We establish the quasireversibility of a certain exponential multiserver queue. The queuing discipline imposes restrictions on the numbers of customers of different classes and subclasses that can be served simultaneously. Customer loss is also allowed. Due to its quasireversibility the queue can be used as a node in a product form network. Our work generalizes work of Le Boudec, Krzesinski, and Crosby.

We characterize the probabilistic nature of the maximum queue length and the maximum waiting time in a multiserver G/G/c queue. We assume a general i.i.d. interarrival process and a general i.i.d. service time process for each server with the possibility of having different service time distributions for different servers. Under a weak additional condition, we will prove that the maximum queue length and waiting time grow asymptotically in probability as logωn−1 and log n1/θ, respectively, where ω < 1 and θ > 0 are parameters of the queuing system. Furthermore, it is shown that the maximum waiting time – when appropriately normalized – converges in distribution to the extreme distribution Λ(x) = exp(−e−x). The maximum queue length exhibits similar behavior, except that some oscillation caused by the discrete nature of the queue length must be taken into account. The first results of this type were obtained for the G/M/l queue by Heyde and for the G/G/l queue by Iglehart. Our analysis is similar to that of Heyde and Iglehart. The generalization to c > 1 servers is made possible due to the recent characterization of the tail of the stationary queue length and waiting time in a G/G/c queue (cf. Sadowsky and Szpankowski [17]).

We analyze the optimal preemptive sequencing of n jobs on M + 1 parallel identical machines to minimize expected total flowtime. The running times of the jobs are independent samples from the distribution Pr(X = H) = p, Pr(X = H + T) = 1 − p, where H, T are random variables of general distribution. Preemption of a job is allowed when H is completed. This problem does not have a simple optimal solution. We show that the scheme of shortest expected remaining processing time first (SERPT) is close to optimal in two senses. The expected flowtime under SERPT and under the optimal policy differ by no more than a constant, independent of the number of jobs, and the expected number of optimal decisions that are not according to SERPT is bounded by a constant, independent of the number of jobs.

We consider the problem of scheduling jobs on parallel machines to minimize flowtime, where all decisions about loading and sequencing must be made before any processing is done. We find policies that minimize flowtime stochastically or in the increasing convex sense under various assumptions.

We are concerned with a burst arrival single-server queue, where arrivals of cells in a burst are synchronized with a constant service time. The main concern is with the loss probability of cells for the queue with a finite buffer. We analyze an embedded Markov chain at departure instants of cells and get a kind of lumpability for its state space. Based on these results, this paper proposes a computation algorithm for its stationary distribution and the loss probability. Closed formulas are obtained for the first two moments of the numbers of cells and active bursts when the buffer size is infinite.

We analyze gated, exhaustive service of an infinite-server system with vacations. Customers enter a queue in a Poisson stream. The servers, working in parallel, serve customers in stages. A stage begins with all customers transferred from the queue to the servers (the gate opens). The servers then begin serving these customers, all simultaneously. The stage ends when their services are completed. Service is exhaustive because the servers must again examine the queue to see if any new customers arrived during the last stage. If there are any, a new stage begins. If there are none, the servers move on to other work. The time spent away from the queue is called vacation time. The queue may represent a node or station in a data transmission network and the servers may be communication channels.

We analyze the equilibrium behavior of the number of requests served during a stage for general service and vacation time distributions. This analysis leads to the solution of a Fredholm integral equation of the second kind. We find conditions under which the system is stable and compute bounds on performance metrics of interest. Approximate techniques are introduced and tested. Finally, an extension to polling systems is studied.

In this paper we give a short proof of a general version of Theorem 3.3 of Benkherouf [2].

Some monotonicity properties and bounds on expectations of certain functions of minimal repair times and interminimal repair times are obtained under assumptions on the aging characteristics of the equipment under maintenance.

In this paper we present two examples of physically realizable processes whose spectral density is not “physically realizable.”

In 1987, Sheldon M. Ross [4] gave an interesting recurrent relation method for approximations of important functions in renewal theory. Using his idea and Feller's technique for inverting Laplace transforms, we obtain a general method for recurrent approximations to the solutions of renewal-type integral equations.