In this paper, a batch arrival general bulk service queueing system with interruptedvacation (secondary job) is considered. At a service completion epoch, if the server findsat least ‘a’ customers waiting for service say ξ, heserves a batch of min (ξ, b) customers, whereb ≥ a. On the other hand, if the queue length is atthe most ‘a-1’, the server leaves for a secondary job (vacation) ofrandom length. It is assumed that the secondary job is interrupted abruptly and the serverresumes for primary service, if the queue size reaches ‘a’, during thesecondary job period. On completion of the secondary job, the server remains in the system(dormant period) until the queue length reaches ‘a’. For the proposedmodel, the probability generating function of the steady state queue size distribution atan arbitrary time is obtained. Various performance measures are derived. A cost model forthe queueing system is also developed. To optimize the cost, a numerical illustration isprovided.

A discrete-time model is presented for a system of two queues competing for the service attention of a single server with infinite buffer capacity. The service requirements are geometrically distributed and independent from customer to customer as well as from the arrivals. The allocation of service attention is governed by feedback policies which are based on past decisions and buffer content histories. The cost of operation per unit time is a linear function of the queue sizes. Under the model assumptions, a fixed prioritization scheme, known as the μc-rule, is shown to be optimal for the expected long-run average criterion and for the expected discounted criterion, over both finite and infinite horizons. Two different approaches are proposed for solving these problems. One is based on the dynamic programming methodology for Markov decision processes, and assumes the arrivals to be i.i.d. The other is valid under no additional assumption on the arrival stream and uses direct comparison arguments. In both cases, the sample path properties of the adopted state-space model are exploited.

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.