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29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. 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.

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.

$Q=\left[{\begin{matrix}D_{0}&D_{1}&0&0&\dots \\0&D_{0}&D_{1}&0&\dots \\0&0&D_{0}&D_{1}&\dots \\\vdots &\vdots &\ddots &\ddots &\ddots \end{matrix}}\right]\;.$

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

${\begin{aligned}0\leq [D_{1}]_{i,j}&<\infty \\0\leq [D_{0}]_{i,j}&<\infty \quad i\neq j\\\,[D_{0}]_{i,i}&<0\\(D_{0}+D_{1}){\boldsymbol {1}}&={\boldsymbol {0}}\end{aligned}}$

Special cases

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. If each of the m Poisson processes has rate λ_i and the modulating continuous-time Markov has has m × m transition rate matrix R, then the MAP representation is

${\begin{aligned}D_{1}&=\operatorname {diag} \{\lambda _{1},\dots ,\lambda _{m}\}\\D_{0}&=R-D_{1}.\end{aligned}}$

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$({\boldsymbol {\alpha }},S)$ with an exit vector denoted ${\boldsymbol {S}}^{0}=-S{\boldsymbol {1}}$, the arrival process has generator matrix,

$Q=\left[{\begin{matrix}S&{\boldsymbol {S}}^{0}{\boldsymbol {\alpha }}&0&0&\dots \\0&S&{\boldsymbol {S}}^{0}{\boldsymbol {\alpha }}&0&\dots \\0&0&S&{\boldsymbol {S}}^{0}{\boldsymbol {\alpha }}&\dots \\\vdots &\vdots &\ddots &\ddots &\ddots \\\end{matrix}}\right]$

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. The homogeneous case has rate matrix,

$Q=\left[{\begin{matrix}D_{0}&D_{1}&D_{2}&D_{3}&\dots \\0&D_{0}&D_{1}&D_{2}&\dots \\0&0&D_{0}&D_{1}&\dots \\\vdots &\vdots &\ddots &\ddots &\ddots \end{matrix}}\right]\;.$

An arrival of size $k$ occurs every time a transition occurs in the sub-matrix $D_{k}$. Sub-matrices $D_{k}$ have elements of $\lambda _{i,j}$, the rate of a Poisson process, such that,

$0\leq [D_{k}]_{i,j}<\infty \;\;\;\;1\leq k$

$0\leq [D_{0}]_{i,j}<\infty \;\;\;\;i\neq j$

$[D_{0}]_{i,i}<0\;$

and

$\sum _{k=0}^{\infty }D_{k}{\boldsymbol {1}}={\boldsymbol {0}}$

Fitting

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

Software

• KPC-toolbox a series of MATLAB scripts to fit a MAP to data.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.