In this paper we consider an independent and identically distributed sequence {Yn} with common distribution function F(x) and a random variable X0, independent of the Yi's, and define a Markovian sequence {Xn} as Xi = X0, if i = 0, Xi = k max{Xi− 1, Yi}, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

Lower and upper bounds of degree m for the probability of the union of n not necessarily exchangeable events are established. These bounds may be constructed to improve the Bonferroni and the Sobel–Uppuluri bounds.

An application to equi-correlated multivariate normal distribution is given.

The theory of insensitivity within generalised semi-Markov processes is extended to cover classes of models in which the generally distributed lifetimes can be terminated prematurely by the deaths of negative exponentially distributed lifetimes. As a consequence of this approach it is shown that there exist classes of processes which are insensitive with respect to characteristics of the general distributions other than the mean. Two examples are given. The first is an analysis of networks of queues in which the generally distributed service times can be interrupted with resulting changes in routing probabilities. The second is a model for the effect of disturbances on the evolution of a vegetation community.

An asymptotic expression is given for the expected number of vertices of the convex hull of a uniform sample from the interior of a d-dimensional simple polytope. This extends a result derived by Rényi and Sulanke for sample points in the plane.

This paper gives the necessary and sufficient condition for the second-order stationarity of lower triangular bilinear models; this condition is in terms of norms of higher-order transfer functions. The spectral density function of the process is evaluated in terms of the transfer functions, and it is shown that the second-order structure is similar to a linear ARMA model with uncorrelated errors. The best linear predictor is obtained and it is shown that the variance of the prediction error is always greater than the optimal prediction error variance obtained from the bilinear model. An expression for the bispectral density function is also obtained.

A direct approach to derive dependence properties among the hitting times of bivariate processes has been initiated by Ebrahimi (1987) and explored further by Ebrahimi and Ramallingam (1988). In this paper, new results are obtained for multivariate processes, which help us to identify positive and negative dependence structures among the hitting times of the processes. Applications of our theorems to reliability of systems are given.

We regard the standard stochastic model of renewal theory as a functional which associates with a lifetime distribution μ the corresponding renewal measure v = Σ0∞μ ∗n. The behaviour of this functional near exponential distributions is investigated. The literature mainly deals with the asymptotic behaviour of v([0, t]) and v[t, t + 1]) as t → ∞— this new approach leads to expansions which are valid uniformly in t ∈ R.

A manuscript with an unknown random number M of misprints is subjected to a series of proofreadings in an effort to detect and correct the misprints. On the nthproofreading, each remaining misprint is detected independently with probability pn– 1. Each proofreading costs an amount CP > 0, and if one stops after n proofreadings, each misprint overlooked costs an amount cn > 0. Two models are treated based on the distribution of M. In the Poisson model, the optimal stopping rule is seen to be a fixed sample size rule. In the binomial model, the myopic rule is optimal in many important cases. A generalization is made to problems in which individual misprints may have distinct probabilities of detection and distinct overlook costs.

The resolvent operators of the following two processes are obtained: (a) the bivariate Markov process W = (X, Y), where Y(t) is an irreducible Markov chain and X(t) is its integral, and (b) the geometric Wiener process G(t) = exp{B(t} where B(t) is a Wiener process with non-negative drift μ and variance parameter σ2. These results are then used via a limiting procedure to determine the long-run average cost per unit time of operating a finite dam where the input process is either X(t) or G(t). The system is controlled by a policy (Attia [1], Lam [6]).

Let V be a countably infinite set, and let {Xn, n = 0, 1, ·· ·} be random vectors in which satisfy Xn = AnXn– 1 + ζn, for i.i.d. random matrices {An} and i.i.d. random vectors {ζ n}. Interpretation: site x in V is occupied by Xn(x) particles at time n; An describes random transport of existing particles, and ζ n(x) is the number of ‘births' at x. We give conditions for (1) convergence of the sequence {Xn} to equilibrium, and (2) a central limit theorem for n–1/2(X1 + · ·· + Xn), respectively. When the matrices {An} consist of 0's and 1's, these conditions are checked in two classes of examples: the ‘drip, stick and flow model' (a stochastic flow with births), and a neural network model.

A wide class of multichannel queueing models appears to be useful in practice where the input stream of customers can be controlled at the moments preceding the customers' departures from the source (e.g. airports, transportation systems, inventories, tandem queues). In addition, the servicing facility can govern the intensity of the servicing process that further improves flexibility of the system. In such a multichannel queue with infinite waiting room the queueing process {Zt; t ≧ 0} is under investigation. The author obtains explicit formulas for the limiting distribution of (Zt) partly using an approach developed in previous work and based on the theory of semi-regenerative processes. Among other results the limiting distributions of the actual and virtual waiting time are derived. The input stream (which is not recurrent) is investigated, and distribution of the residual time from t to the next arrival is obtained. The author also treats a Markov chain embedded in (Zt) and gives a necessary and sufficient condition for its existence. Under this condition the invariant probability measure is derived.

We consider networks with a very large or infinite number of nodes, linked by cable channels. The request which comes to a node is ordered to occupy a certain route of successive channels. The functioning of the system is regulated by the reserving of channels in order of the arrivals of the requests. Under some general conditions the existence of an ergodic region for such networks is proved. The practical value of the result lies in the fact that these conditions do not depend on the size of the graph.

Queueing systems are studied with a last-come, first-served queueing discipline and batch arrivals generated by a finite number of non-exponential sources. A closed-form expression is derived for the steady-state queue length distribution. This expression has a scaled geometric form and is insensitive to the input distribution. Moreover, an algorithm for the recursive computation of the normalizing constant and the busy source distribution is presented. The results are of both practical and theoretical interest as an extension of the standard Poisson batch input case.

We consider a single-server queue with a periodic and ergodic input. It is shown that if the traffic intensity is less than 1, then the waiting time process is asymptotically periodic. Limit theorems associated with the asymptotic behavior of the queue are also proven. The results are then extended to acyclic networks of queues with periodic inputs. Particular cases of these results had been previously obtained for a single queue with periodic Poisson arrival input process and with independent and identically distributed service times.

This paper develops moment formulas for asymptotic workload and waiting time in a single-server queue with periodic Poisson input and general service distribution. These formulas involve the corresponding moments of waiting-time (workload) for the M/G/1 system with the same average arrival rate and service distribution. In certain cases, all the terms in the formulas can be computed exactly, including moments of workload at each ‘time of day.' The approach makes use of an asymptotic version of the Takács [12] integro-differential equation, together with representation results of Harrison and Lemoine [3] and Lemoine [6].

Let {Zφ (t)} be a multitype finite-mean supercritical (CMJ) process with general characteristic φ and malthusian parameter α. It is shown that there exist some constants {c(t} such that {c–1(t)Zφ (t)} converges almost surely to a non-degenerate limit Wφ as t →∞. Characterizations of {c(t)} and Wφ are also given.

It is shown that, in a renewal process with inter-arrival distribution F, an observation from the asymptotic (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and W are independent random variables with U ~ uniform(0, 1) and W distributed according to the length-biased version of F.

This note gives the solution of the following problem concerning geometric probabilities. What is the probability p(Bd; 2) that the circumference determined by three points P, P1 and P2 chosen independently and uniformly at random in the interior of a d-dimensional unit ball Bd in Euclidean space Ed (d ≧ 2) is entirely contained in Bd? From our result we conclude that p(Bd; 2) →π /(3√3) as d →∞.

We develop a representation for general queueing networks, and study some stochastic monotonicity properties that have been previously established in Jackson networks (e.g. Shanthikumar and Yao (1986), (1987)).

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.