Single–server–single-queue–FIFO-discipline queueing systems are considered in which at most a finite number of customers N can be present in the system. Service and arrival rates are taken to be dependent upon that state of the system. Interarrival intervals, service intervals, waiting times and busy periods are studied, and the results obtained are used to investigate the features of a special queueing model characterized by parameters (λ (Ν –n), μn). This model retains the qualitative features of the C-model proposed by Conolly [2] and Chan and Conolly [1]. However, quite unlike the latter, it also leads to closed-form expressions for the transient probabilities, the interarrival and service probability density functions and their moments, as well as the effective interarrival and service densities and their moments. Finally, some computational results are given to compare the model discussed in this paper with the C-model.

Emitted particles arrive at the counter with prolonging dead time so that the interarrival times and the lengths of impulses in any dead time are independent but not necessarily identically distributed random variables, and whenever the counter is idle then the following evolution starts from the beginning. For this class of counters we derive the probability laws of the numbers of particles arriving at the counters during their dead times, and the Laplace transform of the cycle, respectively.

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

Conservative estimates for the probability that open queueing networks are not overloaded during any finite set of time points are derived. The queueing networks have infinitely many servers at each queue and general service times. A key preliminary result is that for multidimensional Markov population processes, the correspondence between initial distributions and their temporal development preserves stochastic order.

The busy-period distributions of M/G/1/K queues with state-dependent arrival rates are considered. Two recursion formulas for the Laplace–Stieltjes transforms of the busy periods under the FCFS and preempt resume LCFS service disciplines are obtained. It is shown that the busy-period distributions for the two service disciplines are, in general, different, in contrast to the fact that they coincide for ordinary M/G/1 queues. For deterministic service times and arrival rates non-increasing in the number of customers in the system, stochastic ordering between these two busy periods is also established.

A non-negative random vector T is said to have a multivariate increasing failure rate distribution (MIFR) if and only if E[h(x, T)] is log concave in x for all functions h(x, t) which are log concave in (x, t) and are non-decreasing and continuous in t for each fixed x. This class of distributions is closed under deletion, conjunction, convolution and weak limits. It contains the multivariate exponential distribution of Marshall and Olkin and those distributions having a log concave density. Also, it follows that if T is MIFR and ψ is non-decreasing, non-negative and concave then ψ (T) is IFR.

A queueing situation often encountered in practice is that in which customers wait for service for a limited time only and leave the system if not served during that time. This paper considers a single-server queueing system with Poisson input and general service times, where a customer becomes a lost customer when his service has not begun within a fixed time after his arrival. For performance measures like the fraction of customers who are lost and the average delay in queue of a customer we obtain exact and approximate results that are useful for practical applications.

The Moran model in population genetics may also be regarded as an invasion process (or voter model) on the complete n-graph. Tracing lines of ancestry is equivalent to studying the dual process. Using this technique certain results concerning lines of ancestry can be derived very rapidly.

In a class of population-size-dependent Galton-Watson processes where extinction does not occur with probability 1 we describe the rate of decay of qi (the probability that the process starting from i ancestors will become extinct) as the number i of ancestors increases.

The results are related to the asymptotic behavior of the Green's function of the critical Galton-Watson process with immigration.

For the queueing system M/G/∞ some distributions connected with the associated busy periods are derived.

The problem is what to do with a slow server in a service facility which has fast and slow servers. Should the slow server be used to render service, or is it better not to use it at all? Simple models for answering this question are formulated and studied.

For a model of a rumour, first given a non-rigorous treatment by Maki and Thompson, it is shown that the proportion of the population never hearing the rumour converges in probability to 0.203 as the population size tends to∞.

In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.

This is an extension of our previous paper [4] on the queue GI/Hm/s. In that paper we have derived results for the actual waiting time, the number of customers in the system at arrival epochs and the number of customers during a busy cycle. Here we obtain results for the virtual waiting time and the number of customers in the system at arbitrary times.

We give necessary and sufficient conditions for various results connecting the tail behaviour of a distribution with that of its right Wiener–Hopf factor.

Weak convergence to reflected Brownian motion is deduced for certain upwardly drifting random walks by coupling them to a simple reflected random walk. The argument is quite elementary, and also gives the right conditions on the drift. A similar argument works for a corresponding continuous-time problem.

We choose a uniform random point in a given convex body K in n-dimensional Euclidean space and through that point the secant of K with random direction chosen independently and isotropically. Given the volume of K, the expectation of the length of the resulting random secant of K was conjectured by Enns and Ehlers [5] to be maximal if K is a ball. We prove this, and we also treat higher-dimensional sections defined in an analogous way. Next, we consider a finite number of independent isotropic uniform random flats meeting K, and we prove that certain geometric probabilities connected with these again become maximal when K is a ball.

Let X(t), t≧0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function σ2(t) is convex and σ2(t) = o(t) for t → 0, then P(max[o,t]X(s) > u) ~ P(X(t) > u) for u → ∞ and each t > 0.

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).

We examine a Markov manpower system in continuous time where demotion rates are 0 and promotion rates are time-dependent. The transient and limiting behaviours of the model are discussed, and illustrated with examples.