# zbMATH — the first resource for mathematics

A single-server retrial queue with server vacations and a finite number of input sources. (English) Zbl 0912.90139
Summary: We consider a single-server retrial queueing system with no waiting space and finite population of customers. When a customer arrives and the server is idle then, according to a probability distribution, the server either immediately serves the arriving customer or takes a vacation during which it serves no customers. This model is a generalization of a few well-known systems including the classical $$M/G/1$$ retrial queue and the $$M/G/1$$ queue with setup times. Using the method of supplementary variables, we derive formulas for the limiting probability distribution of the system and some major performance measures including system throughput and mean delay. Through a numerical example, we discuss its application to the analysis of a communication protocol.

##### MSC:
 90B22 Queues and service in operations research
Full Text:
##### References:
 [1] Carrier, G.F.; Pearson, C.E., Partial differential equations, (1976), Academic Press New York · Zbl 0165.40601 [2] Choi, B.D.; Shin, Y.W.; Ahn, W.C., Retrial queues with collision arising from unslotted CSMA/CD protocol, Queueing systems, 11, 335-356, (1992) · Zbl 0762.60088 [3] Cox, D.R.; Miller, H.D., The theory of stochastic processes, (1965), Methuen London · Zbl 0149.12902 [4] Lin, B.; Sousa, E.S., A modified CSMA-CD protocol for high speed channels, (), 253-259 [5] de Kok, A.G., Algorithmic methods for single server systems with repeated attempts, Statistica neerlandica, 38, I, 23, (1984) · Zbl 0547.60098 [6] Falin, G., Retrial queues, Queueing systems, 7, 127-168, (1990) · Zbl 0709.60097 [7] Keilson, J.; Cozzolino, J.; Young, H., A service system with unfilled requests repeated, Operations research, 16, 1126-1137, (1968) · Zbl 0165.52703 [8] Kleinrock, L., () [9] Kleinrock, L.; Lam, S.S., Packet switching in a multiaccess broadcast channel: performance evaluation, IEEE transactions on communications, 23, 4, 410-422, (1975) · Zbl 0347.94003 [10] Kornyshev, Y.N., Design of a fully accessible switching system with repeated calls, Telecommunications, 23, 46-52, (1969) [11] Kulkarni, V.G., Expected waiting times in a multiclass batch arrival retrial queue, Journal of probability, 23, 144, (1986) · Zbl 0589.60073 [12] Kulkarni, V.G.; Choi, B.D., Retrial queue with server subject to breakdowns and repairs, Queueing systems, 7, 191-208, (1990) · Zbl 0727.60110 [13] Lam, S.S.; Kleinrock, L., Packet switching in a multiaccess broadcast channel: dynamic control procedures, IEEE transactions on communications, 23, 9, 891-904, (1975) · Zbl 0403.94008 [14] Rom, R.; Sidi, M., Multiple access protocols: performance and analysis, (1991), Springer Verlag New York [15] Saaty, T.L., Elements of queueing theory with applications, (1961), McGraw-Hill New York · Zbl 0108.30902 [16] Tasaka, S., Performance analysis of multiple access protocols, (1986), MIT Press Cambridge, MA [17] Tobagi, F.A.; Hunt, V.B., Performance analysis of carrier sence multiple access with collision detection, Computer networks, 4, 245-259, (1980) [18] Yang, T.; Templeton, J.G.C., A survey on retrial queues, Queueing systems, 2, 201-233, (1987) · Zbl 0658.60124 [19] Yang, T.; Posner, M.J.M.; Templeton, J.G.C.; Li, H., An approximation method for the M/G/1 retrial queue with general retrial times, European journal of operational research, 76, 552-562, (1994) · Zbl 0802.60089
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.