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Showing papers in "Journal of Applied Probability in 1980"


Journal ArticleDOI
TL;DR: In this paper, three types of coherence based on the strength of the relevancy axiom are studied, and the effect of component improvement on system performance is studied using a generalization of Birnbaum's reliability importance.
Abstract: In this paper an axiomatic development of multistate systems is presented. Three types of coherence based on the strength of the relevancy axiom are studied. The strongest of these has been investigated previously by El-Neweihi, Proschan, and Sethuraman [3]. One of the weaker types of coherence permits wider applicability to real life situations without sacrificing any of the mathematical results obtained by El-Neweihi, Proschan, and Sethuraman. The concept of system performance is formalized through expected utility and the effect of component improvement on system performance is studied using a generalization of Birnbaum's reliability importance. RELIABILITY; MULTISTATE COMPONENTS; MULTISTATE COHERENT SYSTEMS; COMPONENT IMPORTANCE

315 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider closed networks of interconnected service centers with multiple types of customers and multiple classes, whose stationary state probabilities at arbitrary times have a product form, where a customer can change its class but not its type as it traverses the network.
Abstract: We consider closed networks of interconnected service centers with multiple types of customers and multiple classes, whose stationary state probabilities at arbitrary times have a product form. A customer can change its class but not its type as it traverses the network. We show that the stationary state probabilities at instants at which customers of a particular type arrive at a particular service center and enter a particular class are equal to the stationary state probabilities at arbitrary times for the network with one less customer of that type. Applications of this result are given.

169 citations


Journal ArticleDOI
TL;DR: The policies of assigning at every moment the task with shortest (longest) expected processing time among those not yet completed to the fastest processor available, 2nd shortest to the 2nd fastest etc., are examined, and shown to minimize expected values of various cost functions.
Abstract: : We consider preemptive scheduling of N tasks on m processors; processors have different speeds, tasks require amounts of work which are exponentially distributed, with different parameters. The policies of assigning at every moment the task with shortest (longest) expected processing time among those not yet completed to the fastest processor available, 2nd shortest (longest) to the 2nd fastest etc., are examined, and shown to minimize expected values of various cost functions. As special cases we obtain policies which minimize expected flowtime, expected makespan and expected lifetime of a series system with m component locations and N spares. (Author)

145 citations


Journal ArticleDOI
TL;DR: In this article, the main version of UCSO is shown to be equivalent to the MLR (monotone likelihood ratio) property when densities or probability mass functions exist on the real line.
Abstract: One probability measure is less than or equal to another in the sense of UCSO (uniform conditional stochastic order) if a standard form of stochastic order holds for each pair of conditional probability measures obtained by conditioning on appropriate subsets. UCSO can be applied to the comparison of lifetime distributions or the comparison of decisions under uncertainty when there may be reductions in the set of possible outcomes. When densities or probability mass functions exist on the real line, then the main version of UCSO is shown to be equivalent to the MLR (monotone likelihood ratio) property. UCSO is shown to be preserved by some standard probability operations and not by others.

125 citations



Journal ArticleDOI
TL;DR: For the distribution function of the busy period in the M/G/1 queueing system with traffic intensity less than one, it was shown that the tail varies regularly at infinity.
Abstract: For the distribution function of the busy period in the M/G/1 queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.

101 citations



Journal ArticleDOI
TL;DR: In this article, a linear process is generated by applying a linear filter to independent, identically distributed random variables, and only the modulus of the frequency response function can be estimated if only the linear processes is observed and if the independent identically-distributed random variables are Gaussian.
Abstract: A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

75 citations


Journal ArticleDOI
TL;DR: In this article, the transition matrix of a finite discrete Markov chain is assembled from overlays of matrices which are defined only by the serial correlation coefficients and marginal distribution of the chain to be modelled.
Abstract: By assembling the transition matrix of a finite discrete Markov chain from overlays of matrices which are defined only by the serial correlation coefficients and marginal distribution of the chain to be modelled, a considerable saving is made in the number of parameters required to define a multilag Markov chain. This parsimony is achieved without detriment to the marginal distribution or serial correlation structure of the modelled chain. Applications to daily precipitation sequences and reservoir reliability are outlined to demonstrate the model's versatility.

74 citations


Journal ArticleDOI
TL;DR: In this paper, a class of increasing processes with independent increments that reduce to gamma processes under the further condition of stationarity is introduced, and each such process can be reduced to a simple gamma process by a stochastic integral transformation.
Abstract: We introduce a class of increasing processes with independent increments that reduce to gamma processes under the further condition of stationarity. Each such process can be reduced to a simple gamma process by a stochastic integral transformation. Applications to deformation laws of materials such as concrete are mentioned.

