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]

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

Special cases

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 with an exit vector denoted , the arrival process has generator matrix,

Generalizations

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.[6] [7] The homogeneous case has rate matrix,

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

and

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

Fitting

A MAP can be fitted using an expectation–maximization algorithm.[9]

Software

See also

References

  1. Asmussen, S. R. (2003). "Markov Additive Models". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 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. S2CID 122810934.
  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. doi:10.2307/3213143. JSTOR 3213143. S2CID 123525892.
  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. Lucantoni, D. M. (1993). "The BMAP/G/1 queue: A tutorial". Performance Evaluation of Computer and Communication Systems. Lecture Notes in Computer Science. Vol. 729. pp. 330–358. doi:10.1007/BFb0013859. ISBN 3-540-57297-X. S2CID 35110866.
  7. Singh, Gagandeep; Gupta, U. C.; Chaudhry, M. L. (2016). "Detailed computational analysis of queueing-time distributions of the BMAP/G/1 queue using roots". Journal of Applied Probability. 53 (4): 1078–1097. doi:10.1017/jpr.2016.66. S2CID 27505255.
  8. 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.
  9. 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. Vol. 2794. pp. 218–236. doi:10.1007/978-3-540-45232-4_14. ISBN 978-3-540-40814-7.
  10. Casale, G.; Zhang, E. Z.; Smirni, E. (2008). "KPC-Toolbox: Simple Yet Effective Trace Fitting Using Markovian Arrival Processes" (PDF). 2008 Fifth International Conference on Quantitative Evaluation of Systems. p. 83. doi:10.1109/QEST.2008.33. ISBN 978-0-7695-3360-5. S2CID 252444.
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