# 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]

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

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

{\displaystyle {\begin{aligned}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{aligned}}}

## 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

{\displaystyle {\begin{aligned}D_{1}&=\operatorname {diag} \{\lambda _{1},\dots ,\lambda _{m}\}\\D_{0}&=R-D_{1}.\end{aligned}}}

### 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${\displaystyle ({\boldsymbol {\alpha }},S)}$ with an exit vector denoted ${\displaystyle {\boldsymbol {S}}^{0}=-S{\boldsymbol {1}}}$, the arrival process has generator matrix,

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

## 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,

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

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

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

and

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

## Fitting

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