Markovian Arrival Processes - Batch Markov Arrival Process

Batch Markov Arrival Process

The batch Markovian arrival process (BMAP) is a generalisation of the Markovian arrival process by having arrivals of size greater than one. 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 occurs every time a transition occurs in the sub-matrix . Sub-matrices have elements of, the rate of a Poisson process, such that,

$0\leq _{i,j}<\infty\;\;\;\; 1\leq k$

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

$_{i,i}<0\;$

and

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

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