# Markovian arrival process

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In queueing theory, a discipline within the mathematical theory of probability, a Markovian arrival process (MAP or MArP[1]) is a mathematical model for the time between job arrivals to a system. The simplest such process is a Poisson process where the time between each arrival is exponentially distributed.[2][3]

The processes were first suggested by Neuts in 1979.[2][4]

## Definition

A Markov arrival process is defined by two matrices D0 and D1 where elements of D0 represent hidden transitions and elements of D1 observable transitions. The block matrix Q below is a transition rate matrix for a continuous-time Markov chain.[5]

$Q=\left[\begin{matrix} D_{0}&D_{1}&0&0&\dots\\ 0&D_{0}&D_{1}&0&\dots\\ 0&0&D_{0}&D_{1}&\dots\\ \vdots & \vdots & \ddots & \ddots & \ddots \end{matrix}\right]\; .$

The simplest example is a Poisson process where D0 = −λ and D1 = λ where there is only one possible transition, it is observable and occurs at rate λ. For Q to be a valid transition rate matrix, the following restrictions apply to the Di

\begin{align} 0\leq [D_{1}]_{i,j}&<\infty \\ 0\leq [D_{0}]_{i,j}&<\infty \quad i\neq j \\ \, [D_{0}]_{i,i}&<0 \\ (D_{0}+D_{1})\boldsymbol{1} &= \boldsymbol{0} \end{align}

## Special cases

### Markov-modulated Poisson process

The Markov-modulated Poisson process or MMPP where m Poisson processes are switched between by an underlying continuous-time Markov chain.[6] If each of the m Poisson processes has rate λi and the modulating continuous-time Markov has m × m transition rate matrix R, then the MAP representation is

\begin{align} D_{1} &= \operatorname{diag}\{\lambda_{1},\dots,\lambda_{m}\}\\ D_{0} &=R-D_1. \end{align}

### Phase-type renewal process

The phase-type renewal process is a Markov arrival process with phase-type distributed sojourn between arrivals. For example if an arrival process has an interarrival time distribution PH$(\boldsymbol{\alpha},S)$ with an exit vector denoted $\boldsymbol{S}^{0}=-S\boldsymbol{1}$, the arrival process has generator matrix,

$Q=\left[\begin{matrix} S&\boldsymbol{S}^{0}\boldsymbol{\alpha}&0&0&\dots\\ 0&S&\boldsymbol{S}^{0}\boldsymbol{\alpha}&0&\dots\\ 0&0&S&\boldsymbol{S}^{0}\boldsymbol{\alpha}&\dots\\ \vdots&\vdots&\ddots&\ddots&\ddots\\ \end{matrix}\right]$

## Batch Markov arrival process

The batch Markovian arrival process (BMAP) is a generalisation of the Markovian arrival process by allowing more than one arrival at a time.[7] The homogeneous case has rate matrix,

$Q=\left[\begin{matrix} D_{0}&D_{1}&D_{2}&D_{3}&\dots\\ 0&D_{0}&D_{1}&D_{2}&\dots\\ 0&0&D_{0}&D_{1}&\dots\\ \vdots & \vdots & \ddots & \ddots & \ddots \end{matrix}\right]\; .$

An arrival of size $k$ occurs every time a transition occurs in the sub-matrix $D_{k}$. Sub-matrices $D_{k}$ have elements of $\lambda_{i,j}$, the rate of a Poisson process, such that,

$0\leq [D_{k}]_{i,j}<\infty\;\;\;\; 1\leq k$
$0\leq [D_{0}]_{i,j}<\infty\;\;\;\; i\neq j$
$[D_{0}]_{i,i}<0\;$

and

$\sum^{\infty}_{k=0}D_{k}\boldsymbol{1}=\boldsymbol{0}$

## Fitting

A MAP can be fitted using an expectation–maximization algorithm.[8]