Synchronized negative customers in an unreliable server queue

Negative customers are an accepted means of removing jobs from systems in queueing theory. A lack of synchronization with causal events can adversely affect results. We introduce a formulation for a queue with unreliable servers in which the arrival of negative customers is synchronized with the processor breakdown. Probabilistic re-sampling on repair can also be modelled, resulting in an interesting transition structure in the Markov chain representing the joint states of queue length and number of active processors. This corresponds to a MAP-like process in which one arrival class is of negative customers. Unlike MAP/MAP or MAP/PH queues, job addition/removal processes are co-modulated. We demonstrate that the introduction of synchronization affects the resulting queue solution when compared to the use of independent negative customer arrivals. We suggest that the synchronised negative customer may be of use in modelling systems in which the processor has local storage, and the job information is moved rather than copied. A synchronised re-sample may correspond to the processor having been suspended, perhaps for cooling rather than replacement, then re-activating.