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{{About|arrival processes to queues|bivariate processes|Markov additive process}}
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>
 
The processes were first suggested by Neuts in 1979.<ref>{{cite jstor|3213143}}</ref><ref name="asmussen" />
 
==Definition==
 
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>
 
:<math>
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]\; .</math>
 
The simplest example is a Poisson process where ''D''<sub>0</sub>&nbsp;=&nbsp;−''λ'' and ''D''<sub>1</sub>&nbsp;=&nbsp;''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>
 
:<math>\begin{align}
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{align}</math>
 
==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]].<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''&nbsp;×&nbsp;''m'' transition rate matrix ''R'', then the MAP representation is
 
:<math>\begin{align}
D_{1} &= \operatorname{diag}\{\lambda_{1},\dots,\lambda_{m}\}\\
D_{0} &=R-D_1.
\end{align}</math>
 
===Phase-type renewal process===
 
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,
 
:<math>
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]
</math>
 
==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.<ref>{{cite doi|10.1007/BFb0013859}}</ref> The homogeneous case has rate matrix,
 
:<math>
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]\; .</math>
 
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,
 
:<math>
0\leq [D_{k}]_{i,j}<\infty\;\;\;\; 1\leq k
</math>
 
:<math>
0\leq [D_{0}]_{i,j}<\infty\;\;\;\; i\neq j
</math>
 
:<math>
[D_{0}]_{i,i}<0\;
</math>
 
and
:<math>
\sum^{\infty}_{k=0}D_{k}\boldsymbol{1}=\boldsymbol{0}
</math>
 
==Fitting==
 
A MAP can be fitted using an [[expectation–maximization algorithm]].<ref>{{cite doi|10.1007/978-3-540-45232-4_14}}</ref>
 
===Software===
 
* [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>
 
==References==
{{Reflist}}
 
{{Queueing theory}}
 
[[Category:Queueing theory]]
[[Category:Markov processes]]

Revision as of 18:14, 21 January 2014

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[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 Neuts in 1979.[4][2]

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.[5]

Q=[D0D1000D0D1000D0D1].

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

0[D1]i,j<0[D0]i,j<ij[D0]i,i<0(D0+D1)1=0

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

D1=diag{λ1,,λm}D0=RD1.

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(α,S) with an exit vector denoted S0=S1, the arrival process has generator matrix,

Q=[SS0α000SS0α000SS0α]

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.[7] The homogeneous case has rate matrix,

Q=[D0D1D2D30D0D1D200D0D1].

An arrival of size k occurs every time a transition occurs in the sub-matrix Dk. Sub-matrices Dk have elements of λi,j, the rate of a Poisson process, such that,

0[Dk]i,j<1k
0[D0]i,j<ij
[D0]i,i<0

and

k=0Dk1=0

Fitting

A MAP can be fitted using an expectation–maximization algorithm.[8]

Software

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.

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