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An approximation method for the $$M/G/1$$ retrial queue with general retrial times. (English) Zbl 0802.60089
Summary: We consider a single-server queueing system with no waiting space. Customers arrive according to a Poisson process with rate $$\lambda$$. An arriving customer receives immediate service if he finds the server idle; otherwise he will retry for service after a certain amount of time. Blocked customers will repeatedly retry for service until they get served. The times between consecutive attempts are assumed to be independently distributed with common distribution function $$T(\cdot)$$. The service times of customers are drawn from a common distribution function $$B(\cdot)$$.
We show that the number of customers in the system in steady-state can be decomposed into two independent random variables: the number of customers in the corresponding ordinary $$M/G/1$$ queue (with unlimited waiting space) and the number of customers in the retrial queue given that the server is idle. Applying this decomposition property, an approximation method for the calculation of the steady-state queue size distribution is proposed and some properties of the approximation are discussed. It is demonstrated through numerical results that the approximation works very well for models of practical interest.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
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##### References:
 [1] Aleksandrov, A.M., A queueing system with repeated orders, Engineering cybernetics review, 12, 1-4, (1974) [2] Falin, G.I., Single-line repeated orders queueing systems, Optimization, 17, 649-667, (1986) · Zbl 0618.90033 [3] Kapyrin, V.A., A study of the stationary distributions of a queueing system with recurring demands, Cybernetics, 13, 584-590, (1977) [4] Keilson, J.; Cozzolino, J.; Young, H., A service system with unfilled requests repeated, Operations research, 16, 1126-1137, (1968) · Zbl 0165.52703 [5] Kleinrock, L., () [6] Pourbabai, B., Analysis of a G/M/K/O queueing loss system with heteregeneous servers and retrials, International journal of systems science, 18, 985-992, (1987) · Zbl 0611.90050 [7] Ross, S.M., Stochastic processes, (1983), Wiley New York · Zbl 0555.60002 [8] Yang, T.; Templeton, J.G.C., A survey on retrial queues, Queueing systems: theory and applications, 2, 201-233, (1987) · Zbl 0658.60124 [9] Yang, T.; Li, H., The M/G/1 retrial queue with server vacations, ()
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