70 citations



Journal ArticleDOI
TL;DR: Investigations of stability show that the mean strategy in use in the population can prove stable but that neutrally stable and gradually changing polymorphisms in strategy are to be expected when the contest affects reproductive success.
Abstract: In biological populations, several strategies for engaging in intraspecific contests may be present. Investigations of stability show that the mean strategy in use in the population can prove stable but that neutrally stable and gradually changing polymorphisms in strategy are to be expected when the contest affects reproductive success.

Journal ArticleDOI
TL;DR: In this paper, the authors consider an N-server queue with arbitrary arrivals and service times which are random but with differing rates for different servers, and show that the policy of always assigning an arrival to that free server whose service rate is largest (smallest) stochastically minimises (maximises) the number in the system.
Abstract: We consider an N-server queue with arbitrary arrivals and service times which are random but with differing rates for different servers. Customers arriving when all servers are occupied do not enter the system. We show that the policy of always assigning an arrival to that free server whose service rate is largest (smallest) stochastically minimises (maximises) the number in the system. We then show that in a particular component-repair context with exponential repair times the policy of repairing failed components with the smallest failure rate stochastically maximises the number of working components.

Journal ArticleDOI
TL;DR: In this article, it was shown that in a supercritical Galton-Watson process which does not become extinct, there cannot be a sequence {τ n } of particles, each descended from the preceding one, such that the fraction of all particles which are descendants of τ n does not converge to zero as n →∞.
Abstract: Let Yn be the maximum of n independent positive random variables with common distribution function F and let Sn be their sum. Then converges to zero in probability if and only if is slowly varying. This result implies that in a supercritical Galton-Watson process which does not become extinct, there cannot be a sequence {τ n } of particles, each descended from the preceding one, such that the fraction of all particles which are descendants of τ n does not converge to zero as n →∞. Weakly m-adic trees, which behave to some extent like sample Galton-Watson trees, can have such sequences of particles.

Journal ArticleDOI
TL;DR: In this article, a multigrade population with semi-Markov transitions between grades, Poisson arrivals to each grade, and departures from each grade is considered and the joint distribution of the numbers in each grade at any time is found, and the limiting distributions shown to be independently Poisson.
Abstract: We consider a multigrade population with semi-Markov transitions between grades, Poisson arrivals to each grade, and departures from each grade. For this model the joint distribution of the numbers in each grade at any time is found, and the limiting distributions shown to be independently Poisson; this extends a previous result for a multigrade population with Markov transitions and Poisson recruitment. This model is particularly applicable to manpower planning. The inclusion of semi-Markov transitions allows us to take into account existing knowledge of the distribution of length of service until an individual leaves his firm.

Journal ArticleDOI
TL;DR: In this article, a Dirichlet-multinomial urn model for describing the above phenomena from a stochastic point of view is presented, under certain regularity conditions.
Abstract: Zipf's laws are probability distributions on the positive integers which decay algebraically. Such laws have been shown empirically to describe a large class of phenomena, including frequency of words usage, populations of cities, distributions of personal incomes, and distributions of biological genera and species, to mention only a few. In this paper we present a Dirichlet-multinomial urn model for describing the above phenomena from a stochastic point of view. We derive the Zipf's law under certain regularity conditions; some limit theorems are also obtained for the urn model under consideration.

Journal ArticleDOI
TL;DR: This paper considers the case in which the only memory allowed at any time is the ordering of the elements at that time and allows the decision-maker to utilize such rules as 'only make a change if the same element has been requested k times in a row.
Abstract: : The problem of interest is to determine the optimal ordering so as to minimize the long run average cost Clearly if the P sub i were known the optimal ordering would simply be to order the elements in decreasing order of the P sub i's In fact even if the P sub i's were unknown we could do as well asymptotically by ordering the elements at each unit of time in decreasing order of the number of previous requests for them In this paper we first consider the case in which the only memory allowed at any time is the ordering of the elements at that time; in other words, the only type of reordering rules we allow are ones in which the reordered permutation of elements at any time is only allowed to depend on the present ordering and the position of the element requested We also consider the above problem under the previse that additional memory is allowed In particular we allow the decision-maker to utilize such rules as 'only make a change (according to some preassigned rule) if the same element has been requested k times in a row' We show that as k approaches infinity we can do as well as if we knew the values of the P sub i, and in addition we show that the convergence is monotone



