Markovian arrival 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]

See also

• Rational arrival process

References

1. 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.
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)).
3. Chakravarthy, S. R. (2011). "Markovian Arrival Processes". Wiley Encyclopedia of Operations Research and Management Science. doi:10.1002/9780470400531.eorms0499. ISBN 9780470400531.
4. 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)).
5. Casale, G. (2011). "Building accurate workload models using Markovian arrival processes". ACM SIGMETRICS Performance Evaluation Review. 39: 357. doi:10.1145/2007116.2007176.
6. 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.
7. 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.
8. 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.
9. 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.