Join the Shortest Queue (JSQ) is a popular routing policy for server farms. However, until now all analysis of JSQ has been limited to First-Come-First-Serve (FCFS) server farms, whereas it is known that web server farms are better modeled as Processor Sharing (PS) server farms. We provide the first approximate analysis of JSQ in the PS server farm model for general job size distributions, obtaining the distribution of queue length at each queue. To do this, we approximate the queue length of each queue in the server farm by a one-dimensional Markov chain, in a novel fashion. We also discover some interesting insensitivity properties for PS server farms with JSQ routing, and discuss the near-optimality of JSQ.

We study the tail asymptotics of the r.v. X(T) where fX(t)g is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We nd that the tail of X(T) is sensitive to whether or not T has a heavier or lighter tail than a Weibull distribution with tail e p x. This leads to two distinct cases, heavy-tailed and moderately heavy-tailed, but also some results for the classical light-tailed case are given. The results are applied via distributional Little's law to establish tail asymptotics for steady{state queue length in GI/GI/1 queues with subexponential service times. Further applications are given for queues with vacations, and M/G/1 busy periods.

We explore the performance of an M/GI/1 queue under various scheduling policies from the perspective of a new metric: the slowdown experienced by largest jobs. We consider scheduling policies that bias against large jobs, towards large jobs, and those that are fair, e.g., Processor-Sharing. We prove that as job size increases to infinity, all work conserving policies converge almost surely with respect to this metric to no more than 1=(1 ? ), where denotes load. We also find that the expected slowdown under any work conserving policy can be made arbitrarily close to that under Processor-Sharing, for all job sizes that are sufficiently large.

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 service for unstable preemptive LIFO M/G/1 queues.

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A duality is presented for real-valued stochastic sequences (V n ) defined by a general recursion of the form V n + t =f(V n ,U n ),wth [U n ] a stationary driving sequence and/nonnegative, continuous, and monotone in its first variable. The duality is obtained by defining a dual function g of/, which if used recursively on the time reversal of [U n ] defines a dual risk process. As a consequence, we prove that steady-state probabilities for V n can always be expressed as transient probabilities of the dual risk process. The construction is related to duality of stochastically monotone Markov processes as studied by Siegmund (1976, The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes, Annals of Probability 4: 914-924). Our method of proof involves an elementary sample-path analysis. A variety of examples are given, including random walks with stationary increments and two reflecting barriers, reservoir models, autoregressive processes, and branching processes. Finally, general stability issues of the content process are dealt with by expressing them in terms of the dual risk process.

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