## Bayesian Analysis

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

### Bayesian Analysis of the Stationary MAP_{2}

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

**Full-text: Open access**

#### Abstract

In this article we describe a method for carrying out Bayesian estimation for the two-state stationary Markov arrival process (${\mathit{MAP}}_{2}$), 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 ${\mathit{MAP}}_{2}$ is compared against that of the well-known ${\mathit{MMPP}}_{2}$. As an extension of the method, we estimate the queue length and virtual waiting time distributions of a stationary ${\mathit{MAP}}_{2}/G/1$ 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

**Source**

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

**Dates**

First available in Project Euclid: 24 October 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ba/1477321094

**Digital Object Identifier**

doi:10.1214/16-BA1026

**Subjects**

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

**Keywords**

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

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

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. https://projecteuclid.org/euclid.ba/1477321094

#### References

- 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.Mathematical Reviews (MathSciNet): MR3085880

Digital Object Identifier: doi:10.1007/s11009-011-9266-3 - 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.Mathematical Reviews (MathSciNet): MR2758340

Digital Object Identifier: doi:10.1214/10-AOAS336

Project Euclid: euclid.aoas/1287409385 - 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.Mathematical Reviews (MathSciNet): MR2835230

Digital Object Identifier: doi:10.1007/s11134-011-9221-6 - 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.

### More like this

- Bayesian inference for double Pareto lognormal
queues

Ramirez-Cobo, Pepa, Lillo, Rosa E., Wilson, Simon, and Wiper, Michael P., The Annals of Applied Statistics, 2010 - Bandwidth Estimation for Best-Effort Internet Traffic

Cao, Jin, Cleveland, William S., and Sun, Don X., Statistical Science, 2004 - The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

Hwang, Gang Uk, Choi, Bong Dae, and Kim, Jae-Kyoon, Journal of Applied Probability, 2002

- Bayesian inference for double Pareto lognormal
queues

Ramirez-Cobo, Pepa, Lillo, Rosa E., Wilson, Simon, and Wiper, Michael P., The Annals of Applied Statistics, 2010 - Bandwidth Estimation for Best-Effort Internet Traffic

Cao, Jin, Cleveland, William S., and Sun, Don X., Statistical Science, 2004 - The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1

Hwang, Gang Uk, Choi, Bong Dae, and Kim, Jae-Kyoon, Journal of Applied Probability, 2002 - Tightness of the stationary waiting time in heavy traffic

Szczotka, Władysław, Advances in Applied Probability, 1999 - Real-Time Bayesian Parameter Estimation for Item Response Models

Weng, Ruby Chiu-Hsing and Coad, D. Stephen, Bayesian Analysis, 2018 - Censoring Technique Applied to a Map/G/1 Queue with Set-up Time and Multiple Vacations

Zhang, Zhenzhong and Tong, Jinying, Taiwanese Journal of Mathematics, 2011 - Stationary Waiting-Time Distributions for Single-Server Queues

Loynes, R. M., The Annals of Mathematical Statistics, 1962 - Nonparametric Estimation of the Stationary Waiting Time Distribution Function for the $GI/G/1$ Queue

Pitts, Susan M., The Annals of Statistics, 1994 - A BMAP/SM/1 queueing system with Markovian arrival input of disasters

Dudin, Alexander and Nishimura, Shoichi, Journal of Applied Probability, 1999 - Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain

Kendall, David G., The Annals of Mathematical Statistics, 1953