Classes
struct	AsymVarResult
	Second-order description of a steady-state estimator. More...
struct	QuestInterval
	A confidence interval, asymmetric about the point estimate in general. More...
struct	QuestOptions
	Procedure constants shared by sim_fquest and sim_firquest. More...
struct	QuestResult
	Point estimate and interval delivered by the QUEST procedures. More...
struct	RunLengthPlan
	The plan of sim_runlength_plan: what the run should have been. More...
struct	RunLengthResult
	Outcome of the run-length plan. More...
struct	ShapiroWilkResult
	Outcome of the Shapiro-Wilk normality test. More...
struct	StsQuantileStats
	Batched-quantile statistics of one sample path. More...
struct	VonNeumannResult
	Outcome of the von Neumann randomness test. More...

Functions
double	sim_normcdf (double z)
	Standard normal cumulative distribution function.
double	sim_norminv (double p)
	Standard normal quantile function.
double	sim_tinv (double p, double nu)
	Quantile function of Student's t distribution.
std::vector< long >	sim_firquest_batchcounts (std::size_t R)
	The article's batch counts as a function of the replication count, chosen so that R*b pooled statistics remain enough to test while every replication still contributes at least one batch.
QuestOptions	sim_firquest_options (std::size_t R)
	The FIRQUEST defaults at R replications: b0 = 25 and the R-dependent ladder, every other constant as in FQUEST.
template<class T>
QuestResult< T >	sim_firquest (const std::vector< std::vector< T > > &Y, double p, double alpha=0.05, const QuestOptions *options=nullptr)
	Fixed-sample-size quantile interval from independent replications.
template<class T>
QuestResult< T >	sim_fquest (const std::vector< T > &Y, double p, double alpha=0.05, const QuestOptions &options=QuestOptions())
	Fixed-sample-size confidence interval for a steady-state quantile.
template<class T>
QuestInterval< T >	sim_quest_heuristic_ci (const std::vector< T > &bqe, const T &centre, const T &Ap, const T &Np, std::size_t nstar, double alpha, bool useAutocorr)
	Fallback interval used when a QUEST stage test fails.
QuestOptions	sim_quest_options (const QuestOptions &options=QuestOptions())
	Validates an option set and returns it.
template<class T>
AsymVarResult< T >	sim_asymvar_mm1 (const T &lambda, const T &mu)
	Asymptotic variance of the M/M/1 number-in-system process.
template<class T>
AsymVarResult< T >	sim_asymvar_ctmc (const Matrix< T > &A, const std::vector< T > &f, const std::vector< T > &pi=std::vector< T >())
	Asymptotic variance of a reward on a CTMC: 2 sum_x pi(x)g(x)d(x) with g = f - E_pi[f] and A d = -g, pi d = 0.
template<class T>
RunLengthResult< T >	sim_runlength (const T &mean, const T &asymVar, const T &relPrecision=num_traits< T >::from_rational(1, 20), const T &confidence=num_traits< T >::from_rational(19, 20), const T &runLength=num_traits< T >::from_int(0))
	Run length for a steady-state estimate of a given relative precision.
template<class T>
RunLengthPlan< T >	sim_runlength_plan (const Matrix< T > &means, const Matrix< T > &ciHalfWidth, const T &samplesUsed, const T &relPrecision=num_traits< T >::from_rational(1, 20), const T &confidence=num_traits< T >::from_rational(19, 20))
	How long a simulation run should have been, from the one it already did.
template<class T>
ShapiroWilkResult< T >	sim_shapirowilk (const std::vector< T > &x, double alpha=0.05)
	Shapiro-Wilk test for univariate normality.
template<class T>
StsQuantileStats< T >	sim_sts_quantile_areas (const std::vector< T > &Y, std::size_t b, std::size_t m, double p, double weight=std::sqrt(12.0))
	Standardized time series areas of the batched quantile process.
template<class T>
VonNeumannResult< T >	sim_vonneumann (const std::vector< T > &x, double alpha=0.05)
	Von Neumann ratio test for randomness of a sequence.

Function Documentation

◆ sim_asymvar_ctmc()

template<class T>

AsymVarResult< T > line::sim::sim_asymvar_ctmc	(	const Matrix< T > &	A,
		const std::vector< T > &	f,
		const std::vector< T > &	pi = std::vector<T>() )

Asymptotic variance of a reward on a CTMC: 2 sum_x pi(x)g(x)d(x) with g = f - E_pi[f] and A d = -g, pi d = 0.

