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We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

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In this paper, we are concerned with the analysis of the queue length and waiting time distributions in a batch arrival M X /G/1 retrial queue. Necessary and sufficient conditions are obtained for the existence of the moments of the queue length and waiting time distributions. We also provide recursive formulas for the higher order moments of the queue length and waiting time distributions.

In this paper, we investigate how fast the stationary distribution π (K) of an embedded Markov chain (time-stationary distribution q (K) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q ) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values lim K→∞σ-K ||π(K) - π|| and lim K→∞σ-K || q (K) - q || explicitly.

We consider gated polling systems with two special features: (i) retrials, and (ii) glue or reservation periods. When a type-i customer arrives, or retries, during a glue period of station i, it will be served in the next visit period of the server to that station. Customers arriving at station i in any other period join the orbit of that station and will retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems.For the case of exponentially distributed glue periods, we present an algorithm to obtain the moments of the number of customers in each station. For generally distributed glue periods, we consider the distribution of the total workload in the system, using it to derive a pseudo conservation law which in its turn is used to obtain accurate approximations of the individual mean waiting times.We also consider the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times.

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