Markovian Arrival Processes - Markov Arrival Process

The Markov arrival process (MAP) is a generalization of the Poisson process by having non-exponential distribution sojourn between arrivals. The homogeneous case has rate matrix,

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]\; .

An arrival is seen every time a transition occurs that increases the level (a marked transition), e.g. a transition in the sub-matrix. Sub-matrices and have elements of, the rate of a Poisson process, such that,

0\leq _{i,j}<\infty
0\leq _{i,j}<\infty\;\;\;\; i\neq j
_{i,i}<0\;

and

(D_{0}+D_{1})\boldsymbol{1}=\boldsymbol{0}

There are several special cases of the Markov arrival process.

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