Showing papers in "Journal of Applied Probability in 1997"
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TL;DR: In this article, large deviation results for the random sum S(t)=N~ X,_ t > 0, where (N(t)),?o are nonnegative integer-valued random variables and (X,),,N are i.i.d. non-negative random variables with common distribution function F.
Abstract: We prove large deviation results for the random sum S(t)=N~ X,_ t > 0, where (N(t)),?o are non-negative integer-valued random variables and (X,),,N are i.i.d. non-negative random variables with common distribution function F, independent of (N(t)),>o. Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.
156 citations
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TL;DR: In this paper, an explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived, which is used to obtain approximations to the crossing probabilities for general boundaries.
Abstract: An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.
135 citations
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TL;DR: In this paper, the authors consider an M/G/1 queue with Bernoulli vacations and server breakdowns and obtain a transient solution for both queueing and reliability measures of interest.
Abstract: This note introduces reliability issues to the analysis of queueing systems. We consider an M/G/1 queue with Bernoulli vacations and server breakdowns. The server uptimes are assumed to be exponential, and the server repair times are arbitrarily distributed. Using a supplementary variable method we obtain a transient solution for both queueing and reliability measures of interest. These results provide insight into the effect of server breakdowns and repairs on system performance.
96 citations
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TL;DR: In this article, the M/G/1 queue with repeated requests is studied and the performance characteristics can be expressed in terms of hypergeometric functions, where the service time distribution is exponential.
Abstract: Queueing systems with repeated requests have many useful applications in communications and computer systems modeling. In the majority of previous work the repeat requests are made individually by each unsatisfied customer. However, there is in the literature another type of queueing situation, in which the time between two successive repeated attempts is independent of the number of customers applying for service. This paper deals with the M/G/1 queue with repeated orders in its most general setting, allowing the simultaneous presence of both types of repeat requests. We first study the steady state distribution and the partial generating functions. When the service time distribution is exponential we show that the performance characteristics can be expressed in terms of hypergeometric functions.
88 citations
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TL;DR: In this paper, a general resolvent approach is used to derive occupation probabilities and high-order moments for an M/M/1 queue subject to mass exodus at rate β and mass immigration at rate {α r; r ≥ 1} when idle.
Abstract: An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate {αr; r≥ 1} when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.
85 citations
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TL;DR: In this paper, the applicability of long-range dependence has been defined in terms of covariance properties relevant only to second-order stationary processes, which are useful for processes which may not be secondorder stationary, or indeed have infinite variances.
Abstract: Long-range dependence has usually been defined in terms of covariance properties relevant only to second-order stationary processes. Here we provide new definitions, almost equivalent to the original ones in that domain of applicability, which are useful for processes which may not be second-order stationary, or indeed have infinite variances. The ready applicability of this formulation for categorizing the behaviour for various infinite variance models is shown.
78 citations
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TL;DR: In this paper, the authors show that Braess's paradox can occur in loss networks, where the addition of an extra link leads to worse performance than the deletion of a link.
Abstract: Braess's paradox is said to occur in a network if the addition of an extra link leads to worse performance. It has been shown to occur in transportation networks (such as road networks) and also in queueing networks. Here, we show that it can occur in loss networks.
72 citations
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TL;DR: In this paper, the authors proved strong convergence of the proportions U n / T n of balls in a multitype generalized polya urn model, using martingale arguments, and characterized the limit as a convex combination of left dominant eigenvectors of the replacement matrix R with random Dirichlet coefficients.
Abstract: We prove strong convergence of the proportions U n / T n of balls in a multitype generalized Polya urn model, using martingale arguments. The limit is characterized as a convex combination of left dominant eigenvectors of the replacement matrix R, with random Dirichlet coefficients.
62 citations
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TL;DR: In this paper, an alternative proof of a point-process version of the FKG-Holley-Preston inequality is given, which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.
Abstract: We give an alternative proof of a point-process version of the FKG-Holley-Preston inequality which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.
61 citations
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TL;DR: In this article, a Volterra integral equation of the second kind, including two arbitrary continuous functions, was used to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries.
Abstract: In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutierrez et al. [4], [5].
