In queueing theory, a discipline within the mathematical theory of probability, a Markovian arrival process (MAP or MArP[1]) is a mathematical model for the time between job arrivals to a system.
[2][3] The processes were first suggested by Marcel F. Neuts in 1979.
[2][4] 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.
For Q to be a valid transition rate matrix, the following restrictions apply to the Di 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
[6] [7] The homogeneous case has rate matrix, An arrival of size
[8] If each of the m Poisson processes has rate λi and the modulating continuous-time Markov has m × m transition rate matrix R, then the MAP representation is A MAP can be fitted using an expectation–maximization algorithm.