Showing papers in "Journal of Applied Probability in 1983"
TL;DR: In this article, it was shown that the Hurst effect is present in weakly dependent random variables with a small monotonic trend of the form f(n) = c (m + n), where m is an arbitrary non-negative parameter and c is not 0.
Abstract: Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the form f(n) = c (m + n)", where m is an arbitrary non-negative parameter and c is not 0. For - 1/2 0 it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.
222 citations
TL;DR: In this article, a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process is presented.
Abstract: We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.
182 citations
TL;DR: In this paper, the authors define and analyze a general shock model associated with a correlated pair (Xn, Yn ) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level.
Abstract: In this paper we define and analyze a general shock model associated with a correlated pair (Xn, Yn ) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level. Two models, depending on whether the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length Yn of the subsequent interval until the next shock, are considered. The transform results, an exponential limit theorem, and properties of the associated renewal process of the failure times are obtained. An application in a stochastic clearing system with numerical results is also given.
165 citations
TL;DR: For a stochastic epidemic of the type considered by Bailey and Kendall as mentioned in this paper, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.
Abstract: : For a stochastic epidemic of the type considered by Bailey (1) and Kendall (3), Daniels (2) showed that 'when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels' assumption that the original number of infectives is 'small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description. (Author)
149 citations
TL;DR: In this article, the authors give conditions under which the stationary distribution of a Markov chain admits moments of the general form ff(x)ir(dx), where f is a general function.
Abstract: We give conditions under which the stationary distribution ir of a Markov chain admits moments of the general form ff(x)ir(dx), where f is a general function; specific examples include f(x)=x' and f(x)=es". In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, c0).
149 citations
TL;DR: The class of discrete life distributions for which for 0 ≦ p ≦ 1, where, is also studied in this paper, showing that this class is larger than the HNBUE (HNWUE) class.
Abstract: The class of life distributions for which , where , and , is studied. We prove that this class is larger than the HNBUE (HNWUE) class (consisting of those life distributions for which for x ≧ 0) and present results concerning closure properties under some usual reliability operations. We also study some shock models and a certain cumulative damage model. The class of discrete life distributions for which for 0 ≦ p ≦ 1, where , is also studied.
103 citations
TL;DR: In this paper, the authors prove that performance measures such as expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.
Abstract: Convexity of performance measures of queueing systems is important in solving control problems of multi-facility systems. This note proves that performance measures such as the expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.
90 citations
TL;DR: In this paper, the authors show that the expected number in an M/M/c queue is convex with respect to the traffic intensity, by expressing the second derivative of the expected queue size as the sum of non-negative terms.
Abstract: In this paper, we show that the expected number in an M/M/c queue is convex with respect to the traffic intensity. The proof is conducted by expressing the second derivative of the expected queue size as the sum of non-negative terms.
85 citations
TL;DR: In this article, the Weibull distribution for the strength of a long fiber is reviewed, with emphasis on its derivation via extreme value theory, and it is shown that a corresponding distribution, but with different scale and shape parameters, is appropriate for the lower tail distribution of a bundle of parallel fibers under general load-sharing assumptions.
Abstract: The Weibull distribution for the strength of a long fiber is reviewed, with emphasis on its derivation via extreme-value theory. It is shown that a corresponding Weibull distribution, but with different scale and shape parameters, is appropriate for the lower tail distribution of a bundle of parallel fibers under general load-sharing assumptions. This result is applied to series-parallel systems. The paper contains discussion of the application of these results to the strength of fibrous materials. It is concluded that the Weibull approximation, developed for series-parallel systems, is appropriate for long thin materials consisting of a small number of parallel fibers.
76 citations
TL;DR: An asymptotic convolution property for the generalized inverse Gaussian distribution with λ < 0 is proved in this paper, which is applied to calculate the probability of ruin in the general risk model when these distributions are used to model claim sizes.
Abstract: An asymptotic convolution property for the generalized inverse Gaussian distribution with λ < 0 is proved. This result is applied to calculate the probability of ruin in the general risk model when these distributions are used to model claim sizes. Some related applications are discussed.
74 citations
TL;DR: In this article, a discrete-time network with multiple transitions is defined, characterized by the independence of the service processes at different nodes and shown to have the familiar quasi-reversibility properties.
Abstract: A discrete-time network with multiple transitions is defined. It is characterized by the independence of the service processes at different nodes. That network is shown to have the familiar quasi-reversibility properties.
TL;DR: In this paper, a biological population with local random mating, migration, and mutation is studied, and an expression for the equilibrium probability of genetic relatedness between any two individuals as a function of their clustering distance is given.
Abstract: A biological population with local random mating, migration, and mutation is studied that exhibits clustering at several different levels. The migration is determined by the clustering rather than actual geographic or physical distance. Darwinian selection is assumed to be absent, and population densities are such that nearby individuals have a probability of being related. An expression is found for the equilibrium probability of genetic relatedness between any two individuals as a function of their clustering distance. Asymptotics for a small mutation rate u are discussed for both a finite number of clustering levels (and of total population size), and for an infinite number of levels. A natural example is discussed in which the probability of heterozygosity varies as u to a power times a periodic function of log(1/u).
TL;DR: In this paper, a general result for queueing systems with retrials is presented, which relates the expected number of retrials conducted by an arbitrary customer to the expected total number of retrial customers that take place during an arbitrary service time.
Abstract: A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.
TL;DR: In this paper, it was shown that the ratio of the return on investment under optimal proportional betting to the return of constant betting converges to an exponential distribution with mean 2 as the advantage tends to 0.
