We consider a single server queueing system in which customers arrive at the instants t0, t1, …, tm, …. We write τm = tm+1 − tm, m ≧ 0. There is a single server with distribution of service times B(x) given by where k is an integer not less than unity.

We consider the case of a genetic population for which the selective advantages of the various genotypes are not constant but for each generation depend linearly on the gene frequencies in the population in the previous generation. For such populations, the effect of competition between similar genotypes may be allowed for by suitable choice of the frequency-dependent selective advantages, or, by a reversal of sign, the case where genotypes are favoured by the presence of similar genotypes may also be considered. favoured by the presence of similar genotypes may also be considered. All populations are finite and of constant size so that eventually only one type of gene will survive. The probabilities of survival for each gene are found and compared with the case where there are no frequency-dependent factors. If a small amount of mutation is allowed, gene fixation will not occur and a steady-state distribution of gene frequency will appear. The form of this distribution may be derived simply from the survival probabilities in the corresponding cases where there is no mutation. The main result is that in some cases, frequency-dependent factors have a marked effect on survival probabilities, while in other cases they can be completely ignored. The latter wifi only occur in certain cases where there exists competition between similar genotypes.

Lagrange's theorem on the reversion of power series may be stated in the following form (e.g. Whittker and Watson [3]).

We consider the following problem. Calls arrive at a telephone exchange at the instants t0, t1, … tm …. The telephone exchange contains a denumerable infinity of channels. The holding times of calls are non-negative random variables distributed independently of the times at which calls arrive, independently of which channel a call engages and independently of each other with a common distribution function B(x). Takacs [3Τm = tm+1 − tm, m ≧ are identically and independently distributed non-negative random variables with common distribution function, A (x). Finch [1] has studied the transient behaviour in the case of a recurrent arrival process and exponential holding time, that is when the common distribution of holding time is given by In this paper we make no assumption about the arrival process {tm}. The underlying principle of this paper is the same as that of Finch [2]. We consider the instants of arrival t0, t1,…, tm,… as given and determine various probabilities of interest conditionally as functions of the inter-arrival intervals Τ1,…, Τm,… When the arrival process is a stochastic process we can then determine the relevant unconditional probabilities by integration.

We consider the genetic model introduced by Moran [3] of a haploid population of fixed size M with two genotypes A and a for which the possibility of selection is allowed. In this model an individual is randomly chosen to die and is replaced by a new individual whose probability of being a depends on the selective advantages of the two genotypes and on the number of a individuals before the birth-death event. The probability of eventual elimination of the genotype a, both with and without selection, has been found by Moran [3], while Watterson [4] has found the mean time for absorption and the variance in the case where no selection is allowed. We derive here the mean time and the variance in the case where selection is allowed, thus extending Watterson's result. A diffusion approximation is available for the mean time; it is shown that this gives a very close approximation to the exact value. Comparison is made with the non-overlapping generation model due to Wright [5], and finally some numerical results are exhibited.

In this paper priority queues with K classes of customers with a preemptive repeat and a preemptive resume policy are considered. Customers arrive in independent Poisson processes, are served, within classes, in order of arrival, and have general requirements for service. Transforms of stationary waiting time and queue size distributions and busy period distributions are obtained for individual classes and for the system; the moments of the distributions are considered.

A queue at which arrivals occur randomly in batches of fixed size r and for which the service times are independent negative exponential variates is considered. Expressions are obtained for the moments of the transient waiting time distribution and the distribution of the number of customers in the system just before the nth batch arrives. The distribution of the number of customers served in a busy period is also determined.

Various authors have studied the transient behaviour of single-server queues. Notably, Takacs [13], [14] has analysed a queue with recurrent input and exponential service time distributions, Keilson and Kooharian [9], [10] and Finch [5] have considered a queue with general independent input and service times, Finch [6] has analysed a queue with non-recurrent input and Erlang service, and Jaiswal [8] has considered the bulk-service queue with Poisson input and Erlang service.

We consider a single server queue for which the interarrival times are identically and independently distributed with distribution function A(x) and whose service times are distributed independently of each other and of the interarrival times with distribution function B(x) = 1 − e−x, x ≧ 0. We suppose that the system starts from emptiness and use the results of P. D. Finch [2] to derive an explicit expression for qnj, the probability that the (n + 1)th arrival finds more than j customers in the system. The special cases M/M/1 and D/M/1 are considerend and it is shown in the general case that qnj is a partial sum of the usual Lagrange series for the limiting probability .