This class is a subclass of HypoExponentialDist and also implements the hypoexponential distribution. More...

Inheritance diagram for umontreal.ssj.probdist.HypoExponentialDistQuick:

Public Member Functions
	HypoExponentialDistQuick (double[] lambda)
	Constructs a HypoExponentialDistQuick object, with rates.
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 \(x = F^{-1}(u)\).
void	setLambda (double[] lambda)
	Sets the values \(\lambda_i = \)lambda[ \(i-1\)], \(i = 1,…,k\) for this object.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.HypoExponentialDist
	HypoExponentialDist (double[] lambda)
	Constructs a HypoExponentialDist object, with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
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.
double[]	getParams ()
	Same as getLambda.
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	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 Public Member Functions inherited from umontreal.ssj.probdist.HypoExponentialDist
static double	cdf2 (double[] lambda, double x)
	Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
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 is a subclass of HypoExponentialDist and also implements the hypoexponential distribution.

It uses different algorithms to compute the probabilities. The formula ( tail-hypoexp ) for the complementary distribution is mathematically equivalent to (see [pROS07b] (page 299) and

[66] (Appendix B))

\[ \bar{F}(x) = \mathbb P\left[X_1 + \cdots+ X_k > x \right] = \sum_{i=1}^k e^{-\lambda_i x} \prod_{\substack {j=1\\j\not i}}^k \frac{\lambda_j}{\lambda_j - \lambda_i}. \tag{convolution-hypo} \]

The expression ( convolution-hypo ) is much faster to compute than the matrix exponential formula ( tail-hypoexp ), but it becomes numerically unstable when \(k\) gets large and/or the differences between the \(\lambda_i\) are too small, because it is an alternating sum with relatively large terms of similar size. When the \(\lambda_i\) are close, many of the factors \(\lambda_j - \lambda_i\) in ( convolution-hypo ) are small, and the effect of this is amplified when \(k\) is large. This gives rise to large terms of opposite sign in the sum and the formula becomes unstable due to subtractive cancellation. For example, with the computations done in standard 64-bit floating-point arithmetic, if the \(\lambda_i\) are regularly spaced with differences of \(\lambda_{i+1} - \lambda_i = 0.1\) for all \(i\), the formula ( convolution-hypo ) breaks down already for \(k \approx15\), while if the differences \(\lambda_{i+1} - \lambda_i = 3\), it gives a few decimal digits of precision for \(k\) up to \(\approx300\).

The formula ( fhypoexp ) for the density is mathematically equivalent to the much faster formula

\[ f(x) = \sum_{i=1}^k\lambda_i e^{-\lambda_i x} \prod_{\substack {j=1\\j\not i}}^k \frac{\lambda_j}{\lambda_j - \lambda_i}, \tag{fhypoexp2} \]

which is also numerically unstable when \(k\) gets large and/or the differences between the \(\lambda_i\) are too small.

Definition at line 82 of file HypoExponentialDistQuick.java.

Constructor & Destructor Documentation

◆ HypoExponentialDistQuick()

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

Constructs a HypoExponentialDistQuick object, with rates.

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

Parameters

lambda rates of the hypoexponential distribution

Definition at line 170 of file HypoExponentialDistQuick.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.HypoExponentialDistQuick.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.HypoExponentialDist.

Definition at line 186 of file HypoExponentialDistQuick.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.HypoExponentialDistQuick.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: value of complementary distribution at \(x\)

Reimplemented from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 230 of file HypoExponentialDistQuick.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.HypoExponentialDistQuick.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

Reimplemented from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 182 of file HypoExponentialDistQuick.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.HypoExponentialDistQuick.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: value of distribution at \(x\)

Reimplemented from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 216 of file HypoExponentialDistQuick.java.

◆ density() [1/2]

double umontreal.ssj.probdist.HypoExponentialDistQuick.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.HypoExponentialDist.

Definition at line 178 of file HypoExponentialDistQuick.java.

◆ density() [2/2]

double umontreal.ssj.probdist.HypoExponentialDistQuick.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 from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 202 of file HypoExponentialDistQuick.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.HypoExponentialDistQuick.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.HypoExponentialDist.

Definition at line 190 of file HypoExponentialDistQuick.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.HypoExponentialDistQuick.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 from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 244 of file HypoExponentialDistQuick.java.

◆ setLambda()

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

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

Reimplemented from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 276 of file HypoExponentialDistQuick.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Reimplemented from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 283 of file HypoExponentialDistQuick.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ HypoExponentialDistQuick()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setLambda()

◆ toString()