Markovian arrival process

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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. The simplest such process is a Poisson process where the time between each arrival is exponentially distributed.[2][3]

The processes were first suggested by Marcel F. Neuts in 1979.[2][4]

Definition

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Perspective

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]

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

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Perspective

Batch Markov arrival process

The batch Markovian arrival process (BMAP) is a generalisation of the Markovian arrival process by allowing more than one arrival at a time.[6] [7] 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

Markov-modulated Poisson process

The Markov-modulated Poisson process or MMPP where m Poisson processes are switched between by an underlying continuous-time Markov chain.[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

Fitting

A MAP can be fitted using an expectation–maximization algorithm.[9]

Software

See also

References

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