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A survey on retrial queues. (English) Zbl 0658.60124
A systematized survey of the analytic results for retrival queues is given by discussing a generalized model A/B/s/m/O/H. Here, A and B describe the interarrival and service time distribution, respectively, s denotes the number of waiting positions in the queue, O is the capacity of an orbit for the retrying customers and $$H=\{H_ k$$, $$k\geq 0\}$$ denotes the loss model by which a customer joins to the orbit with probability $$1-H_ k$$ after the k th unsuccessful retrial.
The survey concentrates an single-server models of type M/G/i/m/O/H. For $$m=1$$, $$O=\infty$$, $$H=NL=\{H_ k=1$$, $$k\geq 0\}$$, the number N of customers in the system, the waiting time w, the number $$\eta$$ of retrials, the length L of busy periods, the number M of customers served in such periods, the server idle time i and the distribution $$\pi$$ of time intervals between consecutive departures are examined. For M/M/1/m/O/H with finite m, O and $$H=GL=\{H_ k=\alpha \leq 1$$, $$k\geq 0\}$$ the exact values of state probabilities p(i,j), i,j$$\geq 0$$, are derived, where i and j are the numbers of customers in the waiting room and in the orbit respectively. With $$O=\infty$$ and $$H=NL$$ an iterative procedure to approximate p(i,j) is reported.
In the case of batch arrivals the M/G/1 models are discussed. For the multi-server retrial queues only some major models, techniques dealing with the problems arising, and some main results are presented. Different full-availability and non-full-availability systems are reported. It is shown that the decomposition property, which is a common feature of queues with server vacations, is also true in retrial queues of the model M/G/s/m.
Reviewer: J.Tanko

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
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