Showing papers in "Journal of Applied Probability in 1981"
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TL;DR: The probability that the above so-called random graph is connected is determined and a recursive formula is developed for the distribution of C, the number of connected components it contains, which derives expressions for the mean and variance of C.
Abstract: : Let X(1),X(2), ..., X(n) be independent random variables such that P(X(i) = j) = P sub j , j = 1,2, ..., n, sum from j = 1 to n of P sub j = 1 and consider a graph with n nodes numbered 1,2, ..., n and the arcs (i,X(i)), i = 1,2, ..., n. We determine the probability that the above so-called random graph is connected and then develop a recursive formula for the distribution of C, the number of connected components it contains. We also derive expressions for the mean and variance of C. (Author)
224 citations
TL;DR: In this article, a one-dimensional random walk is studied in which, at each stage, the probabilities of continuing in the same direction or of changing direction are p and 1 -p, respectively.
Abstract: A one-dimensional random walk is studied in which, at each stage, the probabilities of continuing in the same direction or of changing direction are p and 1 - p, respectively. Exact expressions are derived for the n-step transition probabilities, and various limiting distributions are investigated. CORRELATED RANDOM WALK; OCCUPATION PROBABILITIES; DIFFUSION; BESSEL FUNCTIONS
103 citations
TL;DR: In this article, it was shown that distributions F (called dispersive distributions) exist, e.g. the exponential, such that if G, H are ordered in dispersion then so also are the convolutions F ∗G, H ∗H.
Abstract: Two continuous distributions, G, H so related that any two quantiles of H are more widely separated than the corresponding quantiles of G may be said to be ‘ordered in dispersion'; Saunders and Moran have given examples. It is shown here that distributions F (called ‘dispersive' distributions) exist, e.g. the exponential, such that if G, H are ordered in dispersion then so also are the convolutions F ∗G, F ∗H. The class of dispersive distributions is determined, and shown to coincide with the class of strongly unimodal distributions.
102 citations
TL;DR: In this paper, the authors investigated the properties of the first-order bilinear time series model: stationarity and invertibility, and obtained estimates of the parameters by a modified least squares method and shown to be strongly consistent.
Abstract: The paper investigates some properties of the first-order bilinear time series model: stationarity and invertibility. Estimates of the parameters are obtained by a modified least squares method and shown to be strongly consistent.
88 citations
TL;DR: This work presents the stationary joint generating function of the two queue sizes explicitly in the form of an elliptic integral.
Abstract: An exponential server splits its service capacity between two independent Poisson streams of customers, unless one queue is empty, in which case the full service capacity is granted to the other queue. We present the stationary joint generating function of the two queue sizes explicitly in the form of an elliptic integral.
78 citations
TL;DR: For regular Markov processes on a countable space, this paper provided criteria for the forms of ergodicity in the title in terms of the existence of solutions to inequalities involving the Q-matrix of the process.
Abstract: For regular Markov processes on a countable space, we provide criteria for the forms of ergodicity in the title in terms of the existence of solutions to inequalities involving the Q-matrix of the process. An application to birth-death processes is given.
71 citations
TL;DR: In this paper, the authors considered a queueing system with Poisson arrivals and exponential services and showed that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates.
Abstract: Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable.
71 citations
TL;DR: In this article, a modified storage process with state space [0, ∞] was considered, and it was shown that W can be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift.
Abstract: The paper considers a modified storage process with state space [0,∞). Away from the origin, W behaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls to It is shown that W can be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift. When these same two transformations are applied to Brownian motion, one obtains sticky (or slowly reflected) Brownian motion W∗ on [0,∞). Thus W∗ is the natural diffusion approximation for W, and it is shown that W converges in distribution to W∗ under appropriate conditions. The boundary behavior of W∗ is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.
64 citations
TL;DR: In this article, the authors explore best choice problems which allow both recall of applicants and uncertainty of a current applicant accepting an offer of employment, and derive properties of optimal selection procedures for the general case.
Abstract: This paper explores best choice problems which allow both recall of applicants and uncertainty of a current applicant accepting an offer of employment. Properties of optimal selection procedures are derived for the general case. Optimal procedures and the associated probabilities of obtaining the best applicant are found in two special cases. The results unify and extend those of Yang (1974) and Smith (1975). OPTIMAL STOPPING; SECRETARY PROBLEM: RELATIVE RANKS
60 citations
TL;DR: In this paper, it was shown that if all lifetime distributions are non-atomic and share the same essential extrema, and if the incidence matrix of the minimal cut sets has rank equal to the number of parts, then the joint distribution of Z and I determines uniquely the lifetime distribution of each part.
