An unreliable discrete-time retrial queue with probabilistic preemptive priority, balking customers and replacements of repair times

Shaojun Lan; Yinghui Tang; Shaojun Lan; Yinghui Tang

doi:10.3934/math.2020276

AIMS Mathematics

2020, Volume 5, Issue 5: 4322-4344. doi: 10.3934/math.2020276

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Research article

An unreliable discrete-time retrial queue with probabilistic preemptive priority, balking customers and replacements of repair times

Shaojun Lan ^{1
,
,},
Yinghui Tang ²

1.
School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, 643000, Sichuan, China
2.
School of Fundamental Education, Sichuan Normal University, Chengdu, 610066, Sichuan, China

Received: 17 December 2019 Accepted: 28 April 2020 Published: 09 May 2020
MSC : 60K25, 68M20, 90B22

This paper deals with a discrete-time $Geo/G/1$ retrial queueing system with probabilistic preemptive priority and balking customers, in which the server is subject to starting failures and replacements in the repair times may occur with some probability. If the server is found busy at an arrival epoch, the newly arriving customer either interrupts the customer in service to begin its own service with probability $p$ or enters the orbit with probability $1-p$. When an arriving customer (external or repeated) finds the server free, he must turn on the server. If the server is activated successfully, the customer receives service immediately. Otherwise, the server undergoes a repair process. If an external arrival finds that the server is under repair, he decides either to join the orbit with probability $q$ or leaves the system completely (balking) with probability $1-q$. Applying the supplementary variable method and the generating function technique, we analyze the Markov chain underlying the considered queueing model and derive the stationary distributions under different system states, the generating functions for the number of customers in the orbit and in the system, as well as some crucial performance measures in steady state. Especially, some corresponding results under special cases are directly obtained by setting appropriate parameter values. Further, some numerical examples are provided to examine the effect of various system parameters on queueing characteristics. Finally, an operating cost function is formulated to discuss numerically a cost optimization problem.
- discrete-time retrial queue,
- probabilistic preemptive priority,
- balking customers,
- starting failures,
- replacement
Citation: Shaojun Lan, Yinghui Tang. An unreliable discrete-time retrial queue with probabilistic preemptive priority, balking customers and replacements of repair times[J]. AIMS Mathematics, 2020, 5(5): 4322-4344. doi: 10.3934/math.2020276

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Abstract

This paper deals with a discrete-time $Geo/G/1$ retrial queueing system with probabilistic preemptive priority and balking customers, in which the server is subject to starting failures and replacements in the repair times may occur with some probability. If the server is found busy at an arrival epoch, the newly arriving customer either interrupts the customer in service to begin its own service with probability $p$ or enters the orbit with probability $1-p$. When an arriving customer (external or repeated) finds the server free, he must turn on the server. If the server is activated successfully, the customer receives service immediately. Otherwise, the server undergoes a repair process. If an external arrival finds that the server is under repair, he decides either to join the orbit with probability $q$ or leaves the system completely (balking) with probability $1-q$. Applying the supplementary variable method and the generating function technique, we analyze the Markov chain underlying the considered queueing model and derive the stationary distributions under different system states, the generating functions for the number of customers in the orbit and in the system, as well as some crucial performance measures in steady state. Especially, some corresponding results under special cases are directly obtained by setting appropriate parameter values. Further, some numerical examples are provided to examine the effect of various system parameters on queueing characteristics. Finally, an operating cost function is formulated to discuss numerically a cost optimization problem.

