Markovian arrival process

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

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

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

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

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

• Rational arrival process