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| {{About|arrival processes to queues|bivariate processes|Markov additive process}}
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| In [[queueing theory]], a discipline within the mathematical [[probability theory|theory of probability]], a '''Markovian arrival process''' ('''MAP''' or '''MArP'''<ref>{{cite doi|10.1007/0-387-21525-5_11}}</ref>) 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 [[exponential distribution|exponentially distributed]].<ref name="asmussen">{{cite doi|10.1111/1467-9469.00186}}</ref><ref>{{cite doi|10.1002/9780470400531.eorms0499}}</ref>
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| The processes were first suggested by Neuts in 1979.<ref>{{cite jstor|3213143}}</ref><ref name="asmussen" />
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| ==Definition==
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| A Markov arrival process is defined by two matrices ''D''<sub>0</sub> and ''D''<sub>1</sub> where elements of ''D''<sub>0</sub> represent hidden transitions and elements of ''D''<sub>1</sub> observable transitions. The [[block matrix]] ''Q'' below is a [[transition rate matrix]] for a [[continuous-time Markov chain]].<ref>{{cite doi|10.1145/2007116.2007176}}</ref>
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| :<math>
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| Q=\left[\begin{matrix}
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| D_{0}&D_{1}&0&0&\dots\\
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| 0&D_{0}&D_{1}&0&\dots\\
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| 0&0&D_{0}&D_{1}&\dots\\
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| \vdots & \vdots & \ddots & \ddots & \ddots
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| \end{matrix}\right]\; .</math>
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| The simplest example is a Poisson process where ''D''<sub>0</sub> = −''λ'' and ''D''<sub>1</sub> = ''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''<sub>''i''</sub>
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| :<math>\begin{align}
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| 0\leq [D_{1}]_{i,j}&<\infty \\
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| 0\leq [D_{0}]_{i,j}&<\infty \quad i\neq j \\
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| \, [D_{0}]_{i,i}&<0 \\
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| (D_{0}+D_{1})\boldsymbol{1} &= \boldsymbol{0}
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| \end{align}</math>
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| ==Special cases==
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| === Markov-modulated Poisson process ===
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| The '''Markov-modulated Poisson process''' or '''MMPP''' where ''m'' Poisson processes are switched between by an underlying [[continuous-time Markov chain]].<ref>{{cite doi|10.1016/0166-5316(93)90035-S}}</ref> If each of the ''m'' Poisson processes has rate ''λ''<sub>''i''</sub> and the modulating continuous-time Markov has has ''m'' × ''m'' transition rate matrix ''R'', then the MAP representation is
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| :<math>\begin{align}
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| D_{1} &= \operatorname{diag}\{\lambda_{1},\dots,\lambda_{m}\}\\
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| D_{0} &=R-D_1.
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| \end{align}</math>
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| ===Phase-type renewal process===
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| The '''phase-type renewal process''' is a Markov arrival process with [[phase-type distribution|phase-type distributed]] sojourn between arrivals. For example if an arrival process has an interarrival time distribution PH<math>(\boldsymbol{\alpha},S)</math> with an exit vector denoted <math>\boldsymbol{S}^{0}=-S\boldsymbol{1}</math>, the arrival process has generator matrix,
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| :<math> | |
| Q=\left[\begin{matrix}
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| S&\boldsymbol{S}^{0}\boldsymbol{\alpha}&0&0&\dots\\
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| 0&S&\boldsymbol{S}^{0}\boldsymbol{\alpha}&0&\dots\\
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| 0&0&S&\boldsymbol{S}^{0}\boldsymbol{\alpha}&\dots\\
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| \vdots&\vdots&\ddots&\ddots&\ddots\\
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| \end{matrix}\right]
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| </math>
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| ==Batch Markov arrival process==
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| The '''batch Markovian arrival process''' (''BMAP'') is a generalisation of the Markovian arrival process by allowing more than one arrival at a time.<ref>{{cite doi|10.1007/BFb0013859}}</ref> The homogeneous case has rate matrix,
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| :<math>
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| Q=\left[\begin{matrix}
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| D_{0}&D_{1}&D_{2}&D_{3}&\dots\\
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| 0&D_{0}&D_{1}&D_{2}&\dots\\
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| 0&0&D_{0}&D_{1}&\dots\\
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| \vdots & \vdots & \ddots & \ddots & \ddots
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| \end{matrix}\right]\; .</math>
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| An arrival of size <math>k</math> occurs every time a transition occurs in the sub-matrix <math>D_{k}</math>. Sub-matrices <math>D_{k}</math> have elements of <math>\lambda_{i,j}</math>, the rate of a [[Poisson process]], such that,
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| :<math>
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| 0\leq [D_{k}]_{i,j}<\infty\;\;\;\; 1\leq k
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| </math>
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| :<math>
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| 0\leq [D_{0}]_{i,j}<\infty\;\;\;\; i\neq j
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| </math>
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| :<math>
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| [D_{0}]_{i,i}<0\;
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| </math>
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| and
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| :<math>
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| \sum^{\infty}_{k=0}D_{k}\boldsymbol{1}=\boldsymbol{0}
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| </math>
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| ==Fitting==
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| A MAP can be fitted using an [[expectation–maximization algorithm]].<ref>{{cite doi|10.1007/978-3-540-45232-4_14}}</ref>
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| ===Software===
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| * [http://www.cs.wm.edu/MAPQN/kpctoolbox.html KPC-toolbox] a series of [[MATLAB]] scripts to fit a MAP to data.<ref>{{cite doi|10.1109/QEST.2008.33}}</ref>
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| ==References==
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| {{Reflist}}
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| {{Queueing theory}}
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| [[Category:Queueing theory]]
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| [[Category:Markov processes]]
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