We present a solution to a special system of Kolmogorov forward equations. We use this result to present a useful expression for the transition probabilities of the extended simple stochastic epidemic model and an epidemic model involving cross-infection between two otherwise isolated groups.

A well-known result in the theory of branching processes provides an asymptotic expression for the population size (valid for large times) in terms of a single random variable, multiplied by a deterministic exponential growth factor. In the present paper this is generalized to a class of size-dependent population models. The work is based on the series of sojourn times. An essential tool is the use of probabilities conditional upon non-extinction (taboo probabilities).

The ratio of the number Xn of different words (types) in a text of length n (token) words to n has received considerable attention in the literature of statistical linguistics. The present note contains two stochastic models for Xn based on an inhomogeneous discrete Markov process of the pure birth type where the transition probabilities take certain forms depending only upon n. These models are then tested against data obtained from the plays of William Shakespeare.

Equations are derived describing a central limit type large population approximation for continuous time Markov lattice processes in one or more dimensions, such as are commonly encountered in biological models. A method of solving the equations using only the deterministic solution of the process is explained, and it is extended by the use of a martingale argument to provide more detailed information about the process.

A stationary doubly stochastic Poisson sequence is considered. Asymptotic properties of the best linear estimates of the intensities are investigated.

We discuss the potential theory of optimal stopping for a standard process and an unbounded reward function. This is applied to Brownian motion constrained to a N(m, σ2) distribution at time 1. Boyce [2] has discovered, via computer, various interesting features of this example. We provide direct proofs for some of them, in particular for the qualitative jump of the optimal strategy as the variance σ2 passes the critical value 1.

It is shown that, under certain conditions, satisfied by stable distributions, symmetric distributions, distributions with zero mean and finite second moment and other distributions, the distribution function of the maxima of successive partial sums of identically distributed random variables has an asymptotic property. This property implies the regular variation of the tail of the distribution of the hitting times of the associated random walk, and hence that these hitting times belong to the domain of attraction of a stable law.

Laha and Lukacs (1960) have studied distributions with the property that a quadratic statistic has quadratic regression on the sample mean. In doing this, they have arrived at some interesting characterizations for the normal, Poisson, gamma, binomial and negative binomial distributions. Starting with an exponential-type probability density function, the present paper investigates all the distributions for which the 3 × 3 Bhattacharya matrix is diagonal. It is found that the normal, Poisson, gamma, binomial and negative binomial distributions can be characterized by this property. Further, it is observed that for these distributions an s × s Bhattacharya matrix is defined for all s and is also diagonal.

Consider the following queuing system: A sequence of customers arrive at a service unit in a recurrent stream. A customer is of priority k with probability πk, k = 1, …, n. A class i customer preempts service of class k, k > i. Interrupted service is resumed without loss or gain in service time. Service is FIFO within classes. Service times for class k are drawn from a general distribution function Bk(t).

Using the method of phases and a resolution technique from the theory of Markov processes we obtain Laplace transforms of various distributions.

The object of this paper is to study the actual waiting time of a customer in a GI/G/1 queue. This is an important criterion from the viewpoint of both the customers and the efficient functioning of the counter. Suitable point processes in the product space of load and time parameters for any general inter-arrival and service time distributions are defined and integral equations governing the correlation functions are set up. Solutions of these equations are obtained and with the help of these, explicit expressions for the first two moments of the number of customers who have waited for a time longer than w in a given time interval (0, T) are calculated.

A dam model is proposed for which it is assumed that the conditional distributions of the inputs, given the past, are known only to lie in some class M. For selected M, bounds are derived on various quantities of interest such as the mean time to first emptiness. The case of normal inputs is treated in greater detail and a release rule is discussed. The techniques used are similar to those used in the theory of gambling as developed by Dubins and Savage (1965).

This paper considers a fixed-cycle and a semi vehicular-actuated traffic light queue with strictly stationary arrivals and independent and identically distributed departure headways and lost times. These queues are reduced to a generalized model of Loynes and sufficient conditions are derived under which these queues have stationary distributions. Two typical examples of semi vehicular-actuated traffic light queues are discussed.

A generalized queueing system with (N + 2) types of triplets (delay, service time, probability of joining the queue) and with uniformly bounded sojourn times is considered. An expression for the generating function of the Laplace-Stieltjes transforms of the waiting time distributions is derived analytically, in a case where some of the random variables defining the model have a rational Laplace-Stieltjes transform.

The standard Kl/Km/1 queueing system with uniformly bounded sojourn times is considered in particular.

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

Most epidemic models previously studied have assumed a homogeneously mixing population. Instead of making this assumption, a population divided into classes is considered; and it is assumed that the degree of mixing between classes is less than that within classes. The stochastic model in this form is intractable and approximations are made, yielding results in reasonable agreement with simulation trials.

This paper derives an expression for the correlation function of a continuous parameter branching process, and obtains conditions for the convergence of this function to a zero or positive value.

Consider a semi-Markov process X(t) defined on a subset of the non-negative integers with zero as an absorbing state and the non-zero states forming an irreducible class with exit to zero being possible. Conditions are given for the existence of the limits: where Xj(t) is the amount of time prior to time t spent in state j.

The limits (which are independent of the initial state) are evaluated when the sufficient conditions are satisfied.

A stationary independent increment process is the continuous time analogue of the discrete random walk, and, as such, has a wide variety of applications. In this paper we consider M(t), the maximum value that such a process attains by time t. By using renewal theoretic methods we obtain results about M(t). In particular we show that if μ, the mean drift of the process, is positive, then M(t)/t converges to μ, and E[M(t + h) – M(t)] → hμ.

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.

In this note a characterization for a U-shaped probability density function is given using a well-known result for unimodal distributions due to Khinchine. As a corollary of this result, a characteristic property based on moments is found.