The normalization is what pins d: A alone is singular, since a constant may be added without changing sigma^2.

Parameters

A	the generator, rows summing to zero
f	the reward attached to each state
pi	the stationary distribution; solved for when empty

Definition at line 105 of file sim_runlength.h.

References line::sim::AsymVarResult< T >::asymptoticVariance, line::Matrix< T >::cols(), line::sim::AsymVarResult< T >::deviation, line::InputError::InputError(), line::sim::AsymVarResult< T >::mean, line::num_abs(), line::sim::AsymVarResult< T >::relaxationTime, line::Matrix< T >::rows(), sim_asymvar_ctmc(), line::solve(), and line::sim::AsymVarResult< T >::variance.

Referenced by line::ctmc::ctmc_get_asymptotic_variance(), and sim_asymvar_ctmc().

◆ sim_asymvar_mm1()

template<class T>

AsymVarResult< T > line::sim::sim_asymvar_mm1	(	const T &	lambda,
		const T &	mu )

Asymptotic variance of the M/M/1 number-in-system process.

Parameters

lambda	arrival rate
mu	service rate

Definition at line 78 of file sim_runlength.h.

References line::sim::AsymVarResult< T >::asymptoticVariance, line::InputError::InputError(), line::sim::AsymVarResult< T >::mean, line::sim::AsymVarResult< T >::relaxationTime, sim_asymvar_mm1(), and line::sim::AsymVarResult< T >::variance.

Referenced by sim_asymvar_mm1().

◆ sim_firquest()

template<class T>

QuestResult< T > line::sim::sim_firquest	(	const std::vector< std::vector< T > > &	Y,
		double	p,
		double	alpha = 0.05,
		const QuestOptions *	options = nullptr )

Fixed-sample-size quantile interval from independent replications.

Parameters

Y	R replicate sample paths of equal length, finite
p	quantile order in (0,1)
alpha	nominal non-coverage, 0.05 by default
options	procedure constants, nullptr for the FIRQUEST defaults at this R

Definition at line 176 of file sim_firquest.h.

Referenced by sim_firquest().

◆ sim_firquest_batchcounts()

std::vector< long > line::sim::sim_firquest_batchcounts ( std::size_t R )

inline

The article's batch counts as a function of the replication count, chosen so that R*b pooled statistics remain enough to test while every replication still contributes at least one batch.

Parameters

R	number of replications

Returns: the descending batch-count ladder

Definition at line 87 of file sim_firquest.h.

References sim_firquest_batchcounts().

Referenced by sim_firquest_batchcounts(), and sim_firquest_options().

◆ sim_firquest_options()

QuestOptions line::sim::sim_firquest_options ( std::size_t R )

inline

The FIRQUEST defaults at R replications: b0 = 25 and the R-dependent ladder, every other constant as in FQUEST.

Parameters

R	number of replications

Returns: the option set sim_firquest uses when none is supplied

Definition at line 105 of file sim_firquest.h.

References sim_firquest_batchcounts(), and sim_firquest_options().

Referenced by sim_firquest(), and sim_firquest_options().

◆ sim_fquest()

template<class T>

QuestResult< T > line::sim::sim_fquest	(	const std::vector< T > &	Y,
		double	p,
		double	alpha = 0.05,
		const QuestOptions &	options = QuestOptions() )

Fixed-sample-size confidence interval for a steady-state quantile.

Parameters

Y	one simulation sample path, finite
p	quantile order in (0,1)
alpha	nominal non-coverage, 0.05 by default
options	procedure constants, see sim_quest_options

Definition at line 130 of file sim_fquest.h.

Referenced by sim_fquest().

◆ sim_normcdf()

double line::sim::sim_normcdf ( double z )

inline

Standard normal cumulative distribution function.

Parameters

z	the argument

Returns: Phi(z)

Definition at line 52 of file sim_dist.h.

References sim_normcdf().

Referenced by line::trace::mtrace_bootstrap(), sim_normcdf(), sim_shapirowilk(), and sim_vonneumann().

◆ sim_norminv()

double line::sim::sim_norminv ( double p )

inline

Standard normal quantile function.

