Showing papers in "Journal of Applied Probability in 1984"
TL;DR: In this paper, the authors consider the model where φ 1, φ 2 are real coefficients, not necessarily equal, and the at,'s are a sequence of i.i.d. random variables with mean 0.
Abstract: We consider the model where φ 1, φ 2 are real coefficients, not necessarily equal, and the at ,'s are a sequence of i.i.d. random variables with mean 0. Necessary and sufficient conditions on the φ 's are given for stationarity of the process. Least squares estimators of the φ 's are derived and, under mild regularity conditions, are shown to be consistent and asymptotically normal. An hypothesis test is given to differentiate between an AR(1) (the case φ 1 = φ 2) and this threshold model. The asymptotic behavior of the test statistic is derived. Small-sample behavior of the estimators and the hypothesis test are studied via simulated data.
282 citations
TL;DR: Certain first-order autoregressive processes are shown not to be strong mixing and a direct proof is given that gives considerably more insight into the nature of the result than do proofs by contradiction.
Abstract: Certain first-order autoregressive processes are shown not to be strong mixing. A direct proof is given. This proof gives considerably more insight into the nature of the result than do proofs by contradiction. The result and proof help to clarify the relation between the autoregressive and strong mixing conditions.
259 citations
TL;DR: This work extends Erlang work to general networks and shows that if for each pair of nodes there is a unique route, then the blocking probabilities are in product form and are insensitive to the call holding-time distribution, which means that they depend on the call duration only through its mean.
Abstract: Consider a network of nodes (switches) and connecting links. Each link consists of a group of channels (trunks). A call instantaneously seizes channels along a route between the originating and terminating node, holds them for a randomly distributed length of time and frees them instantaneously at the end of the call. If no channels are available, the call is blocked. For special networks with exponential call holding times, Erlang has shown that the steady-state probabilities are in product form. In this paper, we extend this work to general networks and show that if for each pair of nodes there is a unique route, then the blocking probabilities are in product form and are insensitive to the call holding-time distribution, which means that they depend on the call duration only through its mean.
180 citations
146 citations
TL;DR: In this paper, Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties, such as maximum, local time and area distributions.
Abstract: Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple The original purpose of this paper was to find a workable expression for the transform of the Brownian excursion area. We wanted to use two well-known results: Kac's formula for Brownian functionals and Levy's local time decompos- ition. We actually found that these two powerful tools lead also to alternative and sometimes simpler proofs for other results on Brownian excursion prob- abilities: maximum and local time distributions. We were also able to derive a useful general result on Brownian excursion symmetric additive functionals. Finally, we also wanted to understand some curious relations that had been observed by earlier authors: connections between hitting-time densities and maximum distributions. Using simple probabilistic arguments and a formula for Jacobi's third 0 function, we can explain these connections in direct terms. The paper is organized as follows. In Section 1, we summarize basic notations and known results. Sections 2 and 3 are short surveys of Kac's and Levy's results. These tools are applied in Sections 4 and 5 to Brownian excursion maximum and local time. Section 6 contains the main results of the paper: a useful new form for the transform of the Brownian excursion area density and a simple relation for functionals based on a symmetric positive function. Section 7 explains those curious relations alluded to earlier between Brownian excursion probabilities. 1. Basic notations and known results
123 citations
TL;DR: In this paper, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in a general M/G/1 queue with processor sharing (M/G 1/PS) is given, in the form of Laplace-Stieltjes transforms and generating functions.
Abstract: This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS). Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.
122 citations
TL;DR: In this article, the authors investigated the asymptotic stability of queues with time-varying arrival rates and proved limit theorems for the M, G/c queue with a Poisson arrival process with a general deterministic intensity.
