Bayesian predictive inference and threshold decisions for a discrete-time $Geo/DPH/1$ vacation queue under system-size observations

Abstract

This paper develops a Bayesian predictive and decision-oriented framework for a discretetime $Geo/DP H/1$ queue with discrete phase-type service and vacation times under a single-vacation policy, observed only through a dependent system-size trajectory. A fixedorder, fully supported, labelled DPH working model and finite truncation yield a hidden Markov representation. Within this family, the parameter-to-latent-kernel mapping is injective; this result does not establish identifiability from the observed process alone. Posterior inference combines state-and-event augmentation, forward-filtering backwardsampling, conjugate parameter proposals, and an exact Metropolis–Hastings correction for the parameter-dependent stationary initial distribution. Posterior uncertainty is propagated to multi-step system-size distributions, exceedance probabilities, predictive quantiles, and threshold decisions under asymmetric loss. The numerical study combines repeated-sample experiments across three traffic levels and two observation lengths with truncation and prior sensitivity, phase-order and representation comparisons, controlled misspecification, posterior predictive checks, and plug-in and dependence-ignoring benchmarks. Longer observation records generally improve recovery and prediction, although the mean vacation duration remains comparatively difficult to estimate. Predictive and decision performance remain stable within the evaluated local perturbations, whereas ignoring temporal dependence substantially degrades both. The results support the framework as a coherent and auditable basis for prediction and cost-sensitive decision support from partially observed queueing dynamics.

Keywords

• Bayesian predictive inference
• discrete-time queue
• discrete phase-type distribution
• hidden Markov model
• threshold decision
• vacation queue

Project Number

None

References

1. [1] C. Armero, Bayesian inference in Markovian queues, Queueing Syst. 15 (1–4), 419– 426, 1994.
2. [2] C. Armero and M. J. Bayarri, Dealing with uncertainties in queues and networks of queues: A Bayesian approach, in: S. Ghosh (Ed.), Multivariate Analysis, Design of Experiments, and Survey Sampling, 579–608, Marcel Dekker, New York, 1999.
3. [3] C. Armero and M. J. Bayarri, A Bayesian analysis of a queueing system with unlimited service, J. Statist. Plann. Inference 58 (2), 241–261, 1997.
4. [4] C. Armero and M. J. Bayarri, Prior assessments for prediction in queues, Statistician 43 (1), 139–153, 1994.
5. [5] C. Armero and M. J. Bayarri, Bayesian prediction in M/M/1 queues, Queueing Syst. 15 (1–4), 401–417, 1994.
6. [6] A. Asanjarani, Y. Nazarathy and P. G. Taylor, A survey of parameter and state estimation in queues, Queueing Syst. 97 (1–2), 39–80, 2021.
7. [7] S. Asmussen, Applied Probability and Queues, 2nd ed., Springer, New York, 2003.
8. [8] S. Asmussen, O. Nerman and M. Olsson, Fitting phase-type distributions via the EM algorithm, Scand. J. Statist. 23 (4), 419–441, 1996.

