
		const T &	Gamma_s )

Finite-variance case, alpha_a = alpha_s = 2.

Definition at line 46 of file qsys_gig1_rqt.h.

References qsys_gig1_rqt(), and qsys_gigk_rqt().

qsys_gig1_rqt() [2/2]

template<class T>

GigkRqtResult< T > line::qsys::qsys_gig1_rqt	(	const T &	lambda,
		const T &	mu,
		const T &	Gamma_a,
		const T &	Gamma_s,
		const T &	alpha_a,
		const T &	alpha_s )

Robust Queueing Theory (RQT) worst-case system time of a G/G/1 FCFS queue, the single-server case of qsys_gigk_rqt (Theorem 2 and eq.

12).

Parameters

lambda	arrival rate
mu	service rate
Gamma_a	variability parameter of the arrival uncertainty set
Gamma_s	variability parameter of the service uncertainty set
alpha_a	arrival tail coefficient in (1,2]
alpha_s	service tail coefficient in (1,2]

Definition at line 39 of file qsys_gig1_rqt.h.

References qsys_gig1_rqt(), and qsys_gigk_rqt().

Referenced by qsys_gig1_rqt(), and qsys_gig1_rqt().

qsys_gig1_ubnd_kingman()

template<class T>

QsysResult< T > line::qsys::qsys_gig1_ubnd_kingman	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs )

Kingman upper bound on the mean waiting time of a G/G/1 queue.

Parameters

lambda	arrival rate
mu	service rate
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time

Definition at line 41 of file qsys_gig1_ubnd_kingman.h.

References line::num_pow_int(), and qsys_gig1_ubnd_kingman().

Referenced by qsys_gig1_ubnd_kingman(), and line::mva::solver_mva_qsys_analyzer().

qsys_gigk_approx()

template<class T>

QsysResult< T > line::qsys::qsys_gigk_approx	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs,
		unsigned	k )

Default G/I/G/k approximation of the mean response time.

Parameters

lambda	arrival rate
mu	service rate of a single server
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
k	number of servers, k >= 1

Definition at line 47 of file qsys_gigk_approx.h.

References line::InputError::InputError(), line::num_pow_int(), and qsys_gigk_approx().

Referenced by qsys_gigk_approx(), and line::mva::solver_mva_qsys_analyzer().

qsys_gigk_approx_cosmetatos()

template<class T>

QsysResult< T > line::qsys::qsys_gigk_approx_cosmetatos	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs,
		unsigned	k )

Cosmetatos / Page interpolation approximation for the GI/G/k queue.

Parameters

lambda	arrival rate
mu	service rate of a single server
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
k	number of servers, k >= 1

Definition at line 54 of file qsys_gigk_approx_cosmetatos.h.

References line::num_pow_int(), qsys_gigk_approx_cosmetatos(), and qsys_mmk().

Referenced by qsys_gigk_approx_cosmetatos().

qsys_gigk_approx_kingman()

template<class T>

QsysResult< T > line::qsys::qsys_gigk_approx_kingman	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs,
		unsigned	k )

Kingman (Lee-Longton) scaling of the exact M/M/k waiting time.

Parameters

lambda	arrival rate
mu	service rate of a single server
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
k	number of servers, k >= 1

Definition at line 44 of file qsys_gigk_approx_kingman.h.

References line::num_pow_int(), qsys_gigk_approx_kingman(), and qsys_mmk().

Referenced by qsys_gigk_approx_kingman(), and line::mva::solver_mva_qsys_analyzer().

qsys_gigk_approx_whitt()

template<class T>

QsysResult< T > line::qsys::qsys_gigk_approx_whitt	(	const T &	lambda,
		const T &	mu,
		const T &	ca,
		const T &	cs,
		unsigned	k )

Whitt (1993) approximation for the GI/G/k queue, eqs.

(2.16)-(2.25).

Parameters

lambda	arrival rate
mu	service rate of a single server
ca	coefficient of variation of the interarrival time
cs	coefficient of variation of the service time
k	number of servers, k >= 1

Definition at line 61 of file qsys_gigk_approx_whitt.h.

References line::num_abs(), line::num_pow_int(), qsys_gigk_approx_whitt(), and qsys_mmk().

Referenced by qsys_gigk_approx_whitt(), and line::mva::solver_mva_qsys_analyzer().

qsys_gigk_rqt()

template<class T>

GigkRqtResult< T > line::qsys::qsys_gigk_rqt	(	const T &	lambda,
		const T &	mu,
		const T &	Gamma_a,
		const T &	Gamma_s,
		std::size_t	k,
		const T &	alpha_a,
		const T &	alpha_s )

Robust Queueing Theory (RQT) worst-case system time of a G/G/k FCFS queue.

Parameters

lambda	arrival rate
mu	service rate of each server
Gamma_a	variability parameter of the arrival uncertainty set
Gamma_s	variability parameter of the service uncertainty set
k	number of servers
alpha_a	arrival tail coefficient in (1,2]
alpha_s	service tail coefficient in (1,2]

Definition at line 85 of file qsys_gigk_rqt.h.

References line::InputError::InputError(), qsys_gigk_rqt(), line::qsys::GigkRqtResult< T >::rhohat, line::qsys::GigkRqtResult< T >::Sworst, and line::qsys::GigkRqtResult< T >::W.

Referenced by qsys_gig1_rqt(), qsys_gig1_rqt(), qsys_gigk_rqt(), line::mva::solver_mva_qsys_analyzer(), and line::mva::solver_rqt().

qsys_gigk_rqt_gamma()

template<class T>

T line::qsys::qsys_gigk_rqt_gamma	(	const T &	rho,
		const T &	mu,
		const T &	Gamma_a,
		const T &	sigma_s,
		std::size_t	k,
		const T &	alpha_a,
		const std::string &	regime = "independent" )

Service variability parameter of the Robust Queueing Theory (RQT) framework.

Parameters

rho	traffic intensity lambda/(k mu)
mu	service rate of each server, which sets the time unit
Gamma_a	variability parameter of the arrival uncertainty set
sigma_s	standard deviation of the service time
k	number of servers
alpha_a	effective arrival tail coefficient in (1,2]
regime	adaptation regime of Table 1: "independent" (service distribution unknown), "normal" or "pareto"

Definition at line 72 of file qsys_gigk_rqt_gamma.h.

References line::InputError::InputError(), and qsys_gigk_rqt_gamma().

Referenced by qsys_gigk_rqt_gamma(), line::mva::solver_mva_qsys_analyzer(), and line::mva::solver_rqt().

qsys_gm1()

template<class T>

T line::qsys::qsys_gm1	(	const T &	sigma,
		const T &	mu )

Exact mean response time of the G/M/1 queue.

Parameters

sigma	load at arrival instants, 0 <= sigma < 1
mu	service rate

Returns

mean response time W

Definition at line 40 of file qsys_gm1.h.

References qsys_gm1().

Referenced by qsys_gg1(), qsys_gm1(), and line::mva::solver_mva_qsys_analyzer().

qsys_gtmtst_fluid()

template<class Tv>

QsysTvFluidResult< Tv > line::qsys::qsys_gtmtst_fluid	(	const std::function< Tv(const Tv &)> &	lambdaFun,
		const std::function< Tv(const Tv &)> &	sFun,
		const std::function< Tv(const Tv &)> &	muFun,
		const std::function< Tv(const Tv &)> &	patienceCcdf,
		const Tv &	T,
		const TvFluidOptions< Tv > &	opts = TvFluidOptions<Tv>() )

The Gt/Mt/st+GI many-server fluid queue, and the network of them.

Parameters

lambdaFun	arrival rate lambda(t)
sFun	staffing s(t), positive
muFun	service rate mu(t), positive
patienceCcdf	F^c(x) = P(patience > x)
T	horizon; the model is solved on [0,T]
opts	grid step, initial condition and the optional derivatives

Definition at line 149 of file qsys_gtmtst_fluid.h.

References line::qsys::QsysTvFluidResult< T >::alpha, line::qsys::QsysTvFluidResult< T >::arrivalRate, line::qsys::QsysTvFluidResult< T >::B, line::qsys::QsysTvFluidResult< T >::capacityRate, line::InputError::InputError(), line::qsys::QsysTvFluidResult< T >::Q, qsys_gtmtst_fluid(), line::qsys::QsysTvFluidResult< T >::regime, line::qsys::QsysTvFluidResult< T >::sigma, line::qsys::QsysTvFluidResult< T >::staffing, line::qsys::QsysTvFluidResult< T >::times, line::qsys::QsysTvFluidResult< T >::utilization, line::qsys::QsysTvFluidResult< T >::v, line::qsys::QsysTvFluidResult< T >::w, and line::qsys::QsysTvFluidResult< T >::X.

Referenced by npfqn_gtmtst_fluid(), qsys_gtmtst_fluid(), and line::fluid::solver_fluid_qsys().

qsys_hh1_lindley() [1/2]

template<class T>

LindleyResult< T > line::qsys::qsys_hh1_lindley	(	const std::vector< T > &	lambda,
		const std::vector< T > &	pa,
		const std::vector< T > &	mu,
		const std::vector< T > &	ps,
		const std::vector< T > &	Wn,
		unsigned	mmax = 2 )

Conditional waiting-time moments of the Hl/Hn/1 Lindley recursion.

Parameters

lambda	arrival phase rates, positive
pa	arrival phase probabilities, nonnegative and summing to 1
mu	service phase rates, positive
ps	service phase probabilities, nonnegative and summing to 1
Wn	waiting times of customer n, finite and nonnegative
mmax	highest moment order; raised to 2 when smaller, as in MATLAB

Definition at line 62 of file qsys_hh1_lindley.h.

References line::qsys::LindleyResult< T >::analyzer, line::InputError::InputError(), line::Matrix< T >::Matrix(), line::qsys::LindleyResult< T >::mean, line::qsys::LindleyResult< T >::mmax, line::qsys::LindleyResult< T >::moments, line::num_abs(), qsys_hh1_lindley(), qsys_lindley_moment(), and line::qsys::LindleyResult< T >::var.

Referenced by qsys_hh1_lindley(), and qsys_hh1_lindley().

qsys_hh1_lindley() [2/2]

template<class T>

LindleyResult< T > line::qsys::qsys_hh1_lindley	(	const std::vector< T > &	lambda,
		const std::vector< T > &	pa,
		const std::vector< T > &	mu,
		const std::vector< T > &	ps,
		const T &	Wn,
		unsigned	mmax = 2 )

Scalar-Wn overload.

Definition at line 121 of file qsys_hh1_lindley.h.

References qsys_hh1_lindley().

qsys_ldps_workload() [1/2]

template<class T>

LdpsWorkloadResult< T > line::qsys::qsys_ldps_workload	(	const T &	lambda,
		const WorkloadServiceLaw< T > &	B,
		const std::vector< T > &	alpha,
		std::size_t	N )

qsys_ldps_workload on the automatic grid with the reference default ngrid = 2001.

Definition at line 249 of file qsys_ldps_workload.h.

References qsys_ldps_workload().

qsys_ldps_workload() [2/2]

template<class T>

LdpsWorkloadResult< T > line::qsys::qsys_ldps_workload	(	const T &	lambda,
		const WorkloadServiceLaw< T > &	B,
		const std::vector< T > &	alpha,
		std::size_t	N,
		const std::vector< T > &	t,
		std::size_t	ngrid )

Workload distribution of the load-dependent PS station with blocking.