Parameters

p	probability in [0,1]

Returns: Phi^{-1}(p), infinite at the endpoints

Definition at line 62 of file sim_dist.h.

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

Referenced by line::trace::mtrace_bootstrap(), and sim_norminv().

◆ sim_quest_heuristic_ci()

template<class T>

QuestInterval< T > line::sim::sim_quest_heuristic_ci	(	const std::vector< T > &	bqe,
		const T &	centre,
		const T &	Ap,
		const T &	Np,
		std::size_t	nstar,
		double	alpha,
		bool	useAutocorr )

Fallback interval used when a QUEST stage test fails.

Parameters

bqe	the K batched quantile estimators, pooled over replications for sim_firquest
centre	the full-sample empirical quantile
Ap	STS area variance-parameter estimator
Np	NBQ variance-parameter estimator, may be NaN
nstar	number of observations Ap and Np were computed from
alpha	nominal non-coverage
useAutocorr	apply the lag-1 correction to the asymmetric arms

Definition at line 109 of file sim_quest_heuristic_ci.h.

References line::InputError::InputError(), line::sim::QuestInterval< T >::lower, line::num_abs(), sim_quest_heuristic_ci(), sim_tinv(), and line::sim::QuestInterval< T >::upper.

Referenced by sim_firquest(), sim_fquest(), and sim_quest_heuristic_ci().

◆ sim_quest_options()

QuestOptions line::sim::sim_quest_options ( const QuestOptions & options = QuestOptions() )

inline

Validates an option set and returns it.

Parameters

options the constants to check

Returns: the same constants, once every admissibility condition holds

Definition at line 57 of file sim_quest_options.h.

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

Referenced by sim_firquest(), sim_fquest(), and sim_quest_options().

◆ sim_runlength()

template<class T>

RunLengthResult< T > line::sim::sim_runlength	(	const T &	mean,
		const T &	asymVar,
		const T &	relPrecision = num_traits<T>::from_rational(1, 20),
		const T &	confidence = num_traits<T>::from_rational(19, 20),
		const T &	runLength = num_traits<T>::from_int(0) )

Run length for a steady-state estimate of a given relative precision.

Parameters

mean	the steady-state mean being estimated
asymVar	sigma^2 of that estimator
relPrecision	the target half-width as a fraction of the mean
confidence	the confidence level of the interval
runLength	an actual run length, to report the precision it buys; non-positive to skip

Definition at line 179 of file sim_runlength.h.

References line::sim::RunLengthResult< T >::achievedRelPrecision, line::sim::RunLengthResult< T >::halfWidth, line::sim::RunLengthResult< T >::hasRun, line::InputError::InputError(), line::num_abs(), line::sim::RunLengthResult< T >::requiredRunLength, sim_runlength(), and line::sim::RunLengthResult< T >::z.

Referenced by sim_runlength(), and sim_runlength_plan().

◆ sim_runlength_plan()

template<class T>

RunLengthPlan< T > line::sim::sim_runlength_plan	(	const Matrix< T > &	means,
		const Matrix< T > &	ciHalfWidth,
		const T &	samplesUsed,
		const T &	relPrecision = num_traits<T>::from_rational(1, 20),
		const T &	confidence = num_traits<T>::from_rational(19, 20) )

How long a simulation run should have been, from the one it already did.

A batch-means half-width H at confidence 1-alpha over a run of N samples pins the ASYMPTOTIC variance of the estimator,

sigma^2 = (H/z)^2 N, z = Phi^-1((1+confidence)/2),

and that is the quantity a run length is planned from – NOT the stationary variance, which on M/M/1 differs from it by a factor blowing up like (1-rho)^-2. sim_runlength then turns it into the sample count that reaches a requested RELATIVE precision.

An entry with a non-positive mean or half-width is left NaN, since there is nothing to plan from there.

Reference: W. Whitt (1989). Planning queueing simulations. Management Science 35(11), 1341-1366.

Definition at line 249 of file sim_runlength.h.

References line::sim::RunLengthPlan< T >::asymptoticVariance, line::Matrix< T >::cols(), line::sim::RunLengthPlan< T >::confidence, line::InputError::InputError(), line::Matrix< T >::Matrix(), line::sim::RunLengthPlan< T >::relPrecision, line::sim::RunLengthPlan< T >::requiredSamples, line::Matrix< T >::rows(), line::sim::RunLengthPlan< T >::samplesUsed, sim_runlength(), and sim_runlength_plan().

