Package jline.api.qsys
package jline.api.qsys
Queueing system analysis algorithms.
This package contains algorithms for analyzing individual queueing systems, including M/M/1, M/M/c, M/G/1, and other classical queueing models.
- Since:
- LINE 2.0
-
ClassDescriptionResult of D/M/c queue analysis.Result of M/D/c queue analysis via Crommelin's embedded DTMC.The patience (time-to-abandon) law of a queue with customer abandonment, in the three forms accepted by
Qsys_mgisrgi_whitt.How the abandonment rates are read off this law.Result of PH/M/1 queue analysis.Result of PH/M/c queue analysis.Analyzes a BMAP/M/1 queue by the matrix-analytic (M/G/1-type) method.Exact D/D/c/K/K finite-source queue: K sources with a deterministic operating time EO, c servers with a deterministic service time ES.Exact Engset loss system M/G/c/c/K: K sources with Exp(alpha) idle times and c servers without waiting room; a request finding all servers busy is lost.Exact analysis of the Erlang A model M/M/s/r+M.Truncated Gaussian approximation (TGA-G) for the G/GI/n+GI queue.Steady state of the G/GI/s+GI fluid model.Diffusion approximation for the G/GI/n/m queue.Extremal two-moment bounds for the GI/GI/1 queue.Exact analysis of the GI/M/c queue (Takacs 1962; Sztrik 2021, Table 21): renewal arrivals with interarrival LST A*(s), c exponential servers of rate mu, FCFS.G/M/1 Queueing System Analysis.The Gt/Mt/st+GI many-server fluid queue.Conditional waiting-time moments of the Hl/Hn/1 Lindley recursion.The MAP/G/1/K queue with tail drop: Markovian arrivals, an arbitrary service law F, and a buffer of K packets counting the one in transmission.Per-flow throughput and loss ratio of a FIFO buffer with tail drop fed by N flows of arbitrary and mutually different statistical character.The MAP/MAP/c FCFS queue whose service MAP is CARRIED in service-start order.The MAP/PH/c FCFS queue, solved exactly.Two-moment approximation for the maximum of n iid non-negative variables.Exact M/D/1 queue: the M/G/1 measures with cs = 0 and E[S^3] = 1/mu^3, plus the queue-length distribution from the embedded-chain recursion P_n = (P_0 Abar_{n-1} + sum_{j=1}^{n-1} P_j Abar_{n-j}) / a_0, where a_0 = exp(-rho) and Abar_k = P(Poisson(rho) > k).Exact M/E_r/1 queue by the method of phases: the number of phases in the system has p_n = sum_k c_k x_k^n, where x_1..x_r are the roots inside the unit disk of mu x^(r+1) - (lambda + mu) x^r + lambda = 0 and c_k = (1 - rho) / prod_{j != k} (1 - x_j / x_k).M/G/1 FCFS queue whose server fails after an exponential up time of rate alpha, busy or idle, and is restored after a general repair time R, with preemptive resume and arrivals continuing while down: the semantics of Queue.setBreakdown without down service.Exact busy-period measures of the M/G/1 queue (Takacs): the moments of the busy period and of the number served in it, the idle period, and, given the service LST, the busy-period LST from the fixed point B*(s) = S*(s + lambda - lambda B*(s)).Kleinrock's conservation law for work-conserving non-preemptive M/G/1 disciplines whose order does not use the service times: sum_i rho_i Wq_i = rho * sum_i lambda_i E[S_i^2] / (2 (1 - rho)).Exact M/G/1 queue with exceptional first service (Welch 1964): the job that starts a busy period has service law Bf (rate muf, scv csf^2), the others B (rate mu, scv cs^2).Exact M/G/1 queue with preemptive-resume priorities (class 1 highest).Sojourn time distribution of the M/G/1 processor-sharing queue.Service Laplace-Stieltjes transform, which must accept complex arguments.Service density.Exact M/G/1 queue with setup times: a server that empties switches off and needs a setup time U, drawn afresh, before serving the next arrival.Exact