Abstract: Suppose you repeatedly play a game of chance in which you have the advantage. Your return on investment is your net gain divided by the total amount that you have bet. It is shown that the ratio of your return on investment under optimal proportional betting to your return on investment under constant betting converges to an exponential distribution with mean 2 as your advantage tends to 0. The case of non-optimal proportional betting is also treated.
TL;DR: Binomial moments of the time-dependent and limiting distributions of inventory deficit are derived for a continuous-review (s, S) inventory system under the assumption that interarrival times, demand sizes, and lead times are independent sequences of independent, identically distributed random variables.
Abstract: Binomial moments of the time-dependent and limiting distributions of inventory deficit are derived for a continuous-review (s, S) inventory system under the assumption that interarrival times, demand sizes, and lead times are independent sequences of independent, identically distributed random variables. Explicit expressions for the limiting distribution are also given in some special cases of practical interest. The approach used follows that of Finch [2] who investigated the system under the additional assumption of unit demands. INVENTORY THEORY
TL;DR: In this article, a formula for the expected number of clumps minus enclosed voids is derived for bounded laminae homeomorphic to a closed disc with isotropic random direction.
Abstract: When two-dimensional figures, called laminae, are randomly placed on a plane domains result that can either be aggregates or individual laminae. The intersection of the union, U, of these domains with a specified field of view, F, in the plane is considered. The separate elements of the intersection are called clumps; they may be laminae, aggregates or partial laminae and aggregates. A formula is derived for the expected number of clumps minus enclosed voids. For bounded laminae homeomorphic to a closed disc with isotropic random direction the formula contains only their mean area and mean perimeter, the area and perimeter of F, and the intensity of the Poisson process.
TL;DR: In this article, the authors give an elementary proof of Orey's theorem in the recurrent case, and establish rate results for tendency towards equilibrium under moment conditions on the speed measure and the initial distributions.
Abstract: The coupling method is well fitted to be used in the study of the asymptotics of one-dimensional diffusion processes. We give an elementary proof of Orey's theorem in the recurrent case, and establish rate results for tendency towards equilibrium under moment conditions on the speed measure and the initial distributions.
TL;DR: In this paper, the so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process, which is the same as in this paper.
Abstract: In this paper we deal with the Wiener–Hermite expansion of a process generated by an Ito stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.
TL;DR: In this article, the authors prove a threshold theorem for the Reed-Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic.
Abstract: We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.
TL;DR: In this paper, the expected response time of a job that requires processing time t and meets n jobs on arrival in the MIGI1 processor-sharing system is derived, where t is the number of jobs that need to be processed.
Abstract: The expected response time of a job that requires processing time t and meets n jobs on arrival in the MIGI1 processor-sharing system is derived. JOB; PROCESSING TIME; OPERATING SYSTEM; RESPONSE TIME
TL;DR: In this paper, the authors show that there is no a priori reason for supposing that there are no more than one set of ARMA model parameters minimising the one-step-ahead prediction error when the true system is not in the model set.
Abstract: In this paper we answer the following question. Is there any a priori reason for supposing that there is no more than one set of ARMA model parameters minimising the one-step-ahead prediction error when the true system is not in the model set?
TL;DR: In this article, a simple proof of Norton's theorem for multiclass quasireversible networks is proposed, and the proof is proved in terms of quasI-reversibility.
Abstract: We propose a simple proof of Norton's theorem for multiclass quasireversible networks. QUASI-REVERSIBILITY
TL;DR: For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customerstationary characteristics of the number of customers in the system such as idle or loss probabilities, it follows that the arrival process is Poisson as discussed by the authors.
Abstract: For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.
TL;DR: In this paper, the order statistics of a set of independent gamma variables, in general not identically distributed, may serve as a basis for ordering players in a hypothetical game and lower and upper bounds are given for the probability that the sample variances occur in their expected order.
Abstract: The order statistics of a set of independent gamma variables, in general not identically distributed, may serve as a basis for ordering players in a hypothetical game. An alternative formulation in terms of negative binomial variables leads to an expression for the probability that the random gammas are in a given order. This expression may contain rather many terms and some approximations are discussed - firstly as the gamma parameters a, tend to equality with all n, the same, and secondly when the probability of an inversion is small. In another interpretation the probabilities discussed arise in the statement of confidence limits for the ratios of population variances, and here the inversion probability is small enough usually that lower and upper bounds may be given for the probability that the sample variances occur in their expected order. These bounds are calculated from the probability that two variables are in expected order, and for gamma variables this probability is obtained from the F-distribution.
TL;DR: In this paper, it was shown that Zn converges to ∞ almost surely on a set of positive probability, and there are no constants cn such that n/cn converges in probability to a non-degenerate limit.
Abstract: The process we consider is a binary splitting, where the probability of division, , depends on the population size, 2i. We show that Zn converges to ∞ almost surely on a set of positive probability. Zn /n converges in distribution to a proper limit, diverges almost surely on converges almost surely on and there are no constants cn such that Zn /cn converges in probability to a non-degenerate limit.
TL;DR: It is demonstrated that the mean strategy of a diploid sexual population at evolutionary equilibrium can be expected to be an evolutionarily stable strategy (Ess) in the formal sense.
Abstract: A simple argument demonstrates that the mean strategy of a diploid sexual population at evolutionary equilibrium can be expected to be an evolutionarily stable strategy (Ess) in the formal sense. This result follows under a wide set of models of genetic inheritance of strategy (including sexual selection) provided
TL;DR: In this article, a procedure is indicated to estimate first-passage-time p.d. through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations.
Abstract: A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.