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Discrete-time $\mathrm{Geo}^X/G/1$ queue with unreliable server and multiple adaptive delayed vacations. (English) Zbl 1147.60056
Summary: We consider a discrete-time Geo$^X/G/1$ queue with unreliable server and multiple adaptive delayed vacations policy in which the vacation time, service time, repair time and the delayed time all follow arbitrary discrete distribution. By using a concise decomposition method, the transient and steady-state distributions of the queue length are studied, and the stochastic decomposition property of steady-state queue length has been proved. Several common vacation policies are special cases of the vacation policy presented in this study. The relationship between the generating functions of steady-state queue length at departure epoch and arbitrary epoch is obtained. Finally, we give some numerical examples to illustrate the effect of the parameters on several performance characteristics.

60K25Queueing theory
90B22Queues and service (optimization)
Full Text: DOI
[1] Alfa, A. S.: Vacation models in discrete time, Queueing systems 44, 5-30 (2003) · Zbl 1023.90009 · doi:10.1023/A:1024028722553
[2] Atencia, I.; Moreno, P.: A discrete-time geox/G/1 retrial queue with general retrial time, Queueing systems 4, 5-12 (2004) · Zbl 1059.60092 · doi:10.1023/B:QUES.0000039885.12490.02
[3] Baba, Y.: On the MX/G/1 queue with vacation time, Oper. res. Lett. 5, 93-98 (1986) · Zbl 0595.60094 · doi:10.1016/0167-6377(86)90110-0
[4] Bruneel, H.; Kim, B. G.: Discrete-time models for communication system including ATM, (1993)
[5] Chang, S. H.; Choi, D. W.: Performances analysis of a finite-buffer discrete-time queue with bulk arrival, bulk service and vacations, Comput. oper. Res. 32, 2213-2234 (2005) · Zbl 1067.60090 · doi:10.1016/j.cor.2004.01.004
[6] Choudhury, G.: An mx/G/1 queueing system with a setup period and a vacation period, Queueing systems 36, 23-38 (2000) · Zbl 0966.60100 · doi:10.1023/A:1019170817355
[7] Choudhury, G.: A batch arrival queue with a vacation time under single vacation policy, Comput. oper. Res. 29, 1941-1955 (2002) · Zbl 1010.90010 · doi:10.1016/S0305-0548(01)00059-4
[8] Cox, D. R.: The analyses of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Cambridge philos. Soc. 51, 433-441 (1965) · Zbl 0067.10902
[9] Doshi, B. T.: Queueing systems with vacations --- a survey, Queueing systems 1, 29-66 (1986) · Zbl 0655.60089 · doi:10.1007/BF01149327
[10] Fiems, D.; Bruneel, H.: Analysis of a discrete-time queueing system with timed vacation, Queueing systems 42, 243-254 (2002) · Zbl 1011.60073 · doi:10.1023/A:1020571814186
[11] Gaver, D. P.: A waiting line with interrupted service, including priorities, J. roy. Statist. soc. Ser. B 24, 73-96 (1962) · Zbl 0108.31403
[12] Hunter, J. J.: Mathematical techniques of applied probability, Mathematical techniques of applied probability 2 (1983) · Zbl 0539.60065
[13] Ke, J. C.: Modified T vacation policy for an M/G/1 queueing system with an unreliable server and startup, Math. comput. Modelling 41, 1267-1277 (2005) · Zbl 1082.90018 · doi:10.1016/j.mcm.2004.08.009
[14] Ke, J. C.: Operating characteristic analysis on the mx/G/1 system with a variant vacation policy and balking, Appl. math. Modelling 31, 1321-1337 (2007) · Zbl 1129.60083 · doi:10.1016/j.apm.2006.02.012
[15] Levy, Y.; Yechiali, U.: Utilization of the idle time in an M/G/1 queue, Management sci. 22, 202-211 (1975) · Zbl 0313.60067 · doi:10.1287/mnsc.22.2.202
[16] Li, W.; Shi, D.; Chao, X.: Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations, J. appl. Probab. 34, 546-555 (1997) · Zbl 0894.60084 · doi:10.2307/3215393
[17] Neuts, M. F.: Matrix-geometric solutions in stochastic models --- an algorithmic approach, (1981) · Zbl 0469.60002
[18] Rosenberg, E.; Yechiali, U.: The mx/G/1 queue with single and multiple vacations under the FIFO service regime, Oper. res. Lett. 14, 171-179 (1993) · Zbl 0792.60094 · doi:10.1016/0167-6377(93)90029-G
[19] Samanta, S. K.; Chaudhry, M. L.; Gupta, U. C.: Discrete-time $GeoX/G(a,b)/1/N$ queues with single and multiple vacations, Math. comput. Modeling 45, 93-108 (2007) · Zbl 1138.60338 · doi:10.1016/j.mcm.2006.04.008
[20] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority Systems, Part I, vol. I, North-Holland, Amsterdam, 1991. · Zbl 0744.60114
[21] Takagi, H.: Discrete-time systems, Discrete-time systems 3 (1993)
[22] Tang, Y. H.: A single-server M/G/1 queueing system subject to breakdowns --- some reliability and queueing problem, Microelectron. reliability 37, 315-321 (1997)
[23] Tang, Y. H.; Tang, X. W.: The generalized $Mx/G(M/G)/1$ repairable queueing system (I): some queueing indices, J. systems sci. Math. sci. 20, 385-397 (2000) · Zbl 0968.60088
[24] Tang, Y. H.; Tang, X. W.: The queue length distribution for mx/G/1 queue with single server vacation, Acta math. Sci. 21, 397-408 (2001) · Zbl 0984.60097
[25] Tang, Y. H.; Tang, X. W.; Zhao, W.: Analysis of the $Mx/G(M/G)/1$ repairable queueing system with single delay vacation (I): some queueing indices, Systems engrs. Theory pract. 20, 41-50 (2000)
[26] Tian, N.; Zhang, Z. G.: The discrete-time GI/geo/1 queue with multiple vacations, Queueing systems 40, 283-294 (2002) · Zbl 0993.90029 · doi:10.1023/A:1014711529740
[27] Woodward, M. E.: Communication and computer networks: modelling with discrete-time queues, (1994)
[28] Xu, X. Z.; Zhu, Y. X.: Exhaustive geomx/G/1 queue with unreliable server and multiple adaptive vacations, Oper. res. Management science 14, 8-12 (2005)
[29] Yu, H. B.; Nie, Z. K.: The $MAP/PH(PH/PH)/1$ discrete-time queing system with repairable server, Chinese quart. J. math. 16, 60-63 (2001) · Zbl 0986.60093
[30] Zhang, Z. G.; Tian, N. S.: Discrete-time geo/G/1 queue with multiple adaptive vacation, Queueing systems 38, 419-429 (2001) · Zbl 1079.90525 · doi:10.1023/A:1010947911863