Parameters

lambda	Poisson arrival rate
B	required service time: mean, SCV and CDF
alpha	rate scaling alpha(n) = n f(n) for n = 1..N, at least N entries
N	blocking capacity
t	grid at which the CDF is wanted; empty for an automatic grid
ngrid	points of the internal uniform quadrature grid

Definition at line 131 of file qsys_ldps_workload.h.

References line::qsys::WorkloadServiceLaw< T >::cdf, line::qsys::LdpsWorkloadResult< T >::F, line::InputError::InputError(), line::qsys::WorkloadServiceLaw< T >::mean, line::qsys::LdpsWorkloadResult< T >::p, qsys_ldps_workload(), line::qsys::WorkloadServiceLaw< T >::scv, and line::qsys::LdpsWorkloadResult< T >::t.

Referenced by qsys_ldps_workload(), and qsys_ldps_workload().

qsys_lindley_moment() [1/2]

template<class T>

std::vector< T > line::qsys::qsys_lindley_moment	(	const T &	lambda,
		const T &	mu,
		const std::vector< T > &	Wn,
		unsigned	m )

One conditional Lindley moment for exponential primitives.

Parameters

lambda	arrival rate, positive
mu	service rate, positive
Wn	current waiting times, nonnegative
m	moment order, at least 1

Returns

E[max(Wn + S - A, 0)^m] evaluated at every entry of Wn

Definition at line 58 of file qsys_lindley_moment.h.

References line::InputError::InputError(), line::num_factorial(), line::num_pow_int(), and qsys_lindley_moment().

Referenced by qsys_hh1_lindley(), qsys_lindley_moment(), qsys_lindley_moment(), and qsys_mm1_lindley().

qsys_lindley_moment() [2/2]

template<class T>

T line::qsys::qsys_lindley_moment	(	const T &	lambda,
		const T &	mu,
		const T &	Wn,
		unsigned	m )

Scalar overload of the same expression.

Definition at line 97 of file qsys_lindley_moment.h.

References qsys_lindley_moment().

qsys_mapd1() [1/3]

template<class T>

MapD1Result< T > line::qsys::qsys_mapd1	(	const mam::Map< T > &	arrival,
		const T &	s )

qsys_mapd1 with 100 materialized levels, an arrival-count cap of 4096, a level cap of 20000 and tolerance 1e-14.

Definition at line 558 of file qsys_mapd1.h.

References qsys_mapd1().

qsys_mapd1() [2/3]

template<class T>

MapD1Result< T > line::qsys::qsys_mapd1	(	const mam::Map< T > &	arrival,
		const T &	s,
		std::size_t	dist_size,
		unsigned	max_arrivals,
		std::size_t	max_levels,
		const T &	tol )

qsys_mapd1 with the reference's numSteps = 1.

Definition at line 547 of file qsys_mapd1.h.

References qsys_mapd1().

qsys_mapd1() [3/3]

template<class T>

MapD1Result< T > line::qsys::qsys_mapd1	(	const mam::Map< T > &	arrival,
		const T &	s,
		std::size_t	dist_size,
		unsigned	max_arrivals,
		std::size_t	max_levels,
		const T &	tol,
		std::size_t	num_steps )

MAP/D/1 by the exact embedded M/G/1-type chain.

Parameters

arrival	arrival MAP (D0, D1)
s	deterministic service time, s > 0
dist_size	how many entries of queueLengthDist to materialize
max_arrivals	cap on K, the number of arrivals per service that is tracked; K is grown from a Poisson-tail estimate until the missing counting mass falls below tol
max_levels	cap on the number of embedded levels generated by Ramaswami's recursion
tol	tolerance on the counting-mass truncation, on the G iteration and on the level-tail truncation
num_steps	the reference's numSteps: waitingTimeDist is evaluated at {0, s/numSteps, 2 s/numSteps, ...} (default 1)

Definition at line 330 of file qsys_mapd1.h.

References line::qsys::MapD1Result< T >::analyzer, line::mam::Map< T >::D0, line::mam::Map< T >::D1, line::eye(), line::InputError::InputError(), line::inverse(), line::mam::map_lambda(), line::mam::map_prob(), line::matmul(), line::Matrix< T >::Matrix(), line::qsys::MapD1Result< T >::meanQueueLength, line::qsys::MapD1Result< T >::meanSojournTime, line::qsys::MapD1Result< T >::meanWaitingTime, line::mulvec(), line::num_abs(), line::ones(), qsys_mapd1(), line::qsys::MapD1Result< T >::queueLengthDist, line::qsys::MapD1Result< T >::utilization, line::vecmul(), and line::qsys::MapD1Result< T >::waitingTimeDist.

Referenced by qsys_mapd1(), qsys_mapd1(), and qsys_mapd1().

qsys_mapdc() [1/3]

template<class T>

MapDcResult< T > line::qsys::qsys_mapdc	(	const mam::Map< T > &	arrival,
		const T &	s,
		unsigned	c )

qsys_mapdc with 100 materialized levels, an arrival-count cap of 4096, a super-level cap of 20000 and tolerance 1e-14, matching the qsys_mapd1 defaults.

Definition at line 347 of file qsys_mapdc.h.

References qsys_mapdc().

qsys_mapdc() [2/3]

template<class T>

MapDcResult< T > line::qsys::qsys_mapdc	(	const mam::Map< T > &	arrival,
		const T &	s,
		unsigned	c,
		std::size_t	dist_size,
		unsigned	max_arrivals,
		std::size_t	max_levels,
		const T &	tol )

qsys_mapdc with the reference's numSteps = 1.

Definition at line 334 of file qsys_mapdc.h.

References qsys_mapdc().

qsys_mapdc() [3/3]

template<class T>

MapDcResult< T > line::qsys::qsys_mapdc	(	const mam::Map< T > &	arrival,
		const T &	s,
		unsigned	c,
		std::size_t	dist_size,
		unsigned	max_arrivals,
		std::size_t	max_levels,
		const T &	tol,
		std::size_t	num_steps )

MAP/D/c by Crommelin's exact embedded lattice chain.

Parameters

arrival	arrival MAP (D0, D1)
s	deterministic service time, s > 0
c	number of servers, c >= 1
dist_size	how many entries of queueLengthDist to materialize
max_arrivals	cap on K, the number of arrivals per interval tracked
max_levels	cap on the number of super-levels generated
tol	tolerance on the counting-mass truncation, on the G iteration and on the level-tail truncation
num_steps	the reference's numSteps: waitingTimeDist is evaluated at {0, s/numSteps, 2 s/numSteps, ...} (default 1)

Definition at line 157 of file qsys_mapdc.h.

References line::qsys::MapDcResult< T >::analyzer, line::mam::Map< T >::D0, line::mam::Map< T >::D1, line::eye(), line::InputError::InputError(), line::inverse(), line::mam::map_lambda(), line::mam::map_prob(), line::matmul(), line::Matrix< T >::Matrix(), line::qsys::MapDcResult< T >::meanQueueLength, line::qsys::MapDcResult< T >::meanSojournTime, line::qsys::MapDcResult< T >::meanWaitingTime, line::num_abs(), qsys_mapdc(), line::qsys::MapDcResult< T >::queueLengthDist, line::qsys::MapDcResult< T >::utilization, line::vecmul(), and line::qsys::MapDcResult< T >::waitingTimeDist.

Referenced by qsys_mapdc(), qsys_mapdc(), and qsys_mapdc().

qsys_mapg1() [1/3]

template<class T>

MapG1Result< T > line::qsys::qsys_mapg1	(	const mam::Map< T > &	arrival,
		const std::vector< T > &	moments )

qsys_mapg1 with the reference's default of 100 materialized levels.

Definition at line 379 of file qsys_mapg1.h.

References qsys_mapg1().

qsys_mapg1() [2/3]

template<class T>

MapG1Result< T > line::qsys::qsys_mapg1	(	const mam::Map< T > &	arrival,
		const std::vector< T > &	moments,
		std::size_t	dist_size )

qsys_mapg1 with the reference's defaults numQLMoms = 3, numSTMoms = 3.

Definition at line 371 of file qsys_mapg1.h.

References qsys_mapg1().

qsys_mapg1() [3/3]

template<class T>

MapG1Result< T > line::qsys::qsys_mapg1	(	const mam::Map< T > &	arrival,
		const std::vector< T > &	moments,
		std::size_t	dist_size,
		std::size_t	numQLMoms,
		std::size_t	numSTMoms )

The MAP/G/1 FCFS queue.

Parameters

arrival	arrival MAP (D0, D1)
moments	the first one, two or three raw moments of the service time
dist_size	how many entries of queueLengthDist to materialize (the reference's numQLProbs, default 100)
numQLMoms	how many raw queue-length moments to return (numQLMoms)
numSTMoms	how many raw sojourn-time moments to return (numSTMoms)

Definition at line 343 of file qsys_mapg1.h.

References line::qsys::MapG1Result< T >::analyzer, line::qsys::MapG1Result< T >::fitKind, line::mam::map_lambda(), line::qsys::MapG1Result< T >::meanQueueLength, line::qsys::MapMap1Result< T >::meanQueueLength, line::qsys::MapG1Result< T >::meanSojournTime, line::qsys::MapMap1Result< T >::meanSojournTime, line::qsys::MapG1Result< T >::meanWaitingTime, qsys_mapg1(), qsys_mapg1_service_fit(), qsys_mapmap1(), line::qsys::MapG1Result< T >::queueLengthDist, line::qsys::MapMap1Result< T >::queueLengthDist, line::qsys::MapG1Result< T >::queueLengthMoments, line::qsys::MapMap1Result< T >::queueLengthMoments, line::qsys::MapG1Result< T >::serviceFit, line::qsys::MapG1Result< T >::servicePhases, line::qsys::MapG1Result< T >::sojournTimeMoments, line::qsys::MapMap1Result< T >::sojournTimeMoments, and line::qsys::MapG1Result< T >::utilization.

Referenced by qsys_mapg1(), qsys_mapg1(), and qsys_mapg1().

qsys_mapg1_service_fit()

template<class T>

mam::Map< T > line::qsys::qsys_mapg1_service_fit	(	const std::vector< T > &	moments,
		MapG1ServiceFit &	kind )

Fit a general service time to a PH, following qsys_mapg1.m's fitServiceToPH.

Exposed separately because the branch selection is the part of the reference that is transcribed verbatim, and a caller may want the fitted process without solving a queue.

Parameters

moments	the first one, two or three raw moments of the service time
kind	out: which branch was taken

Definition at line 280 of file qsys_mapg1.h.

References Acyclic, line::mam::aph_fit(), Erlang, Exponential, Hyperexponential, line::InputError::InputError(), and qsys_mapg1_service_fit().

Referenced by qsys_mapg1(), and qsys_mapg1_service_fit().

qsys_mapg1k() [1/2]

template<class T>

MapG1kResult< T > line::qsys::qsys_mapg1k	(	const mam::Map< T > &	arrival,
		const ServiceLaw< T > &	svc,
		std::size_t	K )

qsys_mapg1k with the reference defaults tol = 1e-12, nmax = 200000.

Definition at line 711 of file qsys_mapg1k.h.

