Showing papers in "Journal of Applied Probability in 1985"
TL;DR: In this article, the NBUE aging property is shown to be inherited by the distribution of the time between successive complete repairs under suitable monotonicity of the function p. This is equivalent to assuming that maintenance is executed in negligible time.
Abstract: : The failure process studied in this paper models the following maintained system setting. A piece of equipment is put in operation at time t=0. Each time it fails, a maintenance action is taken which, with probability p(t), is a complete repair or, with probability q(t)=1-p(t), is a minimal repair, where t is the age at failure of the equipment under maintenance. Since availability results are not pursued in this paper, only operating time will be recorded. This is equivalent to assuming that maintenance is executed in negligible time. It is also assumed that complete repairs restore failed items to their good as new condition in such a way that the times between successive complete repairs are independent and identically distributed. The formal development of the model is given in an appendix where the basic facts are established. In section 2, we show that some aging properties of the equipment's life distribution are inherited by the distribution of the time between successive complete repairs under suitable monotonicity of the function p. A counterexample is also given to the conjecture of Brown and Proschan (1980) that the NBUE aging property is also inherited when the function p is constant. In Section 3, some inequalities and further properties of the model are developed which, as in Section 2, extend results obtained by Brown and Proschan.
412 citations
TL;DR: In this paper, an inequality concerning products of NBU probability measures is derived, which has as a consequence that if μ 1, μ 2, ···, μn are NBU probabilities on ℝ+, then the product-measure μ = μ × μ 2 × ··· × μn onℝn + is SNBU.
Abstract: A probability measure μ on ℝn + is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ 1, μ 2, ···, μn are NBU probability measures on ℝ+, then the product-measure μ = μ × μ 2 × ··· × μn on ℝn + is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds. Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.
338 citations
TL;DR: Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary which is computationally simple and exact in the limit as the boundary becomes increasingly remote.
Abstract: Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.
185 citations
TL;DR: In this paper, the model Zt = φ (0, k)+ φ(1, k)Zt −1 + at (k) was considered and the least squares estimators of the model parameters were derived and, under mild regularity conditions, were shown to be strongly consistent and asymptotically normal.
Abstract: We consider the model Zt = φ (0, k)+ φ(1, k)Zt –1 + at (k) whenever r k−1
174 citations
TL;DR: For a model of a rumour, first given a non-rigorous treatment by Maki and Thompson, it was shown that the proportion of the population never hearing the rumour converges in probability to 0.203 as the population size tends to oo.
Abstract: For a model of a rumour, first given a non-rigorous treatment by Maki and Thompson, it is shown that the proportion of the population never hearing the rumour converges in probability to 0.203 as the population size tends to oo. EPIDEMIC A model of a rumour appeared in Maki and Thompson (1973) and was reproduced in Frauenthal (1980). Both books recognised that the treatment they gave was not rigorous and that there were no estimates of errors. This short note shows how these problems may be overcome, chiefly by identifying some of the large number of martingales embedded in the process. Assume n + 1 villages of three types: (1) Susceptible (have not heard the rumour) (2) Infective (have heard it and wish to spread it) (3) Removed (have heard it and no longer wish to spread it). A telephone call is placed between two villages chosen at random. If an infective calls a susceptible the rumour is spread and the susceptible becomes an infective. If an infective calls an infective or a removed the caller loses interest and becomes a removed. All other calls have no effect. We show that the proportion of villages never hearing the rumour converges in probability to 0.203 as n-- oo. We say a time epoch occurs whenever a change takes place. This does not correspond to any normal use of time. Let sk, ik, and rk be the number of susceptibles, infectives and removed at epoch k. When ik > 0, P{sk+l, = Sk - 1, ik+i = i5k + 1, rk+1 = r } = s/n P{sk+, = Sk, i5k+ = ik - 1, rk+1 = rk + 1} = 1 - sin.
156 citations
TL;DR: In this paper, the authors prove stochastic decomposition results for variations of the GI/G/1 queue, where the server, when it becomes idle, goes on a vacation for a random length of time.
