This note introduces reliability issues to the analysis of queueing systems. We consider an M/G/1 queue with Bernoulli vacations and server breakdowns. The server uptimes are assumed to be exponential, and the server repair times are arbitrarily distributed. Using a supplementary variable method we obtain a transient solution for both queueing and reliability measures of interest. These results provide insight into the effect of server breakdowns and repairs on system performance.

Using Palm-martingale calculus, we derive the workload characteristic function and queue length moment generating function for the BMAP/GI/1 queue with server vacations. In the queueing system under study, the server may start a vacation at the completion of a service or at the arrival of a customer finding an empty system. In the latter case we will talk of a server set-up time. The distribution of a set-up time or of a vacation period after a departure leaving a non-empty system behind is conditionally independent of the queue length and workload. Furthermore, the distribution of the server set-up times may be different from the distribution of vacations at service completion times. The results are particularized to the M/GI/1 queue and to the BMAP/GI/1 queue (without vacations).

This expository paper deals with the right tail behaviour of a class of Poisson mixtures. An Abelian-type result is obtained using basic theory of regular variation. Applications to compound distributions in insurance risk theory and queue length distributions under various queue disciplines in the case of Poisson arrivals are then discussed.