Abstract: Given a coherent reliability system, let Z be the age of the machine at breakdown, and I the set of parts dead by time Z. We prove that if all lifetime distributions are non-atomic and share the same essential extrema, and if the incidence matrix of the minimal cut sets has rank equal to the number of parts, then the joint distribution of Z and I determines uniquely the lifetime distribution of each part. We present a Newton–Kantorovic iterative method for the computation of those distributions. We deal informally with the relaxation of the assumptions and with the statistical problem where instead of the joint distribution of Z and I we have an empirical estimate of this joint distribution.
53 citations
TL;DR: In this article, it was shown that a 0n/n converges in probability as n → ∞ to a finite constant μ (U) called the time constant whenever Uk converges weakly to U.
Abstract: Let U be the distribution function of the non-negative passage time of an individual edge of the square lattice, and let a0n be the minimal passage time from (0, 0) to (n, 0). The process a0n/n converges in probability as n → ∞to a finite constant μ (U) called the time constant. It is proven that μ (Uk )→ μ(U) whenever Uk converges weakly to U.
TL;DR: In this article, it was shown that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network.
Abstract: The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady-state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network. We consider closed systems in which a fixed and finite number of customers circulate through the network and no external arrivals or departures are permitted, and open systems in which customers originate from an external source according to a Poisson process, possibly non-homogeneous, and each customer eventually leaves the system.
TL;DR: In this article, an asymptotic expression for the variance of the sample correlation between stochastically independent, stationary Gaussian lattice processes with zero means was derived for a wide class of domains in '2.
Abstract: Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u E '2} and {Y(u), u E 2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and { Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in '2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.
TL;DR: In this paper, the limit distribution P { S n = k } is obtained when n → ∞, np → λ, and S n is Σ i = 1 n X i.
Abstract: Let X 1 X 2 , · ·· be a Markov Bernoulli sequence with initial probabilities p of success and q = 1 – p of failure, and probabilities 1 – (1 – π ) p , (1 – π ) p in the first row and (1 – π ) (1 – p ), (1 – π ) p + πin the second row of the transition matrix. If we define S n = Σ i =1 n X i , then the limit distribution P { S n = k } is obtained when n → ∞, np → λ.
TL;DR: In this article, the use of alternating renewal processes to model the behaviour of a repairable system is discussed, and properties of two measures of availability are reviewed, and the introduction of forward recurrence times to
Abstract: The use of alternating renewal processes to model the behaviour of a repairable system is discussed, and properties of two measures of availability are reviewed. It is shown how the introduction of forward recurrence times to
TL;DR: In this paper, it was shown that the survival function H(t) is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).
Abstract: Suppose that a device is subjected to shocks and that P sub(k), k = 0, 1, 2,..., denotes the probability of surviving k shocks. Then H(t) = Sigma super( infinity )@)dk sub(=) sub(0) P(N(t)=k)P sub(k) is the probability that the device will survive beyond t, where N = (N(t):t greater than or equal to 0) is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the pure birth shock model. In this paper we shall prove that H(t) is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).
TL;DR: In this article, a non-linear threshold autoregressive model for time series models of nonlinear random vibrations is proposed. But the model is not suitable for the analysis of real data.
Abstract: Time series models for non-linear random vibrations are discussed from the viewpoint of the specification of the dynamics of the damping and restoring force of vibrations, and a non-linear threshold autoregressive model is introduced. Typical non-linear phenomena of vibrations are demonstrated using the models. Stationarity conditions and some structural aspects of the model are briefly discussed. Applications of the model in the statistical analysis of real data are also shown with numerical results.
TL;DR: In this article, a vector X of patient prognostic variables is modeled as a linear diffusion process with time-dependent, non-random, continuous coefficients, and the instantaneous force of mortality (hazard function) operating on the patient is assumed to be a timedependent, continuous quadratic functional of the prognostic vector.
