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 Marcel F. 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 D_i

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

Special cases

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,

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

Generalizations

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.^{[6] [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

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

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.^[8] If each of the m Poisson processes has rate λ_i and the modulating continuous-time Markov has m × m transition rate matrix R, then the MAP representation is

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

Fitting

A MAP can be fitted using an expectation–maximization algorithm.^[9]

Software

• KPC-toolbox a library of MATLAB scripts to fit a MAP to data.^[10]