The queues being studied here are of the type GI/G/k in statistical equilibrium (with traffic intensity less than one). The exponential limiting formula for the waiting time distribution function in heavy traffic, conjectured by Kingman (1965) and established by Köllerström (1974), is extended here. The asymptotic properties of the moments are investigated as well as further approximations for the characteristic function and error bounds for the limiting foemulae.

We introduce a model for the spreading of fake news in a community of size n. There are $j_n = \alpha n - g_n$ active gullible persons who are willing to believe and spread the fake news, the rest do not react to it. We address the question ‘How long does it take for $r = \rho n - h_n$ persons to become spreaders?’ (The perturbation functions $g_n$ and $h_n$ are o(n), and $0\le \rho \le \alpha\le 1$ .) The setup has a straightforward representation as a convolution of geometric random variables with quadratic probabilities. However, asymptotic distributions require delicate analysis that gives a somewhat surprising outcome. Normalized appropriately, the waiting time has three main phases: (a) away from the depletion of active gullible persons, when $0< \rho < \alpha$ , the normalized variable converges in distribution to a Gumbel random variable; (b) near depletion, when $0< \rho = \alpha$ , with $h_n - g_n \to \infty$ , the normalized variable also converges in distribution to a Gumbel random variable, but the centering function gains weight with increasing perturbations; (c) at almost complete depletion, when $r = j -c$ , for integer $c\ge 0$ , the normalized variable converges in distribution to a convolution of two independent generalized Gumbel random variables. The influence of various perturbation functions endows the three main phases with an infinite number of phase transitions at the seam lines.

A Zipf's law is a probability distribution on the positive integers which decays algebraically. Such laws describe (approximately) a large class of phenomena. We formulate a model for such phenomena and, in terms of our model, give necessary and sufficient conditions for a Zipf's law to hold.

Suppose the n × n matrix A gives the payoffs for some evolutionary game, and its entries are the values of independent, identically distributed, continuous random variables. The distribution of the pattern of evolutionarily stable strategies for A will depend, if n ≧ 3, on this underlying distribution. A fairly complete picture for n = 3 is found, and some results are obtained for n ≧ 4.

We use an interacting particle system on ℤ to model an epidemic. Each site of ℤ can be in either one of three states: empty, healthy or infected. An empty site x gets occupied by a healthy individual at a rate βn1(x) where n1(x) is the number of healthy nearest neighbors of x. A healthy individual at x gets infected at rate αn2(x) where n2(x) is the number of infected nearest neighbors of x. An infected individual dies at rate δ independently of everything else. We show that for all α, β and δ> 0 and all initial configurations, all the sites of a fixed finite set remain either all empty or all healthy after an almost surely finite time. Moreover, if the initial configuration has infinitely many healthy individuals then the process converges almost surely (in the sense described above) to the all healthy state. We also consider a model introduced by Durrett and Neuhauser where healthy individuals appear spontaneously at rate β > 0 and for which coexistence of 1's and 2's was proved in dimension 2 for some values of α and β. We prove that coexistence may occur in any dimension.

Let w1, w12 and w2 be the fitnesses of genotypes A1A1, A1A2 and A2A2 in an infinite diploid population, and let pn be the A1, gene frequency in the nth generation. If fitness varies independently from generation to generation, then pn is a Markov process with a continuum of states. If E[In(wi/w12)] < 0 for i = 1, 2, then there is a unique stationary probability, and the distribution of pn converges to it as n → ∞.

The asymptotic behaviour of the sequence (𝒞n(ω), wc,n(ω)/n), is studied where 𝒞n(ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n) until time n and wc,n(ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ωc) which represents uniquely the chain (ξ n) as a circuit chain, and ω c is given a probabilistic interpretation.

This paper contains a detailed study of the Poisson cluster process on the real line, concentrating on two aspects; first, the asymptotic distribution of the number of points in [0,t) as t→ ∞ for both transient and equilibrium cluster processes and, secondly, a general formula for the probability generating function of the equilibrium process. Asymptotic formulae for cumulants of the process are also derived. The results obtained generalize those of previous writers. The approach is analytical, in contrast to the probabilistic treatment of P. A. W. Lewis.