M/G/1 queue with non-preemptive shortest-job-first scheduling (Phipps 1956): a job of size x waits W(x) = (lambda E[S^2] / 2) / (1 - rho(x))^2, rho(x) = lambda int_0^x t dF(t), and the class waiting time is W(x) averaged over its size law.Analyzes an M/G/1 queue with SRPT (Shortest Remaining Processing Time) scheduling using the Schrage-Miller class-conditional response-time formula.M/G/1 (and M^X/G/1) FCFS queue with exhaustive service and server vacations.Exact mean performance of the M/G/1/1 retrial queue.Heavy-traffic approximation of the orbit size of the M/G/1/1 retrial queue.Gamma parameters of the orbit size and its mean and variance.Mean performance of the M/G/1/1 retrial queue.Exact M/G/1/K/K finite-source queue (Takacs 1962): K sources with Exp(alpha) operating times and one FCFS server with a general service law.Exact M/G/c-PS queue (c servers shared in processor sharing): insensitive, so the number in system has the M/M/c law.Synchronous approximation of the M/G/c/K queue with dual bounds (German, EPEW 2025).Engineering solution of the call-center model M/GI/s/r+GI.Exact M/H2/1 queue: service Exp(mu1) w.p.M/M/1 queueing system analysis.Conditional waiting-time moments of the M/M/1 Lindley recursion.Exact sojourn-time moments of the multiclass M/M/1-PS queue.Conditional downstream waiting time in an M/M/1 tandem.Exact stationary distribution of the M/M/1/1 retrial queue.Stationary probabilities and exact means.Exact per-class throughput and loss ratio of an MMAP[K]/G/1/K queue with tail drop: marked Markovian arrivals, an arbitrary service law F common to all classes, and a buffer of K packets counting the one in transmission.The MMAP[K]/G[K]/1 FCFS queue: K customer types with class-dependent GENERAL service, fed by a marked Markovian arrival process.The MMAP[K]/G[K]/1 queue with preemptive-resume last-come-first-served service: K customer types with class-dependent GENERAL service, fed by a marked Markovian arrival process.A service law given by its Laplace-Stieltjes transform and raw moments, the MATLAB struct formstruct('lst', @(s) ..., 'moments', [...]): the transform must admit a COMPLEX argument, and at least numWMoms+1 moments are needed.The MMAP[K]/MAP[K]/1 FCFS queue in which each class's service phase is CARRIED.Return value of qsys_mmapmap1, mirroring the MATLAB struct.Exact M/M/c queue with non-preemptive (HOL) priorities and a common service rate mu (Cobham 1954): with C the Erlang-C probability of the aggregate load and sigma_i = sum_{j<=i} lambda_j / (c mu), Wq_i = (C / (c mu)) / ((1 - sigma_{i-1}) (1 - sigma_i)).Expected tardiness at arrival (ETA) E[(T - d)^+], tardiness probability P(T > d) and mean response time E[T] of a class-r order in a multiclass priority M/M/c queue with a common service rate, conditional on the state the order observes on arrival (Hafizoglu, Gel and Keskinocak, INFORMS J.Results at each due date of the input.Exact stationary expected tardiness E[(T_i - d_i)^+] and tardiness probability P(T_i > d_i) of each class of a multiclass M/M/c queue with a common exponential service rate, FCFS within a class and static priorities (class 1 highest), under head-of-line ("np") or preemptive-resume ("pr") priority (Hafizoglu, Gel and Keskinocak, INFORMS J.Per-class results, in the order of the input rates.Exact Erlang loss system M/M/c/c (insensitive: also M/G/c/c with mean service 1/mu): truncated Poisson distribution of busy servers and the Erlang-B blocking probability from the stable recursion B_k = A B_{k-1} / (k + A B_{k-1}).Fixed-point approximation for the M/M/c/c retrial queue.Blocking probability, orbit-induced rate and iteration count.Result container for the M/M/c/K metrics.Exact M/M/c/K/K