Abstract: This paper discusses the asymptotic behavior of the M,/ G/c queue having a Poisson arrival process with a general deterministic intensity. Since traditional equilibrium does not always exist, other notions of asymptotic stability are introduced and investigated. For the periodic case, limit theorems are proved complementing Harrison and Lemoine (1977) and Lemoine (1981). PERIODIC QUEUE; NON-STATIONARY QUEUE; MULTISERVER QUEUE; PERIODIC POISSON PROCESS; REGENERATIVE PROCESS; WAITING TIME 0. Introduction and summary The purpose of this paper is to contribute to the theory of queues with time-varying arrival rates. We assume that the arrival process is a Poisson process with a general deterministic intensity A (t). We are interested in periodic arrival processes, which have an 'embedded stationarity' and can be represented as stationary point processes with the proper initial conditions, but we are also interested in arrival processes that are fundamentally non-stationary, that cannot be put in the framework of Franken et al. (1981). Most of the work on queues with time-varying arrival rates has been concerned with describing the time-dependent behavior of the queue. Early papers by Luchak (1956) and Clarke (1956) focused on solving the Kolmogorov equations for the queue-length process. Since then, considerable progress has been made by developing approximations for the time-dependent behavior; see Rothkopf and Oren (1979), Clark (1981), and Taafe (1982) for closure approximations; see Newell (1968), (1971), McClish (1979), Keller (1982) and Massey (1981) for asymptotic expansions; see these sources for earlier work. Received 18 January 1983; revision received 20 April 1983. * Postal address: Bell Laboratories, Crawfords Corner Road, Holmdel, NJ 07733, U.S.A.
83 citations
TL;DR: In this article, it was shown that for a diffusion process, the first passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents.
Abstract: We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.
74 citations
TL;DR: In this article, the search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t.
Abstract: Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation. Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.
71 citations
TL;DR: In this paper, the first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier.
Abstract: The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein-Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed. EXIT TIMES; DIFFUSION PROCESS; NEURAL FIRING; ORNSTEIN-UHLENBECK PROCESS
64 citations
TL;DR: In this article, sufficient conditions for subexponential convergence of a distribution function on [0,4 [, with finite mean] were investigated and some applications to risk theory and renewal theory were given.
Abstract: We investigate sufficient conditions so that F,(x)= foF(y)dy is subexponential. Here F is a distribution function on [0,4 [, with finite mean. Some applications to risk theory and rates of convergence in renewal theory are given.
TL;DR: In this article, extinction criteria for the Galton-Watson process with arbitrary mating functions in terms of the averaged reproduction mean per mating unit were discussed, and a satisfactory answer to a question put forward by Hull (1982) was given.
Abstract: This note deals with extinction criteria for bisexual Galton–Watson processes with arbitrary mating functions in terms of the averaged reproduction mean per mating unit. It gives a satisfactory answer to a question put forward by Hull (1982).
TL;DR: In this article, a generalisation of multistate coherent structures is proposed where the state of each component in a binary coherent structure can take any value in the unit interval, as can the structure function.
Abstract: A generalisation of multistate coherent structures is proposed where the state of each component in a binary coherent structure can take any value in the unit interval, as can the structure function. The notions of duality, critical elements and strong coherency for such a structure are discussed and the functional form of the structure function is analysed. An expression is derived for the distribution function of the state of the system, given the distributions of the states of the components, and generalisations of the Moore–Shannon and IFRA and NBU closure theorems are proved. The states of the components are then permitted to vary with time and a first-passage-time distribution is discussed. A simple model for such a process, based on the concept of partial availability, is then proposed. Lastly, an alternative continuum structure function is introduced and discussed.
TL;DR: In this paper, the authors generalize Jackson's theorem to the non-ergodic case and show that, despite the fact that the entire Jackson network will not achieve steady state, it is still possible to determine the maximal subnetwork that does.
Abstract: We generalize Jackson's theorem to the non-ergodic case. Here, despite the fact that the entire Jackson network will not achieve steady state, it is still possible to determine the maximal subnetwork that does. We do so by formulating and algorithmically solving a new non-linear throughput equation. These results, together with the ergodic results and the ones for closed networks, completely characterize the large-time behavior of any Jackson network.
TL;DR: The Poisson shot-noise process discussed in this article takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s), s) are independent stochastic processes.
Abstract: The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.
TL;DR: An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network, and Expressions are obtained for the expected width conditioned on N, N, M, and (N, D).
Abstract: An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x ? 0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is = x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random. NETWORKS; BRANCHING PROCESS
TL;DR: In this paper, the authors considered the boundary value problem in two queues in parallel and proposed a procedure for the solution in the context of finite waiting-room size and some comparisons are made with the single queue system and an independent two-queue system.
