Analysis of a single server queue in a multi-phase random environment with working vacations and customers’ impatience

• Autorzy: Bouchentouf Amina Angelika , Guendouzi Abdelhak , Houalef Meriem , Majid Shakir

Identyfikatory

DOI

10.37190/ord220202

• Języki publikacji: EN

Abstrakty

EN

In this paper, we analyze an M/M/1 queueing system under both single and multiple working vacation policies, multiphase random environment, waiting server, balking and reneging. When the system is in operative phase j = 1, 2, . . . , K, customers are served one by one. Whenever the system becomes empty, the server waits a random amount of time before taking a vacation, causing the system to move to working vacation phase 0 at which new arrivals are served at a lower rate. Using the probability generating function method, we obtain the distribution for the steady-state probabilities of the system. Then, we derive important performance measures of the queueing system. Finally, some numerical examples are illustrated to show the impact of system parameters on performance measures of the queueing system.

Słowa kluczowe

EN

queueing model; multi-phase random environment; working vacation policies; impatient customer; probability generating function

• Wydawca: Oficyna Wydawnicza Politechniki Wrocławskiej
• Czasopismo: Operations Research and Decisions
• Rocznik: 2022
• Tom: Vol. 32, No. 2
• Strony: 16--33
• Opis fizyczny: Bibliogr. 48 poz., rys.

Twórcy

autor

Bouchentouf Amina Angelika

• Laboratory of Mathematics, Djillali Liabes University of Sidi Bel Abbes, 22000 Sidi Bel Abbes, Algeria

• https://orcid.org/0000-0001-8972-4221

autor

Guendouzi Abdelhak

• Laboratory of Stochastic Models, Statistic and Applications, Doctor Tahar Moulay University of Saida, En-nasr, 20000 Saida, Algeria

autor

Houalef Meriem

• Laboratory of Mathematics, University of Sidi Bel Abbes, Ecole Supérieure En Sciences Appliquées, Tlemcen 13000, Algeria