References qsys_mapg1k().

qsys_mapg1k() [2/2]

template<class T>

MapG1kResult< T > line::qsys::qsys_mapg1k	(	const mam::Map< T > &	arrival,
		const ServiceLaw< T > &	svc,
		std::size_t	K,
		const T &	tol,
		std::size_t	nmaxCap )

MAP/G/1/K with tail drop.

Parameters

arrival	arrival MAP (D0, D1)
svc	service law
K	buffer size in packets, K >= 1, the one in service included
tol	uniformization truncation tolerance
nmaxCap	cap on the uniformization order

Definition at line 467 of file qsys_mapg1k.h.

References line::qsys::SvclawCoeffsResult< T >::cn, line::qsys::MapG1kResult< T >::countingResidual, line::mam::Map< T >::D0, line::mam::Map< T >::D1, line::mc::dtmc_solve(), line::eye(), line::inverse(), line::qsys::MapG1kResult< T >::lambda, line::qsys::MapG1kResult< T >::lossProbability, line::mam::map_lambda(), line::matmul(), line::qsys::MapG1kResult< T >::meanQueueLength, line::qsys::MapG1kResult< T >::meanServiceTime, line::mulvec(), line::qsys::MapG1kResult< T >::nmax, line::num_abs(), line::ones(), line::qsys::MapG1kResult< T >::p0, line::qsys::MapG1kResult< T >::p0vec, line::qsys::MapG1kResult< T >::pK, line::qsys::MapG1kResult< T >::pKvec, line::qsys::MapG1kResult< T >::plevel, qsys_mapg1k(), qsys_svclaw_coeffs(), line::qsys::SvclawCoeffsResult< T >::residual, line::qsys::MapG1kResult< T >::rho, line::qsys::MapG1kResult< T >::sigma, line::qsys::SvclawCoeffsResult< T >::Smean, line::qsys::MapG1kResult< T >::throughput, line::qsys::MapG1kResult< T >::utilization, and line::vecmul().

Referenced by qsys_mapg1k(), qsys_mapg1k(), qsys_mapg1k_perflow(), and qsys_mmapg1k().

qsys_mapg1k_perflow() [1/2]

template<class T>

MapG1kPerflowResult< T > line::qsys::qsys_mapg1k_perflow	(	const std::vector< mam::Map< T > > &	flows,
		const ServiceLaw< T > &	svc,
		std::size_t	K )

qsys_mapg1k_perflow with the qsys_mapg1k defaults tol = 1e-12, nmax = 200000.

Definition at line 145 of file qsys_mapg1k_perflow.h.

References qsys_mapg1k_perflow().

qsys_mapg1k_perflow() [2/2]

template<class T>

MapG1kPerflowResult< T > line::qsys::qsys_mapg1k_perflow	(	const std::vector< mam::Map< T > > &	flows,
		const ServiceLaw< T > &	svc,
		std::size_t	K,
		const T &	tol,
		std::size_t	nmaxCap )

Per-flow analysis of a MAP-fed tail-drop buffer.

Parameters

flows	the N arrival MAPs, whose modulating orders may differ
svc	service law, shared by all flows
K	buffer size in packets, the one in transmission included
tol	convergence tolerance
nmaxCap	cap on the level truncation

Definition at line 90 of file qsys_mapg1k_perflow.h.

References line::mam::Map< T >::D0, line::mam::Map< T >::D1, line::InputError::InputError(), line::qsys::MapG1kPerflowResult< T >::lambda, line::qsys::MapG1kPerflowResult< T >::lambdaAggregate, line::qsys::MapG1kPerflowResult< T >::lossAggregate, line::qsys::MapG1kPerflowResult< T >::lossRatio, line::mam::map_lambda(), line::qsys::MapG1kPerflowResult< T >::meanServiceTime, line::qsys::MapG1kResult< T >::meanServiceTime, line::qsys::MapG1kPerflowResult< T >::p0, line::qsys::MapG1kResult< T >::p0, line::qsys::MapG1kPerflowResult< T >::pK, line::qsys::MapG1kResult< T >::pK, qsys_mapg1k(), qsys_mapg1k_perflow(), line::qsys::MapG1kPerflowResult< T >::rho, line::qsys::MapG1kPerflowResult< T >::throughput, and line::qsys::MapG1kPerflowResult< T >::throughputAggregate.

Referenced by qsys_mapg1k_perflow(), and qsys_mapg1k_perflow().

qsys_mapm1() [1/2]

template<class T>

MapMcResult< T > line::qsys::qsys_mapm1	(	const mam::Map< T > &	arrival,
		const T &	mu )

qsys_mapm1 with 100 materialized levels.

Definition at line 85 of file qsys_mapm1.h.

References qsys_mapm1().

qsys_mapm1() [2/2]

template<class T>

MapMcResult< T > line::qsys::qsys_mapm1	(	const mam::Map< T > &	arrival,
		const T &	mu,
		std::size_t	dist_size )

MAP/M/1 by the matrix-geometric solution, i.e.

qsys_mapmc at c = 1.

Parameters

arrival	arrival MAP (D0, D1)
mu	exponential service rate
dist_size	how many entries of queueLengthDist to materialize

Definition at line 77 of file qsys_mapm1.h.

References qsys_mapm1(), and qsys_mapmc().

Referenced by qsys_mapm1(), and qsys_mapm1().

qsys_mapmap1() [1/3]

template<class T>

MapMap1Result< T > line::qsys::qsys_mapmap1	(	const mam::Map< T > &	arrival,
		const mam::Map< T > &	service )

qsys_mapmap1 with 100 materialized levels, the reference's numQLProbs.

Definition at line 251 of file qsys_mapmap1.h.

References qsys_mapmap1().

qsys_mapmap1() [2/3]

template<class T>

MapMap1Result< T > line::qsys::qsys_mapmap1	(	const mam::Map< T > &	arrival,
		const mam::Map< T > &	service,
		std::size_t	dist_size )

qsys_mapmap1 with the reference's defaults numQLMoms = 3, numSTMoms = 3.

Definition at line 243 of file qsys_mapmap1.h.

References qsys_mapmap1().

qsys_mapmap1() [3/3]

template<class T>

MapMap1Result< T > line::qsys::qsys_mapmap1	(	const mam::Map< T > &	arrival,
		const mam::Map< T > &	service,
		std::size_t	dist_size,
		std::size_t	numQLMoms,
		std::size_t	numSTMoms )

MAP/MAP/1 as the reference solves it: the service MAP collapsed to its PH marginal (mapToPh), then the MAP/PH/1 queue by the exact QBD solution.

Parameters

arrival	arrival MAP (C0, C1)
service	service MAP (D0, D1); its correlation is discarded, as in the reference (see the file comment)
dist_size	how many entries of queueLengthDist to materialize (the reference's numQLProbs)
numQLMoms	how many raw queue-length moments to return
numSTMoms	how many raw sojourn-time moments to return

Definition at line 177 of file qsys_mapmap1.h.

References line::mam::Map< T >::D0, line::mam::Mmap< T >::D0, line::mam::Map< T >::D1, line::mam::Mmap< T >::D1, line::mam::Mmap< T >::Dc, line::InputError::InputError(), line::mam::map_lambda(), line::Matrix< T >::Matrix(), line::qsys::MapMap1Result< T >::meanQueueLength, line::qsys::MapMap1Result< T >::meanSojournTime, line::qsys::MapMap1Result< T >::meanWaitingTime, line::mam::mmapph1fcfs_stdistr_ph(), line::mam::QbdMapMap1Result< T >::pi0, line::mam::QbdMapMap1Result< T >::pqueue, line::mam::qbd_mapmap1(), line::mam::qbd_qlen_moment(), qsys_mapmap1(), line::qsys::MapMap1Result< T >::queueLengthDist, line::qsys::MapMap1Result< T >::queueLengthMoments, line::mam::QbdMapMap1Result< T >::R, line::Matrix< T >::rows(), line::qsys::MapMap1Result< T >::sojournTimeMoments, and line::qsys::MapMap1Result< T >::utilization.

Referenced by qsys_mapg1(), qsys_mapmap1(), qsys_mapmap1(), qsys_mapmap1(), qsys_mapph1(), and qsys_phph1().

qsys_mapmapc()

template<class T>

MapPhcResult< T > line::qsys::qsys_mapmapc	(	const mam::Map< T > &	arrival,
		const mam::Map< T > &	service,
		unsigned	c,
		std::size_t	dist_size = 500,
		std::size_t	num_w_moms = 3 )

MAP/MAP/c FCFS with the service phase carried in start order, exactly.

Parameters

arrival	arrival MAP (D0, D1) of order ma
service	service MAP (S0, S1) of order p
c	number of servers, c >= 1
dist_size	cap on the queue length probabilities materialized
num_w_moms	how many waiting-time moments to return (0: E[Wq] by Little)

Returns the MapPhcResult struct; phaseCount is the repeating configuration count.

Definition at line 65 of file qsys_mapmapc.h.

References line::Matrix< T >::cols(), line::mam::Map< T >::D0, line::mam::Map< T >::D1, line::eye(), line::InputError::InputError(), line::inverse(), line::mam::ldqbd_mphc(), line::mam::map_lambda(), line::matmul(), line::qsys::MapPhcResult< T >::meanQueueLength, line::qsys::MapPhcResult< T >::meanSojournTime, line::qsys::MapPhcResult< T >::meanWaitingTime, line::mulvec(), line::ones(), line::qsys::MapPhcResult< T >::phaseCount, line::qsys::MapPhcResult< T >::probWait, line::mam::LdqbdMphcBlocks< T >::Q0, line::mam::LdqbdMphcBlocks< T >::Q1, line::mam::LdqbdMphcBlocks< T >::Q2, line::mam::qbd_R_logred(), qsys_mapmapc(), line::qsys::MapPhcResult< T >::queueLengthDist, line::Matrix< T >::rows(), line::solve(), line::qsys::MapPhcResult< T >::utilization, and line::qsys::MapPhcResult< T >::waitingTimeMoments.

Referenced by qsys_mapmapc(), and line::mam::solver_mam_basic_mmap_inner().

qsys_mapmc() [1/2]

template<class T>

MapMcResult< T > line::qsys::qsys_mapmc	(	const mam::Map< T > &	arrival,
		const T &	mu,
		unsigned	c )

qsys_mapmc with 500 materialized levels, the reference's maxNumComp default.

Definition at line 224 of file qsys_mapmc.h.

References qsys_mapmc().

qsys_mapmc() [2/2]

template<class T>

MapMcResult< T > line::qsys::qsys_mapmc	(	const mam::Map< T > &	arrival,
		const T &	mu,
		unsigned	c,
		std::size_t	dist_size )

MAP/M/c by the matrix-geometric solution.

Parameters

arrival	arrival MAP (D0, D1)
mu	exponential service rate of one server
c	number of servers, c >= 1
dist_size	how many entries of queueLengthDist to materialize

Definition at line 104 of file qsys_mapmc.h.

References line::mam::Map< T >::D0, line::mam::Map< T >::D1, line::InputError::InputError(), line::inverse(), line::mam::map_lambda(), line::matmul(), line::qsys::MapMcResult< T >::meanQueueLength, line::qsys::MapMcResult< T >::meanSojournTime, line::qsys::MapMcResult< T >::meanWaitingTime, line::mulvec(), line::ones(), line::mam::qbd_fundmat(), qsys_mapmc(), line::qsys::MapMcResult< T >::queueLengthDist, line::solve(), line::qsys::MapMcResult< T >::utilization, and line::vecmul().