Abstract: A vector X of patient prognostic variables is modeled as a linear diffusion process with time-dependent, non-random, continuous coefficients. The instantaneous force of mortality (hazard function) operating on the patient is assumed to be a time-dependent, continuous quadratic functional of the prognostic vector. Conditional on initial data X 0, the probability of surviving T units of time is expressed in terms of the solution of a Riccati equation, which can be evaluated in closed form if the coefficients of the process and the hazard are constant. This conditional expectation does not preserve proportional hazards.
TL;DR: In this article, it was shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it was then pointed out that Hooke's theorem leads to the extension of a related theorem of Takaics.
Abstract: This paper is concerned with asymptotic results for a single-server queue having periodic Poisson input and general service-time distribution, and carries forward the analysis of this model initiated in Harrison and Lemoine [3]. First, it is shown that a theorem of Hooke relating the stationary virtual and actual waiting-time distributions for the GI/G/1 queue extends to the periodic Poisson model; it is then pointed out that Hooke's theorem leads to the extension (developed in [3]) of a related theorem of Takaics. Second, it is demonstrated that the asymptotic distribution for the server-load process at a fixed 'time of day' coincides with the distribution for the supremum, over the time horizon [0, o0), of the sum of a stationary compound Poisson process with negative drift and a continuous periodic function. Some implications of this characterization result for the computation and approximation of the asymptotic distributions are then discussed, including a direct proof, for the periodic Poisson case, of a recent result of Rolski [6] comparing mean asymptotic customer waiting time with that of a corresponding M/G/1 system.
TL;DR: In this paper, the problem of Monte Carlo estimation of M(t) = EN(t), the expected number of renewals in [0, t] for a renewal process with known interarrival time distribution F, is considered.
Abstract: The problem of Monte Carlo estimation of M(t) = EN(t), the expected number of renewals in [0, t] for a renewal process with known interarrival time distribution F, is considered. Several unbiased estimators which compete favorably with the naive estimator, N(t), are presented and studied. An approach to reduce the variance of the Monte Carlo estimator is developed and illustrated.
TL;DR: A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment including instability and extinction conditions and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.
Abstract: A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions and when suitable absorbing barriers have been defined methods are given for the calculation of extinction probabilities and the expected duration of the process. (EXCERPT)
TL;DR: In this article, an iterative solution is derived to the time-dependent Kolmogorov equations of queueing networks, and is shown to be convergent, where the time periods during which the equilibrium assumption can and should not be made are identified; for example in terms of a time constant which is easily computed to a first-order approximation.
Abstract: In most contemporary queueing network analysis, the assumption is made that a network is in a state of equilibrium. That is, the network's state space probabilities are assumed to be time independent. It is therefore important to be able to quantify precisely when this assumption is valid. Furthermore there are also situations in which it is desirable to model the transient behaviour of networks which occur in practice, such as computer and communication systems. For example, the immediate effects of component failure or instantaneous alteration of system status may be predicted. In this paper an iterative solution is derived to the time-dependent Kolmogorov equations of queueing networks, and is shown to be convergent. From the solution, modelling of transient situations becomes possible and the time periods during which the equilibrium assumption can and should not be made may be identified; for example in terms of a time constant which is easily computed to a first-order approximation.
TL;DR: The probability generating functional representation of a multidimensional Poisson cluster process is utilized to derive a formula for its likelihood function, but the prohibitive complexity of this formula precludes its practical application to statistical inference.
Abstract: The probability generating functional representation of a multidimensional Poisson cluster process is utilized to derive a formula for its likelihood function, but the prohibitive complexity of this formula precludes its practical application to statistical inference. In the case of isotropic processes, it is however feasible to compute functions such as the probability Q(r) of finding no point in a disc of radius r and the probability Q(r | 0) of nearest-neighbor distances greater than r, as well as the expected number C(r | 0) of points at a distance less than r from a given point. Explicit formulas and asymptotic developments are derived for these functions in the n-dimensional case. These can effectively be used as tools for statistical analysis.
TL;DR: In this paper, the authors give fairly elementary proofs for the threshold theorem due to Williams (1971) and Whittle (1955) based on an application of the reflection principle through the ballot problem.
Abstract: We give here fairly elementary proofs for the threshold theorems due to Williams (1971) and Whittle (1955). Our proofs are based on an application of the reflection principle through the ballot problem and the exact distribution of the size of the epidemic as derived by Foster (1955). Williams's threshold theorem is extended to an epidemic with multiple introduction of cases.
TL;DR: In this paper, it was shown that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering.
Abstract: Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.