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A Pollaczek-Khintchine formula for $$M/G/1$$ queues with disasters. (English) Zbl 0867.60082
Summary: A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here, we relax this assumption and derive a Pollaczek-Khintchine-like formula for $$M/G/1$$ queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in servce for unstable preemptive LIFO $$M/G/1$$ queues.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.) 60G10 Stationary stochastic processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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