Bayesian Analysis

Bayesian Analysis of the Stationary MAP2

P. Ramírez-Cobo, R. E. Lillo, and M. P. Wiper

Full-text: Open access


In this article we describe a method for carrying out Bayesian estimation for the two-state stationary Markov arrival process (MAP2), which has been proposed as a versatile model in a number of contexts. The approach is illustrated on both simulated and real data sets, where the performance of the MAP2 is compared against that of the well-known MMPP2. As an extension of the method, we estimate the queue length and virtual waiting time distributions of a stationary MAP2/G/1 queueing system, a matrix generalization of the M/G/1 queue that allows for dependent inter-arrival times. Our procedure is illustrated with applications in Internet traffic analysis.

Article information

Bayesian Anal., Volume 12, Number 4 (2017), 1163-1194.

First available in Project Euclid: 24 October 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference 62M05: Markov processes: estimation 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx]

phase-type distributions Markov modulated Poisson process (MMPP) Identifiability canonical representation Gibbs sampler steady-state distributions

Creative Commons Attribution 4.0 International License.


Ramírez-Cobo, P.; Lillo, R. E.; Wiper, M. P. Bayesian Analysis of the Stationary MAP 2. Bayesian Anal. 12 (2017), no. 4, 1163--1194. doi:10.1214/16-BA1026.