Abstract: In this note we prove some stochastic decomposition results for variations of the GI/G/1 queue. Our main model is a GI/G/1 queue in which the server, when it becomes idle, goes on a vacation for a random length of time. On return from vacation, if it finds customers waiting, then it starts serving the first customer in the queue. Otherwise it takes another vacation and so on. Under fairly general conditions the waiting time of an arbitrary customer, in steady state, is distributed as the sum of two independent random variables: one corresponding to the waiting time without vacations and the other to the stationary forward recurrence time of the vacation. This extends the decomposition result of Gelenbe and Iasnogorodski [5]. We use sample path arguments, which are also used to prove stochastic decomposition in a GI/G/1 queue with set-up time.
137 citations
TL;DR: In this article, the first-passage time of the Ornstein-Uhlenbeck process through a constant boundary is investigated for large boundaries, and it is shown that an exponential p.d. arises, whose mean is the average first-Passage time from 0 to the boundary.
Abstract: The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.
96 citations
TL;DR: A direct and general proof of the equivalence of partial balance and insensitivity is given in this paper, where the authors also give a proof for the existence of the inverse of partial imbalance.
Abstract: A direct and general proof is given of the equivalence of partial balance and insensitivity.
94 citations
TL;DR: In this paper, a rearrangement inequality is applied to solve an optimal permutation problem for consecutive-k-out-of-n: F networks, and its implications on a recent conjecture of Derman et al. are discussed.
Abstract: Let X l, · ··, Χ n be independent binary variables with parameters θ l, · ··, θ n respectively, and let R denote the length of the longest run of 1's. This note concerns a new expression for and a rearrangement inequality. The inequality is applied to solve an optimal permutation problem for consecutive-k-out-of-n: F networks, and its implications on a recent conjecture of Derman et al. (1982) are discussed.
93 citations
TL;DR: In this article, a Poisson stream of customers arrives at a service center which consists of two single-server queues in parallel, and the service times of the customers are exponentially distributed, and both servers serve at the same rate.
Abstract: A Poisson stream of customers arrives at a service center which consists of two single-server queues in parallel. The service times of the customers are exponentially distributed, and both servers serve at the same rate. Arriving customers join the shortest of the two queues, with ties broken in any plausible manner. No jockeying between the queues is allowed. Employing linear programming techniques, we calculate bounds for the probability distribution of the number of customers in the system, and its expected value in equilibrium. The bounds are asymptotically tight in heavy traffic.
77 citations
TL;DR: In this article, the authors considered a single-server queueing system with Poisson input and general service times, where a customer becomes a lost customer when his service has not begun within a fixed time after his arrival.
Abstract: A queueing situation often encountered in practice is that in which customers wait for service for a limited time only and leave the system if not served during that time. This paper considers a single-server queueing system with Poisson input and general service times, where a customer becomes a lost customer when his service has not begun within a fixed time after his arrival. For performance measures like the fraction of customers who are lost and the average delay in queue of a customer we obtain exact and approximate results that are useful for practical applications.
TL;DR: Some upper (lower) bounds on E max1,_ X, (E min_,1 Xi), where the X 's are real-valued random variables are given.
Abstract: In this paper, we give some upper (lower) bounds on E max1,_ X, (E min_,1 Xi), where the X 's are real-valued random variables. Some applications are given.
TL;DR: In this article, variability orderings are considered that are preserved under conditioning on a common subset, where the variability order is determined by the expectation of all convex functions and can be easily checked by checking if f(x)/g(x) is log-concave.
Abstract: Variability orderings indicate that one probability distribution is more spread out or dispersed than another. Here variability orderings are considered that are preserved under conditioning on a common subset. One density f on the real line is said to be less than or equal to another, g, in uniform conditional variability order (UCVO) if the ratio f(x)/g(x) is unimodal with the model yielding a supremum, but f and g are not stochastically ordered. Since the unimodality is preserved under scalar multiplication, the associated conditional densities are ordered either by UCVO or by ordinary stochastic order. If f and g have equal means, then UCVO implies the standard variability ordering determined by the expectation of all convex functions. The UCVO property often can be easily checked by seeing if f(x)/g(x) is log-concave. This is illustrated in a comparison of open and closed queueing network models.
TL;DR: For the queueing system M/G/oo some distributions connected with the associated busy periods are derived as discussed by the authors, where the distribution is based on the distribution of queues with unrestricted service.
