This class implements the hypoexponential distribution, also called the generalized Erlang distribution. More...

Inheritance diagram for umontreal.ssj.probdist.HypoExponentialDist:

Public Member Functions
	HypoExponentialDist (double[] lambda)
	Constructs a HypoExponentialDist object, with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
double[]	getLambda ()
	Returns the values \(\lambda_i\) for this object.
void	setLambda (double[] lambda)
	Sets the values \(\lambda_i = \)lambda[ \(i-1\)], \(i = 1,…,k\) for this object.
double[]	getParams ()
	Same as getLambda.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (double[] lambda, double x)
	Computes the density function \(f(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
static double	cdf (double[] lambda, double x)
	Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
static double	cdf2 (double[] lambda, double x)
	Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
static double	barF (double[] lambda, double x)
	Computes the complementary distribution \(\bar{F}(x)\), with.
static double	inverseF (double[] lambda, double u)
	Computes the inverse distribution function \(F^{-1}(u)\), with.
static double	getMean (double[] lambda)
	Returns the mean, \(E[X] = \sum_{i=1}^k 1/\lambda_i\), of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
static double	getVariance (double[] lambda)
	Returns the variance, \(\mbox{Var}[X] = \sum_{i=1}^k 1/\lambda_i^2\), of the hypoexponential distribution with rates.
static double	getStandardDeviation (double[] lambda)
	Returns the standard deviation of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)],.

Detailed Description

This class implements the hypoexponential distribution, also called the generalized Erlang distribution.

Let the \(X_j\), \(j=1,…,k\), be

\(k\) independent exponential random variables with different rates \(\lambda_j\), i.e. assume that \(\lambda_j \neq\lambda_i\) for \(i \neq j\). Then the sum \(\sum_{j=1}^kX_j\) is called a hypoexponential random variable.

Let the \(k\times k\) upper triangular bidiagonal matrix

\[ \tag{tail-hypomatrix} \mathbf{A}= \begin{pmatrix} -\lambda_1 & \lambda_1 & 0 & … & 0 \\ 0 & -\lambda_2 & \lambda_2 & … & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & … & 0 & -\lambda_{k-1} & \lambda_{k-1} \\ 0 & … & 0 & 0 & -\lambda_k \end{pmatrix} \]

with \(\lambda_j\) the rates of the \(k\) exponential random variables; then the cumulative complementary probability of the hypoexponential distribution is given by [pNEU81a], [112]

\[ \tag{tail-hypoexp} \bar{F}(x) = \mathbb P \left[X_1 + \cdots+ X_k > x \right] = \sum_{j=1}^k \left(e^{\mathbf{A}x}\right)_{1j}, \]

i.e., it is the sum of the elements of the first row of matrix \(e^{\mathbf{A}x}\). The density of the hypoexponential distribution is

\[ f(x) = \left(-e^{\mathbf{A}x}\mathbf{A}\right)_{1k} = \lambda_k \left(e^{\mathbf{A}x}\right)_{1k}, \tag{fhypoexp} \]

i.e., it is element \((1,k)\) of matrix \(-e^{\mathbf{A}x}\mathbf{A}\). The distribution function is as usual \(F(x) = 1 - \bar{F}(x)\).

See the class HypoExponentialDistQuick for alternative formulae for the probabilities.

Definition at line 74 of file HypoExponentialDist.java.

Constructor & Destructor Documentation

◆ HypoExponentialDist()

umontreal.ssj.probdist.HypoExponentialDist.HypoExponentialDist ( double[] lambda )

Constructs a HypoExponentialDist object, with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).

Parameters

lambda rates of the hypoexponential distribution

Definition at line 120 of file HypoExponentialDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.HypoExponentialDist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistEqual, and umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 133 of file HypoExponentialDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.HypoExponentialDist.barF	(	double[]	lambda,
		double	x )

static

Computes the complementary distribution \(\bar{F}(x)\), with.

\(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).

Parameters

lambda	rates of the hypoexponential distribution
x	value at which the complementary distribution is evaluated

Returns: complementary distribution at \(x\)

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 227 of file HypoExponentialDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.HypoExponentialDist.cdf ( double x )

Returns the distribution function \(F(x)\).

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Implements umontreal.ssj.probdist.Distribution.

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistEqual, and umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 129 of file HypoExponentialDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.HypoExponentialDist.cdf	(	double[]	lambda,
		double	x )

static

Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).

