In queueing theory, a discipline within the mathematical theory of probability, a Markovian arrival process (MAP or MArP) 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.
The processes were first suggested by Marcel F. Neuts in 1979.
A Markov arrival process is defined by two matrices, D<sub>0</sub> and D<sub>1</sub> where elements of D<sub>0</sub> represent hidden transitions and elements of D<sub>1</sub> observable transitions. The block matrix Q below is a transition rate matrix for a continuous-time Markov chain.
The simplest example is a Poisson process where D<sub>0</sub> = âÂÂû and D<sub>1</sub> = û where there is only one possible transition, it is observable, and occurs at rate û. For Q to be a valid transition rate matrix, the following restrictions apply to the D<sub>i</sub>
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 with an exit vector denoted , the arrival process has generator matrix,
The batch Markovian arrival process (BMAP) is a generalisation of the Markovian arrival process by allowing more than one arrival at a time. The homogeneous case has rate matrix,
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,
and
The Markov-modulated Poisson process or MMPP where m Poisson processes are switched between by an underlying continuous-time Markov chain. If each of the m Poisson processes has rate û<sub>i</sub> and the modulating continuous-time Markov has m àm transition rate matrix R, then the MAP representation is
A MAP can be fitted using an expectationâÂÂmaximization algorithm.