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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]



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 casesEdit

Markov-modulated Poisson processEdit

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 m × m transition rate matrix R, then the MAP representation is


Phase-type renewal processEdit

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,


Batch Markov arrival processEdit

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,


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,





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


See alsoEdit


  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. ^ a b 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)). Cite uses deprecated parameter |registration= (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)). Cite uses deprecated parameter |registration= (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.