Referenced by line::ldes::engine::batch_means_ci(), and sim_runlength_plan().

◆ sim_shapirowilk()

template<class T>

ShapiroWilkResult< T > line::sim::sim_shapirowilk	(	const std::vector< T > &	x,
		double	alpha = 0.05 )

Shapiro-Wilk test for univariate normality.

Parameters

x	the sample, 3 to 5000 finite observations, order irrelevant
alpha	significance level in (0,1), 0.05 by default

Definition at line 122 of file sim_shapirowilk.h.

References line::InputError::InputError(), line::sim::ShapiroWilkResult< T >::nobs, line::sim::ShapiroWilkResult< T >::pvalue, line::sim::ShapiroWilkResult< T >::reject, sim_normcdf(), sim_shapirowilk(), line::sim::ShapiroWilkResult< T >::W, and line::sim::ShapiroWilkResult< T >::zscore.

Referenced by sim_firquest(), sim_fquest(), and sim_shapirowilk().

◆ sim_sts_quantile_areas()

template<class T>

StsQuantileStats< T > line::sim::sim_sts_quantile_areas	(	const std::vector< T > &	Y,
		std::size_t	b,
		std::size_t	m,
		double	p,
		double	weight = std::sqrt(12.0) )

Standardized time series areas of the batched quantile process.

Parameters

Y	exactly b*m finite observations, in sample-path order
b	batch count, positive
m	batch size, positive
p	quantile order in (0,1)
weight	constant STS weight function, sqrt(12) by default

Definition at line 131 of file sim_sts_quantile_areas.h.

References line::sim::StsQuantileStats< T >::Ap, line::sim::StsQuantileStats< T >::areas, line::sim::StsQuantileStats< T >::b, line::sim::StsQuantileStats< T >::bqe, line::InputError::InputError(), line::sim::StsQuantileStats< T >::m, line::sim::StsQuantileStats< T >::n, line::sim::StsQuantileStats< T >::Np, line::sim::StsQuantileStats< T >::quantile, sim_sts_quantile_areas(), and line::sim::StsQuantileStats< T >::Vp.

Referenced by sim_firquest(), sim_fquest(), and sim_sts_quantile_areas().

◆ sim_tinv()

double line::sim::sim_tinv	(	double	p,
		double	nu )

inline

Quantile function of Student's t distribution.

Parameters

p	probability in [0,1]
nu	degrees of freedom, positive

Returns: the p-quantile of t with nu degrees of freedom

Definition at line 77 of file sim_dist.h.

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

Referenced by sim_firquest(), sim_fquest(), sim_quest_heuristic_ci(), and sim_tinv().

◆ sim_vonneumann()

template<class T>

VonNeumannResult< T > line::sim::sim_vonneumann	(	const std::vector< T > &	x,
		double	alpha = 0.05 )

Von Neumann ratio test for randomness of a sequence.

Parameters

x	the sequence, at least 3 finite observations
alpha	significance level in (0,1), 0.05 by default

Definition at line 70 of file sim_vonneumann.h.

References line::InputError::InputError(), line::sim::VonNeumannResult< T >::nobs, line::sim::VonNeumannResult< T >::pvalue, line::sim::VonNeumannResult< T >::ratio, line::sim::VonNeumannResult< T >::reject, sim_normcdf(), sim_vonneumann(), and line::sim::VonNeumannResult< T >::zscore.

Referenced by sim_firquest(), sim_fquest(), and sim_vonneumann().

Classes

Functions

Function Documentation

◆ sim_asymvar_ctmc()

◆ sim_asymvar_mm1()

◆ sim_firquest()

◆ sim_firquest_batchcounts()

◆ sim_firquest_options()

◆ sim_fquest()

◆ sim_normcdf()

◆ sim_norminv()

◆ sim_quest_heuristic_ci()

◆ sim_quest_options()

◆ sim_runlength()

◆ sim_runlength_plan()

◆ sim_shapirowilk()

◆ sim_sts_quantile_areas()

◆ sim_tinv()

◆ sim_vonneumann()