In 1989, Parekh and Walrand introduced a method to efficiently estimate the probability of a rare event in a single queue or network of queueus. The event they consider is that the total number of customers in the system reaches some level N in a busy cycle. Parekh and Walrand introduce a simple change of measure, which is stateindependent, in order to estimate this probability efficiently usin...

We consider a Lévy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper we consider only light-tailed inputs. For the Brownian input case, we derive exact tail a...

We consider a Lévy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper we consider only light-tailed inputs, and derive exact asymptotics for the tail distribut...

This paper presents a novel technique to derive asymptotic expressions for rare event probabilities for random walks in the quarter plane. For concreteness, we study a tandem queue with Poisson arrivals, exponential service times and coupled processors. The service rate for one queue is only a fraction of the global service rate when the other queue is non empty; when one queue is empty, the ot...

We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate λ at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of the next queue n + 1. The question addressed is how steady-state queues scale as N →∞. We show that the answer depends on whether λ is below or above the critical v...