Abstract: The model considered in this note has been referred to by Haight (1958), Kingman (1961) and Flatto and McKean (1977) as two queues in parallel. Customers choose the shorter of the two queues which are otherwise independent. This system is known to be inferior to a single queue feeding the two servers, but how much? Some elementary considerations provide a fresh perspective on this awkward boundary-value problem. A procedure is proposed for the solution in the context of finite waiting-room size and some comparisons are made with the single-queue system and an independent two-queue system.
TL;DR: In this paper, the authors considered a branching process model where the law of the offspring distribution depends on the population size and gave necessary and sufficient conditions for convergence in L 1 and L 2 of Wn = Zn/mn.
Abstract: We consider a branching-process model {Zn }, where the law of offspring distribution depends on the population size. We consider the case when the means mn (mn is the mean of offspring distribution when the population size is equal to n) tend to a limit m > 1 as n →∞. For a certain class of processes {Zn } necessary conditions for convergence in L 1 and L 2 and sufficient conditions for almost sure convergence and convergence in L 2 of Wn = Zn/mn are given.
TL;DR: In this article, the similarity between order statistics and record values motivated the authors to investigate some relationships between the order statistic and the record values, and these relationships are employed to characterize a continuous distribution by some moment properties of the spacings of record values.
Abstract: The similarity between order statistics and record values motivated us, in this paper, to investigate some relationships between the order statistics and the record values. These relationships are employed to characterize a continuous distribution by some moment properties of the spacings of the record values, and hence obtain characterizations of the exponential distribution. Some of the well-known results follow trivially. EXPONENTIAL DISTRIBUTION
TL;DR: In this paper, a simple method is proposed for the generation of successive "nearest neighbours" to a given origin in an n-dimensional Poisson process, which provides efficient simulation of random Voronoi polytopes.
Abstract: A simple method is proposed for the generation of successive 'nearest neighbours' to a given origin in an n-dimensional Poisson process. It is shown that the method provides efficient simulation of random Voronoi polytopes. Results are given of simulation studies in two and three dimensions.
TL;DR: In this article, the authors derived the mean and the variance of the equilibrium sojourn time of the GI/M/1/PS queue, and deduced that the variance is larger for the processor-sharing model than for the corresponding FCFS model.
Abstract: A queueing model of considerable interest in computer engineering is the processor-sharing model in which the server shares its fixed capacity equally among all units present in the system. Here, we derive the mean and the variance of the equilibrium sojourn time, and deduce that the variance of the sojourn time is larger for the processor-sharing model than for the corresponding FCFS model. A queueing model of considerable interest in computer engineering is the processor-sharing model in which the server shares its fixed capacity equally among all units present in the system. The service discipline in such a model is the limiting case of the 'round robin discipline' as the quantum of service At given to each customer shrinks to 0. The sojourn time in the M/M/1 queue with processor sharing (MIM/1/IPS) was studied by Coffman, Muntz and Trotter [1]. A recent paper by Ott [5] obtains the distribution of the sojourn time in the M/G/1/PS queue. Other previous papers on the MI/G/1/PS queue have restricted their attention to the conditional mean sojourn time only, and we refer to O'Donovan [4] for details. The subject of this paper is the GI/M/1/PS queue, and it is hoped that our results will be helpful in examining the effect of traffic variability in a processorsharing context. The GI/M/1/PS model is obtained by assuming that arrivals form a renewal process, that successive service requirements are i.i.d. exponential random variables and that the server depletes the residual service requirement of each customer present at the rate 1/n when n customers are present. An equivalent description would be that our model is an infinite-server queue with renewal input and state-dependent exponential servers each of whom serves at the rate t /n when n customers are present. The sojourn time of a customer is the
TL;DR: In this article, it was shown that if two successive jth record values and associated to an i.i.d. sequence are such that and are independent, then the sequence has to derive from an exponential distribution (in the continuous case).
Abstract: It is shown that, in some particular cases, it is equivalent to characterize a continuous distribution by properties of records and by properties of order statistics. As an application, we give a simple proof that if two successive jth record values and associated to an i.i.d. sequence are such that and are independent, then the sequence has to derive from an exponential distribution (in the continuous case). The equivalence breaks up for discrete distributions, for which we give a proof that the only distributions such that Xk, n and Xk +1, n – Xk, n are independent for some k ≧ 2 (where Xk, n is the kth order statistic of X 1, ···, Xn ) are degenerate.
