This class implements the hypoexponential distribution for the case of equidistant \(\lambda_i = (n+1-i)h\). More...

Inheritance diagram for umontreal.ssj.probdist.HypoExponentialDistEqual:

Public Member Functions
	HypoExponentialDistEqual (int n, int k, double h)
	Constructor for equidistant 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 \(x = F^{-1}(u)\).
double[]	getParams ()
	Returns the three parameters of this hypoexponential distribution as array \((n, k, h)\).
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.
void	setLambda (double[] lambda)
	Sets the values \(\lambda_i = \)lambda[ \(i-1\)], \(i = 1,…,k\) for this object.
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 (int n, int k, double h, double x)
	Computes the density function \(f(x)\), with the same arguments as in the constructor.
static double	cdf (int n, int k, double h, double x)
	Computes the distribution function \(F(x)\), with arguments as in the constructor.
static double	barF (int n, int k, double h, double x)
	Computes the complementary distribution \(\bar{F}(x)\), as in formula ( `conv-hypo-equal` ).
static double	inverseF (int n, int k, double h, double u)
	Computes the inverse distribution \(x=F^{-1}(u)\), with arguments as in the constructor.
Static Public Member Functions inherited from umontreal.ssj.probdist.HypoExponentialDist
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 for the case of equidistant \(\lambda_i = (n+1-i)h\).

We have \(\lambda_{i+1} - \lambda_i = h\), with \(h\) a constant, and \(n \ge k\) are integers.

The formula ( convolution-hypo ) becomes

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

The formula ( fhypoexp2 ) for the density becomes

\[ f(x) = \sum_{i=1}^k (n+1-i)h e^{-(n+1-i)h x} \prod_{\substack {j=1\\j\not i}}^k \frac{n+1-j}{i - j}. \tag{fhypoexp3} \]

Definition at line 54 of file HypoExponentialDistEqual.java.

Constructor & Destructor Documentation

◆ HypoExponentialDistEqual()

umontreal.ssj.probdist.HypoExponentialDistEqual.HypoExponentialDistEqual	(	int	n,
		int	k,
		double	h )

Constructor for equidistant rates.

The rates are \(\lambda_i = (n+1-i)h\), for \(i = 1,…,k\).

Parameters

n	largest rate is \(nh\)
k	number of rates
h	difference between adjacent rates

Definition at line 67 of file HypoExponentialDistEqual.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.HypoExponentialDistEqual.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 80 of file HypoExponentialDistEqual.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.HypoExponentialDistEqual.barF	(	int	n,
		int	k,
		double	h,
		double	x )

static

Computes the complementary distribution \(\bar{F}(x)\), as in formula ( conv-hypo-equal ).

Parameters

n	max possible number of \(\lambda_i\)
k	effective number of \(\lambda_i\)
h	step between two successive \(\lambda_i\)
x	value at which the complementary distribution is evaluated

Returns: value of complementary distribution at \(x\)

Definition at line 135 of file HypoExponentialDistEqual.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.HypoExponentialDistEqual.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 76 of file HypoExponentialDistEqual.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.HypoExponentialDistEqual.cdf	(	int	n,
		int	k,
		double	h,
		double	x )

static

Computes the distribution function \(F(x)\), with arguments as in the constructor.

Parameters

n	max possible number of \(\lambda_i\)
k	effective number of \(\lambda_i\)
h	step between two successive \(\lambda_i\)
x	value at which the distribution is evaluated

Returns: value of distribution at \(x\)

Definition at line 116 of file HypoExponentialDistEqual.java.

◆ density() [1/2]

double umontreal.ssj.probdist.HypoExponentialDistEqual.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 72 of file HypoExponentialDistEqual.java.

◆ density() [2/2]

double umontreal.ssj.probdist.HypoExponentialDistEqual.density	(	int	n,
		int	k,
		double	h,
		double	x )

static

Computes the density function \(f(x)\), with the same arguments as in the constructor.

Parameters

n	max possible number of \(\lambda_i\)
k	effective number of \(\lambda_i\)
h	step between two successive \(\lambda_i\)
x	value at which the distribution is evaluated

Returns: density at \(x\)

Definition at line 98 of file HypoExponentialDistEqual.java.

◆ getParams()

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

Returns the three parameters of this hypoexponential distribution as array \((n, k, h)\).

Returns: parameters of the hypoexponential distribution

Reimplemented from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 171 of file HypoExponentialDistEqual.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.HypoExponentialDistEqual.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 84 of file HypoExponentialDistEqual.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.HypoExponentialDistEqual.inverseF	(	int	n,
		int	k,
		double	h,
		double	u )

static

Computes the inverse distribution \(x=F^{-1}(u)\), with arguments as in the constructor.

Parameters

n	max possible number of \(\lambda_i\)
k	effective number of \(\lambda_i\)
h	step between two successive \(\lambda_i\)
u	value at which the inverse distribution is evaluated

Returns: inverse distribution at \(u\)

Definition at line 153 of file HypoExponentialDistEqual.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Reimplemented from umontreal.ssj.probdist.HypoExponentialDist.

Definition at line 186 of file HypoExponentialDistEqual.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ HypoExponentialDistEqual()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getParams()

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ toString()