finite-source (machine repair) queue: K sources, each operating for an Exp(alpha) time before requesting an Exp(mu) service at one of c servers, FCFS.Halfin-Whitt QED approximation for the M/M/s queue, and the square-root staffing rule that inverts it.Exact time-varying analysis of the Mt/G/infinity queue.Modified-offered-load and pointwise-stationary approximations for a time-varying multiserver system.M^X/G/1 FCFS queue with batch Poisson arrivals.Stability (positive recurrence) of the M/M/c/c retrial queue.Verdict, slack of the condition (positive iff stable) and the rule applied.Uniformization coefficients of a service-time descriptor, shared by the qsys functions that take the law exactly rather than as a PH fit (Qsys_mapg1k,Qsys_mgck_sync).Tandem network Lindley recursion on a sample path.Tail bounds for a GI/Hn/1 -> ./Hn/1 tandem of two FCFS single servers.Steady-state measures of a multiserver queue with customer abandonment, as produced byQsys_mgisrgi_whittandQsys_erlanga.Result of the matrix-analytic (M/G/1-type) analysis of a BMAP/M/1 queue.Steady state of the G/GI/s+GI fluid model, as produced byQsys_ggisgi_fluid.One conditional Lindley moment for exponential primitives.Conditional waiting-time moments of one Lindley step.Result of MAP/D/c queue analysis.Return value ofQsys_mapg1k_perflow, mirroring the MATLAB result struct ofqsys_mapg1k_perflow.m.Return value ofQsys_mapg1k, mirroring the MATLAB result struct ofqsys_mapg1k.mfield for field.Return value of qsys_mapphc, mirroring the MATLAB struct.Result of MAP/PH type queue analysis.The outputs [W, Wq, U, Q, res] of a single-server closed form whose MATLAB twin also returns a struct of extra measures (qsys_mg1_setup, qsys_mg1_exceptional): the four mean values plus that struct, with the MATLAB field names as keys.Mean-value metrics of a single-server queue: the four outputs [W, Wq, U, Q] of the MATLAB qsys_mxg1, qsys_mg1_vacation and qsys_mg1_breakdown.Result ofQsys_mg1_ps.qsys_mg1_ps(double, double[], double[][], double[], double[], double[], int): the sojourn time distribution of the M/G/1 processor-sharing queue.Return value ofQsys_mgck_sync, mirroring the MATLAB result struct ofqsys_mgck_sync.mfield for field.Sojourn-time moments of the multiclass M/M/1-PS queue.Return value ofQsys_mmapg1k: the per-class throughputs and loss ratios of an MMAP[K]/G/1/K buffer, mirroring the MATLAB result struct ofqsys_mmapg1k.m.Return value of qsys_mmapgk1_lcfspr, mirroring the MATLAB struct.Return value of qsys_mmapgk1, mirroring the MATLAB struct.Time-varying measures of the Mt/G/infinity queue, as produced byQsys_mtginf.The outputs [W, rho, res] of a multiclass closed form whose MATLAB twin returns the per-class sojourn times W, the M/M/1-equivalent load rho = Q/(1+Q) and a struct of extra measures (qsys_mg1_prio_pr, qsys_mmc_prio, qsys_mg1_sjf).Result of BMAP/PH/N/N retrial queue analysis.Service-time descriptor of the MAP/G/1/K family, the JAR form of the MATLABsvcstruct ofqsys_mapg1k.mand of the C++line::qsys::ServiceLaw.Which family the service law belongs to.Return value ofQsys_svclaw_coeffs: the outputs [cn, Smean] ofqsys_svclaw_coeffs.m, plus the truncation residual |1 - sum(cn)|.Conditional downstream waiting time in a two-station tandem.Waiting times and epochs of a tandem sample path.Tail bounds of a two-station tandem and the coefficients that produced them.Trajectory of the Gt/Mt/st+GI many-server fluid queue, as produced byQsys_gtmtst_fluid.Holder for the stationary distribution of the quantity of work in a single service stage, together with the grid it is reported on and the stationary distribution of the number of requests in the system.Information about a valid retrial queue topology.