Tools for Markovian Arrival Processes (butools.map)

To load this package, either start the BuToolsInit script, or execute

addpath('butools/map') in Matlab,
<<BuTools`MAP` in Mathematica,
from butools.map import * in Python/Numpy.

Markovian arrival processes and rational arrival processes

Continuous time Markovian arrival processes (MAPs) are characterized by two matrices, \(D_0\) and \(D_1\). MAPs have a continuous time Markov chain in the background with generator \(D=D_0+D_1\). Every time this Markov chain changes state through a transition belonging to \(D_1\), an arrival event is generated. As the state of the background process is not re-initiated after the arrival events, MAPs are capable of generating correlated arrivals.

Rational arrival processes (RAPs, also known as matrix-exponential processes, MEPs) are the generalizations of MAPs. Formally, all formulas for the statistical quantities are very similar to the ones of MAPs. However, \(D_0\) and \(D_1\) can hold general numbers, the entries do not have to be valid transition rates. RAPs therefore lack the simple stochastic interpretation that MAPs have.

Both MAPs and RAPs can be generalized to multi-type arrival processes. If there are K different arrival types, marked MAPs (MMAPs) and marked RAPs (MRAPs) defined by matrices \(D_0,\dots,D_K\) are able to describe the multi-type arrival process.

BuTools provides several tools for MAPs, RAPs and their marked variants in the maps package.

Simple statistical properties and tools

MarginalDistributionFromMAP Returns the phase type marginal distribution of a Markovian arrival process.
MarginalDistributionFromRAP Returns the matrix-exponential marginal distribution of a rational arrival process.
MarginalDistributionFromMMAP Returns the phase type marginal distribution of a marked Markovian arrival process.
MarginalDistributionFromMRAP Returns the matrix-exponential marginal distribution of a marked rational arrival process.
MarginalMomentsFromMAP Returns the moments of the marginal distribution of a Markovian arrival process.
MarginalMomentsFromRAP Returns the moments of the marginal distribution of a rational arrival process.
MarginalMomentsFromMMAP Returns the moments of the marginal distribution of a marked Markovian arrival process.
MarginalMomentsFromMRAP Returns the moments of the marginal distribution of a marked rational arrival process.
LagCorrelationsFromMAP Returns the lag autocorrelations of a Markovian arrival process.
LagCorrelationsFromRAP Returns the lag autocorrelations of a rational arrival process.
LagkJointMomentsFromMAP Returns the lag-k joint moments of a Markovian arrival process.
LagkJointMomentsFromRAP Returns the lag-k joint moments of a rational arrival process.
LagkJointMomentsFromMMAP Returns the lag-k joint moments of a marked Markovian arrival process.
LagkJointMomentsFromMRAP Returns the lag-k joint moments of a marked rational arrival process.
RandomMAP Returns a random Markovian arrival process with given mean value.
RandomMMAP Returns a random marked Markovian arrival process with given mean value.
CheckMAPRepresentation Checks if the input matrixes define a continuous time MAP.
CheckRAPRepresentation Checks if the input matrixes define a continuous time RAP.
CheckMMAPRepresentation Checks if the input matrixes define a continuous time MMAP.
CheckMRAPRepresentation Checks if the input matrixes define a continuous time MRAP.
SamplesFromMAP Generates random samples from a Markovian arrival process.
SamplesFromMMAP Generates random samples from a marked Markovian arrival process.
ImageFromMAP Depicts the given Markovian arrival process, and either displays it or saves it to file.
ImageFromMMAP Depicts the given marked Markovian arrival process, and either displays it or saves it to file.

Inverse characterization tools

RAPFromMoments Creates a rational arrival process that has the same marginal and lag-1 joint moments as given.
MRAPFromMoments Creates a marked rational arrival process that has the same marginal and lag-1 joint moments as given.
RAPFromMomentsAndCorrelations Returns a rational arrival process that has the same moments and lag autocorrelation coefficients as given.
MAP2FromMoments Returns a MAP(2) which has the same 3 marginal moments and lag-1 autocorrelation as given.
MAP2CorrelationBounds Returns the upper and lower correlation bounds for a MAP(2) given the three marginal moments.
MAPFromFewMomentsAndCorrelations Returns a MAP that matches the given 2 or 3 moments and the lag-1 autocorrelation.

Representation transformation methods

CanonicalFromMAP2 Returns the canonical form of an order-2 Markovian arrival process.
MAPFromRAP Obtains a Markovian representation of a rational arrival process of the same size, if possible.
MMAPFromMRAP Obtains a Markovian representation of a marked rational arrival process of the same size, if possible.
MinimalRepFromRAP Returns the minimal representation of a rational arrival process.
MinimalRepFromMRAP Returns the minimal representation of a marked rational arrival process.