61 citations
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TL;DR: In this paper, two notions of stochastic comparisons of nonnegative random variables via ratios that are determined by their Laplace transforms are studied, and various properties of them are derived.
Abstract: The purpose of this paper is to study two notions of stochastic comparisons of nonnegative random variables via ratios that are determined by their Laplace transforms. Some interpretations of the new orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. Finally, some applications in reliability theory are described.
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TL;DR: In this paper, the authors provided a more elementary proof of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a).
Abstract: This paper consists of two parts. The first part provides a more elementary proof of the asymptotic theorem of the refusals stream for an M/GI/1/n queueing system discussed in Abramov (1991a). The central property of the refusals stream discussed in the second part of this paper is that, if the expectations of interarrival and service time of an M/GI/1/n queueing system are equal to each other, then the expectation of the number of refusals during a busy period is equal to 1. This property is extended for a wide family of single-server queueing systems with refusals including, for example, queueing systems with bounded waiting time.
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TL;DR: In this paper, an extended class of time-continuous branching processes, motivated by the study of stochastic control theory and interacting particle systems, are studied and the uniqueness, extinction, recurrence and positive recurrence criteria for the processes are presented.
Abstract: This paper is devoted to studying an extended class of time-continuous branching processes, motivated by the study of stochastic control theory and interacting particle systems. The uniqueness, extinction, recurrence and positive recurrence criteria for the processes are presented. The main new point in our proofs is the use of several different comparison methods. The resulting picture shows that the methods are effective and hence should also be meaningful in other situations.
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TL;DR: In this paper, the authors obtained some stochastic comparison results involving min {X 1, X 2,…, XN ] and max{X 1, X 2,…, XN }.
Abstract: Let X 1, X 2,… be a sequence of independent random variables and let N be a positive integer-valued random variable which is independent of the Xi. In this paper we obtain some stochastic comparison results involving min {X 1, X 2,…, XN ) and max{X 1, X 2,…, XN }.
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TL;DR: In this paper, the authors assume that the state of a system forms a continuous-time Markov chain or a higher-dimensional Markov process after introducing some supplementary variables, and derive a formula for evaluating the rate of occurrence of failures for the system.
Abstract: In this article, we assume that the state of a system forms a continuous-time Markov chain or a higher-dimensional Markov process after introducing some supplementary variables. A formula for evaluating the rate of occurrence of failures for the system is derived. As an application of the theory, a maintenance model for a two-component system is also studied.
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TL;DR: In this paper, the authors considered a single-server queue with Poisson arrivals and multiple customer feedbacks, and studied the joint queue length process of new and old customers, as well as the waiting time distribution of customers.
Abstract: This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution He keeps trying in this manner (as an `old' customer) until his service is successful The server operates according to the `gated vacation' strategy: when it returns from a vacation to find $K$ (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers Some extensions are also discussed
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TL;DR: In this article, the authors considered queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain and studied the effect that the transition matrix has on the waiting time of the n th customer as well as on the stationary waiting time.
Abstract: This paper considers queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain. We study the effect that the transition matrix has on the waiting time of the n th customer as well as on the stationary waiting time. The main theorem generalizes results of Szekli et al. (1994a) and partly confirms their conjecture. In this context we show the importance of a new stochastic ordering concept.
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TL;DR: It is proved that for a k-out-of-n system where components and spares have independent and identical life distributions active spare allocation at the component level is superior to active spare allocations at the system level in likelihood ratio ordering, which is stronger than hazard rate ordering.
Abstract: Design engineers are well aware that a system where active spare allocation is made at the component level has a lifetime stochastically larger than the corresponding system where active spare allocation is made at the system level. In view of the importance of hazard rate ordering in reliability and survival analysis, Boland and El-Neweihi (1995) recently investigated this principle in hazard rate ordering and demonstrated that it does not hold in general. They showed that for a 2-out-of-n system with independent and identical components and spares, active spare allocation at the component level is superior to active spare allocation at the system level. They conjectured that such a principle holds in general for a k-out-of-n system when components and spares are independent and identical. We prove that for a k-out-of-n system where components and spares have independent and identical life distributions active spare allocation at the component level is superior to active spare allocation at the system level in likelihood ratio ordering. This is stronger than hazard rate ordering, thus establishing the conjecture of Boland and El-Neweihi (1995). ACTIVE SPARE ALLOCATION; HAZARD RATE ORDERING; ORDER STATISTICS; LIKELIHOOD RATIO ORDERING AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 62N05 SECONDARY 90B25
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TL;DR: In this article, the joint large deviation properties of traffic streams are altered when the traffic passes through a shared buffer according to a FCFS service policy with stochastic service capacity.
