Showing papers in "Journal of Applied Probability in 1977"
TL;DR: In this paper, the authors considered a queuing system consisting of a finite number of identical exponential servers, each server has its own queue, and upon arrival each customer must be assigned to some server's queue.
Abstract: We consider a queuing system consisting of a finite number of identical exponential servers. Each server has its own queue, and upon arrival each customer must be assigned to some server's queue. Under the assumption that no jockeying between queues is permitted, it is shown that the intuitively satisfying rule of assigning each arrival to the shortest line maximizes, with respect to stochastic order, the discounted number of customers to complete their service in any time t.
443 citations
TL;DR: In this article, the ergodicity problem for backwards products of stochastic matrices was studied and conditions for ergodic convergence were derived and their relation to the consensus problem was considered.
Abstract: The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.
363 citations
TL;DR: In this article, a result like the Kesten-stigum theorem is obtained for certain martingales associated with the branching random walk and a special case, when a Malthusian parameter exists, is considered in greater detail.
Abstract: A result like the Kesten-Stigum theorem is obtained for certain martingales associated with the branching random walk. A special case, when a ‘Malthusian parameter’ exists, is considered in greater detail and a link with some known results about the Crump-Mode model for a population is established.
359 citations
TL;DR: Basic theory is developed for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure.
Abstract: : The vast majority of reliability analyses assume that components and system are in either of two states: functioning or failed. The present paper develops basic theory for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure. Axioms are laid down extending the standard notion of a coherent system to the new notion of a multistate coherent system. For such systems deterministic and probabilistic properties are obtained for system performance which are analogous to well-known results for coherent system reliability.
331 citations
TL;DR: In this article, the authors considered the maxima of independent Weiner processes spatially normalized with time scales compressed and showed that a weak limit process exists, and their one-dimensional distributions are of standard extreme-value type.
Abstract: The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.
307 citations
Abstract: If Fn∗ is the n-fold Stieltjes convolution of the increasing function F, then a version of Chernoff's theorem, on the limiting behaviour of (Fn∗ (na))1/n , is established for Fn∗ . If Z (n)(t) is the number of the nth-generation people to the left of t in a supercritical branching random walk then an analogous result is proved for Z (n).
197 citations
TL;DR: In this paper, necessary and sufficient conditions for a distribution function in ℝ2 to be max-infinitely divisible are given for a multivariate extremal process and an approach to the study of the range of an i.i.d. sample.
Abstract: Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if F t is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.
137 citations
TL;DR: In this article, the points of a homogeneous Poisson process within a compact convex set are observed, and the authors consider how to reconstruct this domain from the observations and show that it is possible to reconstruct the Poisson domain from observations.
Abstract: The points of a homogeneous Poisson process within a compact convex set are observed. We consider how to reconstruct this domain from the observations.
115 citations
TL;DR: In this article, the authors investigated the properties of the number of up-and down-crossings of the stationary virtual waiting-time process of the GI/G/1 queueing system during a busy cycle.
Abstract: For the sample functions of the stationary virtual waiting-time process v, of the GI/G/1 queueing system some properties of the number of up- and downcrossings of level v by the v,-process during a busy cycle are investigated. It turns out that the simple fact that this number of upcrossings is equal to that of downcrossings leads in a rather easy way to basic relations for the waiting-time distributions. This approach to the study of the v,-process of the GI G /1 system seems to be applicable to many other types of stochastic processes. As another example of this approach the infinite dam with non-constant release rate is briefly discussed.
109 citations
TL;DR: In this article, a construction for a stationary sequence of random variables {Xi } which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence is given.
Abstract: A construction is given for a stationary sequence of random variables {Xi } which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence {e i }. The joint and trivariate exponential distributions of Xi −1, Xi and Xi + 1 are studied, as well as the intensity function, point spectrum and variance time curve for the point process which has the {Xi } sequence for successive times between events. Initial conditions to make the point process count stationary are given, and extensions to higher-order moving averages and Gamma point processes are discussed.
100 citations
TL;DR: In this paper, the least squares estimators of the autoregressive constants for a stationary auto-regression model were considered. And the least square estimators P,3(N), j = 1, - − - -, p, from N data points, of the autoregression constants for the stationary autoregression model are considered.
Abstract: The least squares estimators P,3(N), j = 1, - - -, p, from N data points, of the autoregressive constants for a stationary autoregressive model are considered
TL;DR: In this article, a rigorous treatment is given for a construction via Markov chains of a binary (0, 1) stationary homogeneous Markov random field on Z × Z.
Abstract: A rigorous treatment is given for a construction via Markov chains of a binary (0–1) stationary homogeneous Markov random field on Z × Z. The resulting process possesses rather interesting properties. For example, its correlation structure is geometric and it may be easily simulated. Some of the results are rather unintuitive — indeed counter-intuitive — but their demonstration is straightforward involving only the most elementary properties of Markov chains.
TL;DR: In this article, the moments of the delay distribution and other measures of performance for a multi-channel queue are bounded by corresponding corresponding corresponding moments of delay distribution for a single channel queue.
Abstract: Moments of the delay distribution and other measures of performance for a multi-channel queue are shown to be bounded above by corresponding
TL;DR: In this paper, an asymptotic evaluation of the probability that the Wiener path first crosses a square root boundary is given, which is applied to estimate the moments of the first-passage time distribution of the Ornstein-Uhlenbeck process to a constant boundary.
Abstract: This paper gives an asymptotic evaluation of the probability that the Wiener path first crosses a square root boundary. The result is applied to estimate the moments of the first-passage time distribution of the Ornstein–Uhlenbeck process to a constant boundary.