Referenced by qsys_mapm1(), qsys_mapmc(), and qsys_mapmc().

qsys_mapph1() [1/2]

template<class T>

MapMap1Result< T > line::qsys::qsys_mapph1	(	const mam::Map< T > &	arrival,
		const std::vector< T > &	sigma,
		const Matrix< T > &	S )

qsys_mapph1 with 100 materialized levels, the reference's numQLProbs.

Definition at line 115 of file qsys_mapph1.h.

References qsys_mapph1().

qsys_mapph1() [2/2]

template<class T>

MapMap1Result< T > line::qsys::qsys_mapph1	(	const mam::Map< T > &	arrival,
		const std::vector< T > &	sigma,
		const Matrix< T > &	S,
		std::size_t	dist_size )

MAP/PH/1 by the exact QBD solution of the equivalent MAP/MAP/1 queue.

Parameters

arrival	arrival MAP (D0, D1)
sigma	PH service entry vector, length m
S	PH service sub-generator, m x m
dist_size	how many entries of queueLengthDist to materialize

Definition at line 106 of file qsys_mapph1.h.

References qsys_mapmap1(), and qsys_mapph1().

Referenced by qsys_mapph1(), and qsys_mapph1().

qsys_mapphc() [1/2]

template<class T>

MapPhcResult< T > line::qsys::qsys_mapphc	(	const mam::Map< T > &	arrival,
		const std::vector< T > &	alpha,
		const Matrix< T > &	S,
		unsigned	c )

Definition at line 445 of file qsys_mapphc.h.

References qsys_mapphc().

qsys_mapphc() [2/2]

template<class T>

MapPhcResult< T > line::qsys::qsys_mapphc	(	const mam::Map< T > &	arrival,
		const std::vector< T > &	alpha,
		const Matrix< T > &	S,
		unsigned	c,
		std::size_t	dist_size,
		std::size_t	num_w_moms,
		const std::vector< T > &	w_points )

MAP/PH/c FCFS, exactly.

Parameters

arrival	arrival MAP (D0, D1) of order ma
alpha	PH service initial vector of order ms
S	PH service sub-generator of order ms
c	number of servers, c >= 1
dist_size	cap on the queue length probabilities materialized
num_w_moms	how many waiting-time moments to return
w_points	times at which to evaluate P(Wq > t)

Definition at line 142 of file qsys_mapphc.h.

References line::Matrix< T >::cols(), line::mam::Map< T >::D0, line::mam::Map< T >::D1, line::expm(), line::eye(), line::InputError::InputError(), line::inverse(), line::mam::map_lambda(), line::matmul(), line::qsys::MapPhcResult< T >::meanQueueLength, line::qsys::MapPhcResult< T >::meanSojournTime, line::qsys::MapPhcResult< T >::meanWaitingTime, line::mulvec(), line::ones(), line::qsys::MapPhcResult< T >::phaseCount, line::qsys::MapPhcResult< T >::probWait, line::mam::qbd_R_logred(), qsys_mapphc(), line::qsys::MapPhcResult< T >::queueLengthDist, line::Matrix< T >::rows(), line::solve(), line::qsys::MapPhcResult< T >::utilization, line::qsys::MapPhcResult< T >::waitingTimeCCDF, line::qsys::MapPhcResult< T >::waitingTimeMoments, and line::qsys::MapPhcResult< T >::waitingTimePoints.

Referenced by qsys_mapphc(), and qsys_mapphc().

qsys_maxima_twomoment()

template<class T>

MaximaResult< T > line::qsys::qsys_maxima_twomoment	(	std::size_t	n,
		const T &	mean,
		const T &	cs2,
		const T &	q = num_traits<T>::from_int(0),
		bool	exactFitted = true )

Two-moment approximation for the maximum of n iid non-negative variables.

Parameters

n	the number of samples
mean	the mean of the underlying law
cs2	its squared coefficient of variation
q	a quantile level in (0,1); non-positive returns the mean
exactFitted	also compute the maximum exactly from the fitted law

Definition at line 76 of file qsys_maxima_twomoment.h.

References line::qsys::MaximaResult< T >::eta, line::qsys::MaximaResult< T >::exactFittedValue, line::qsys::MaximaResult< T >::family, line::qsys::MaximaResult< T >::hasFitted, line::InputError::InputError(), qsys_maxima_twomoment(), line::qsys::MaximaResult< T >::reliable, line::qsys::MaximaResult< T >::slope, line::qsys::MaximaResult< T >::threshold, and line::qsys::MaximaResult< T >::value.

Referenced by qsys_maxima_twomoment().

qsys_mdc_crommelin()

template<class T>

MDcCrommelinResult< T > line::qsys::qsys_mdc_crommelin	(	const T &	lambda_arr,
		const T &	s,
		unsigned	c,
		long	truncation = -1 )

M/D/c by Crommelin's embedded chain.

Parameters

lambda_arr	Poisson arrival rate
s	deterministic service time
c	number of servers
truncation	state-space cap; <= 0 selects the reference's automatic level

Definition at line 69 of file qsys_mdc_crommelin.h.

References line::InputError::InputError(), line::qsys::MDcCrommelinResult< T >::meanQueueLength, line::qsys::MDcCrommelinResult< T >::meanSojournTime, line::qsys::MDcCrommelinResult< T >::meanWaitingQueue, line::qsys::MDcCrommelinResult< T >::meanWaitingTime, qsys_mdc_crommelin(), line::solve(), and line::qsys::MDcCrommelinResult< T >::utilization.

Referenced by qsys_mdc_crommelin().

qsys_mg1()

template<class T>

QsysResult< T > line::qsys::qsys_mg1	(	const T &	lambda,
		const T &	mu,
		const T &	cs )

Exact mean response time of the M/G/1 queue (Pollaczek-Khinchine).

Parameters

lambda	arrival rate
mu	service rate
cs	coefficient of variation of the service time

Returns

W = mean response time, rhohat = Q/(1+Q)

Definition at line 39 of file qsys_mg1.h.

References line::num_pow_int(), and qsys_mg1().

Referenced by qsys_gg1(), qsys_gig1_approx_myskja2(), qsys_mg1(), and line::mva::solver_mva_qsys_analyzer().

qsys_mg1_fb()

template<class T>

Mg1DisciplineResult< T > line::qsys::qsys_mg1_fb	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu,
		const std::vector< T > &	cs )

M/G/1 under FB (feedback), also called LAS (least attained service).

Parameters

lambda	per-class arrival rates
mu	per-class service rates
cs	per-class coefficients of variation of the service time

Definition at line 82 of file qsys_mg1_fb.h.

References line::num_abs(), line::NumericError::NumericError(), qsys_mg1_fb(), line::qsys::Mg1DisciplineResult< T >::rhohat, and line::qsys::Mg1DisciplineResult< T >::W.

Referenced by qsys_mg1_fb(), and line::mva::solver_mva_qsys_sizebased_analyzer().

qsys_mg1_lrpt()

template<class T>

Mg1DisciplineResult< T > line::qsys::qsys_mg1_lrpt	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu,
		const std::vector< T > &	cs )

M/G/1 under LRPT (longest remaining processing time).

Parameters

lambda	per-class arrival rates
mu	per-class service rates
cs	per-class coefficients of variation of the service time

Definition at line 70 of file qsys_mg1_lrpt.h.

References line::num_abs(), qsys_mg1_lrpt(), line::qsys::Mg1DisciplineResult< T >::rhohat, and line::qsys::Mg1DisciplineResult< T >::W.

Referenced by qsys_mg1_lrpt(), and line::mva::solver_mva_qsys_sizebased_analyzer().

qsys_mg1_prio()

template<class T>

Mg1PrioResult< T > line::qsys::qsys_mg1_prio	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu,
		const std::vector< T > &	cs )

M/G/1 with non-preemptive head-of-line priorities: per-class mean response times from the Cobham/Kleinrock formula.

Parameters

lambda	per-class arrival rates, highest priority first
mu	per-class service rates
cs	per-class coefficients of variation of the service time

Definition at line 58 of file qsys_mg1_prio.h.

References line::InputError::InputError(), qsys_mg1_prio(), line::qsys::Mg1PrioResult< T >::rhohat, and line::qsys::Mg1PrioResult< T >::W.

Referenced by qsys_mg1_prio(), and line::mva::solver_mva_qsys_prio_analyzer().

qsys_mg1_ps() [1/2]

Mg1PsResult line::qsys::qsys_mg1_ps ( double lambda, const std::function< Cplx(const Cplx &)> & bhat, double m1, const Mg1PsOptions & opt )

inline

Sojourn-time distribution of the M/G/1-PS queue for a service law given by its Laplace-Stieltjes transform, the reference's second form qsys_mg1_ps(lambda, bhat, m1, ...).

f(tau; s) is inverted in tau numerically on a Bromwich contour placed 0.25 (1-rho)/m1 to the right of the dominant root tau*(s). The service density opt.pdf is needed to remove the conditioning (lstUncond, lstUncondVal, m2Uncond, varUncond are NaN or unavailable without it), m2 is NaN, and a time grid opt.t is refused, as in the reference: inverting a numerically inverted transform is unstable in double precision.

Parameters

lambda	arrival rate, finite and positive
bhat	service LST, accepting complex arguments
m1	mean service time
opt	grids and tuning

Definition at line 731 of file qsys_mg1_ps.h.

References line::InputError::InputError(), and qsys_mg1_ps().

qsys_mg1_ps() [2/2]

Mg1PsResult line::qsys::qsys_mg1_ps ( double lambda, const std::vector< double > & alpha, const Matrix< double > & Tmat, const Mg1PsOptions & opt )

inline

Sojourn-time distribution of the M/G/1 processor-sharing queue.

Parameters

lambda	arrival rate, finite and positive
alpha	phase-type initial probability vector
Tmat	phase-type subgenerator
opt	grids and tuning

Definition at line 485 of file qsys_mg1_ps.h.

References line::Matrix< T >::cols(), line::expm(), line::InputError::InputError(), line::ones(), qsys_mg1_ps(), line::Matrix< T >::rows(), and line::solve().

Referenced by qsys_mg1_ps(), and qsys_mg1_ps().

qsys_mg1_psjf()

template<class T>

Mg1DisciplineResult< T > line::qsys::qsys_mg1_psjf	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu,
		const std::vector< T > &	cs )

M/G/1 under PSJF (preemptive shortest job first).

Parameters

lambda	per-class arrival rates
mu	per-class service rates
cs	per-class coefficients of variation of the service time

Definition at line 79 of file qsys_mg1_psjf.h.

References line::num_abs(), line::NumericError::NumericError(), qsys_mg1_psjf(), line::qsys::Mg1DisciplineResult< T >::rhohat, and line::qsys::Mg1DisciplineResult< T >::W.

Referenced by qsys_mg1_psjf(), and line::mva::solver_mva_qsys_sizebased_analyzer().

qsys_mg1_setf()

template<class T>

Mg1DisciplineResult< T > line::qsys::qsys_mg1_setf	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu,
		const std::vector< T > &	cs )

M/G/1 under SETF (shortest elapsed time first), the non-preemptive counterpart of FB/LAS.

