Markovian arrival process
In queueing theory, a discipline within the mathematical theory of probability, a Markovian arrival process 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.
Definition
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.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 Di
Special cases
Phase-type renewal 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 with an exit vector denoted, the arrival process has generator matrix,Generalizations
Batch Markov arrival process
The batch Markovian arrival process 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