Abstract
A queueing system with a single server providing two stages of service in succession is considered. Every customer receives service in the first stage and in the sequel he decides whether to proceed to the second phase of service or to depart and join a retrial box from where he repeats the demand for a special second stage service after a random amount of time and independently of the other customers in the retrial box. When the server becomes idle, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service times are arbitrarily distributed. For such a system the stability conditions and the system state probabilities are investigated both in a transient and in a steady state. A stochastic decomposition result is also presented. Numerical results are finally obtained and used to investigate system performance.
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Artalejo, J.R.: A classified bibliography of research on retrial queues: Progress in 1990–1999. Top 7(2), 187–211 (1999)
Choi, D.I., Kim, T.: Analysis of a two-phase queueing system with vacations and Bernoulli feedback. Stoch. Anal. Appl. 21(5), 1009–1019 (2003)
Choudhury, G.: Steady state analysis of a M/G/1 queue with linear retrial policy and two-phase service under Bernoulli vacation schedule. Appl. Math. Model. (2007). doi:10.1016/j.apm.2007.09.020
Choudhury, G., Madan, K.C.: A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy. Math. Comput. Model. 42, 71–85 (2005)
Cinlar, E.: Introduction to Stochastic Processes. Prentice Hall, New York (1975)
Doshi, B.T.: Analysis of a two-phase queueing system with general service times. Oper. Res. Lett. 10, 265–272 (1991)
Falin, G.I., Fricker, C.: On the virtual waiting time in an M/G/1 retrial queue. J. Appl. Probab. 28, 446–460 (1991)
Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman and Hall, London (1997)
Falin, G.I., Artalejo, J.R., Martin, M.: On the single server retrial queue with priority customers. Queueing Syst. 14, 439–455 (1993)
Katayama, T., Kobayashi, K.: Sojourn time analysis of a queueing system with two-phase service and server vacations. Nav. Res. Logist. 54(1), 59–65 (2006)
Krishna, C.M., Lee, Y.H.: A study of a two-phase service. Oper. Res. Lett. 9, 91–97 (1990)
Kulkarni, V.G., Liang, H.M.: Retrial queues revisited. In: Dshalalow, J.H. (ed.) Frontiers in Queueing, pp. 19–34. CRC Press, Boca Raton (1997)
Krishna Kumar, B., Vijayakumar, A., Arivudainambi, D.: An M/G/1 retrial queueing system with two-phase service and preemptive resume. Ann. Oper. Res. 113, 61–79 (2002)
Krishna Kumar, B., Arivudainambi, D., Vijayakumar, A.: On the M (x)/G/1 retrial queue with Bernoulli schedule and general retrial times. Asia-Pac. J. Oper. Res. 19, 117–194 (2002)
Langaris, C., Katsaros, A.: Time dependent analysis of a queue with batch arrivals and N levels of non-preemptive priority. Queueing Syst. 19, 269–288 (1995)
Madan, K.C.: On a single server queue with two-stage heterogeneous service and deterministic server vacations. Int. J. Syst. Sci. 32(7), 837–844 (2001)
Moutzoukis, E., Langaris, C.: Two queues in tandem with retrial customers. Probab. Eng. Inf. Sci. 15, 311–325 (2001)
Pakes, A.G.: Some conditions of ergodicity and recurrence of Markov chains. Oper. Res. 17, 1058–1061 (1969)
Takacs, L.: Introduction to the Theory of Queues. Oxford Univ. Press, New York (1962)
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Dimitriou, I., Langaris, C. Analysis of a retrial queue with two-phase service and server vacations. Queueing Syst 60, 111–129 (2008). https://doi.org/10.1007/s11134-008-9089-2
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DOI: https://doi.org/10.1007/s11134-008-9089-2