Abstract: In this paper we describe how the joint large deviation properties of traffic streams are altered when the traffic passes through a shared buffer according to a FCFS service policy with stochastic service capacity. We also consider the stationary case, proving large deviation principles for the state of the system in equilibrium and for departures from an equilibrium system. IThis is a revised version of 'Large deviations in queueing networks', DIAS Technical Report DIAS-APG-9413, and has been submitted to J. Appl. Prob.
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TL;DR: In this paper, a generalization of the house-selling problem to selling k houses is considered, where the offers, X1, X2, ⋯, are independent, identically distributed k-dimensional random vectors having a known distribution with finite second moments.
Abstract: We consider a generalization of the house-selling problem to selling k houses. Let the offers, X1, X2, ⋯, be independent, identically distributed k-dimensional random vectors having a known distribution with finite second moments. The decision maker is to choose simultaneously k stopping rules, N1, ⋯, Nk, one for each component. The payoff is the sum over j of the jth component of XN, minus a constant cost per observation until all stopping rules have stopped. Simple descriptions of the optimal rules are found. Extension is made to problems with recall of past offers and to problems with a discount.
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TL;DR: In this article, the extremal index is computed and any value in (0, 1) is possible for a class of shot noise processes with heavy tailed amplitudes, and the joint limiting distribution of high local maxima is obtained.
Abstract: Extreme value results for a class of shot noise processes with heavy tailed amplitudes are considered. For a process of the form, , where { τ k } are the points of a renewal process and { A k } are i.i.d. with d.f. having a regularly varying tail, the limiting behavior of the maximum is determined. The extremal index is computed and any value in (0, 1) is possible. Two-dimensional point processes of the form are shown to converge to a compound Poisson point process limit. As a corollary to this result, the joint limiting distribution of high local maxima is obtained.
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TL;DR: In this paper, it was shown that for a certain storage network the backward content process is increasing and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments.
Abstract: We show that for a certain storage network the backward content process is increasing, and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments. Moreover, for the feedforward case, we show that under some minimal conditions, two content processes with net input processes which differ only by initial conditions can be coupled in finite time and that the difference of two content processes vanishes in the limit if the difference of the net input processes monotonically approaches a constant. As a consequence, it is shown that for the natural stability conditions, when the net input process has stationary increments, the distribution of the content process converges in total variation to a proper limit, independent of initial conditions.
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TL;DR: In this paper, the authors consider a continuous polling system in heavy traffic and show that the steady-state number of waiting customers has approximately a gamma distribution, given their total number, the configuration of these customers is approximately deterministic.
Abstract: We consider a continuous polling system in heavy traffic. Using the relationship between such systems and age-dependent branching processes, we show that the steady-state number of waiting customers in heavy traffic has approximately a gamma distribution. Moreover, given their total number, the configuration of these customers is approximately deterministic.
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TL;DR: In this paper, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected value of the sum of all elements with first digit d is approximately constant.
Abstract: The accountant Nigrini remarked that in tables of data distributed according to Benford's law, the sum of all elements with first digit d (d = 1, 2,..., 9) is approximately constant. In this note, a mathematical formulation of Nigrini's observation is given and it is shown that Benford's law is the unique probability distribution such that the expected
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TL;DR: In this paper, the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value is discussed.
Abstract: Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.
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TL;DR: In this paper, upper and lower bounds for the tail probabilities of compound distributions using simple properties of the claim size distribution were derived, and general bounds for various classes of claim size distributions were obtained.
Abstract: Upper and lower bounds are derived for the tail probabilities of compound distributions using simple properties of the claim size distribution. General bounds are then obtained for various classes of claim size distributions. Some examples are given.