Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z2} and {Y(u), u (Z2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.

Each point with integer coordinates in d dimensions is occupied by one individual. These individuals produce offspring at a Poisson rate 1, and these offspring migrate and displace other individuals. With probability u (the mutation rate) an offspring is of an entirely new type. A number of points N0 will be occupied by the same type as the individual at the origin. It is shown that the distribution of N0 arising from an ancient mutation does not differ greatly from the distribution of N0 when the mutation is recent. However, the geographical spread is shown to be important, and a central limit theorem is proved for the age of the mutant clone given that a representative is present at a large distance from the origin.

In most contemporary queueing network analysis, the assumption is made that a network is in a state of equilibrium. That is, the network's state space probabilities are assumed to be time independent. It is therefore important to be able to quantify precisely when this assumption is valid. Furthermore there are also situations in which it is desirable to model the transient behaviour of networks which occur in practice, such as computer and communication systems. For example, the immediate effects of component failure or instantaneous alteration of system status may be predicted.

In this paper an iterative solution is derived to the time-dependent Kolmogorov equations of queueing networks, and is shown to be convergent. From the solution, modelling of transient situations becomes possible and the time periods during which the equilibrium assumption can and should not be made may be identified; for example in terms of a time constant which is easily computed to a first-order approximation.

A target is assumed to choose its starting position in a search at an unknown position in a finite search space. No prior probability distribution for the target's initial location is assumed. During the search the target is assumed to move from position to position in the search space according to a Markov process. A search is defined to be the observation of a sequence of random variables. Representations for the minimax estimator for target location at any stage of the search, the least favorable prior distribution for the target, and the value of the estimation game are presented. An example is computed in which Bayes estimators are compared with minimax estimators for target location.

A non-linear stochastic model for the single-server machine interference problem is investigated. The service rate is a rational function of the number of failed machines, the numerator as well as the denominator of the function being linear in n. Depending on the relative values of the parameters involved, the service rate may increase or decrease with n, exhibiting the incentive or disincentive effect. The analysis is carried out by the usual diffusion approximation technique. A notable feature of this model is the threshold effect exhibited in the mean queue size.

Some recent constructions for the generation of dependent sequences of identically distributed negative exponential random variables with specific correlation structures are generalized. This is achieved by attributing a correlation structure to the binary sequence which controls the generation of the exponentials. The procedure causes the autocorrelation function of the exponential sequence to copy that of the binary sequence and thus be extended to include negative values and other values beyond the usual range.

Normal and abnormal cells are positioned at the vertices of a regular two-dimensional lattice. Abnormal cells divide k times as fast as normal cells. Whenever a cell divides, the daughter is the same type as the parent and replaces an adjacent cell. The Kolmogorov forwards and backwards equations are derived, and then used to obtain bounds for the distribution function of the time when all the abnormal cells are forced from the plane. These bounds are used to comment on the non-asymptotic variance of the number of abnormal cells at a given time and on a method of estimating k.

Let X1, X2, · ·· be i.i.d. and Yk = max{X1, X2, · ··, Xk}. Then Xj is a record value of the sequence if Yj > Yj-1). Denoting record value times by Ln, n = 1, 2, ···, we prove that the independence of XLj − XLi and XLi with Lj > characterizes the exponential distribution.

We examine the geometry of regions of maintainable structures arising in a Markov manpower model. The regions are described in terms of convex hulls, and it is shown that for systems divided into two or three grades these regions form an increasing sequence. It is also shown that the monotonie property fails quite drastically for a four-graded system.

An open problem is discussed for a related model.

Exact expressions are derived for the transition probabilities of the birth-death process with parameters and which serves as a queueing model where potential customers are discouraged by queue length.

An exponential server splits its service capacity between two independent Poisson streams of customers, unless one queue is empty, in which case the full service capacity is granted to the other queue. We present the stationary joint generating function of the two queue sizes explicitly in the form of an elliptic integral.

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated.

The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

The possibility that interspecific contest tactics might be able to stabilize two species predator-prey populations is investigated using a matrix model of the population. It is found that if the interspecific tactics have no influence on intraspecific encounters, the non-trivial equilibrium population composition is necessarily unstable. The result provides some insight into a reported instability of a predator-prey population with ‘stable' strategies.

Suppose that n persons each know a different piece of information, and that, whenever a pair of them talk on the telephone, each tells the other all the information he knows at that time. If the calls are made at random, we show that the expected number of calls required for everyone to know all n pieces of information is asymptotically 1.5 n log n + O(n). This sharpens an earlier result of D. W. Boyd and J. M. Steele. Some numerical comparisons are given.

We study a production-inventory system in which input is deterministic, and its rate is the controlled parameter. The output is a compound Poisson process with exponential jump-size distribution. The parameters of the output process are unknown. The following adaptive policy is used: At each time t the unknown parameters are estimated by maximum likelihood method and the input rate which would be optimal if these were the true values of the parameter is used. The issues investigated are (1) the convergence of maximum likelihood estimates to the true value, and (2) the asymptotic properties of the cost under the adaptive policy.

The paper considers a modified storage process with state space [0,∞). Away from the origin, W behaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls to It is shown that W can be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift. When these same two transformations are applied to Brownian motion, one obtains sticky (or slowly reflected) Brownian motion W∗ on [0,∞). Thus W∗ is the natural diffusion approximation for W, and it is shown that W converges in distribution to W∗ under appropriate conditions. The boundary behavior of W∗ is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.

For a time-homogeneous continuous-parameter Markov chain we show that as t → 0 the transition probability pn,j (t) is at least of order where r(n, j) is the minimum number of jumps needed for the chain to pass from n to j. If the intensities of passage are bounded over the set of states which can be reached from n via fewer than r(n, j) jumps, this is the exact order.

Félix Pollaczek was born on 1 December 1892 in Vienna, Austria; he was the descendant of a well-known mid-European Jewish family which for several centuries had counted many learned men amongst its members. He became a Frenchman by naturalization in 1947. He obtained his scientific education and training at the Latin grammar school of Vienna, and at the Technical Universities of Vienna and Brno (Czechoslovakia); he later studied mathematics at the University of Berlin. A master's degree in electrical engineering in 1920 from Brno was followed by a doctoral degree in mathematics in 1922 from Berlin with a thesis on number theory. This was concerned with the fields generated by the Ith and 12th roots of one, I being an irregular prime number. His first paper, on number theory, dates from 1917.

Conditions for the non-sure extinction of Galton-Watson processes with generation dependence are obtained. Also a condition is given for such processes to have a strictly increasing probability generating function.