A reasonably general but greatly simplified model for a finite sexually reproducing population is considered. We restrict ourselves to the case where there are distinct, non-overlapping generations, but there is hope that the techniques will also be applicable to some very simple minded continuous time models with overlapping generations (see also Section 5, Example 3). There are t∞ < ∞ different types of females and t1 < ∞ types of males and our main concern is the asymptotic behavior of the variables and

Some results for a general Markov branching-diffusion process are presented, and applied to a model recently considered by Bailey. Moments of the limiting distributions of certain natural measures of the spatial location and dispersion of the population are shown to be expressible in terms of the Lauricella FD-type hypergeometric functions, when the population multiplies according to the simple birth and death process with λ > μ.

This paper discusses the transient and limiting behavior of a system of queues, consisting of two service units in tandem in which the second unit has finite capacity. When the second unit reaches full capacity, a phenomenon termed “blocking” occurs. A wide class of rules to resolve blocking is defined and studied in a unified way.

The input to the first unit is assumed to be Poisson, the service times in the first unit are independent with a general, common distribution. When the system is not blocked, the second unit releases its customers according to a state-dependent, death process.

The analysis of the time-dependence relies heavily on several imbedded Markov renewal processes. In particular, the analog of the busy period for the M/G/1 queue is modeled here as a “Markov renewal branching process”. The study of this process requires the definition of a class of matrix functions which generalizes some classical definitions of matrix function. In terms of these “matrix functions” we are led to consider functional iterates and a matrix analog of Takács' functional equation for the transform of the distribution of the busy period in the M/G/1 model.

We further discuss the joint distribution of the queue lengths in units I and II and its marginal and limiting distributions.

A final section is devoted to an informal discussion on how the numerical analysis of this system of queues may be organized.

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers ≡ service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let λi denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and μj the service rate (reciprocal of the mean service time) in the jth service channel. Then is the total arrival rate to the system and is the maximum service rate of the system. As a measure of congestion we define the traffic intensity ρ = λ/μ.