Abstract: For the queueing system M/G/oo some distributions connected with the associated busy periods are derived. QUEUES WITH UNLIMITED SERVICE
TL;DR: In this article, records from the sequence Yj = Xj, + cj, j ≧ 1 where c > 0 if upper records are of interest and c < 0 if lower records are studied.
Abstract: We consider records from the sequence Yj = Xj, + cj, j ≧ 1 where c > 0 if upper records are of interest and c < 0 if lower records are studied. If {Xj } is stationary the record rate is asymptotically constant and if {Xj } is i.i.d. the record rate is, in addition, asymptotically normal. These results are illustrated by analysis of the times in the mile run.
TL;DR: In this article, the total gain in N successive Petersburg games is considered, and a limit theorem for SN/N-n when N = 2 n and n → ∞ is proved, which provides a simple rule of thumb for determining a premium for the game, which is safe from the point of view of the casino.
Abstract: The total gain, SN , in N successive Petersburg games is considered, and a limit theorem for SN/N – n when N = 2 n and n → ∞is proved. The limit distribution can be determined numerically with good accuracy, and this fact provides a simple rule of thumb for determining a premium for the game, which is safe from the point of view of the casino.
TL;DR: In this paper, the authors studied the asymptotic behavior of the first-passage-time p.f. through a constant boundary when the boundary approaches the endpoints of the diffusion interval.
Abstract: The asymptotic behavior of the first-passage-time p.d.f. through a constant boundary is studied when the boundary approaches the endpoints of the diffusion interval. We show that for a class of diffusion processes possessing a steady-state distribution this p.d.f. is approximately exponential, the mean being the average first-passage time to the boundary. The proof is based on suitable recursive expressions for the moments of the first-passage time. PASSAGE-TIME MOMENTS
TL;DR: In this paper, the intensity conservation laws for stationary processes with jump points are derived for queues with randomly changed service rate. But they are not applicable to queues with queue invariance relations.
Abstract: K6nig et al. (1978) have derived the so-called intensity conservation law in a stationary process connected with a marked point process (PMP). That law has been shown to be useful in obtaining invariance relations in queues (cf. Franken et al. (1981)). In this paper, somewhat different versions of the intensity conservation laws are derived for a stationary process with jump points. These laws are applied to queues with randomly changed service rate. As special cases, most of equations obtained by K6nig et al.'s law can be derived from this law. Also, we derive some inequalities between characteristic quantities in a queue with a simple type of randomly changed service rate. STATIONARY PROCESS WITH JUMP POINTS; POINT PROCESS; QUEUE WITH RANDOM SERVICE RATE
TL;DR: In this article, the authors generalized discrete renewal theory to study the occurrence of a collection of patterns in random sequences, where a renewal is defined to be an occurrence of one of the patterns in the collection which does not overlap an earlier renewal.
Abstract: Discrete renewal theory is generalized to study the occurrence of a collection of patterns in random sequences, where a renewal is defined to be the occurrence of one of the patterns in the collection which does not overlap an earlier renewal. The action of restriction enzymes on DNA sequences provided motivation for this work. Related results of Guibas and Odlyzko are discussed.
TL;DR: In this paper, an asymptotic formula for the tail distribution of the maximum for a class of Gaussian processes with stationary increments is given, where the incremental variance function o2(t) is convex.
Abstract: Let X(t), t 0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function o2(t) is convex and r2(t) = o(t) for t -0, then P(maxIojX(s)> u)- P(X(t)> u) for u ---0 and each t >0. SAMPLE FUNCTION MAXIMUM 1. Introduction and summary The main result of this paper is an asymptotic formula for the tail of the distribution of the maximum for a class of Gaussian processes X(t) with stationary increments. We assume that X(0) = 0 almost surely and EX(t) = 0 for t > 0; and put o2(t)= EX2(t). If o-2(t)= t, then X(t) is the standard Brownian motion, and the classical result of L6vy states that P(maxto. ,X(s)> u)= 2P(X(t) > u) for all u > 0 and t >0. If o2(t)= t2, then the process is of the trivial form X(t)= (t, where ( is a standard normal random variable, and so P(maxro,,tX(s)> u)= P(X(t)> u). These results suggest a similar question about the process with o-2(t)= t", for some 1 u)---P(X(t)>u), for u -> o, for every t > 0. The main result states that the distribution of the maximum is asymptotically equivalent to the distribution of the random variable observed at the terminal time. This signifies that the latter random variable is likely to be the largest in
TL;DR: In this article, the authors studied the problem of what to do with a slow server in a service facility which has fast and slow servers, and whether it is better to use the slow server at all or not to use it at all.
