Markovian arrival processes

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 Neuts in 1979.

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 D_i

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 λ_i and the modulating continuous-time Markov has m × m transition rate matrix R, then the MAP representation is

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,

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