Export citation


  • Abate, J. and Whitt, W. (1995). “Numerical Inversion of Laplace transforms of probability distributions.” ORSA Journal on Computing, 7(1): 36–43.
  • Armero, C. and Bayarri, M. (1994). “Bayesian prediction in $M/M/1$ queues.” Queueing Systems, 15: 401–417.
  • Armero, C. and Bayarri, M. (1999). “Dealing with uncertainties in queues and networks of queues: a Bayesian approach.” Multivariate Analysis, Design of Experiments and Survey Sampling.
  • Asmussen, S. and Albrecher, H. (2010). Ruin probabilities. Advanced Series on Statistical Science & Applied Probability. World Scientific.
  • Asmussen, S. and Koole, G. (1993). “Marked point processes as limits of Markovian arrival streams.” Journal of Applied Probability, 30: 365–372.
  • Banerjee, A., Gupta, U., Horváth, G., and Chakravarthy, S. (2015). “Analysis of a finite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service.” Computers & Operations Research, 60: 138–149.
  • Blackwell, D. (1947). “Conditional expectation and unbiased sequential estimation.” Annals of Mathematical Statistics, 18: 105–110.
  • Bodrog, L., Heindlb, A., Horváth, G., and Telek, M. (2008). “A Markovian canonical form of second-order matrix-exponential processes.” European Journal of Operational Research, 190: 459–477.
  • Casale, G., Z. Zhang, E., and Simirni, E. (2010). “Trace data characterization and fitting for Markov modeling.” Performance Evaluation, 67: 61–79.
  • Çinlar, E. (1975). Introduction to stochastic processes. Usa: Prentice–Hall.
  • Chakravarthy, S. (2001). “The Batch Markovian arrival process: a review and future work.” In et al., A. K. (ed.), Advances in probability and stochastic processes, 21–49.
  • Chaudhry, M., Singh, G., and Gupta, U. (2013). “A simple and complete computational analysis of $MAP/R/1$ queue using roots.” Methodology and Computing in Applied Probability, 15: 563–582.
  • Cheung, E. and Landriault, D. (2010). “A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model.” Insurance: Mathematics and Economics, 46: 127–134.
  • Cheung, E. and Runhuan, F. (2013). “A unified analysis of claims costs up to ruin in a Markovian arrival risk model.” Insurance: Mathematics and Economics, 53: 98–109.
  • Chib, S. (1995). “Marginal likelihood from the Gibbs output.” Journal of the American Statistical Association, 90: 1313–1321.
  • Dudina, O., Kim, C., and Dudin, S. (2013). “Retrial queueing system with Markovian arrival flow and phase-type service time distribution.” Computers & Industrial Engineering, 66(2): 360–370.
  • Fearnhead, P. and Sherlock, C. (2006). “An exact Gibbs sampler for the Markov-modulated Poisson process.” Journal of the Royal Statistical Society: Series B, 65(5): 767–784.
  • Green, P. (1995). “Reversible jump Markov Chain Monte Carlo computation and Bayesian model determination.” Biometrika, 82: 711–732.
  • Gruet, M., Philippe, A., and Robert, C. (1999). “MCMC control spreadsheets for exponential mixture estimation.” Journal of Computational and Graphical Statistics, 8: 298–317.
  • Hervé, L. and Ledoux, J. (2013). “Geometric rho-mixing property of the inter-arrival times of a stationary Markovian Arrival Process.” Journal of Applied Probability, 50: 598–601.
  • Kang, S. and Sung, D. (1995). “Two-state MMPP modeling of ATM superposed traffic streams based on the characterization of correlated interarrival times.” In Global Telecommunications Conference, 1995, GLOBECOM’95, IEEE, volume 2, 1422–1426. IEEE.
  • Kriege, J. and Buchholz, P. (2010). “An empirical comparison of MAP fitting algorithms.” In International GI/ITG Conference on Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance, 259–273. Springer.
  • Latouche, G. and Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling, volume 5. SIAM.
  • Lucantoni, D. (1991). “New results for the single server queue with a Batch Markovian Arrival Process.” Stochastic Models, 7: 1–46.
  • Lucantoni, D. (1993). “The $BMAP/G/1$ queue: A tutorial.” In Donatiello, L. and Nelson, R. (eds.), Models and Techniques for Performance Evaluation of Computer and Communication Systems, 330–358. New York: Springer.
  • Mcgrath, M., Gross, D., and Singpurwalla, N. (1987). “A subjective Bayesian approach to the theory of queues I-Modeling.” Queueing Systems, 1(4): 317–333.
  • Montoro-Cazorla, D. and Pérez-Ocón, R. (2014). “A reliability system under different types of shock governed by a Markovian arrival process and maintenance policy K.” European Journal of Operational Research, 235(3): 636–642.
  • Neuts, M. F. (1974). Probability distributions of phase type. Purdue University. Department of Statistics.
  • Neuts, M. F. (1979). “A versatile Markovian point process.” Journal of Applied Probability, 16: 764–779.
  • Okamura, H., Dohi, T., and Trivedi, K. (2009). “Markovian arrival process parameter estimation with group data.” IEEE/ACM Transactions on Networking, 17(4): 1326–1339.
  • Prabhu, N. (1998). Stochastic storage processes: queues, insurance risk, dams, and data communication. Springer Science & Business Media.
  • Ramaswami, V. (1980). “The N/G/1 queue and its detailed analysis.” Advances in Applied Probability, 222–261.
  • Ramirez, P., Lillo, R., and Wiper, M. (2008). “Bayesian analysis of a queueing system with a long-tailed arrival process.” Communications in Statistics – Simulation and Computation, 37(4): 697–712.
  • Ramírez-Cobo, P., Lillo, R., and Wiper, M. (2010a). “Bayesian inference for double Pareto lognormal queues.” The Annals of Applied Statistics, 4(3): 1533–1557.
  • Ramírez-Cobo, P., Lillo, R., and Wiper, M. (2010b). “Nonidentifiability of the two-state Markovian arrival process.” Journal of Applied Probability, 47(3): 630–649.
  • Ramírez-Cobo, P., Lillo, R., and Wiper, M. (2014a). “Identifiability of the MAP2/G/1 queueing system.” Top, 22(1): 274–289.
  • Ramírez-Cobo, P., Marzo, X., Olivares-Nadal, A., Francoso, J., Carrizosa, E., and Pita, M. (2014b). “The Markovian arrival process: a statistical model for daily precipitation amounts.” Journal of Hydrology, 510: 459–471.
  • Riska, A., Squillante, M., Yu, S.-Z., Liu, Z., and Zhang, L. (2002). “Matrix-analytic analysis of a MAP/PH/1 queue fitted to web server data.” Matrix-Analytic Methods; Theory and Applications, 333–356.
  • Rodríguez, J., Lillo, R., and Ramírez-Cobo, P. (2015). “Failure modeling of an electrical N-component framework by the non-stationary Markovian arrival process.” Reliability Engineering & System Safety, 134: 126–133.
  • Rydén, T. (1996). “On identifiability and order of continous-time aggregated Markov chains, Markov-modulated Poisson processes, and phase-type distributions.” Journal of Applied Probability, 33: 640–653.
  • Scott, S. (1999). “Bayesian analysis of a two-state Markov modulated Poisson process.” Journal of Computational and Graphical Statistics, 8(3): 662–670.
  • Scott, S. (2002). “Bayesian methods for hidden Markov models. Recursive Computing in the 21st Century.” Journal of the American Statistical Association, 457: 337–351.
  • Scott, S. and Smyth, P. (2003). “The Markov Modulated Poisson Process and Markov Poisson Cascade with applications to web traffic modeling.” Bayesian Statistics, 7: 1–10.
  • Telek, M. and Horváth, G. (2007). “A minimal representation of Markov arrival processes and a moments matching method.” Performance Evaluation, 64(9): 1153–1168.
  • Wu, J., Liu, Z., and Yang, G. (2011). “Analysis of the finite source MAP/PH/N retrial G-queue operating in a random environment.” Applied Mathematical Modelling, 35(3): 1184–1193.
  • Xue, J. and Alfa, A. (2011). “Geometric tail of queue length of low-priority customers in a nonpreemptive priority MAP/PH/1 queue.” Queueing Systems, 69(1): 45–76.
  • Zhang, M. and Hou, Z. (2011). “Performance analysis of MAP/G/1 queue with working vacations and vacation interruption.” Applied Mathematical Modelling, 35(4): 1551–1560.