9. [9] M. C. Ausín, R. E. Lillo and M.P. Wiper, Bayesian control of the number of servers in a GI/M/c queueing system, J. Statist. Plann. Inference 137 (10), 3043–3057, 2007.
10. [10] M. C. Ausín, M. P. Wiper and R.E. Lillo, Bayesian estimation for the M/G/1 queue using a phase-type approximation, J. Statist. Plann. Inference 118 (1–2), 83–101, 2004.
11. [11] J. O. Berger, Statistical Decision Theory and Bayesian Analysis, 2nd ed., Springer, New York, 1985.
12. [12] M. Bladt and B. F. Nielsen, Matrix-Exponential Distributions in Applied Probability, Springer, New York, 2017.
13. [13] H. Braham, L. Berdjoudj, M. Boualem and N. Rahmania, Analysis of a non- Markovian queueing model: Bayesian statistics and MCMC methods, Monte Carlo Methods Appl. 25 (2), 147–154, 2019.
14. [14] O. Cappé, E. Moulines and T. Rydén, Inference in Hidden Markov Models, Springer, New York, 2005.
15. [15] C. K. Carter and R. Kohn, On Gibbs sampling for state space models, Biometrika 81 (3), 541–553, 1994.
16. [16] S. Chib, Calculating posterior distributions and modal estimates in Markov mixture models, J. Econometrics 75 (1), 79–97, 1996.
17. [17] A. B. Clarke, Maximum likelihood estimates in a simple queue, Ann. Math. Statist. 28 (4), 1036–1040, 1957.
18. [18] P. Fearnhead, Filtering recursions for calculating likelihoods for queues based on interdeparture time data, Statist. Comput. 14 (3), 261–266, 2004.
19. [19] A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari and D. B. Rubin, Bayesian Data Analysis, 3rd ed., Chapman and Hall/CRC, Boca Raton, 2013.
20. [20] A. Gelman, X.-L. Meng and H. Stern, Posterior predictive assessment of model fitness via realized discrepancies, Statist. Sinica 6 (4), 733–807, 1996.
21. [21] T. Gneiting, Making and evaluating point forecasts, J. Amer. Statist. Assoc. 106 (494), 746–762, 2011.
22. [22] T. Gneiting and A. E. Raftery, Strictly proper scoring rules, prediction, and estimation, J. Amer. Statist. Assoc. 102 (477), 359–378, 2007.
23. [23] C. Goswami and N. Selvaraju, The discrete-time MAP/PH/1 queue with multiple working vacations, Appl. Math. Model. 34 (4), 931–946, 2010.
24. [24] J. K. Jose and M. Manoharan, Bayesian estimation of queue parameters by bivariate prior distributions with order restrictions, J. Indian Statist. Assoc. 56 (1–2), 29–46, 2018.
25. [25] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, SIAM, Philadelphia, 1999.
26. [26] J.-H. Li, N.-S. Tian and W.-Y. Liu, Discrete-time GI/Geo/1 queue with multiple working vacations, Queueing Syst. 56 (1), 53–63, 2007.
27. [27] M. F. Neuts, Matrix-analytic methods in queuing theory, Eur. J. Oper. Res. 15 (1), 2–12, 1984.
28. [28] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins University Press, Baltimore, 1981.
29. [29] P. Ramírez-Cobo, R. E. Lillo and M. P. Wiper, Bayesian analysis of the stationary MAP2, Bayesian Anal. 12 (4), 1163–1194, 2017.
30. [30] D. Ríos Insua, M. Wiper and F. Ruggeri, Bayesian analysis of M/Er/1 and M/Hk/1 queues, Queueing Syst. 30 (3–4), 289–308, 1998.
31. [31] J. V. Ross, T. Taimre and P. K. Pollett, Estimation for queues from queue length data, Queueing Syst. 55 (2), 131–138, 2007.
32. [32] P. Salima, M. Manoharan and J. K. Jose, Bayesian inference on Geo/DPH/1 queueing model with phase-type vacation time, J. Appl. Probab. Stat. 21 (1), 1–20, 2026.
33. [33] P. Salima, M. Manoharan and J. K. Jose, Classical and Bayesian inference on Geo/DPH/1 queueing model with discrete phase-type vacation time, Reliab. Theory Appl. 21 (1), 309–323, 2026, doi:10.24412/1932-2321-2026-190-309-323.
34. [34] S. L. Scott, Bayesian methods for hidden Markov models: Recursive computing in the 21st century, J. Amer. Statist. Assoc. 97 (457), 337–351, 2002.
35. [35] M. Stephens, Dealing with label switching in mixture models, J. Roy. Statist. Soc. Ser. B 62 (4), 795–809, 2000.
36. [36] C. Sutton and M. I. Jordan, Bayesian inference for queueing networks and modeling of internet services, Ann. Appl. Stat. 5 (1), 254–282, 2011.
37. [37] A. Vehtari, A. Gelman, D. Simpson, B. Carpenter and P.-C. Bürkner, Ranknormalization, folding, and localization: An improved bR for assessing convergence of MCMC (with discussion), Bayesian Anal. 16 (2), 667–718, 2021.