We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let Nt denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (Nt, T, X(T)), where X(T) is the jumping contour process of the tree truncated at time T, is equal to that of (M, -IM, Y′M) conditional on {M≠0}. Here M+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; IM is the infimum of the path YM, which in turn is defined as Y killed at its last visit of 0 before e; and Y′M is the Vervaat transform of YM. This identity yields an explanation for the geometric distribution of NT (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {NT=n}, and also on {NT=n,T<a}, the ages and residual lifetimes of the n alive individuals at time T are i.i.d. and independent of n. We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital.

We present an exact simulation algorithm for the stationary distribution of the customer delay D for first-in–first-out (FIFO) M/G/c queues in which ρ=λ/μ<1. We assume that the service time distribution G(x)=P(S≤x),x≥0 (with mean 0<E(S)=1/μ<∞), and its corresponding equilibrium distribution Ge(x)=μ∫0x P(S>y)dy are such that samples of them can be simulated. We further assume that G has a finite second moment. Our method involves the general method of dominated coupling from the past (DCFTP) and we use the single-server M/G/1 queue operating under the processor sharing discipline as an upper bound. Our algorithm yields the stationary distribution of the entire Kiefer–Wolfowitz workload process, the first coordinate of which is D. Extensions of the method to handle simulating generalized Jackson networks in stationarity are also remarked upon.

In this paper we present a stability criterion for processor-sharing queues, in which the throughput may depend on the number of customers in the system (such as in the case of interferences between users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers.

We consider single-server queues with exponentially distributed service times, in which the arrival process is governed by a semi-Markov process (SMP). Two service disciplines, processor sharing (PS) and random service (RS), are investigated. We note that the sojourn time distribution of a type-l customer who, upon his arrival, meets k customers already present in the SMP/M/1/PS queue is identical to the waiting time distribution of a type-l customer who, upon his arrival, meets k+1 customers already present in the SMP/M/1/RS queue. Two sets of system equations, one for the joint transform of the sojourn time and queue size distributions in the SMP/M/1/PS queue, and the other for the joint transform of the waiting time and queue size distributions in the SMP/M/1/RS queue, are derived. Using these equations, the mean sojourn time in the SMP/M/1/PS queue and the mean waiting time in the SMP/M/1/RS queue are obtained. We also consider a special case of the SMP in which the interarrival time distribution is determined only by the type of the customer who has most recently arrived. Numerical examples are also presented.

The basic queueing system considered in this paper is the M/G/1 processor-sharing queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the reduced-service-rate approximation is established when service times are heavy tailed. This result is applied to various models based on M/G/1 processor-sharing queues.

Two service schemes for an M/M/1 time-sharing system with a limited number of service positions are studied. Both schemes possess the equilibrium properties of symmetric queues; however, in the first one, a preempted job is placed at the end of the waiting line; while in the second one, it is placed at the head of the line. Methods for calculating the Laplace transforms and moments of the response times are presented. The variances of the response times are then compared numerically to indicate that the first scheme is superior to the second scheme. It is also indicated that in both cases the response time variance decreases when the number of service positions increases.

A class of M/G/1 time-sharing queues with a finite number of service positions and unlimited waiting space is described. The equilibrium distribution of symmetric queues belonging to this class is invariant under arbitrary service-independent reordering of the customers at instants of arrivals and departures. The delay time distribution, in the special case of one service position where preempted customers join the end of the line, is provided in terms of Laplace transforms and generating functions. It is shown that placing preempted customers at the end of the line rather than at the beginning of the line results in a reduction of the delay time variance. Comparisons with the delay time variance of the case of unlimited number of service positions (processor sharing system) are presented.

An exponential server splits its service capacity between two independent Poisson streams of customers, unless one queue is empty, in which case the full service capacity is granted to the other queue. We present the stationary joint generating function of the two queue sizes explicitly in the form of an elliptic integral.