Parameters

lambda	per-class arrival rates
mu	per-class service rates
cs	per-class coefficients of variation of the service time

Definition at line 101 of file qsys_mg1_setf.h.

References line::NumericError::NumericError(), qsys_mg1_setf(), line::qsys::Mg1DisciplineResult< T >::rhohat, and line::qsys::Mg1DisciplineResult< T >::W.

Referenced by qsys_mg1_setf(), and line::mva::solver_mva_qsys_sizebased_analyzer().

qsys_mg1_srpt()

template<class T>

Mg1DisciplineResult< T > line::qsys::qsys_mg1_srpt	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu,
		const std::vector< T > &	cs )

M/G/1 under SRPT (shortest remaining processing time), by the Schrage-Miller formula.

Parameters

lambda	per-class arrival rates
mu	per-class service rates
cs	per-class coefficients of variation of the service time

Definition at line 183 of file qsys_mg1_srpt.h.

References line::NumericError::NumericError(), qsys_mg1_srpt(), line::qsys::Mg1DisciplineResult< T >::rhohat, and line::qsys::Mg1DisciplineResult< T >::W.

Referenced by qsys_mg1_srpt(), and line::mva::solver_mva_qsys_sizebased_analyzer().

qsys_mg1k_loss()

template<class T, class Density>

Mg1kLossResult< T > line::qsys::qsys_mg1k_loss	(	const T &	lambda,
		Density &&	density,
		unsigned	K )

Exact M/G/1/K loss probability, via the chain embedded at service-start epochs.

Parameters

lambda	Poisson arrival rate
density	service-time density, callable as density(t) -> T
K	system capacity, jobs in service included, K >= 2

Definition at line 105 of file qsys_mg1k_loss.h.

References line::mc::dtmc_solve(), line::InputError::InputError(), line::qsys::Mg1kLossResult< T >::lossProbability, line::num_factorial(), line::num_pow_int(), line::NumericError::NumericError(), qsys_mg1k_loss(), and line::qsys::Mg1kLossResult< T >::utilization.

Referenced by qsys_mg1k_loss().

qsys_mg1k_loss_mgs()

template<class T>

Mg1kLossMgsResult< T > line::qsys::qsys_mg1k_loss_mgs	(	const T &	lambda,
		const T &	mu,
		const T &	mu_scv,
		unsigned	K )

MacGregor Smith's closed-form approximation of the M/G/1/K loss probability.

Parameters

lambda	arrival rate
mu	service rate
mu_scv	squared coefficient of variation of the service time
K	system capacity, jobs in service included

Definition at line 55 of file qsys_mg1k_loss_mgs.h.

References line::NumericError::NumericError(), qsys_mg1k_loss_mgs(), and line::qsys::Mg1kLossMgsResult< T >::utilization.

Referenced by qsys_mg1k_loss_mgs(), and line::mva::solver_mva_qsys_analyzer().

qsys_mgck_sync() [1/2]

template<class T>

MgckSyncResult< T > line::qsys::qsys_mgck_sync	(	const T &	lambda,
		std::size_t	c,
		std::size_t	K,
		const ServiceLaw< T > &	svc,
		const std::string &	variant,
		std::size_t	nphases,
		const T &	tol,
		std::size_t	nmaxCap )

Synchronous approximation of the M/G/c/K queue with dual bounds.

Parameters

lambda	Poisson arrival rate (> 0)
c	number of identical FCFS servers (>= 1)
K	system capacity, jobs in service included (K >= c)
svc	service law, as for qsys_mapg1k
variant	"faster" (lower bounds on the queue) or "slower" (upper bounds)
nphases	0 for the approximation without expansion; N >= 1 for the synchronous phase expansion, deterministic service only
tol	uniformization truncation tolerance
nmaxCap	cap on the uniformization order

Definition at line 425 of file qsys_mgck_sync.h.

References line::qsys::MgckSyncResult< T >::analyzer, line::qsys::ServiceLaw< T >::det, Deterministic, line::qsys::mgck_sync_detail::expanded(), line::InputError::InputError(), line::qsys::ServiceLaw< T >::kind, line::qsys::MgckSyncResult< T >::lossProbability, line::qsys::MgckSyncResult< T >::meanQueueLength, line::qsys::MgckSyncResult< T >::meanQueueLengthQ, line::qsys::MgckSyncResult< T >::meanSojournTime, line::qsys::MgckSyncResult< T >::meanWaitingTime, line::qsys::MgckSyncResult< T >::nphases, line::qsys::MgckSyncResult< T >::p0, line::qsys::MgckSyncResult< T >::pK, line::qsys::mgck_sync_detail::plain(), qsys_mgck_sync(), line::qsys::MgckSyncResult< T >::queueLengthDist, line::qsys::MgckSyncResult< T >::stateSpaceSize, line::qsys::MgckSyncResult< T >::throughput, line::UnsupportedError::UnsupportedError(), line::qsys::MgckSyncResult< T >::utilization, and line::qsys::MgckSyncResult< T >::variant.

Referenced by qsys_mgck_sync(), qsys_mgck_sync(), and line::mam::solver_mam_sync().

qsys_mgck_sync() [2/2]

template<class T>

MgckSyncResult< T > line::qsys::qsys_mgck_sync	(	const T &	lambda,
		std::size_t	c,
		std::size_t	K,
		const ServiceLaw< T > &	svc,
		const std::string &	variant,
		std::size_t	nphases = 0 )

qsys_mgck_sync with the reference defaults tol = 1e-12, nmax = 200000.

Definition at line 487 of file qsys_mgck_sync.h.

References qsys_mgck_sync().

qsys_mginf() [1/2]

template<class T>

MginfResult< T > line::qsys::qsys_mginf	(	const T &	lambda,
		const T &	mu )

Exact solution of the M/G/infinity queue.

Parameters

lambda	arrival rate
mu	service rate, mean service time 1/mu

Definition at line 53 of file qsys_mginf.h.

References line::qsys::MginfResult< T >::has_pk, line::qsys::MginfResult< T >::L, line::qsys::MginfResult< T >::Lq, line::qsys::MginfResult< T >::p0, line::qsys::MginfResult< T >::pk, qsys_mginf(), line::qsys::MginfResult< T >::W, and line::qsys::MginfResult< T >::Wq.

Referenced by qsys_mginf(), and qsys_mginf().

qsys_mginf() [2/2]

template<class T>

MginfResult< T > line::qsys::qsys_mginf	(	const T &	lambda,
		const T &	mu,
		unsigned	k )

Exact solution of the M/G/infinity queue.

Parameters

k	state whose probability is additionally returned
lambda	arrival rate
mu	service rate

Definition at line 78 of file qsys_mginf.h.

References line::qsys::MginfResult< T >::has_pk, line::num_factorial(), line::num_pow_int(), line::qsys::MginfResult< T >::pk, and qsys_mginf().

qsys_mgisrgi_whitt()

template<class T>

QsysAbandonResult< T > line::qsys::qsys_mgisrgi_whitt	(	const T &	lambda,
		const T &	mu,
		unsigned	s,
		double	r,
		const Patience< T > &	patience,
		const MgisrgiOptions &	opts = MgisrgiOptions() )

Engineering solution of the call-center model M/GI/s/r+GI.

Parameters

lambda	arrival rate
mu	service rate of one server, the reciprocal of the mean service time
s	number of servers, s >= 1
r	extra waiting spaces; infinity for an unbounded queue
patience	the patience law
opts	cdf times, truncation controls and inversion settings

Definition at line 228 of file qsys_mgisrgi_whitt.h.

References line::qsys::QsysAbandonResult< T >::abandonRate, line::qsys::QsysAbandonResult< T >::abandonRates, line::qsys::QsysAbandonResult< T >::cdfWait, line::qsys::QsysAbandonResult< T >::cdfWaitAbandon, line::qsys::QsysAbandonResult< T >::cdfWaitServed, Exponential, line::qsys::QsysAbandonResult< T >::exponentialPatience, line::qsys::Patience< T >::form, line::InputError::InputError(), line::lti::laplace_invert(), line::lti::laplace_method(), line::qsys::QsysAbandonResult< T >::meanNumber, line::qsys::QsysAbandonResult< T >::meanQueueLength, line::qsys::QsysAbandonResult< T >::meanWait, line::qsys::QsysAbandonResult< T >::meanWaitAbandon, line::qsys::QsysAbandonResult< T >::meanWaitServed, line::qsys::QsysAbandonResult< T >::numWaitingSpaces, line::qsys::QsysAbandonResult< T >::patienceRate, line::qsys::QsysAbandonResult< T >::probAbandon, line::qsys::QsysAbandonResult< T >::probLoss, line::qsys::QsysAbandonResult< T >::probNoWait, line::qsys::QsysAbandonResult< T >::probServed, qsys_mgisrgi_whitt(), line::qsys::QsysAbandonResult< T >::queueLengthDist, line::qsys::QsysAbandonResult< T >::secondMomentWait, line::qsys::Patience< T >::theta, line::qsys::QsysAbandonResult< T >::throughput, line::qsys::QsysAbandonResult< T >::totalAbandonRates, line::qsys::QsysAbandonResult< T >::utilization, line::qsys::QsysAbandonResult< T >::varNumber, line::qsys::QsysAbandonResult< T >::varQueueLength, line::qsys::QsysAbandonResult< T >::varWaitAbandon, line::qsys::QsysAbandonResult< T >::varWaitServed, and line::qsys::QsysAbandonResult< T >::waitPoints.

Referenced by qsys_erlanga(), qsys_mgisrgi_whitt(), and line::mva::solver_mva_qsys_analyzer().

qsys_mm1()

template<class T>

QsysResult< T > line::qsys::qsys_mm1	(	const T &	lambda,
		const T &	mu )

Exact mean response time of the M/M/1 queue.

Parameters

lambda	arrival rate
mu	service rate

Returns

W = mean response time, rhohat = rho (MATLAB returns the true rho here)

Definition at line 35 of file qsys_mm1.h.

References qsys_mm1().

Referenced by line::fj::fj_respt_2way(), line::fj::fj_rmax(), line::fj::fj_synch_delay(), qsys_gg1(), qsys_mm1(), and line::mva::solver_mva_qsys_analyzer().

qsys_mm1_dps() [1/2]

template<class T>

Mm1DpsResult< T > line::qsys::qsys_mm1_dps	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu,
		const std::vector< T > &	w )

MATLAB defaults: tol = 1e-10, maxCutoff = 2048.

Definition at line 193 of file qsys_mm1_dps.h.

References qsys_mm1_dps().

qsys_mm1_dps() [2/2]

template<class T>

Mm1DpsResult< T > line::qsys::qsys_mm1_dps	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu,
		const std::vector< T > &	w,
		const T &	tol,
		unsigned	maxCutoff )

Multiclass M/M/1 under DPS (discriminatory processor sharing), solved numerically on the truncated population chain.

Parameters

lambda	per-class Poisson arrival rates
mu	per-class exponential service rates
w	per-class DPS weights, positive
tol	convergence tolerance on the mean counts (MATLAB 1e-10)
maxCutoff	hard bound on the truncation level (MATLAB 2048)

Definition at line 148 of file qsys_mm1_dps.h.

References line::InputError::InputError(), line::num_abs(), qsys_mm1_dps(), line::qsys::Mm1DpsResult< T >::rho, and line::qsys::Mm1DpsResult< T >::T_.

