Markovian arrival 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,

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

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

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

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

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

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