Tools for Discrete Markovian Arrival Processes (butools.dmap)

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

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

Discrete Markovian arrival processes and rational arrival processes

Discrete time Markovian arrival processes (DMAPs) are characterized by two matrices, \(D_0\) and \(D_1\). DMAPs have a discrete time Markov chain in the background with transition probability matrix \(D=D_0+D_1\). 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, DMAPs are capable of generating correlated arrival times.

Discrete rational arrival processes (DRAPs) are the generalizations of DMAPs. Formally, all formulas for the statistical quantities are very similar to the ones of DMAPs. However, \(D_0\) and \(D_1\) can hold general numbers, the entries do not have to be valid probabilities. DRAPs therefore lack the simple stochastic interpretation that DMAPs have.

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

BuTools provides several tools for DMAPs, DRAPs and their marked variants in the dmaps package.

Simple statistical properties and tools

MarginalDistributionFromDMAP Returns the discrete phase type distributed marginal distribution of a discrete Markovian arrival process.
MarginalDistributionFromDRAP Returns the matrix-geometrically distributed marginal of a discrete rational arrival process.
MarginalDistributionFromDMRAP Returns the matrix-geometrically distributed marginal of a discrete marked rational arrival process.
MarginalDistributionFromDMMAP Returns the discrete phase type distributed marginal of a discrete marked Markovian arrival process.
MarginalMomentsFromDMAP Returns the moments of the marginal distribution of a discrete Markovian arrival process.
MarginalMomentsFromDRAP Returns the moments of the marginal distribution of a discrete rational arrival process.
MarginalMomentsFromDMMAP Returns the moments of the marginal distribution of a discrete marked Markovian arrival process.
MarginalMomentsFromDMRAP Returns the moments of the marginal distribution of a discrete marked rational arrival process.
LagCorrelationsFromDMAP Returns the lag autocorrelations of a discrete Markovian arrival process.
LagCorrelationsFromDRAP Returns the lag autocorrelations of a discrete rational arrival process.
LagkJointMomentsFromDMAP Returns the lag-k joint moments of a discrete Markovian arrival process.
LagkJointMomentsFromDRAP Returns the lag-k joint moments of a discrete rational arrival process.
LagkJointMomentsFromDMMAP Returns the lag-k joint moments of a discrete marked Markovian arrival process.
LagkJointMomentsFromDMRAP Returns the lag-k joint moments of a discrete marked rational arrival process.
RandomDMAP Returns a random discrete Markovian arrival process.
RandomDMMAP Returns a random discrete marked Markovian arrival process.
CheckDMAPRepresentation Checks if the input matrixes define a discrete time MAP.
CheckDRAPRepresentation Checks if the input matrixes define a discrete time RAP.
CheckDMMAPRepresentation Checks if the input matrixes define a discrete time MMAP.
CheckDMRAPRepresentation Checks if the input matrixes define a discrete time MRAP.
SamplesFromDMAP Generates random samples from a discrete Markovian arrival process.
SamplesFromDMMAP Generates random samples from a discrete marked Markovian arrival process.
ImageFromDMAP Depicts the given discrete Markovian arrival process, and either displays it or saves it to file.
ImageFromDMMAP Depicts the given discrete marked Markovian arrival process, and either displays it or saves it to file.

Inverse characterization tools

DRAPFromMoments Creates a discrete rational arrival process that has the same marginal and lag-1 joint moments as given.
DMRAPFromMoments Creates a discrete marked rational arrival process that has the same marginal and lag-1 joint moments as given.
DMAP2FromMoments Returns a DMAP(2) which has the same 3 marginal moments and lag-1 autocorrelation as given.

Representation transformation methods

CanonicalFromDMAP2 Returns the canonical form of an order-2 discrete Markovian arrival process.
DMAPFromDRAP Obtains a Markovian representation of a discrete rational arrival process of the same size, if possible.
DMMAPFromDMRAP Obtains a Markovian representation of a discrete marked rational arrival process of the same size, if possible.