Markovian arrival process

This article is about arrival processes to queues. For bivariate processes, see Markov additive process.

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.^[2]^[4]

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=\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.^[6] 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

{\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.^[7] 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.^[8]

Software

KPC-toolbox a library of MATLAB scripts to fit a MAP to data.^[9]

References

↑ Asmussen, S. R. (2003). "Markov Additive Models". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 302–339. doi:10.1007/0-387-21525-5_11. ISBN 978-0-387-00211-8.
1 2 Asmussen, S. (2000). "Matrix-analytic Models and their Analysis". Scandinavian Journal of Statistics. 27 (2): 193–226. doi:10.1111/1467-9469.00186. JSTOR 4616600 – via JSTOR. (registration required (help)).
↑ Chakravarthy, S. R. (2011). "Markovian Arrival Processes". Wiley Encyclopedia of Operations Research and Management Science. doi:10.1002/9780470400531.eorms0499. ISBN 9780470400531.
↑ Neuts, Marcel F. (1979). "A Versatile Markovian Point Process". Journal of Applied Probability. Applied Probability Trust. 16 (4): 764–779. JSTOR 3213143 – via JSTOR. (registration required (help)).
↑ Casale, G. (2011). "Building accurate workload models using Markovian arrival processes". ACM SIGMETRICS Performance Evaluation Review. 39: 357. doi:10.1145/2007116.2007176.
↑ Fischer, W.; Meier-Hellstern, K. (1993). "The Markov-modulated Poisson process (MMPP) cookbook". Performance Evaluation. 18 (2): 149. doi:10.1016/0166-5316(93)90035-S.
↑ Lucantoni, D. M. (1993). "The BMAP/G/1 queue: A tutorial". Performance Evaluation of Computer and Communication Systems. Lecture Notes in Computer Science. 729. p. 330. doi:10.1007/BFb0013859. ISBN 3-540-57297-X.
↑ Buchholz, P. (2003). "An EM-Algorithm for MAP Fitting from Real Traffic Data". Computer Performance Evaluation. Modelling Techniques and Tools. Lecture Notes in Computer Science. 2794. pp. 218–236. doi:10.1007/978-3-540-45232-4_14. ISBN 978-3-540-40814-7.
↑ Casale, G.; Zhang, E. Z.; Smirni, E. (2008). "KPC-Toolbox: Simple Yet Effective Trace Fitting Using Markovian Arrival Processes". 2008 Fifth International Conference on Quantitative Evaluation of Systems (PDF). p. 83. doi:10.1109/QEST.2008.33. ISBN 978-0-7695-3360-5.

Queueing theory

Single queueing nodes	D/M/1 queue M/D/1 queue M/D/c queue M/M/1 queue Burke's theorem M/M/c queue M/M/∞ queue M/G/1 queue Pollaczek–Khinchine formula Matrix analytic method M/G/k queue G/M/1 queue G/G/1 queue Kingman's formula Lindley equation Fork–join queue Bulk queue

Arrival processes	Poisson process Markovian arrival process Rational arrival process

Queueing networks	Jackson network Traffic equations Gordon–Newell theorem Mean value analysis Buzen's algorithm Kelly network G-network BCMP network

Service policies	FIFO LIFO Processor sharing Round-robin Shortest job first Shortest remaining time

Key concepts	Continuous-time Markov chain Kendall's notation Little's law Product-form solution Balance equation Quasireversibility Flow-equivalent server method Arrival theorem Decomposition method Beneš method

Limit theorems	Fluid limit Mean field theory Heavy traffic approximation Reflected Brownian motion

Extensions	Fluid queue Layered queueing network Polling system Adversarial queueing network Loss network Retrial queue

Information systems	Data buffer Erlang (unit) Erlang distribution Flow control (data) Message queue Network congestion Network scheduler Pipeline (software) Quality of service Scheduling (computing) Teletraffic engineering

Category

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