In this paper we derive a successive method of solution to a general class of models pertaining to the investigation of the migration process. A specific application of the results to a problem arising in a study of the epidemiology of leprosy has been considered. The application to explorative studies in biological modelling using an interactive computing mode has been discussed.

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

We study the expected time for the work in an M/G/1 system to exceed the level x, given that it started out initially empty, and show that it can be expressed solely in terms of the Poisson arrival rate, the service time distribution and the stationary delay distribution of the M/G/1 system. We use this result to construct an efficient simulation procedure.

An explicit, computable, and sufficient condition for exponential ergodicity of single-birth processes is presented. The corresponding criterion for birth–death processes is proved using a new method. As an application, some sufficient conditions are obtained for exponential ergodicity of an extended class of continuous-time branching processes and of multidimensional Q-processes, by comparison methods.

It is desired to minimize the expected cost of finding an object which moves back and forth between two locations according to an unobservable Markov process. When the object is in location i (i = 1, 2) it resides there for a time which is exponentially distributed with parameter λ1 and then moves to the other location. The location of the object is not known and at each instant until it is found exactly one of the two locations must be searched. Searching location i for time δ costs ciδ and conditional on the object being in location i there is a probability αiδ + o(δ) that this search will find it. The probability that the object starts in location 1 is known to bé p1(0). The location to be searched at time t is to be chosen on the basis of the value of p1(t), the probability that the object is in location 1, given that it has not yet been discovered. We prove that there exists a threshold Π such that the optimal policy may be described as: search location 1 if and only if the probability that the object is in location 1 is greater than Π. Expressions for the threshold Π are given in terms of the parameters of the model.

Let X1,n ≦ … ≦ Xn, n be the order statistics of a sample of size n from a distribution function F. Desu (1971) showed that if for all n ≧ 2, nX1,n is identically distributed as X1, 1, then F is the exponential distribution (or else F degenerates). The purpose of this note is to point out that special cases of known characterization theorems already constitute an improvement over this result. We show that the characterization is preserved if “identically distributed” is weakened to “having identical (finite) expectation”, and “for all n ≧ 2” is weakened to “for a sequence of n's with divergent sum of reciprocals”.

Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n → ∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.

In this note a model for shot noise generated by a semi-Markov process is developed. The moments of the shot noise process are found, and some applications of this model are briefly indicated.

The branching diffusion with immigration is studied. Under general branching and diffusion laws, the process is shown to be mixing, according to Brillinger's definition. Brillinger's central limit theorem for spatially homogeneous mixing processes is generalized to prove that, under a renormalization transformation, the distribution of the branching diffusion with immigration converges to a completely random Gaussian random measure. In addition, the existence of a steady-state distribution is proven in the case of subcritical branching, and this distribution is shown to be mixing. Hence the steady-state random field also obeys a spatial central limit theorem.

Non-exponential queueing networks with blocking are studied under two types of blocking protocols: the ‘stop' and the ‘repeat' protocol under which services are to be either stopped or repeated respectively upon blocking. The two protocols are shown to be stationary equivalent when the system satisfies a partial balance condition. This condition is related to product-form results. The results are illustrated by closed and open network applications with reversible and nonreversible routing.

The signature representation shows that the reliability of the system is a mixture of the reliability functions of the k-out-of-n systems. The first representation was obtained for systems with independent and identically distributed (IID) components and after it was extended to exchangeable (EXC) components. The purpose of the present paper is to extend it to the class of systems with identically distributed (ID) components which have a diagonal-dependent copula. We prove that this class is much larger than the class with EXC components. This extension is used to compare systems with non-EXC components.

We give formulas for updating both the steady-state probability distribution and the fundamental matrices of a singularly perturbed Markov chain. This formula generalizes Schweitzer's regular perturbation formulas to the case of singular perturbations.