Referenced by qsys_mm1_dps(), qsys_mm1_dps(), and line::mva::solver_mva_dps_exact().

qsys_mm1_lindley() [1/2]

template<class T>

LindleyResult< T > line::qsys::qsys_mm1_lindley	(	const T &	lambda,
		const T &	mu,
		const std::vector< T > &	Wn,
		unsigned	mmax = 2 )

Conditional waiting-time moments of the M/M/1 Lindley recursion.

Parameters

lambda	arrival rate, positive
mu	service rate, positive
Wn	waiting times of customer n, finite and nonnegative
mmax	highest moment order; raised to 2 when smaller, as in MATLAB

Definition at line 71 of file qsys_mm1_lindley.h.

References line::qsys::LindleyResult< T >::analyzer, line::InputError::InputError(), line::Matrix< T >::Matrix(), line::qsys::LindleyResult< T >::mean, line::qsys::LindleyResult< T >::mmax, line::qsys::LindleyResult< T >::moments, qsys_lindley_moment(), qsys_mm1_lindley(), and line::qsys::LindleyResult< T >::var.

Referenced by qsys_mm1_lindley(), and qsys_mm1_lindley().

qsys_mm1_lindley() [2/2]

template<class T>

LindleyResult< T > line::qsys::qsys_mm1_lindley	(	const T &	lambda,
		const T &	mu,
		const T &	Wn,
		unsigned	mmax = 2 )

Scalar overload.

Definition at line 106 of file qsys_mm1_lindley.h.

References qsys_mm1_lindley().

qsys_mm1_ps()

template<class T>

Mm1PsResult< T > line::qsys::qsys_mm1_ps	(	const std::vector< T > &	lambda,
		const std::vector< T > &	mu )

Exact sojourn-time moments of the multiclass M/M/1-PS queue.

Parameters

lambda	per-class Poisson arrival rates, nonnegative
mu	per-class exponential service rates, positive

Definition at line 63 of file qsys_mm1_ps.h.

References line::qsys::Mm1PsResult< T >::alpha, line::InputError::InputError(), qsys_mm1_ps(), line::qsys::Mm1PsResult< T >::W, and line::qsys::Mm1PsResult< T >::W2.

Referenced by qsys_mm1_ps().

qsys_mm1_tandem_lindley() [1/2]

template<class T>

Mm1TandemLindleyResult< T > line::qsys::qsys_mm1_tandem_lindley	(	const T &	lambda,
		const T &	mu1,
		const T &	mu2,
		const std::vector< T > &	Wk,
		const std::vector< T > &	Wk1 )

Conditional waiting time at the SECOND station of an M/M/1 -> /M/1 tandem.

Parameters

lambda	arrival rate at station 1, positive
mu1	service rate at station 1, positive
mu2	service rate at station 2, positive
Wk	waiting times at station 1, finite nonnegative
Wk1	waiting times at station 2, same length as Wk

Definition at line 97 of file qsys_mm1_tandem_lindley.h.

References line::qsys::Mm1TandemLindleyResult< T >::analyzer, line::qsys::Mm1TandemLindleyResult< T >::idleProb, line::InputError::InputError(), line::qsys::Mm1TandemLindleyResult< T >::interdepMean, line::qsys::Mm1TandemLindleyResult< T >::mean, line::num_abs(), and qsys_mm1_tandem_lindley().

Referenced by qsys_mm1_tandem_lindley(), and qsys_mm1_tandem_lindley().

qsys_mm1_tandem_lindley() [2/2]

template<class T>

Mm1TandemLindleyResult< T > line::qsys::qsys_mm1_tandem_lindley	(	const T &	lambda,
		const T &	mu1,
		const T &	mu2,
		const T &	Wk,
		const T &	Wk1 )

Scalar overload.

Definition at line 141 of file qsys_mm1_tandem_lindley.h.

References qsys_mm1_tandem_lindley().

qsys_mm1k_loss()

template<class T>

Mm1kLossResult< T > line::qsys::qsys_mm1k_loss	(	const T &	lambda,
		const T &	mu,
		unsigned	K )

Blocking probability of the M/M/1/K queue.

Parameters

lambda	arrival rate
mu	service rate
K	system capacity, jobs in service included, K >= 1

Definition at line 53 of file qsys_mm1k_loss.h.

References line::lang::GlobalConstants::FineTol, line::InputError::InputError(), line::qsys::Mm1kLossResult< T >::lossProbability, line::num_pow_int(), qsys_mm1k_loss(), and line::qsys::Mm1kLossResult< T >::utilization.

Referenced by qsys_mm1k_loss(), and line::nc::solver_nc_solve().

qsys_mmapg1k() [1/2]

template<class T>

MmapG1kResult< T > line::qsys::qsys_mmapg1k	(	const Matrix< T > &	D0,
		const std::vector< Matrix< T > > &	D1c,
		const ServiceLaw< T > &	svc,
		std::size_t	K )

qsys_mmapg1k with the qsys_mapg1k defaults tol = 1e-12, nmax = 200000.

Definition at line 146 of file qsys_mmapg1k.h.

References qsys_mmapg1k().

qsys_mmapg1k() [2/2]

template<class T>

MmapG1kResult< T > line::qsys::qsys_mmapg1k	(	const Matrix< T > &	D0,
		const std::vector< Matrix< T > > &	D1c,
		const ServiceLaw< T > &	svc,
		std::size_t	K,
		const T &	tol,
		std::size_t	nmaxCap )

MMAP[K]/G/1/K with tail drop.

Parameters

D0	hidden transition matrix of the arrival MMAP (M x M)
D1c	per-class arrival matrices; D0 + sum_k D1c[k] must be an irreducible generator
svc	service law, shared by all classes
K	buffer size in packets
tol	convergence tolerance
nmaxCap	cap on the level truncation

Definition at line 89 of file qsys_mmapg1k.h.

References line::Matrix< T >::cols(), line::mam::Map< T >::D0, line::mam::Map< T >::D1, line::InputError::InputError(), line::qsys::MmapG1kResult< T >::lambda, line::qsys::MmapG1kResult< T >::lambdaAggregate, line::qsys::MmapG1kResult< T >::lossAggregate, line::qsys::MapG1kResult< T >::lossProbability, line::qsys::MmapG1kResult< T >::lossRatio, line::mam::map_prob(), line::Matrix< T >::Matrix(), line::qsys::MapG1kResult< T >::meanQueueLength, line::qsys::MmapG1kResult< T >::meanQueueLength, line::qsys::MapG1kResult< T >::meanServiceTime, line::qsys::MmapG1kResult< T >::meanServiceTime, line::mulvec(), line::ones(), line::qsys::MapG1kResult< T >::p0, line::qsys::MmapG1kResult< T >::p0, line::qsys::MapG1kResult< T >::pK, line::qsys::MmapG1kResult< T >::pK, line::qsys::MapG1kResult< T >::pKvec, line::qsys::MmapG1kResult< T >::pKvec, line::qsys::MapG1kResult< T >::plevel, line::qsys::MmapG1kResult< T >::plevel, qsys_mapg1k(), qsys_mmapg1k(), line::qsys::MapG1kResult< T >::rho, line::qsys::MmapG1kResult< T >::rho, line::Matrix< T >::rows(), line::qsys::MmapG1kResult< T >::throughput, line::qsys::MmapG1kResult< T >::throughputAggregate, line::qsys::MapG1kResult< T >::utilization, and line::qsys::MmapG1kResult< T >::utilization.

Referenced by qsys_mmapg1k(), and qsys_mmapg1k().

qsys_mmapgk1() [1/5]

template<class T>

MmapGk1Result< T > line::qsys::qsys_mmapgk1	(	const std::vector< Matrix< T > > &	MMAP,
		const std::vector< lang::Distrib< T > > &	svc )

qsys_mmapgk1 on Distribution objects with the reference's defaults.

Definition at line 828 of file qsys_mmapgk1.h.

References qsys_mmapgk1().

qsys_mmapgk1() [2/5]

template<class T>

MmapGk1Result< T > line::qsys::qsys_mmapgk1	(	const std::vector< Matrix< T > > &	MMAP,
		const std::vector< lang::Distrib< T > > &	svc,
		const std::vector< T > &	w_points,
		std::size_t	num_w_moms,
		double	tol,
		std::size_t	iter_max )

qsys_mmapgk1 on Distribution objects.

Definition at line 815 of file qsys_mmapgk1.h.

References qsys_mmapgk1().

qsys_mmapgk1() [3/5]

template<class T>

MmapGk1Result< T > line::qsys::qsys_mmapgk1	(	const std::vector< Matrix< T > > &	MMAP,
		const std::vector< MmapGk1Service< T > > &	svc )

qsys_mmapgk1 on service variants with the reference's defaults.

Definition at line 836 of file qsys_mmapgk1.h.

References qsys_mmapgk1().

qsys_mmapgk1() [4/5]

template<class T>

MmapGk1Result< T > line::qsys::qsys_mmapgk1	(	const std::vector< Matrix< T > > &	MMAP,
		const std::vector< MmapGk1Service< T > > &	svc,
		const std::vector< T > &	w_points,
		std::size_t	num_w_moms,
		double	tol,
		std::size_t	iter_max )

qsys_mmapgk1 on service variants with the reference's quadNodes = 120.

Definition at line 805 of file qsys_mmapgk1.h.

References qsys_mmapgk1().

qsys_mmapgk1() [5/5]

template<class T>

MmapGk1Result< T > line::qsys::qsys_mmapgk1	(	const std::vector< Matrix< T > > &	MMAP,
		const std::vector< MmapGk1Service< T > > &	svc,
		const std::vector< T > &	w_points,
		std::size_t	num_w_moms,
		double	tol,
		std::size_t	iter_max,
		std::size_t	quad_nodes )

MMAP[K]/G[K]/1 FCFS, per type.

Parameters

MMAP	LINE convention {D0, D1, D^(1), ..., D^(K)}, D1 = sum_k D^(k)
svc	K service laws, one per marked type, each a Distribution object, a (D0, D1) MAP pair or a transform struct
w_points	times at which to evaluate the per-type waiting time CDF
num_w_moms	how many per-type waiting time moments to return
tol	fixed point tolerance on U
iter_max	fixed point iteration cap (the reference's 10000)
quad_nodes	the reference's quadNodes (default 120): the Stieltjes fallback uses max(quadNodes, 20) * 20 CDF increments

Definition at line 623 of file qsys_mmapgk1.h.

References line::qsys::MmapGk1Result< T >::arrivalRate, line::qsys::MmapGk1Result< T >::idleVector, line::InputError::InputError(), line::inverse(), line::qsys::MmapGk1Result< T >::lambdas, line::matmul(), line::qsys::MmapGk1Result< T >::meanQueueLength, line::qsys::MmapGk1Result< T >::meanSojournTime, line::qsys::MmapGk1Result< T >::meanWaitingTime, line::mulvec(), qsys_mmapgk1(), line::Matrix< T >::rows(), line::qsys::MmapGk1Result< T >::utilization, line::qsys::MmapGk1Result< T >::waitCDF, line::qsys::MmapGk1Result< T >::waitMoments, and line::qsys::MmapGk1Result< T >::waitPoints.

