Package jline.api.snc

Class Snc_env_map

java.lang.Object

jline.api.snc.Snc_env_map

public final class Snc_env_map
extends Object

MGF arrival envelope of a MAP/MMPP flow with unit-size jobs.

For a Markovian arrival process (D0,D1) counting N(0,t) unit-work jobs, E[exp(theta*N(0,t))] = pi*expm((D0+D1*exp(theta))*t)*1. With lstar the eigenvalue of maximal real part of A(theta)=D0+D1*e^theta and v > 0 its right Perron eigenvector, bounding 1 <= v/min(v) entrywise gives

   rho(theta)   = lstar/theta,
   sigma(theta) = log(max(v)/min(v))/theta,
 

the standard exponential-form envelope of a Markov-modulated source. The burst term is what the modulating chain contributes: 0 for a one-phase MAP, where this reproduces Snc_env_poisson exactly, and positive for an MMPP.

THE PERRON PAIR IS COMPUTED BY POWER ITERATION ON THE SHIFTED MATRIX A+cI, not by a general eigensolver. A(theta) is essentially nonnegative, so the shift makes it nonnegative with a positive diagonal, hence primitive whenever the MAP is irreducible, and the iteration converges to the pair the bound needs without asking a general solver which of its eigenvectors is the positive one. The MATLAB reference uses eig and agrees to machine precision.

Port of matlab/src/api/snc/snc_env_map.m. Reference: C.-S. Chang, Performance Guarantees in Communication Networks, Springer 2000, Ch. 7.

Method Summary

Modifier and Type	Method	Description
static SncEnvelope	of(Matrix D0, Matrix D1)
static double[]	snc_env_map(Matrix D0, Matrix D1, double theta)

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Details

snc_env_map

public static double[] snc_env_map(Matrix D0,
 Matrix D1,
 double theta)

Parameters:

D0 - hidden-transition generator block of the MAP

D1 - arrival-transition block of the MAP

theta - Chernoff parameter, theta > 0

Returns:

{sigma, rho}

of

public static SncEnvelope of(Matrix D0,
 Matrix D1)

Parameters:

D0 - hidden-transition generator block

D1 - arrival-transition block

Returns:

the envelope as a function of theta