autor

Majid Shakir

• Department of Mathematics, University of Ladakh, India

• https://orcid.org/0000-0002-4140-8753

Bibliografia

• [1] Abou-El-Ata, M. O., and Hariri, A. M. A. The M/M/c/N queue with balking and reneging. Computers & Operations Research 19, 8 (1992), 713–716.
• [2] Afroun, F., Aïssani, D., Hamadouche, D., and Boualem, M. Q–matrix method for the analysis and performance evaluation of unreliable M/M/1/N queueing model. Mathematical Methods in the Applied Sciences 41, 18 (2018), 9152–9163.
• [3] Altman, E., and Yechiali, U. Analysis of customers impatience in queues with server vacation. Queueing Systems 52, 4 (2006), 261–279.
• [4] Altman, E., and Yechiali, U. Infinity-server queues with system’s additional tasks and impatient customers. Probability in the Engineering and Informational Sciences 22, 4 (2008), 477–493.
• [5] Ammar, S. Transient solution of an M/M/1 vacation queue with a waiting server and impatient customers. Journal of the Egyptian Mathematical Society 25, 3 (2017), 337–342.
• [6] Ayyappan, G., and Nirmala, M. An M[X] /M(a, b)/1 queue with breakdown and delay time to two phase repair under multiple vacation. Applications and Applied Mathematics 13, 2 (2018), 639–66.
• [7] Baykal-Gursoy, M., and Xiao, W. Stochastic decomposition in M/M/∞ queues with Markov modulated service rate. Queueing Systems 48, 1 (2004), 75–88.
• [8] Boualem, M. Stochastic analysis of a single server unreliable queue with balking and general retrial time. Discrete and Continuous Models and Applied Computational Science 28, 4 (2020), 319–326.
• [9] Bouchentouf, A. A., Boualem, M., Cherfaoui, M., and Medjahri, L. Variant vacation queueing system with Bernoulli feedback, balking and server’s states-dependent reneging. Yugoslav Journal of Operations Research 31, 4 (2021), 557–575.
• [10] Bouchentouf, A. A., Cherfaoui, M., and Boualem, M. Performance and economic analysis of a single server feedback queueing model with vacation and impatient customers. OPSEARCH 56, 1 (2019), 300–323.
• [11] Bouchentouf, A. A., Cherfaoui, M., and Boualem, M. Analysis and performance evaluation of Markovian feedback multi–server queueing model with vacation and impatience. American Journal of Mathematical and Management Sciences 40, 3 (2021), 261–282.
• [12] Bouchentouf, A. A., and Guendouzi, A. Sensitivity analysis of multiple vacation feedback queueing system with differentiated vacations, vacation interruptions and impatient customers. International Journal of Applied Mathematics and Statistics 57, 6 (2018), 104–121.
• [13] Bouchentouf, A. A., and Guendouzi, A. Cost optimization analysis for an MX/M/c vacation queueing system with waiting servers and impatient customers. SeMA Journal 76, 2 (2019), 309–341.
• [14] Bouchentouf, A. A., and Guendouzi, A. The MX/M/c Bernoulli feedback queue with variant multiple working vacations and impatient customers: Performance and economic analysis. Arabian Journal of Mathematics 9, 2 (2020), 309–327.
• [15] Bouchentouf, A. A., Guendouzi, A., and Kandouci, A. Performance and economic analysis of Markovian Bernoulli feedback queueing system with vacations, waiting server and impatient customers. Acta Universitatis Sapientiae, Mathematica 10, 2 (2018), 218–241.
• [16] Bouchentouf, A. A., Guendouzi, A., and Kandouci, A. Performance and economic study of heterogeneous M/M/2/N feedback queue with working vacation and impatient customer. ProbStat Forum 12 (2019), 15–35.
• [17] Bouchentouf, A. A., and Yahiaoui, L. On feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption. Arabian Journal of Mathematics 6, 1 (2017), 1–11.
• [18] Boxma, O. J., and Kurkova, I. A. The M/G/1 queue with two service speeds. Advances in Applied Probability 33, 2 (2001), 520–540.
• [19] Brusco, M., Jacobs, L., Bongiorno, R., Lyons, D., and Tang, B. Improving personnel scheduling at airline stations. Operations Research 43, 5 (1995), 741–751.
• [20] Choi, B. D., Kim, B., and Zhu, D. MAP /M/c queue with constant impatient time. Mathematics of Operations Research 29, 2 (2004), 309–325.
• [21] Cordeiro, J. D., and Kharoufeh, J. P. The unreliable M/M/1 retrial queue in a random environment. Stochastic Models 28, 1 (2012), 29–48.
• [22] D’Auria, B. m/m/∞ queues in semi-Markovian random environment. Queueing Systems 58, 3 (2008), 221–237.
• [23] Haghighi, A. M., Medhi, J., and Mohanty, S. G. On a multi-server Markovian queueing system with balking and reneging. Computers & Operations Research 13, 4 (1986), 421–425.
