Markovian arrival process

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.{{Cite journal | last1 = Casale | first1 = G. | doi = 10.1145/2007116.2007176 | title = Building accurate workload models using Markovian arrival processes | journal = ACM SIGMETRICS Performance Evaluation Review | volume = 39 | pages = 357 | year = 2011 }}

:$$

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

:$\backslash begin\{align\}$

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}

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$(\backslash boldsymbol\{\backslash alpha\},S)$ with an exit vector denoted $\backslash boldsymbol\{S\}^\{0\}=-S\backslash boldsymbol\{1\}$, 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]

The batch Markovian arrival process (BMAP) is a generalisation of the Markovian arrival process by allowing more than one arrival at a time.{{Cite book | last1 = Lucantoni | first1 = D. M. | chapter = The BMAP/G/1 queue: A tutorial | doi = 10.1007/BFb0013859 | title = Performance Evaluation of Computer and Communication Systems | series = Lecture Notes in Computer Science | volume = 729 | pages = 330–358 | year = 1993 | isbn = 3-540-57297-X | s2cid = 35110866 }}{{cite journal |last1=Singh |first1=Gagandeep |last2=Gupta |first2=U. C. |last3=Chaudhry |first3=M. L. |title=Detailed computational analysis of queueing-time distributions of the BMAP/G/1 queue using roots |journal=Journal of Applied Probability |date=2016 |volume=53 |issue=4 |pages=1078–1097 |doi=10.1017/jpr.2016.66 |s2cid=27505255 |url=https://www.cambridge.org/core/journals/journal-of-applied-probability/article/abs/detailed-computational-analysis-of-queueingtime-distributions-of-the-bmapg1-queue-using-roots/740DBCF255AFE602075EDB174FF0F25D}} 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 $\backslash lambda\_\{i,j\}$, 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^{\infty}_{k=0}D_{k}\boldsymbol{1}=\boldsymbol{0}

The Markov-modulated Poisson process or MMPP where m Poisson processes are switched between by an underlying continuous-time Markov chain.{{Cite journal |last1=Fischer |first1=W. |last2=Meier-Hellstern |first2=K. |year=1993 |title=The Markov-modulated Poisson process (MMPP) cookbook |journal=Performance Evaluation |volume=18 |issue=2 |pages=149 |doi=10.1016/0166-5316(93)90035-S}} 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

:$\backslash begin\{align\}$

D_{1} &= \operatorname{diag}\{\lambda_{1},\dots,\lambda_{m}\}\\

D_{0} &=R-D_1.

\end{align}

A MAP can be fitted using an expectation–maximization algorithm.{{Cite book | last1 = Buchholz | first1 = P. | chapter = An EM-Algorithm for MAP Fitting from Real Traffic Data | doi = 10.1007/978-3-540-45232-4_14 | title = Computer Performance Evaluation. Modelling Techniques and Tools | series = Lecture Notes in Computer Science | volume = 2794 | pages = 218–236 | year = 2003 | isbn = 978-3-540-40814-7 }}

• [https://github.com/kpctoolboxteam/kpc-toolbox KPC-toolbox] a library of MATLAB scripts to fit a MAP to data.{{Cite book | last1 = Casale | first1 = G. | last2 = Zhang | first2 = E. Z. | last3 = Smirni | first3 = E. | author3-link = Evgenia Smirni | doi = 10.1109/QEST.2008.33 | chapter = KPC-Toolbox: Simple Yet Effective Trace Fitting Using Markovian Arrival Processes | title = 2008 Fifth International Conference on Quantitative Evaluation of Systems | pages = 83 | year = 2008 | isbn = 978-0-7695-3360-5 | s2cid = 252444 | chapter-url = http://www.doc.ic.ac.uk/~gcasale/qest08kpctoolbox.pdf}}

• Rational arrival process

{{Reflist}}

{{Queueing theory}}

Category:Queueing theory

Category:Markov processes