Hierarchical Bayesian Parameter Estimation of Queueing Systems using Utilization Data

doi:10.23940/ijpe.22.05.p1.307316

Int J Performability Eng ›› 2022, Vol. 18 ›› Issue (5): 307-316.doi: 10.23940/ijpe.22.05.p1.307316

Hierarchical Bayesian Parameter Estimation of Queueing Systems using Utilization Data

Chen Li^a,*, Junjun Zheng^b, Hiroyuki Okamura^c, and Tadashi Dohi^c

^aDepartment of Computer Science and Systems Engineering, Kyushu Institute of Technology, Iizuka, 8208502, Japan;
^bDepartment of Information Science and Engineering, Ritsumeikan University, Kusatsu, 5258577, Japan;
^cGraduate School of Advanced Science Engineering, Hiroshima University, Higashihiroshima, 7398527, Japan

Submitted on ; Revised on ; Accepted on
Contact: * E-mail address: li260@bio.kyutech.ac.jp
About author:Chen Li is a Post-Doctoral Researcher with the Department of Bioscience and Bioinformatics, Faculty of Computer Science and Systems Engineering, Kyushu Institute of Technology, Japan. His research interests include performance evaluation, data mining and deep learning.
Junjun Zheng is an Assistant Professor with the Department of Information Science and Engineering, Ritsumeikan University, Japan. His research interests include performance evaluation and dependable computing.
Hiroyuki Okamura is a Professor with the Graduate School of Advanced Science and Engineering, Hiroshima University, Japan. His research interests include performance evaluation, dependable computing, and applied statistics.
Tadashi Dohi is a Professor with the Graduate School of Advanced Science and Engineering, Hiroshima University, Japan. His research interests include reliability engineering, software reliability, and dependable computing.

Abstract

Abstract: Utilization data is a kind of time-series data that consists of the proportion of the system's busy time in a fixed time interval. Utilization data can indicate the status of servers in a computer system, such as CPU utilization. Unfortunately, estimating model parameters from utilization data is challenging due to the inability to obtain the exact job arrival time and service time. Moreover, the maximum likelihood estimation (MLE) method tends to be sensitive, which may cause an overfitting problem. In this paper, a hierarchical Bayes (HB) based approach is proposed to estimate the parameters of queueing systems from utilization data. Specifically, a time non-homogeneous queueing system M_t/M/1/K whose job arrival follows a non-homogeneous Poisson process (NHPP) is supposed. Then, a series of homogeneous Poisson processes (HPP) is approximated to simplify the NHPP. Finally, the HB method is applied to estimate parameters of the M_t/M/1/K to address the sensitive issue of the MLE method. In numerical experiments, the effectiveness of the proposed HB-based approach with CPU utilization data is validated. In addition, the statistical properties of estimated parameters with MLE and HB are also studied in experiments.

Key words: parameter estimation, utilization data, queueing systems, non-homogeneous Poisson process, hierarchical Bayes

Chen Li, Junjun Zheng, Hiroyuki Okamura, and Tadashi Dohi. Hierarchical Bayesian Parameter Estimation of Queueing Systems using Utilization Data [J]. Int J Performability Eng, 2022, 18(5): 307-316.