• [24] Haghighi, A. M., and Mishev, D. P. A parallel priority queueing system with finite buffers. Journal of Parallel and Distributed Computing 66, 3 (2006), 379–392.
• [25] Haghighi, A. M., and Mishev, D. P. Stochastic three–stage hiring model as a tandem queueing process with bulk arrivals and Erlang phase–type selection, MX/M(k,k) /1 − MY /Er/1 − ∞. International Journal of Mathematics in Operations Research 5, 5 (2013), 571–603.
• [26] Haghighi, A. M., Mishev, D. P., and Chukova, S. S. Single-server Poisson queueing system with delayed-service. International Journal of Operational Research 3, 4 (2008), 363–383.
• [27] Haghighi, A. M., and Trueblood, R. Busy period analysis of a parallel multi-processor system with task split and feedback. Advances In System Science and Applications (ASSA), Special Issue (1997).
• [28] Huang, L., and Lee, T. T. Generalized Pollaczek-Khinchin formula for Markov channels. IEEE Transactions on Communications 61, 8 (2013), 3530–3540.
• [29] Jiang, T., and Liu, L. The GI/M/1 queue in a multi-phase service environment with disasters and working breakdown. International Journal of Computer Mathematics 94, 4 (2017), 707–726.
• [30] Jiang, T., Liu, L., and Li, J. Analysis of the M/G/1 queue in multi-phase random environment with disasters. Journal of Mathematical Analysis and Applications 430, 2 (2015), 857–873.
• [31] Kadi, M., Bouchentouf, A. A., and Yahiaoui, L. On a multiserver queueing system with customers’ impatience until the end of service under single and multiple vacation policies. Applications and Applied Mathematics: An International Journal (AAM) 15, 2 (2020), 740–763.
• [32] Kim, B., and Kim, J. A single server queue with Markov modulated service rates and impatient customers. Performance Evaluation 83-84 (2015), 1–15.
• [33] Kumaran, S. K., Dogra, D. P., and Roy, P. P. Queuing theory guided intelligent traffic scheduling through video analysis using Dirichlet process mixture model. Expert Systems with Applications 118 (2019), 169–181.
• [34] Li, J., and Liu, L. On the discrete-time Geo/G/1 queue with vacations in random environment. Discrete Dynamics in Nature and Society ID 4029415 (2016), 1–9.
• [35] Li, J., and Liu, L. Performance analysis of a complex queueing system with vacations in random environment. Advances in Mechanical Engineering 9, 8 (2017), 1–9.
• [36] Neuts, M. A queue subject to extraneous phase changes. Advances in Applied Probabability 3, 1 (1971), 78–119.
• [37] Neuts, M. F. Matrix-Geometric Solutions in Stochastic Models: Algorithmic Approach. Johns Hopkins University Press, Baltimore, 1981.
• [38] O’Cinneide, C. A., and Purdue, P. The M/M/∞ queue in a random environment. Journal of Applied Probability 23, 1 (1986), 175–184.
• [39] Padmavathy, R., Kalidass, K., and Ramanath, K. Vacation queues with impatient customers and a waiting server. International Journal of Latest Trends in Software Engineering 1, 1 (2011), 10–19.
• [40] Paz, N., and Yechiali, U. An M/M/1 queue in random environment with disasters. Asia-Pacific Journal of Operational Research 31, 3 (2014), 1–12.
• [41] Sampath, M. I. G. S., and Liu, J. Impact of customers’ impatience on an M/M/1 queueing system subject to differentiated vacations with a waiting server. Quality Technology & Quantitative Management 17, 2 (2020), 125–148.
• [42] Srivatsa Srinivas, S., and Marathe, R. R. Equilibrium in a finite capacity M/M/1 queue with unknown service rates consisting of strategic and non-strategic customers. Queueing Systems 96, 3 (2020), 329–356.
• [43] Sudhesh, R., and Azhagappan, A. Transient analysis of an M/M/1 queue with variant impatient behavior and working vacations. OPSEARCH 55 (2018), 787–806.
• [44] Sudhesh, R., Azhagappan, A., and Dharmaraja, S. Transient analysis of M/M/1 queue with working vacation heterogeneous service and customers’ impatience. RAIRO - Operations Research 51, 3 (2017), 591–606.
• [45] Yechiali, U. A queueing-type birth-and-death process defined on a continuous-time markov chain. Operations Research 21, 2 (1973), 604–609.
• [46] Yechiali, U., and Naor, P. Queueing problems with heterogeneous arrival and service. Operations Research 19, 3 (1971), 722–734.
• [47] Yu, S.-L., and Liu, Z.-M. Analysis of queues in a random environment with impatient customers. Acta Mathematicae Applicatae Sinica, English Series 33, 4 (2017), 837–850.
• [48] Zirem, D., Boualem, M., Adel-Aissanou, K., and Aïssani, D. Analysis of a single server batch arrival unreliable queue with balking and general retrial time. Quality Technology & Quantitative Management 16, 6 (2019), 672–695.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).