Referenced by qsys_mmapgk1(), qsys_mmapgk1(), qsys_mmapgk1(), qsys_mmapgk1(), and qsys_mmapgk1().

qsys_mmapmap1()

template<class T>

MmapMap1Result< T > line::qsys::qsys_mmapmap1	(	const mam::Mmap< T > &	arrival,
		const std::vector< mam::Map< T > > &	svc,
		std::vector< bool >	carry = std::vector<bool>() )

MMAP[K]/MAP[K]/1 FCFS with per-class carried service phases, exactly.

Parameters

arrival	marked arrival process (D0, D1, Dc[0..K-1])
svc	svc[k] = (S0, S1), the service MAP of class k
carry	per-class flags of the carried classes; empty for every arriving class whose MAP is not renewal

Definition at line 96 of file qsys_mmapmap1.h.

References line::qsys::MmapMap1Result< T >::arrivalRate, line::qsys::MmapMap1Result< T >::carry, line::mam::Mmap< T >::D0, line::mam::Mmap< T >::D1, line::mam::Mmap< T >::Dc, line::eye(), line::InputError::InputError(), line::inverse(), line::qsys::mmapmap1detail::is_renewal(), line::mam::kron(), line::mam::map_pie(), line::mam::map_prob(), line::qsys::MmapMap1Result< T >::markCount, line::matmul(), line::Matrix< T >::Matrix(), line::qsys::MmapMap1Result< T >::meanQueueLength, line::qsys::MmapMap1Result< T >::meanSojournTime, line::mam::mmapph1fcfs_ncmean(), and qsys_mmapmap1().

Referenced by qsys_mmapmap1(), and line::mam::solver_mam_basic_mmap_inner().

qsys_mmcc_retrial_fp() [1/2]

template<class T>

MmccRetrialFpResult< T > line::qsys::qsys_mmcc_retrial_fp	(	const T &	lambda,
		const T &	mu,
		unsigned	c )

qsys_mmcc_retrial_fp with the reference defaults tol = 1e-10, maxiter = 10000.

Definition at line 118 of file qsys_mmcc_retrial_fp.h.

References qsys_mmcc_retrial_fp().

qsys_mmcc_retrial_fp() [2/2]

template<class T>

MmccRetrialFpResult< T > line::qsys::qsys_mmcc_retrial_fp	(	const T &	lambda,
		const T &	mu,
		unsigned	c,
		const T &	tol,
		std::size_t	maxiter )

M/M/c/c with retrials by the Cohen fixed point.

Parameters

lambda	fresh arrival rate
mu	service rate of one server
c	number of servers, which is also the capacity
tol	stopping tolerance on |r_{k+1} - r_k|
maxiter	iteration cap

Definition at line 81 of file qsys_mmcc_retrial_fp.h.

References line::qsys::MmccRetrialFpResult< T >::blockingProbability, line::qsys::MmccRetrialFpResult< T >::converged, erlang_b_recursive(), line::InputError::InputError(), line::qsys::MmccRetrialFpResult< T >::iterations, line::num_abs(), qsys_mmcc_retrial_fp(), and line::qsys::MmccRetrialFpResult< T >::retrialRate.

Referenced by qsys_mmcc_retrial_fp(), and qsys_mmcc_retrial_fp().

qsys_mmck()

template<class T>

MmckResult< T > line::qsys::qsys_mmck	(	const T &	lambda,
		const T &	mu,
		unsigned	c,
		unsigned	K )

Exact analysis of the M/M/c/K queue (truncated Erlang form).

Parameters

lambda	Poisson arrival rate
mu	exponential service rate of one server
c	number of servers, c >= 1
K	system capacity, K >= c

Definition at line 63 of file qsys_mmck.h.

References line::InputError::InputError(), line::qsys::MmckResult< T >::lossProbability, line::qsys::MmckResult< T >::meanQueueLength, line::qsys::MmckResult< T >::meanQueueLengthQ, line::qsys::MmckResult< T >::meanSojournTime, line::qsys::MmckResult< T >::meanWaitingTime, line::num_factorial(), line::num_pow_int(), line::NumericError::NumericError(), qsys_mmck(), line::qsys::MmckResult< T >::queueLengthDist, line::qsys::MmckResult< T >::throughput, and line::qsys::MmckResult< T >::utilization.

Referenced by qsys_mmck(), line::mam::solver_mam_basic_mmap_inner(), line::mam::solver_mam_decmmap(), and line::mam::solver_mna_open().

qsys_mmk()

template<class T>

QsysResult< T > line::qsys::qsys_mmk	(	const T &	lambda,
		const T &	mu,
		unsigned	k )

Exact mean response time of the M/M/k queue (Erlang-C).

Parameters

lambda	arrival rate
mu	service rate of a single server
k	number of servers, k >= 1

Returns

W = mean response time, rhohat = rho = lambda/(mu k)

Definition at line 60 of file qsys_mmk.h.

References line::InputError::InputError(), and qsys_mmk().

Referenced by qsys_gigk_approx_cosmetatos(), qsys_gigk_approx_kingman(), qsys_gigk_approx_whitt(), qsys_mmk(), line::mva::solver_mva_qsys_analyzer(), and line::nc::solver_nc().

qsys_mmk_qed()

template<class T>

QsysQedResult< T > line::qsys::qsys_mmk_qed	(	const T &	lambda,
		const T &	mu,
		unsigned	s )

Halfin-Whitt QED approximation for the M/M/s queue, and the square-root staffing rule that inverts it.

Parameters

lambda	arrival rate
mu	service rate of one server
s	number of servers, s >= 1

Definition at line 134 of file qsys_mmk_qed.h.

References line::qsys::QsysQedResult< T >::beta, line::InputError::InputError(), line::qsys::QsysQedResult< T >::meanNumber, line::qsys::QsysQedResult< T >::meanQueueLength, line::qsys::QsysQedResult< T >::meanWait, line::qsys::QsysQedResult< T >::meanWaitDelayed, line::qsys::QsysQedResult< T >::offeredLoad, line::qsys::QsysQedResult< T >::probDelay, qsys_mmk_qed(), qsys_mmk_qed_alpha(), line::qsys::QsysQedResult< T >::trafficIntensity, and line::qsys::QsysQedResult< T >::utilization.

Referenced by qsys_mmk_qed(), qsys_mmk_qed_staffing(), and line::mva::solver_mva_qsys_analyzer().

qsys_mmk_qed_alpha()

template<class T>

T line::qsys::qsys_mmk_qed_alpha

(

const T &

beta

)

The Halfin-Whitt delay-probability function alpha(beta); 1 at beta <= 0.

Parameters

beta	the QED server-slack parameter

Definition at line 91 of file qsys_mmk_qed.h.

References qsys_mmk_qed_alpha().

Referenced by qsys_mmk_qed(), qsys_mmk_qed_alpha(), and qsys_mmk_qed_staffing().

qsys_mmk_qed_erlangc()

template<class T>

T line::qsys::qsys_mmk_qed_erlangc	(	unsigned	s,
		const T &	lambda,
		const T &	mu )

Erlang C by the Erlang B recursion B_j = a B_{j-1}/(j + a B_{j-1}), which never forms a^j/j!

and so never overflows.

Parameters

s	number of servers
lambda	arrival rate
mu	service rate of one server

Definition at line 115 of file qsys_mmk_qed.h.

References qsys_mmk_qed_erlangc().

Referenced by qsys_mmk_qed_erlangc().

qsys_mmk_qed_staffing()

template<class T>

QsysQedStaffingResult< T > line::qsys::qsys_mmk_qed_staffing	(	const T &	lambda,
		const T &	mu,
		const T &	target,
		QedCriterion	crit = QedCriterion::Delay,
		const T &	deadline = num_traits<T>::from_int(0),
		const T &	level = num_traits<T>::from_int(0),
		bool	exact = false )

Square-root staffing of the M/M/s queue.

Parameters

lambda	arrival rate
mu	service rate of one server
target	the largest acceptable P(W>0) for Delay, the largest acceptable E[W] for MeanWait, unused for ServiceLevel
crit	which target to meet
deadline	the deadline of the service-level criterion
level	the probability that deadline must be met with
exact	walk s until the EXACT Erlang C measure meets the target

Definition at line 221 of file qsys_mmk_qed.h.

References line::qsys::QsysQedResult< T >::beta, line::qsys::QsysQedStaffingResult< T >::beta, line::qsys::QsysQedStaffingResult< T >::betaTarget, Delay, line::qsys::QsysQedStaffingResult< T >::exactUsed, line::InputError::InputError(), MeanWait, line::qsys::QsysQedResult< T >::meanWait, line::qsys::QsysQedStaffingResult< T >::meanWait, line::qsys::QsysQedStaffingResult< T >::numServers, line::qsys::QsysQedStaffingResult< T >::offeredLoad, line::qsys::QsysQedResult< T >::probDelay, line::qsys::QsysQedStaffingResult< T >::probDelay, qsys_mmk_qed(), qsys_mmk_qed_alpha(), qsys_mmk_qed_staffing(), ServiceLevel, and line::qsys::QsysQedStaffingResult< T >::serviceLevel.

Referenced by qsys_mmk_qed_staffing().

qsys_mtginf()

template<class T>

QsysMtginfResult< T > line::qsys::qsys_mtginf	(	const std::function< T(const T &)> &	lambdaFun,
		const std::function< T(const T &)> &	serviceCcdf,
		const T &	ES,
		const std::vector< T > &	tvals,
		double	startTime = -std::numeric_limits<double>::infinity(),
		double	ES2 = std::numeric_limits<double>::quiet_NaN(),
		const std::function< T(const T &)> &	servicePdf = std::function<T(const T&)>(),
		double	tol = 1e-12,
		std::size_t	panels = 4000,
		double	maxAge = 1e12 )

Exact time-varying analysis of the Mt/G/infinity queue.

Parameters

lambdaFun	the arrival rate; must accept arguments in the past when startTime is infinite
serviceCcdf	G^c(x) = P(S > x)
ES	the mean service time
tvals	the times at which to evaluate
startTime	time the system started empty; -Inf assumes an infinite past
ES2	the second moment of the service time; NaN skips the lag
servicePdf	the service density for the exact departure rate, or empty
tol	service-tail cut for the age integral
panels	Simpson panels for that integral
maxAge	cap on the age integrated over

Definition at line 166 of file qsys_mtginf.h.

References line::qsys::QsysMtginfResult< T >::arrivalRate, line::qsys::QsysMtginfResult< T >::departureRate, line::qsys::QsysMtginfResult< T >::hasLag, line::InputError::InputError(), line::qsys::QsysMtginfResult< T >::lagApproximation, line::qsys::QsysMtginfResult< T >::meanLag, line::qsys::QsysMtginfResult< T >::meanNumber, line::qsys::QsysMtginfResult< T >::offeredLoadPSA, qsys_mtginf(), line::qsys::QsysMtginfResult< T >::times, and line::qsys::QsysMtginfResult< T >::varNumber.

Referenced by qsys_mtginf(), qsys_mtgs0_mol(), and line::fluid::solver_fluid_qsys().

qsys_mtgs0_mol()

template<class T>

QsysMolResult< T > line::qsys::qsys_mtgs0_mol	(	const std::function< T(const T &)> &	lambdaFun,
		const std::function< T(const T &)> &	serviceCcdf,
		const T &	ES,
		unsigned	s,
		const std::vector< T > &	tvals,
		double	startTime = -std::numeric_limits<double>::infinity(),
		bool	delay = false )

Modified-offered-load and pointwise-stationary approximations for a time-varying multiserver system.

