Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let Sn=ξ1+⋯+ξn and Mn=maxk≤nSk. Let τ=min{n≥1: Sn≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

Several recent papers have shown that for the M/G/1/n queue with equal arrival and service rates, the expected number of lost customers per busy cycle is equal to 1 for every n ≥ 0. We present an elementary proof based on Wald's equation and, for GI/G/1/n, obtain conditions for this quantity to be either less than or greater than 1 for every n ≥ 0. In addition, we extend this result to batch arrivals, where, for average batch size β, the same quantity is either less than or greater than β. We then extend these results to general ways that customers may be lost, to an arbitrary order of service that allows service interruption, and finally to reneging.

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.

Busy period problems in infinite server queues are studied systematically, starting from the batch service time. General relations are given for the lengths of the busy cycle, busy period and idle period, and for the number of customers served in a busy period. These relations show that the idle period is the most difficult while the busy cycle is the simplest of the four random variables. Renewal arguments are used to derive explicit results for both general and special cases.

In this note, the strong limiting behaviour of busy periods in GI/G/1 queues is studied under the condition that the traffic intensity equals unity.

A single- or multiserver queue with work-conserving service discipline and a stationary and ergodic input stream with bounded service times and arbitrarily light traffic intensity may have infinite mean waiting time. We give an example of this paradox and we also give a mixing condition which, in the case of the preemptive-resume LCFS discipline, excludes this phenomenon. Furthermore, the same methods allow to estimate the durations of the first busy period and cycle and the number of customers served in the first busy cycle of a work-conserving queue.

The G/G/1 queueing system is studied by means of the regeneration point method, exploiting the concept of busy cycles. Recurrence relations are set-up for the distribution of the queue length at the arrival epochs. The same method is used to obtain the correlation structure of arrivals and departures for the G/M/1 queue.

This paper studies the GI/G/1 queueing system in which no customer can stay longer than a fixed interval D. This is also a model for the dam with finite capacity, instantaneous water supply and constant release rule. Using analytical method together with the property that the queueing process ‘starts anew’ probabilistically whenever an arriving customer initiates a busy period, we obtain various transient and stationary results for the system.

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.

The general theory for the many server queue due to Pollaczek (1961) and generalized by the author (de Smit (1973)) is applied to the system with exponential service times. In this way many explicit results are obtained for the distributions of characteristic quantities, such as the actual waiting time, the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. Most of these results are new, even for the special case of Poisson arrivals.

Pollaczek's theory for the many server queue is generalized and extended. Pollaczek (1961) found the distribution of the actual waiting times in the model G/G/s as a solution of a set of integral equations. We give a somewhat more general set of integral equations from which the joint distribution of the actual waiting time and some other random variables may be found. With this joint distribution we can obtain distributions of a number of characteristic quantities, such as the virtual waiting time, the queue length, the number of busy servers, the busy period and the busy cycle. For a wide class of many server queues the formal expressions may lead to explicit results.