An approximation method for the \(M/G/1\) retrial queue with general retrial times.

*(English)*Zbl 0802.60089Summary: 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.

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 |

##### Keywords:

stochastic decomposition; single-server queueing system; Poisson process; retrial queue; approximation method
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\textit{T. Yang} et al., Eur. J. Oper. Res. 76, No. 3, 552--562 (1994; Zbl 0802.60089)

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