Parameters

lambdaFun	the arrival rate
serviceCcdf	G^c(x) = P(S > x)
ES	the mean service time
s	number of servers
tvals	times at which to evaluate
startTime	time the system started empty; -Inf assumes an infinite past
delay	use Erlang C rather than Erlang B

Definition at line 112 of file qsys_mtgs0_mol.h.

References line::qsys::QsysMolResult< T >::arrivalRate, line::qsys::QsysMtginfResult< T >::arrivalRate, line::InputError::InputError(), line::qsys::QsysMolResult< T >::instantLoad, line::qsys::QsysMolResult< T >::meanBusyMOL, line::qsys::QsysMtginfResult< T >::meanNumber, line::qsys::QsysMolResult< T >::offeredLoad, line::qsys::QsysMtginfResult< T >::offeredLoadPSA, line::qsys::QsysMolResult< T >::probBlockMOL, line::qsys::QsysMolResult< T >::probBlockPSA, qsys_erlang_b(), qsys_erlang_c(), qsys_mtginf(), qsys_mtgs0_mol(), line::qsys::QsysMolResult< T >::times, and line::qsys::QsysMtginfResult< T >::times.

Referenced by qsys_mtgs0_mol(), and line::fluid::solver_fluid_qsys().

qsys_mxm1()

template<class T>

MxM1Result< T > line::qsys::qsys_mxm1	(	const T &	lambda_batch,
		const T &	mu,
		const T &	E_X,
		const T &	E_X2 )

M^X/M/1: the batch-arrival queue with exponential service.

Parameters

lambda_batch	batch arrival rate
mu	service rate
E_X	mean batch size
E_X2	second raw moment of the batch size

Definition at line 59 of file qsys_mxm1.h.

References line::InputError::InputError(), line::qsys::MxM1Result< T >::Q, qsys_mxm1(), line::qsys::MxM1Result< T >::U, line::qsys::MxM1Result< T >::W, and line::qsys::MxM1Result< T >::Wq.

Referenced by qsys_mxm1(), qsys_mxm1_pmf(), and qsys_mxm1_variance().

qsys_mxm1_pmf()

template<class T>

MxM1Result< T > line::qsys::qsys_mxm1_pmf	(	const T &	lambda_batch,
		const T &	mu,
		const std::vector< T > &	batch_sizes,
		const std::vector< T > &	pmf )

Support-and-pmf form: MATLAB's qsys_mxm1(..., batch_sizes, pmf).

The pmf is renormalized, as in MATLAB, so an unnormalized weight vector is accepted.

Definition at line 85 of file qsys_mxm1.h.

References line::InputError::InputError(), qsys_mxm1(), and qsys_mxm1_pmf().

Referenced by qsys_mxm1_pmf().

qsys_mxm1_variance()

template<class T>

MxM1Result< T > line::qsys::qsys_mxm1_variance	(	const T &	lambda_batch,
		const T &	mu,
		const T &	E_X,
		const T &	Var_X )

Variance form: MATLAB's qsys_mxm1(..., Var_X, 'variance').

Definition at line 75 of file qsys_mxm1.h.

References qsys_mxm1(), and qsys_mxm1_variance().

Referenced by qsys_mxm1_variance().

qsys_phm1() [1/2]

template<class T>

PhM1Result< T > line::qsys::qsys_phm1	(	const std::vector< T > &	alpha,
		const Matrix< T > &	Tm,
		const T &	mu )

qsys_phm1 with the fzero-equivalent default bracket tolerance.

Definition at line 139 of file qsys_phm1.h.

References qsys_phm1().

qsys_phm1() [2/2]

template<class T>

PhM1Result< T > line::qsys::qsys_phm1	(	const std::vector< T > &	alpha,
		const Matrix< T > &	Tm,
		const T &	mu,
		const T &	tol )

Exact PH/M/1, the GI/M/1 queue with phase-type interarrival times.

Parameters

alpha	PH entry probability vector, length k
Tm	PH sub-generator, k x k
mu	exponential service rate
tol	width of the bisection bracket at which to stop (MATLAB's fzero stops at the double round-off level, so 1e-16 is the matching default)

Definition at line 92 of file qsys_phm1.h.

References line::Matrix< T >::cols(), line::InputError::InputError(), line::qsys::PhM1Result< T >::meanQueueLength, line::qsys::PhM1Result< T >::meanSojournTime, line::qsys::PhM1Result< T >::meanWaitingQueue, line::qsys::PhM1Result< T >::meanWaitingTime, line::ones(), qsys_phm1(), line::Matrix< T >::rows(), line::qsys::PhM1Result< T >::sigma, line::solve(), and line::qsys::PhM1Result< T >::utilization.

Referenced by qsys_phm1(), qsys_phm1(), and line::mva::solver_mva_qsys_analyzer().

qsys_phmc() [1/2]

template<class T>

PhMcResult< T > line::qsys::qsys_phmc	(	const std::vector< T > &	alpha,
		const Matrix< T > &	Tm,
		const T &	mu,
		unsigned	c )

qsys_phmc with the MATLAB defaults, 50000 iterations and tolerance 1e-14.

Definition at line 233 of file qsys_phmc.h.

References qsys_phmc().

qsys_phmc() [2/2]

template<class T>

PhMcResult< T > line::qsys::qsys_phmc	(	const std::vector< T > &	alpha,
		const Matrix< T > &	Tm,
		const T &	mu,
		unsigned	c,
		unsigned	maxIter,
		const T &	tol )

Exact PH/M/c by Neuts' matrix-geometric method.

Parameters

alpha	PH entry probability vector, length k
Tm	PH sub-generator, k x k
mu	exponential service rate of one server
c	number of servers, c >= 1
maxIter	iteration budget for the R fixed point (MATLAB 50000)
tol	convergence tolerance on R (MATLAB 1e-14)

Definition at line 97 of file qsys_phmc.h.

References line::Matrix< T >::cols(), line::InputError::InputError(), line::inverse(), line::matmul(), line::qsys::PhMcResult< T >::meanQueueLength, line::qsys::PhMcResult< T >::meanSojournTime, line::qsys::PhMcResult< T >::meanWaitingQueue, line::qsys::PhMcResult< T >::meanWaitingTime, line::mulvec(), line::num_abs(), line::ones(), qsys_phmc(), line::Matrix< T >::rows(), line::solve(), and line::qsys::PhMcResult< T >::utilization.

Referenced by qsys_phmc(), and qsys_phmc().

qsys_phph1() [1/2]

template<class T>

MapMap1Result< T > line::qsys::qsys_phph1	(	const std::vector< T > &	alpha,
		const Matrix< T > &	Tm,
		const std::vector< T > &	beta,
		const Matrix< T > &	S )

qsys_phph1 with 100 materialized levels, the reference's numQLProbs.

Definition at line 88 of file qsys_phph1.h.

References qsys_phph1().

qsys_phph1() [2/2]

template<class T>

MapMap1Result< T > line::qsys::qsys_phph1	(	const std::vector< T > &	alpha,
		const Matrix< T > &	Tm,
		const std::vector< T > &	beta,
		const Matrix< T > &	S,
		std::size_t	dist_size )

PH/PH/1 by the exact QBD solution of the equivalent MAP/MAP/1 queue.

Parameters

alpha	PH arrival entry vector, length n
Tm	PH arrival sub-generator, n x n
beta	PH service entry vector, length m
S	PH service sub-generator, m x m
dist_size	how many entries of queueLengthDist to materialize

Definition at line 78 of file qsys_phph1.h.

References qsys_mapmap1(), and qsys_phph1().

Referenced by qsys_phph1(), and qsys_phph1().

qsys_svclaw_coeffs()

template<class T>

SvclawCoeffsResult< T > line::qsys::qsys_svclaw_coeffs	(	const ServiceLaw< T > &	svc,
		const T &	theta,
		const T &	tol,
		std::size_t	nmaxCap )

qsys_svclaw_coeffs: uniformization coefficients of a service-time descriptor; see qsys_svclaw_coeffs.h.

Defined here because qsys_mapg1k calls it.

Definition at line 437 of file qsys_mapg1k.h.

References line::qsys::SvclawCoeffsResult< T >::cn, line::util::LineConsole::get_verbose(), qsys_svclaw_coeffs(), line::qsys::SvclawCoeffsResult< T >::residual, line::util::SILENT, and line::qsys::SvclawCoeffsResult< T >::Smean.

Referenced by line::qsys::mgck_sync_detail::coefficients(), qsys_mapg1k(), and qsys_svclaw_coeffs().

qsys_tandem_lindley()

template<class T>

TandemLindleyResult< T > line::qsys::qsys_tandem_lindley	(	const std::vector< T > &	A,
		const Matrix< T > &	S,
		const std::vector< T > &	W0 = std::vector<T>() )

Sample-path Lindley recursion along a tandem of single-server FCFS queues.

Parameters

A	(N) interarrival times at the first station, finite nonnegative
S	(N x K) service times, finite nonnegative
W0	(K) initial waiting times, one per station; empty means all zero

Definition at line 73 of file qsys_tandem_lindley.h.

References line::qsys::TandemLindleyResult< T >::analyzer, line::Matrix< T >::cols(), line::qsys::TandemLindleyResult< T >::departure, line::qsys::TandemLindleyResult< T >::G, line::InputError::InputError(), line::Matrix< T >::Matrix(), qsys_tandem_lindley(), line::Matrix< T >::rows(), line::qsys::TandemLindleyResult< T >::T_, and line::qsys::TandemLindleyResult< T >::W.

Referenced by qsys_tandem_lindley().

qsys_tandem_ub_ciucu()

template<class T>

TandemUbResult< T > line::qsys::qsys_tandem_ub_ciucu	(	const std::vector< T > &	x,
		const std::function< T(const T &)> &	lst,
		const std::vector< T > &	p,
		const std::vector< T > &	mu,
		const std::function< T(const T &)> &	dlst = std::function<T(const T&)>() )

Tail bounds for a GI/Hn/1 -> .

/Hn/1 tandem of two FCFS single servers.

Parameters

x	thresholds at which the tails are bounded, nonnegative
lst	interarrival transform, s -> E[e^{-s X}] for s >= 0
p	service phase probabilities, nonnegative and summing to one
mu	service phase rates, positive
dlst	s -> E[X e^{-s X}], minus the derivative of lst; empty to obtain it by a Richardson-extrapolated central difference, which costs four extra transform evaluations and loses roughly four digits

Definition at line 105 of file qsys_tandem_ub_ciucu.h.

References line::qsys::TandemUbResult< T >::A, line::qsys::TandemUbResult< T >::alpha, line::qsys::TandemUbResult< T >::analyzer, line::qsys::TandemUbResult< T >::B, line::qsys::TandemUbResult< T >::C, line::qsys::TandemUbResult< T >::D, line::InputError::InputError(), qsys_tandem_ub_ciucu(), line::RootResult< T >::root, line::root_brent(), line::qsys::TandemUbResult< T >::S, line::qsys::TandemUbResult< T >::theta, and line::qsys::TandemUbResult< T >::W.

Referenced by qsys_tandem_ub_ciucu().