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

References

1. Trivedi, K.S, Probability and Statistics with Reliability, Queueing, and Computer Sciences Applications, John Wiley & Sons, New York, 2nd edition, 2001.
2. Dharmaraja S., Trivedi K.S., andLogothetis D., Performance Modeling of Wireless Networks with Generally Distributed Handoff Interarrival Times. Computer Communications, vol. 26, no. 15, pp. 1747-1755, 2003.
3. Choudhury, A. and Basak, A.Statistical Inference on Traffic Intensity in an M/M/1 Queueing System. International Journal of Management Science and Engineering Management, vol.13, no. 4, pp. 274-279, 2018.
4. Stasiak M., Gabowski M., Wisniewski A., andZwierzykowski P.Modelling and Dimensioning of Mobile Networks: from GSM to LTE, Wiley, 2011.
5. Fischer M.J., Gross D., Masi D.B., andShortle J.F.Analyzing the Waiting Time Process in Internet Queueing Systems with the Transform Approximation Method. The Telecommunications Review, vol. 12, pp. 21-32, 2001.
6. Bhat U. N.An introduction to queueing theory: modeling analysis in applications, Second Edition. Birkhauser Boston, USA, 2008.
7. Clarke A.B.Maximum Likelihood Estimates in a Simple Queue. The Annals of Mathematical Statistics, vol. 28, no. 4, pp. 1036-1040, 1957.
8. Bhat, U.N. and Basawa, I.V.Maximum Likelihood Estimation in Queueing Systems. In Advances on Methodological and Applied Aspects of Probability and Statistics, CRC Press, pp. 13-30, 2019.
9. Dieleman N.A.Data-Driven Fitting of the G/G/1 Queue. Journal of Systems Science and Systems Engineering, vol.30, no. 1, pp. 17-28., 2021.
10. Krishnan, T. and McLachlan, G.J. The EM Algorithm and Extensions, John Wiley and Sons, 1997.
11. Buchholz P.An EM-algorithm for MAP Fitting from Real Traffic Data. In International Conference on Modelling Techniques and Tools for Computer Performance Evaluation. Springer, Berlin, Heidelberg, pp. 218-236, 2003.
12. Okamura H., Dohi T., andTrivedi K.S.Markovian Arrival Process Parameter Estimation with Group Data. IEEE/ACM Transactions on networking, vol. 17, no. 4, pp.1326-1339, 2009.
13. Li C., Okamura H., andDohi T.Parameter Estimation of M_t/M/1/K Queueing Systems with Utilization Data. IEEE Access, vol.7, pp. 42664-42671. 2019.
14. Ross J.V., Taimre T., andPollett P.K.Estimation for Queues from Queue Length Data. Queueing Systems, vol. 55, no. 2, pp.131-138, 2007.
15. Liu H., Liang W., Rai L., Teng K., andWang S.A Real-time Queue Length Estimation Method Based on Probe Vehicles in CV Environment. IEEE Access, vol. 7, pp. 20825-20839, 2019.
16. Rothkopf, M.H. and Oren, S.S.A Closure Approximation for the Nonstationary M/M/s Queue. Management Science, vol. 25, no. 6, pp. 522-534, 1979.
17. Heyman, D.P. and Whitt, W.The Asymptotic Behavior O Queues with Time-varying Arrival Rates. Journal of Applied Probability, vol. 21, no. 1, pp.143-156, 1984.
18. Wang T.Y., Ke J.C., Wang K.H., andHo S.C.Maximum Likelihood Estimates and Confidence Intervals of an M/M/R Queue with Heterogeneous Servers. Mathematical Methods of Operations Research, vol. 63, no. 2, pp. 371-384, 2006.
19. Dempster A.P., Laird N.M., andRubin D.B.Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society: Series B (Methodological), vol. 39, no. 1, pp. 1-22, 1977.
20. Wu C.J.,On the Convergence Properties of the EM Algorithm.The Annals of statistics, pp. 95-103, 1983.
21. Rydén T.An EM Algorithm for Estimation in Markov-modulated Poisson Processes. Computational Statistics & Data Analysis, vol. 21, no. 4, pp. 431-447, 1996.
22. Basawa I., Bhat U., andZhou J.Parameter Estimation in Queueing Systems using Partial Information.Ohio State University, Tech. Rep, 2006.
23. Armero, C. and Bayarri, M.J.A Bayesian Analysis of a Queueing System with Unlimited Service. Journal of Statistical Planning and Inference, vol. 58, no. 2, pp. 241-261, 1997.
24. Armero, C. and Conesa, D.Bayesian Hierarchical Models in Manufacturing Bulk Service Queues. Journal of statistical planning and inference, vol. 136, no. 2, pp. 335-354, 2006
25. Pelikan M.Hierarchical Bayesian Optimization Algorithm. In Hierarchical Bayesian optimization algorithm, Springer, Berlin, Heidelberg, pp. 105-129, 2005.
26. Nilsson H., Rieskamp J., andWagenmakers E.J.Hierarchical Bayesian Parameter Estimation for Cumulative Prospect Theory. Journal of Mathematical Psychology, vol. 55, no. 1, pp. 84-93, 2011.
27. Cruz F.R., Quinino R.C., andHo L.L.Bayesian Estimation of Traffic Intensity based on Queue Length in a Multi-server M/M/s Queue. Communications in Statistics-Simulation and Computation, vol. 46, no. 9, pp. 7319-7331, 2017.
28. Kiapour A.Bayesian Estimation of the Expected Queue Length of a System M/M/1 with Certain and Uncertain Priors.Communications in Statistics-Theory and Methods, pp.1-8, 2020.

[1]	Manu Banga, Abhay Bansal, and Archana Singh. Proposed Hybrid Approach to Predict Software Fault Detection [J]. Int J Performability Eng, 2019, 15(8): 2049-2061.
[2]	Vidhyashree Nagaraju, Thierry Wandji, and Lance Fiondella. Improved Algorithm for Non-Homogeneous Poisson Process Software Reliability Growth Models Incorporating Testing-Effort [J]. Int J Performability Eng, 2019, 15(5): 1265-1272.
[3]	Manu Banga, Abhay Bansal, and Archana Singh. Proposed Intelligent Software System for Early Fault Detection [J]. Int J Performability Eng, 2019, 15(10): 2578-2588.
[4]	Shixiao Xiao and Haiping Ren. Hierarchical Bayesian Reliability Analysis of Binomial Distribution based on Zero-Failure Data [J]. Int J Performability Eng, 2018, 14(9): 2076-2082.

Hierarchical Bayesian Parameter Estimation of Queueing Systems using Utilization Data

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 4

Recommended 0