An expansion in orthogonal polynomials is found for the transition density in a neutral multi-allele diffusion model where the mutation rates of allele types Ai → Aj are assumed to be uj(≥ O). The density is found when the mutation rate is positive for all allele types, and when some or all have zero mutation. The asymptotic conditional density is found for a mixture of positive and zero mutation rates.

The infinite alleles limit with equal mutation is studied. Eigenfunctions of the process are derived and the frequency spectrum found. An important result is that the first eigenfunction depends only on the homozygosity.

A density for the time to fixation with zero mutation is found for the K allele, and infinite alleles model.

The exact transient sampling distribution is found for the infinite neutral alleles model with mutation, thus extending the stationary distribution found by Ewens (1972). The first eigenfunction in the sampling distribution depends only on the homozygosity. In a large sample the expected number of allele types is close to the expected number in a stationary distribution and depends little on the initial frequencies or the time of sampling. The expected number of types in a sample for a non-stationary population is tabulated.

The sampling distribution from a population with no mutation is found. The probability that the population is monomorphic given that the sample contains only one type is tabulated, where initially the population has a large number of alleles with equal frequencies.

A study is made of the joint distribution of the allele frequencies in two samples taken time t apart from a stationary population. Of particular interest is the number of allele types in common in two such samples. The distribution of the number of types in common in a population viewed at time t apart is also found and tabulated for different mutation rates and time. The expected frequency of the allele types in common is tabulated.

The distribution of allele frequencies in two divergent populations of common ancestry is shown to be the same as the distribution of frequencies in a single population at two time points 2t apart. A sufficient statistic for the time of divergence is shown to be the paired frequencies of the allele types in common. The expected waiting time until the populations have no allele types in common is tabulated.

The models under consideration are a class of infinite particle systems which can be written as a superposition of branching random walks. This paper gives some results about the limiting behavior of the number of particles in a compact set as t → ∞ and also gives both sufficient and necessary conditions for the existence of a non-trivial translation-invariant stationary distribution.

The secretary problem refers to a certain class of optimal stopping problems based on relative ranks. To allow a more realistic formulation of the problem, this paper considers an arbitrary loss function. A finite and an infinite problem are defined and the optimal solutions are obtained. The solution for the infinite problem is given by a differential equation while the finite problem is given by a difference equation. Under general conditions, the finite problem tends to the infinite problem. An example involving the secretary problem with interview cost is considered and illustrates the usefulness of the present paper.

The matrix of intensities of a Markov process with discrete state space and continuous time parameter undergoes random changes in time in such a way that it stays constant between random instants. The resulting non-Markovian process is analyzed with the help of supplementary process defined in terms of variations of the intensity matrix. Several examples are presented.

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues.

A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.

We compare two queueing systems with the same number of servers that differ by having stochastically ordered service times and/or interarrival times as well as different waiting room capacities. We establish comparisons for the sequences of actual-arrival and departure epochs, and demonstrate by counterexample that many stochastic comparisons possible with infinite waiting rooms no longer hold with finite waiting rooms.

We compare two queueing systems with identical general arrival streams, but different numbers of servers, different waiting room capacities, and stochastically ordered service time distributions. Under appropriate conditions, it is possible to construct two new systems on the same probability space so that the new systems are probabilistically equivalent to the original systems and each sample path of the stochastic process representing system size in one system lies entirely below the corresponding sample path in the other system. This construction implies stochastic order for these processes and many associated quantities of interest, such as a busy period, the number of customers lost in any interval, and the virtual waiting time.