TL;DR: The scan statistic is defined as the supremum of a particular continuous-time stochastic process, and is used as a test statistic for testing uniformity against a simple clustering type of alternative.
Abstract: The scan statistic is defined as the supremum of a particular continuous-time stochastic process, and is used as a test statistic for testing uniformity against a simple clustering type of alternative. Its distribution under the null hypothesis is investigated and weak convergence of the stochastic process to the appropriate Gaussian process is proved. An interesting link is forged between the circular scan statistic and Kuiper's statistic, which rids us of the trouble of estimating a nuisance parameter. Distributions under the alternative are then derived, and asymptotic power comparisons are made. CLUSTERING; INVARIANCE; KUIPER'S STATISTIC; POISSON NOISE; POWER; RECTANGULAR SIGNAL; SCAN STATISTIC; SUPREMUM
TL;DR: In this article, the authors extended the work of Strauss (1975) on clustering in the two-colour case and compared it with the more general methods of Besag (1974).
Abstract: This paper is concerned with nearest-neighbour systems on the coloured lattice (unordered state space). It extends the paper of Strauss (1975) on clustering in the two-colour case. Comparison is made with the more general methods of Besag (1974). Some tests are developed, and illustrated with an example. NEAREST-NEIGHBOUR SYSTEM; MARKOV RANDOM FIELD; CLUSTERING; QUALITATIVE DATA
TL;DR: In this article, the central limit and iterated logarithm results for B n (S n −S 221E;) where the multipliers B n ↑ ∞ a.s.
Abstract: Let {S n , n ≧ 1} be a zero, mean square integrable martingale for which \(\lim _{n \to \infty } ES_n^2 < \infty \) so that S n →S 221E; a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for B n (S n −S 221E;) where the multipliers B n ↑ ∞ a.s. An example on the P olya urn scheme is given to illustrate the results.
TL;DR: In this paper, a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input is considered and the asymptotic distributions associated with Z and W are related in various ways.
Abstract: Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE ( S ) W = { W 1 , W 2 , · ··} and the server load (or virtual waiting-time process) Z = { Z ( t ), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takacs) relating the limiting virtual and actual waiting-time distributions of a GI / G /1 queue.
TL;DR: In this paper, the authors considered several schemes of epidemic process, e.g., when the infection is delivered according to arc direction, and proved that the probability of infecting all the n points with m = 1 is ∼ e/n, when n → ∞; another result is that m = o(√ n) cannot infect an essential part of the graph (having the size of O(n)).
Abstract: A random graph is a collection of n points and n directed arcs: a directed arc goes equiprobably from each point to one of (n – 1) other points. m points are initially ‘infected'. We consider several schemes of epidemic process, e.g. when the infection is delivered according to arc direction. We prove that the probability of infecting all the n points with m = 1 is ∼ e/n, when n → ∞; another result is that m = o(√ n) cannot infect an essential part of the graph (having the size of O(n)). Possible applications of the models to real world phenomena are briefly discussed.
TL;DR: In this paper, a simple technique is derived for finding the distribution of the distance from a fixed point, chosen independently of the process of figures, to the k th nearest figure.
Abstract: Consider a process of identically-shaped (but not necessarily equal-sized) figures (e.g. points, clusters of points, lines, spheres) embedded at random in n-dimensional space. A simple technique is derived for finding the distribution of the distance from a fixed point, chosen independently of the process of figures, to the k th nearest figure. The technique also shows that the distribution is independent of the distribution of the orientations of the figures. It is noted that the distribution obtained above (for equal-sized figures) is identical to the distribution of the distance from a fixed figure to the k th nearest of a random process of points.
TL;DR: In this article, the authors present a limiting result for the random variable Yn (r) which arises in a clustering model of Strauss (1975) and show that under some sparseness-of-points conditions the process converges weakly to a non-homogeneous Poisson process when n → ∞.
Abstract: In this article we present a limiting result for the random variable Yn (r) which arises in a clustering model of Strauss (1975) The result is that under some sparseness-of-points conditions the process {Yn (r): 0 ≦ r ≦ r ∞} converges weakly to a non-homogeneous Poisson process {Y(r): 0 ≦ r ≦ r ∞} when n → ∞ Simulation results are given to indicate the accuracy of the approximation when n is moderate and applications of the limiting result to tests for clustering are discussed
TL;DR: In the infinite-allele Wright model as mentioned in this paper, a diploid population of N individuals undergoing random mating and mutation is considered, and the distribution of the age of an allelic type now known to have frequency p, and its distribution of frequencies since the allele came into existence.
Abstract: Consider a diploid population of N individuals undergoing random mating and mutation as in the infinite-allele Wright model. Choose a particular gene in generation Q for large Q, and let f t (0 ≦ t ≦ Q) be the frequency of the allelic type of the predecessor of that particular gene (or the gene itself) in generation t. By considering a ‘diffusion approximation' {xt } of {ft }, we find the distribution of the age of an allelic type now known to have frequency p, and of its distribution of frequencies since the allele came into existence. A novelty here is that the process {xt } is not a diffusion, but a process with jumps; it has x = 0 as an inaccessible entrance boundary but periodically jumps to it from the interior of [0, 1]. The formulas obtained are the same as those derived by Maruyama and Kimura, who used a totally different approach.
TL;DR: In this paper, a limiting distribution for the age of a class of Markov processes is found if the present state of the process is known, and this distribution is used to find the ages of branching processes.
Abstract: A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.