In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.

Large stochastic systems of marked particles are considered. These ‘populations' grow according to pairwise mutually catalytic reactions and in addition particles may exchange their type (mark) by a sampling procedure. We are interested in the explosive behavior of the model and its high density properties (law of large numbers).

The constructive solution to the problem of representing a strictly stationary Markov chain ζ with a countable infinity of r-sequences (i1, i2, · ··, ir), r > 1, as states by a class of directed weighted circuits is given. Associating the chain ζ with its dual chain η having reversed states and the same transition law, a connection with physical laws that govern diffusion of electrical current through a directed planar network with r-series-connected nodes is shown.

A Slepian model for the local behaviour near the level upcrossings of a x2-process with dependent Gaussian components is presented. In case of independent components, this model is shown to take on a rather simple form, thereby simplifying earlier results by Aronowich and Adler.

The Slepian model is applied to the envelope of a stationary Gaussian process and used to approximate the probability of ‘empty' envelope upcrossings, i.e. the probability that an envelope upcrossing is not followed by a level crossing in the original process.

The problem of the optimal control of the Wiener process in Rn is considered. The optimal value of the control is obtained from the mathematical expectation of a quantity defined in terms of the moment and the place where the uncontrolled process hits the boundary of the continuation region for the first time. Explicit results are presented.

A distribution F of a non-negative random variable belongs to the subexponential family of distributions S if 1 – F(2)(x) ~ 2(1 – F(x)) as x →∞. This family is of considerable interest in branching processes, queueing theory, transient renewal theory and infinite divisibility theory. Much is known about the kind of distributions that belong to S but the question of whether S is closed under convolution has remained unresolved for some time. This paper contains an example which demonstrates that S is not in fact closed.

Let·p denote the p-norm on . For n points drawn independently and uniformly from the unit d-cube, we obtain the limit distribution of the largest nearest-neighbour link (with respect to | · |p) in the cases ; (2) d = 3, p = 1, 2, ∞and (3) 4, p =∞.

Majority systems are encountered in both decision theory and reliability theory. In decision theory for example a jury or committee employing a majority rule will make the ‘correct' decision if a majority of the individuals do so. In reliability theory some coherent systems function if and only if a majority of the components work properly. In this paper results concerning the reliability of majority systems are developed which are applicable in both areas. Two models incorporating dependence between individuals or components in majority systems are introduced, and various monotonicity results for their reliability functions are established. Comparisons are also made between direct (or simple) and indirect majority systems.

In this paper, we develop general repair models for a repairable system by using the idea of the virtual age process of the system. If the system has the virtual age Vn –1 = y immediately after the (n – l)th repair, the nth failure-time Xn is assumed to have the survival function where is the survival function of the failure-time of a new system. A general repair is represented as a sequence of random variables An taking a value between 0 and 1, where An denotes the degree of the nth repair. For the extremal values 0 and 1, An = 1 means a minimal repair and An= 0 a perfect repair. Two models are constructed depending on how the repair affects the virtual age process: Vn = Vn– 1+ AnXn as Model 1 and Vn = An(Vn– 1 + Xn) as Model II. Various monotonicity properties of the process with respect to stochastic orderings of general repairs are obtained. Using a result, an upper bound for E[Sn] when a general repair is used is derived.

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n(s) the number of vertices in the union of all those components of Γn(p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n(s) and the limit distribution of ρ n(s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

In [1], [4] and [6] simple and intuitively obvious bounds were suggested for various overflow systems and claimed to be insensitive, that is to be valid also in the non-exponential case. This insensitivity was formalized in [4] and [6] when the service distributions have a monotone failure rate. This technical note provides another proof which for the case of a pure overflow system with one customer class extends the insensitivity to the general case.

Little's theorem states that under very general conditions L = λW, where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L)l) = λ lE((W)l) are also true, where (L)l = L(L – 1)· ·· (L – l + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji and Sji, where Nji is the total number of class j jobs at center i and Sji is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji and Wji are related this way, where Qji is the number of class j jobs queued, but not in service at center i and Wji is the waiting time in queue of a class j job at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.

Monotonicity of throughput is established in some non-Markovian queueing networks by means of pathwise comparisons. In a series of · /GI/s/N queues with loss at the first node it is proved that increasing the waiting room and/or the number of servers increases the throughput. For a closed network of · /GI/s queues it is shown that the throughput increases as the total number of jobs increases. The technique used for these results does not apply to blocking systems with finite buffers and feedback. Using a stronger coupling argument we prove throughput monotonicity as a function of buffer size for a series of two ·/M/1/N queues with loss and feedback from the second to the first node.

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

Single-channel queues with Poisson arrivals, general service distributions, and no queue capacity are studied. A customer who finds the server busy either leaves the system for ever or may return to try again after an exponentially distributed time. Steady-state probabilities are approximated and bounded in two different ways. We characterize the service distribution by its Laplace transform, and use this characterization to determine the better method of approximation.

Suppose the payoffs (aij) in the n × n matrix A are drawn independently from some continuous probability distribution. The number of tactics used in an ESS is investigated. Asymptotic results on the size of the ESS with largest support are given, using the work of Karlin and Kingman on the size of polymorphisms in one-locus multi-allele diploid selection models.

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.

In this article we give some characterizations of Poisson processes, the model which we consider is inspired by Kimeldorf and Thall (1983) and we generalize the results of Chandramohan and Liang (1985). More precisely, we consider an arbitrarily delayed renewal process, at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i = 1, ···, k. We shall show that the existence of a pair of uncorrelated processes Di, Dj, i ≠ j, implies the renewal process is Poisson. Some other related characterization results are also obtained.

In this paper we study the maximum of Gaussian Fourier series emerging in the analysis of random vibrations of a finite string. Evaluating the distribution of the maximal displacement corresponds to the analysis of the maximum for the related Gaussian Fourier series with independent coefficients.

When vibrations are triggered by an initial white noise disturbance (where the instantaneous form of the vibrating string is composed of three processes pieced together) we give upper bounds for the maximal displacement.

In the last section we consider forced vibrations at special instants where the instantaneous form of the vibrating string has the structure of a Brownian bridge. This enables us to give the exact distribution of the maximum for the related sine series.

The superposition of independent, discrete, renewal processes produces a counting process which is also a discrete time series. The conditional distribution and correlation structure of this kind of time series may be obtained. In suitable conditions the conditional distribution has a spectrum which is exactly or approximately rational. When this is so, an ARMA can be found which matches the spectrum of the superposition.