Parameters

lambda	rates of the hypoexponential distribution
x	value at which the distribution is evaluated

Returns: distribution at \(x\)

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 178 of file HypoExponentialDist.java.

◆ cdf2()

double umontreal.ssj.probdist.HypoExponentialDist.cdf2	(	double[]	lambda,
		double	x )

static

Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).

Returns \(1 - \)barF(lambda, x), which is much faster than cdf but loses precision in the lower tail.

Parameters

lambda	rates of the hypoexponential distribution
x	value at which the distribution is evaluated

Returns: distribution at \(x\)

Definition at line 211 of file HypoExponentialDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.HypoExponentialDist.density ( double x )

Returns \(f(x)\), the density evaluated at \(x\).

Parameters

x	value at which the density is evaluated

Returns: density function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistEqual, and umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 125 of file HypoExponentialDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.HypoExponentialDist.density	(	double[]	lambda,
		double	x )

static

Computes the density function \(f(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).

Parameters

lambda	rates of the hypoexponential distribution
x	value at which the density is evaluated

Returns: density at \(x\)

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 161 of file HypoExponentialDist.java.

◆ getLambda()

double[] umontreal.ssj.probdist.HypoExponentialDist.getLambda ( )

Returns the values \(\lambda_i\) for this object.

Definition at line 324 of file HypoExponentialDist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.HypoExponentialDist.getMean ( )

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 141 of file HypoExponentialDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.HypoExponentialDist.getMean ( double[] lambda )

static

Returns the mean, \(E[X] = \sum_{i=1}^k 1/\lambda_i\), of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).

Parameters

lambda rates of the hypoexponential distribution

Returns: mean of the hypoexponential distribution

Definition at line 282 of file HypoExponentialDist.java.

◆ getParams()

double[] umontreal.ssj.probdist.HypoExponentialDist.getParams ( )

Same as getLambda.

Implements umontreal.ssj.probdist.Distribution.

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistEqual.

Definition at line 345 of file HypoExponentialDist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.HypoExponentialDist.getStandardDeviation ( )

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 149 of file HypoExponentialDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.HypoExponentialDist.getStandardDeviation ( double[] lambda )

static

Returns the standard deviation of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)],.

\(i = 1,…,k\).

Parameters

lambda rates of the hypoexponential distribution

Returns: standard deviation of the hypoexponential distribution

Definition at line 316 of file HypoExponentialDist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.HypoExponentialDist.getVariance ( )

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 145 of file HypoExponentialDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.HypoExponentialDist.getVariance ( double[] lambda )

static

Returns the variance, \(\mbox{Var}[X] = \sum_{i=1}^k 1/\lambda_i^2\), of the hypoexponential distribution with rates.

\(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).

Parameters

lambda rates of the hypoexponential distribution

Returns: variance of the hypoexponential distribution

Definition at line 299 of file HypoExponentialDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.HypoExponentialDist.inverseF ( double u )

Returns the inverse distribution function \(x = F^{-1}(u)\).

Restrictions: \(u \in[0,1]\).

Parameters

u	value at which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException if \(u\) is not in the interval \([0,1]\)

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistEqual, and umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 137 of file HypoExponentialDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.HypoExponentialDist.inverseF	(	double[]	lambda,
		double	u )

static

Computes the inverse distribution function \(F^{-1}(u)\), with.

\(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).

Parameters

lambda	rates of the hypoexponential distribution
u	value at which the inverse distribution is evaluated

Returns: inverse distribution at \(u\)

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 248 of file HypoExponentialDist.java.

◆ setLambda()

void umontreal.ssj.probdist.HypoExponentialDist.setLambda ( double[] lambda )

Sets the values \(\lambda_i = \)lambda[ \(i-1\)], \(i = 1,…,k\) for this object.

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 333 of file HypoExponentialDist.java.

◆ toString()

String umontreal.ssj.probdist.HypoExponentialDist.toString ( )

Returns a String containing information about the current distribution.

Reimplemented in umontreal.ssj.probdist.HypoExponentialDistEqual, and umontreal.ssj.probdist.HypoExponentialDistQuick.

Definition at line 352 of file HypoExponentialDist.java.

The documentation for this class was generated from the following file:

src/main/java/umontreal/ssj/probdist/HypoExponentialDist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ HypoExponentialDist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ cdf2()

◆ density() [1/2]

◆ density() [2/2]

◆ getLambda()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setLambda()

◆ toString()