Journal ArticleDOI
TL;DR: In this paper, the authors used Veraverbeke's approach to explore another aspect of the asymptotics of Z+ and X+, viz., the existence of moments.
Abstract: A well-known result in the theory of random walks states that E{X2) is finite if and only if E{Z+} and E{Zj are both finite (Z, and Z_ being the ladder heights and X a typical step-length) in which case E{X2} = 2E{Z}E {Z }. This paper contains results relating the existence of moments of X of order P to the existence of the moments of Z+ and Z_ of order p - 1. The main result is that if p3 >2 E{JIXl} 0} and T_ = min{n 1: S, _ 0) iff Pr{X, x } is regularly varying of index - (a + 1) at oo. (When applied in the context of the GI/G/1 queue this yields a similar connection between the tails of the stationary waiting-time distribution and the service-time distribution, which is a result due to Cohen (1973), Theorem 1.) In this note we use Veraverbeke's approach to explore another aspect of the asymptotics of Z+ and X+, viz., the existence of moments. Our aim is to find extensions of the classical result that E (X2) is finite iff both E (Z,) and E(Z_) are finite. Our starting point is the distribution form of the Wiener-Hopf factorization

Journal ArticleDOI
TL;DR: In this paper, it was shown that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and servicetime distributions become more variable.
Abstract: In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and servicetime distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/GIs queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the n th customer. The stochastic comparisons also imply an ordering for the expected waiting times in MIGIs queues.

Journal ArticleDOI
TL;DR: In this article, the best known linear bounds in terms of S, and S2 were obtained for the probability of the occurrence of at least m out of n events, where S, = 1i P(A), and S 2 =
Abstract: Given n events A1, A2, " ? , A,, bounds are obtained for the probability of the occurrence of at least m out of the n events. The bounds are linear in terms of S, and S2 where S, = 1i P(A,) and S2 =

Journal ArticleDOI
TL;DR: In this article, a single device shock model is studied and sufficient conditions on the shocking process are found so that the life distribution will increase failure rate, under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases.
Abstract: A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate. SHOCK PROCESS; INCREASING FAILURE RATE; CUMULATIVE DAMAGE; LIFE DISTRIBUTIONS

Journal ArticleDOI
TL;DR: In this article, a simple version of the Borel-Cantelli lemma is proposed, where the independence condition is replaced by A,,, A, A+ C A+2 for some t E N. This property of (A) may frequently be assumed without loss of generality.
Abstract: The general part of the Borel-Cantelli lemma says that for any sequence of events (A,) defined on a probability space (fl, 1, P), the divergence of Y, P(A, ) is necessary for P(A, i.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the A, are independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced by A,,, A,+ C A+2 for some t E N. We will see that this property of (A,) may frequently be assumed without loss of generality. We also disclose a useful duality which allows straightforward conclusions without selecting independent sequences. A simple random walk example and a new result in the theory of ,-branching processes will show the tractability of the method. BOREL-CANTELLI LEMMA; RANDOM WALK; O-BRANCHING PROCESSES

Journal ArticleDOI
TL;DR: In this paper, the conditional distribution of a positive integervalued random variable with order statistics is shown to be the same as the distribution of an order-statistical random variable under mild conditions.
Abstract: Let X,, X2, ", X,, be independent identically distributed positive integervalued random variables with order statistics X,:,,, X2:, -, X:,. If the X,'s have a geometric distribution then the conditional distribution of X,k,,, X,:,, given Xk, ,,Xk:, >0 is the same as the distribution of Xl:,_k. Also the random variable X2:, X,:,, is independent of the event [X,:, = 1]. Under mild conditions each of these two properties characterizes the geometric distribution. CHARACTERIZATION; GEOMETRIC DISTRIBUTION; ORDER STATISTICS


Journal ArticleDOI
TL;DR: In this paper, the spectral density of a Gaussian stationary process was investigated and the asymptotic properties of its estimator were investigated, where the variance of the estimator was O(n−n −n −1, where n−1 is an appropriate known function.
Abstract: Let g(x) be the spectral density of a Gaussian stationary process. Then, for each continuous function 4u(x) we shall give an estimator of f$", (x)4t(g(x))dx whose asymptotic variance is O(n-), where 1((-) is an appropriate known function. Also we shall investigate the asymptotic properties of its estimator.

Journal ArticleDOI
TL;DR: In this paper, infinite server queues whose input is a Phase Type Renewal Process are discussed, and the problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations.
Abstract: : This paper discusses infinite server queues whose input is a Phase Type Renewal Process. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations are provided with as many as ten phases. The paper contains some useful explicit formulas and also discusses the interesting special case where the service time is also of phase type. The Phase Type Distributions include a wide variety of models such as generalized Erlang, hyperexponential (mixtures of a finite number of exponentials) as very special cases and possess great versatality in modeling a number of interesting qualitative features such as bimodality.

Journal ArticleDOI
TL;DR: Using these reults, two examples of controlled MIG/1 queueing systems are solved and simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer.
Abstract: Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled MIG/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples. QUEUEING SYSTEMS; CONTROL OF QUEUES; UP- AND DOWNCROSSINGS; REGENERATIVE PROCESSES

Journal ArticleDOI
TL;DR: In this article, it was shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable random variables.
Abstract: In this paper it will be shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable random variables.