Abstract: The problem is what to do with a slow server in a service facility which has fast and slow servers. Should the slow server be used to render service, or is it better not to use it at all? Simple models for answering this question are formulated and studied.
TL;DR: In this article, a population-size-dependent Galton-Watson process with transition probability generating functions is treated by means of functional iteration methods and criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t--oc.
Abstract: Some classes of population-size-dependent Galton-Watson processes {Z(t)}, .=,,..., whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t--oc; the results can be stated under conditions on moments of the reproduction distributions.
TL;DR: In this article, the authors consider a particle aggregate in R and derive estimators of the qth moment of the volumeweighted distribution of particle volume, based on point-sampling of particles and measurements on q-flats through sampled particles.
Abstract: In stereology or applied geometric probability quantitative characterization of aggregates of particles from information on lower-dimensional sections plays a major role. Most stereological methods developed for particle aggregates are based on the assumption that the particles are of the same, known (simple) shape. Information on the volume-weighted distribution of particle size may, however, be obtained under fairly general assumptions about particle shape if particle volume is chosen as size parameter. In fact, there exists in this case an unbiased stereological estimator of the first moment under the sole assumption that the particles are convex. In the present paper, we consider a particle aggregate in R" and derive estimators of the qth moment of the volumeweighted distribution of particle volume, based on point-sampling of particles and measurements on q-flats through sampled particles. The estimators are valid for arbitrarily shaped particles but if the particles are non-convex it is necessary for the determination of the estimators to be able to identify the different separated parts on a q-flat through the particle aggregate which belong to the same particle. Explicit forms of the estimators are given for q = 1. For q = 2, an explicit form of one of the estimators is derived for an aggregate of triaxial ellipsoids in three-dimensional space.
TL;DR: In this paper, the same method was applied for proving Bonferroni-Galambos-type inequalities, and the lower and upper bounds of S m were given in terms of S k and S l.
Abstract: Let A 1 , A 2 , · ··, A n be events on a probability space. Denote by S k the k th binomial moment of the number M n of those A 's which occur. Sharp lower and upper bounds of S m will be given in terms of S k and S l . The same method can be applied for proving Bonferroni–Galambos-type inequalities.
TL;DR: In this paper, stopping rules in the context of proofreading are studied, and a computational procedure to obtain an optimal stopping rule is presented, which is the same as in this paper.
Abstract: Stopping rules in the context of proofreading are studied. Previous results by Yang et al. [6] are corrected and a computational procedure to obtain an optimal rule is presented.
TL;DR: An analogue of the Kesten-Stigum theorem, and sufficient conditions for the geometric rate of growth in the rth mean and almost surely, are obtained for population-size-dependent branching processes.
Abstract: An analogue of the Kesten–Stigum theorem, and sufficient conditions for the geometric rate of growth in the rth mean and almost surely, are obtained for population-size-dependent branching processes
TL;DR: On propose une analyse stochastique d'une file d'attente avec des arrivees de Poisson and de two serveurs exponentiels differents.
Abstract: On propose une analyse stochastique d'une file d'attente avec des arrivees de Poisson et deux serveurs exponentiels differents
TL;DR: In this article, the authors propose a systeme dans lequel les tâches arrivent a un tampon a partir duquel il existe plusieurs routes paralleles vers une destination.
Abstract: On considere un systeme dans lequel les tâches arrivent a un tampon a partir duquel il existe plusieurs routes paralleles vers une destination. On cherche a optimiser la politique individuelle qui minimise pour chaque tâche, son propre retard
TL;DR: In this article, a risk-sensitive formulation of the Whittle and Gait optimal control problem is given, and the reduction of an optimally controlled homing problem to the treatment of an uncontrolled process can be achieved in the risk sensitive case.
Abstract: The 'homing' optimal control problem, described in Whittle and Gait (1970), is given a risk-sensitive formulation. It is shown that the reduction of an optimally controlled homing problem to the treatment of an uncontrolled process, demonstrated by Whittle and Gait, can be achieved in the risk-sensitive case. Two scalar problems are analyzed in detail.