×

zbMATH — the first resource for mathematics

A discrete-time \(Geo/G/1\) retrial queue with starting failures and second optional service. (English) Zbl 1172.90368
Summary: We consider a discrete-time \(Geo/G/1\) retrial queue with starting failures in which all the arriving customers require a first essential service while only some of them ask for a second optional service. We study the Markov chain underlying the considered queueing system and its ergodicity condition. Explicit formulae for the stationary distribution and some performance measures of the system in steady state are obtained. We also obtain two stochastic decomposition laws regarding the probability generating function of the system size. Finally, some numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.

MSC:
90B22 Queues and service in operations research
60K25 Queueing theory (aspects of probability theory)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bruneel, H.; Kim, B.G., Discrete-time models for communication systems including ATM, (1993), Kluwer Academic Publishers Boston
[2] Hunter, J.J., ()
[3] Chaudhry, M.L.; Templeton, J.G.C., A first course in bulk queues, (1983), Wiley New York · Zbl 0559.60073
[4] Takagi, H., ()
[5] Woodward, M.E., Communication and computer networks: modelling with discrete-time queues, (1994), IEEE Computer Soc. Press Los Alamitos, CA
[6] Artalejo, J.R., A classified bibliography of research on retrial queues: progress in 1990-1999, Top, 7, 2, 187-211, (1999) · Zbl 1009.90001
[7] Artalejo, J.R., Accessible bibliography on retrial queues, Mathematical and computer modelling, 30, 1-6, (1999) · Zbl 1009.90001
[8] Falin, G.I.; Templeton, J.G.C., Retrial queues, (1997), Chapman & Hall London · Zbl 0944.60005
[9] Yang, T.; Li, H., On the steady-state queue size distribution of the discrete-time \(G e o \text{/} G \text{/} 1\) queue with repeated customers, Queueing systems, 21, 199-215, (1995) · Zbl 0840.60085
[10] Atencia, I.; Moreno, P., Discrete-time \(G e o^{[X]} \text{/} G_H \text{/} 1\) retrial queue with Bernoulli feedback, Computers and operations research, 31, 359-381, (2004)
[11] Choi, B.D.; Kim, J.W., Discrete-time \(G e o_1, G e o_2 \text{/} G \text{/} 1\) retrial queueing system with two types of calls, Computers and mathematics with applications, 33, 10, 79-88, (1997) · Zbl 0878.90041
[12] Li, H.; Yang, T., \(G e o \text{/} G \text{/} 1\) discrete-time retrial queue with Bernoulli schedule, European journal of operate research, 111, 3, 629-649, (1998) · Zbl 0948.90043
[13] Li, H.; Yang, T., Steady-state queue size distribution of discrete-time \(P H \text{/} G e o \text{/} 1\) retrial queues, Mathematical and computer modelling, 30, 51-63, (1999) · Zbl 1042.60543
[14] Takahashi, M.; Osawa, H.; Fujisawa, T., \(G e o^{[X]} \text{/} G \text{/} 1\) retrial queue with non-preemptive priority, Asia-Pacific journal of operational research, 16, 215-234, (1999) · Zbl 1053.90505
[15] Atencia, I.; Moreno, P., A discrete-time \(G e o \text{/} G \text{/} 1\) retrial queue with general retrial times, Queueing systems, 48, 5-21, (2004) · Zbl 1059.60092
[16] Artalejo, J.R.; Atencia, I.; Moreno, P., A discrete-time \(G e o^{[X]} \text{/} G \text{/} 1\) retrial queue with control of admission, Applied mathematical modelling, 29, 1100-1120, (2005) · Zbl 1163.90413
[17] Madan, K.C., An \(M \text{/} G \text{/} 1\) queue with second optional service, Queueing systems, 34, 37-46, (2000) · Zbl 0942.90008
[18] Medhi, J.A., A single server Poisson input queue with a second optional channel, Queueing systems, 42, 239-242, (2002) · Zbl 1011.60072
[19] Wang, J., An \(M \text{/} G \text{/} 1\) queue with second optional service and server breakdowns, Computers and mathematics with applications, 47, 1713-1723, (2004) · Zbl 1061.60102
[20] I. Atencia, P. Moreno, A discrete-time retrial queue with 2nd optional service, in: Proceeding of Fifth International Workshop on Retrial Queues, 2004, pp. 117-121
[21] Aissani, A., On the \(M \text{/} G \text{/} 1 \text{/} 1\) queueing system with repeated order and unreliable server, Journal of technology, 6, 98-123, (1988), (in French)
[22] Kulkarni, V.G.; Choi, B.D., Retrial queues with server subject to breakdowns and repairs, Queueing systems, 7, 191-208, (1990) · Zbl 0727.60110
[23] Artalejo, J.R., New results in retrial queueing systems with breakdown of the servers, Statistica neerlandica, 48, 23-36, (1994) · Zbl 0829.60087
[24] Yang, T.; Li, H., The \(M \text{/} G \text{/} 1\) retrial queue with the server subject to starting failures, Queueing systems, 16, 83-96, (1994) · Zbl 0810.90046
[25] Artalejo, J.R.; Gómez-Corral, A., Unreliable retrial queues due to service interruptions arising from facsimile networks, Belgian journal of operations research, statistics and computer science, 38, 31-41, (1998) · Zbl 1010.90503
[26] Wang, J.; Cao, J.; Li, Q., Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing systems, 38, 363-380, (2001) · Zbl 1028.90014
[27] Li, Q.L.; Ying, Y.; Zhao, Y.Q., A \(B M A P \text{/} G \text{/} 1\) retrial queue with a server subject to breakdowns and repairs, Annals of operations research, 141, 233-270, (2006) · Zbl 1122.90332
[28] Krishna Kumar, B.; Pavai Madheswari, S.; Vijayakumar, A., The \(M \text{/} G \text{/} 1\) retrial queue with feedback and starting failures, Applied mathematical modelling, 26, 1057-1075, (2002) · Zbl 1018.60088
[29] Moreno, P., A discrete-time retrial queue with unreliable server and general server lifetime, Journal of mathematical sciences, 132, 5, 643-655, (2006) · Zbl 1411.60138
[30] Yang, T.; Templeton, J.G.C., A survey on retrial queues, Queueing systems, 2, 201-233, (1987) · Zbl 0658.60124
[31] Artalejo, J.R.; Falin, G.I., Stochastic decomposition for retrial queues, Top, 2, 329-342, (1994) · Zbl 0837.60084
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.