References

[1]	T. Yang, J. G. C. Templeton, A survey on retrial queues, Queueing Syst., 2 (1987), 201-233. doi: 10.1007/BF01158899
[2]	G. I. Falin, A survey of retrial queues, Queueing Syst., 7 (1990), 127-167. doi: 10.1007/BF01158472
[3]	J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7 (1999), 187-211. doi: 10.1007/BF02564721
[4]	J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009, Math. Comput. Modell., 51 (2010), 1071-1081. doi: 10.1016/j.mcm.2009.12.011
[5]	J. Kim, B. Kim, A survey of retrial queueing systems, Ann. Oper. Res., 247 (2016), 3-36. doi: 10.1007/s10479-015-2038-7
[6]	G. I. Falin, J. G. C. Templeton, Retrial Queues, London: Chapman & Hall, 1997.
[7]	J. R. Artalejo, A. Gómez-Corral, Retrial Queueing Systems: A Computational Approach, Berlin: Springer, 2008.
[8]	T. Meisling, Discrete-time queueing theory, Oper. Res., 6 (1956), 96-105.
[9]	J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Discrete Time Models: Techniques and Applications, New York: Academic Press, 1983.
[10]	H. Bruneel, B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, Boston: Kluwer Academic Publishers, 1993.
[11]	M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, California: IEEE Computer Society Press, 1994.
[12]	T. Yang, H. Li, On the steady-state queue size distribution of the discrete-time Geo/G/1 queue with repeated customers, Queueing Syst., 21 (1995), 199-215. doi: 10.1007/BF01158581
[13]	I. Atencia, P. Moreno, A discrete-time Geo/G/1 retrial queue with general retrial times, Queueing Syst., 48 (2004), 5-21. doi: 10.1023/B:QUES.0000039885.12490.02
[14]	J. R. Artalejo, I. Atencia, P. Moreno, A discrete-time Geo^[X]/G/1 retrial queue with control of admission, Appl. Math. Model., 29 (2005), 1100-1120. doi: 10.1016/j.apm.2005.02.005
[15]	A. K. Aboul-Hassan, S. I. Rabia, F. A. Taboly, Performance evaluation of a discrete-time Geo^[X]/G/1 retrial queue with general retrial times, Comput. Math. Appl., 58 (2009), 548-557. doi: 10.1016/j.camwa.2009.03.101
[16]	I. Atencia, I. Fortes, S. Nishimura, et al. A discrete-time retrial queueing system with recurrent customers, Comput. Oper. Res., 37 (2010), 1167-1173. doi: 10.1016/j.cor.2009.03.029
[17]	J. Wang, Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation, J. Syst. Sci. Complex., 25 (2012), 504-513. doi: 10.1007/s11424-012-0254-7
[18]	I. Atencia-Mc.Killop, J. L. Galán-García, G. Aguilera-Venegas, et al. A Geo^[X]/G^[X]/1 retrial queueing system with removal work and total renewal discipline, Appl. Math. Comput., 319 (2018), 245-253.
[19]	S. Gao, X. Wang, Analysis of a single server retrial queue with server vacation and two waiting buffers based on ATM networks, Math. Probl. Eng., 2019 (2019), 1-14.
[20]	I. Atencia, P. Moreno, A discrete-time Geo/G/1 retrial queue with server breakdowns, Asia Pac. J. Oper. Res., 23 (2006), 247-271. doi: 10.1142/S0217595906000929
[21]	I. Atencia, P. Moreno, A discrete-time Geo/G/1 retrial queue with the server subject to starting failures, Ann. Oper. Res., 141 (2006), 85-107. doi: 10.1007/s10479-006-5295-7
[22]	J. Wang, Q. Zhao, A discrete-time Geo/G/1 retrial queue with starting failures and second optional service, Comput. Math. Appl., 53 (2007), 115-127. doi: 10.1016/j.camwa.2006.10.024
[23]	I. Atencia, I. Fortes, S. Sánchez, A discrete-time retrial queueing system with starting failures, Bernoulli feedback and general retrial times, Comput. Ind. Eng., 57 (2009), 1291-1299. doi: 10.1016/j.cie.2009.06.011
[24]	J. Wang, P. Zhang, A discrete-time retrial queue with negative customers and unreliable server, Comput. Ind. Eng., 56 (2009), 1216-1222. doi: 10.1016/j.cie.2008.07.010
[25]	S. Gao, Z. Liu, A repairable Geo^X/G/1 retrial queue with Bernoulli feedback and impatient customers, Acta Math. Appl. Sin. E., 30 (2014), 205-222. doi: 10.1007/s10255-014-0278-y
[26]	I. Atencia, A discrete-time queueing system with server breakdowns and changes in the repair times, Ann. Oper. Res., 235 (2015), 37-49. doi: 10.1007/s10479-015-1940-3
[27]	I. Atencia, A Geo/G/1 retrial queueing system with priority services, Eur. J. Oper. Res., 256 (2017), 178-186. doi: 10.1016/j.ejor.2016.07.011
[28]	S. Lan, Y. Tang, Performance analysis of a discrete-time Geo/G/1 retrial queue with nonpreemptive priority, working vacations and vacation interruption, J. Ind. Manag. Optim., 15 (2019), 1421-1446.
[29]	H. Li, T. Yang, Geo/G/1 discrete time retrial queue with Bernoulli schedule, Eur. J. Oper. Res., 111 (1998), 629-649. doi: 10.1016/S0377-2217(97)90357-X
[30]	B. D. Choi, J. W. Kim, Discrete-time Geo₁,Geo₂/G/1 retrial queueing systems with two types of calls, Comput. Math. Appl., 33 (1997), 79-88.
[31]	J. Wu, Z. Liu, Y. Peng, A discrete-time Geo/G/1 retrial queue with preemptive resume and collisions, Appl. Math. Model., 35 (2011), 837-847. doi: 10.1016/j.apm.2010.07.039
[32]	M. Jain, A. Bhagat, C. Shekhar, Double orbit finite retrial queues with priority customers and service interruptions, Appl. Math. Comput., 253 (2015), 324-344.
[33]	S. Upadhyaya, Performance prediction of a discrete-time batch arrival retrial queue with Bernoulli feedback, Appl. Math. Comput., 283 (2016), 108-119.
[34]	F. A. Haight, Queueing with balking, Biometrika, 44 (1957), 360-369. doi: 10.1093/biomet/44.3-4.360
[35]	A. K. Aboul-Hassan, S. I. Rabia, A. Kadry, Analytical study of a discrete-time retrial queue with balking customers and early arrival scheme, Alex. Eng. J., 44 (2005), 911-917.
[36]	A. K. Aboul-Hassan, S. I. Rabia, F. A. Taboly, A discrete time Geo/G/1 retrial queue with general retrial times and balking customers, J. Korean Stat. Soc., 37 (2008), 335-348. doi: 10.1016/j.jkss.2008.04.006
[37]	M. Lozano, P. Moreno, A discrete time single-server queue with balking: economic applications, Appl. Econ., 40 (2008), 735-748. doi: 10.1080/00036840600749607
[38]	V. Laxmi, V. Goswami, K. Jyothsna, Analysis of discrete-time single server queue with balking and multiple working vacations, Qual. Technol. Quant. Manag., 10 (2013), 443-456. doi: 10.1080/16843703.2013.11673424
[39]	A. G. Pakes, Some conditions for ergodicity and recurrence of Markov chains, Oper. Res., 17 (1969), 1058-1061. doi: 10.1287/opre.17.6.1058
[40]	R. L. Rardin, Optimization in Operations Research, New Jersey: Prentice Hall, 1998.

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