TL;DR: In this paper, it was shown that the median residual life function of a distribution can be determined by solving the Schroder's equation under mild assumptions on φ, where φ is the reliability function.
Abstract: In reliability studies, it is well known that the mean residual life function determines the distribution function uniquely. In this paper we show how closely we can determine a distribution when its median residual life function M[S | t] is known. This amounts to solving the functional equation , where R is the reliability function. We actually study a more general functional equation f(φ(t)) = sf(t) called Schroder's equation. It is shown that, under mild assumptions on φ, the solution is of the form f(t) = f 0(t)k(log f 0 (t)), where f 0 is a well-behaved particular solution which can be constructed and k is a periodic function; thus the solution is not unique. Two examples are solved to illustrate the method. Finally, these examples are used to solve the problem of linear M[S | t] studied by Schmittlein and Morrison. As an extra benefit, all of our results hold equally well for the more general sth percentile residual life function.
TL;DR: The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope and the expected surface area, its expected mean width and its expected number of facets are explicitly determined in this paper.
Abstract: The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope. Its expected surface area, its expected mean width and its expected number of facets are explicitly determined.
TL;DR: In this paper, the authors study the problem of optimal admission to multiserver queues in a semi-Markovian way, and establish optimality of a generalized control-limit rule depending on the actual environment.
Abstract: We study the problem of optimal customer admission to multiserver queues. These queues are assumed to live in an extraneous environment which changes in a semi-Markovian way. Arrivals, service mechanism and random reward/cost structure may all depend on these surroundings. Included as special cases are SM/M/c queues, in particular G/M/c queues, in a random environment. By a direct inductive approach we establish optimality of a generalized control-limit rule depending on the actual environment. Particular emphasis is laid on different applications that show the versatility of the proposed setup.
TL;DR: In this paper, it was shown that life distribution properties of the threshold right tail probability are inherited as corresponding properties of survival probability, under suitable conditions on the parameters of the damage process.
Abstract: A device is subject to damage. The damage occurs randomly in time according to a pure jump process. The device has a threshold and it fails once the damage exceeds the threshold. We show that life distribution properties of the threshold right tail probability are inherited as corresponding properties of the survival probability, under suitable conditions on the parameters of the damage process. Moreover we discuss an optimal replacement problem for such devices.
TL;DR: In this paper, it was shown that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution.
Abstract: We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions. EXPONENTIAL DISTRIBUTIONS
TL;DR: Operator methods are used in this paper to systematically analyze the behavior of the Jackson network, and rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system are considered.
Abstract: Operator methods are used in this paper to systematically analyze the behavior of the Jackson network. Here, we consider rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system. By deriving the equations that govern an arbitrary subnetwork, we can see how the mean and variance for the queue length of one node as well as the covariance for two nodes vary in time. We can estimate the transient behavior by deriving a stochastic upper bound for the joint distribution of the network in terms of a judicious choice of independent M/M/1 queue-length processes. The bound we derive is one that cannot be derived by a sample-path ordering of the two processes. Moreover, we can stochastically bound from below the process for the total number of customers in the network by an M/M/1 system also. These results allow us to approximate the network by the known transient distribution of the M/M/1 queue. The bounds are tight asymptotically for large-time behavior when every node exceeds heavy-traffic conditions.
TL;DR: In this paper, a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond-percolation process.
Abstract: We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.
TL;DR: In this paper, the steady-state behavior of multistate monotone systems of multi-state components is considered by applying the theory for stationary and synchronous processes with an embedded point process, and an explicit formula is given for the mean time which the system in steady state sojourns in states not below a fixed critical level.
Abstract: In this paper the steady-state behaviour of multistate monotone systems of multistate components is considered by applying the theory for stationary and synchronous processes with an embedded point process. After reviewing some general results on stationary availability, stationary interval availability and stationary mean interval performance probabilities, we concentrate on systems with independently working and separately maintained components. For this case an explicit formula is given for the mean time which the system in steady state sojourns in states not below a fixed critical level.