Showing papers in "Journal of Applied Probability in 1979"
TL;DR: In this article, a versatile class of point processes on the real line, which are closely related to finite-state Markov processes, is introduced, and the treatment of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.
Abstract: : A versatile class is introduced of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper. Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.
708 citations
TL;DR: In this article, it was shown that if the constants an and bn are chosen in an optimal way, then the rate of convergence of (Yn − bn )/an to the extreme value distribution exp(e −x ), as measured by the supremum metric or the Levy metric, is proportional to 1/log n.
Abstract: Let Yn denote the largest of n independent N(0, 1) variables. It is shown that if the constants an and bn are chosen in an optimal way then the rate of convergence of (Yn – bn )/an to the extreme value distribution exp(–e–x ), as measured by the supremum metric or the Levy metric, is proportional to 1/log n.
236 citations
TL;DR: In this paper, a definition of evolutionary stable strategy (ES) is proposed for games in which there are two types of player, each with its own set of strategies, and the fitness of any strategy depends upon the strategy mix, of both types, in the population as a whole.
Abstract: A definition of ESS (evolutionarily stable strategy) is suggested for games in which there are two types of player, each with its own set of strategies, and the fitness of any strategy depends upon the strategy mix, of both types, in the population as a whole. We check that the standard ESS results hold for this definition and discuss a mate-desertion model which has appeared in the literature in which the two types are male and female.
178 citations
TL;DR: In this article, the authors introduced a convex supremum of M in the class of all joint distributions of the Xi's with specified marginals, and of the bottleneck probability of each path.
Abstract: 1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all . 2. Let H be the Hardy–Littlewood maximal function HY (x) = E(Y – X | Y > x). Then HY (Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y. 3. Let X 1 X 2 · ·· Xn be random variables with given marginal distributions, let I 1, I 2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij , and delay times Xi .) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi 's with specified marginals, and of the ‘bottleneck probability' of each path.
151 citations
TL;DR: In this article, the joint probability density of the maximum value Y = max [X(t),0-5 t - T], its location 6 = 0(X), 0-0 T; and the endpoint X(T), where X(t) = X, (t) is a Wiener process with drift, X,(t)= W(t)+ ct, 0O-t 5 T
Abstract: We give a simple expression for the joint probability density of: (a) the maximum value Y = max [X(t),0-5 t - T]; (b) its location 6 = 0(X), 0- 0 T; (c) the endpoint X(T), where X(t) = X, (t) is a Wiener process with drift, X,(t)= W(t)+ ct, 0O- t 5 T That is, we find the density p(0, y,x)= p(O, y,x; c, T) of 0, Y, X(T), p(O, y,x; c, T)dOdydx = P(6 EdO, X,(6)E dy, X,(T)E dx) is given by, 0 < 0 < T, x - y, 0 < y, 7r 3/2(T - 0)2ex 20 2(T- 0) e
108 citations
TL;DR: In this paper, the Chung-Smirnov property is shown to hold for nonparametric estimators of a sequence of d.i.d. functions with common distribution function (d.f.).
Abstract: Given an i.i.d. sequence X 1,X 2, … with common distribution function (d.f.) F, the usual non-parametric estimator of F is the e.d.f. Fn ; where Uo is the d.f. of the unit mass at zero. An admissible perturbation of the e.d.f., say , is obtained if Uo is replaced by a d.f. , where is a sequence of d.f.'s converging weakly to Uo. Such perturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as . It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then ‘has the Chung–Smirnov property'; i.e., with probability one, 1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution and Hn is the d.f. of the unit mass at n–α then the above lim sup is ≦ 1 or = ∞, depending on whether or
72 citations
TL;DR: In this article, the authors consider the problem of scheduling n tasks on to several identical processors to meet the objective of minimizing the expected flow-time and show that the strategy which always serves those tasks whose processing-time distributions have the highest hazard rates is shown to be optimal when these distributions are all exponential.
Abstract: We consider the problem of how to schedule n tasks on to several identical processors to meet the objective of minimising the expected flow-time. The strategy which always serves those tasks whose processing-time distributions have the highest hazard rates is shown to be optimal when these distributions are all exponential.
70 citations
TL;DR: Using the theory of point processes, Miyazawa et al. as discussed by the authors gave a formal treatment to queueing processes in the steady state, and obtained invariance relations between several quantities in GIG/c queues.
Abstract: Using the theory of point processes, we give a formal treatment to queueing processes in the steady state. Based on this result, we obtain invariance relations between several quantities in GIG/c queues. As applications, the finiteness of their moments is discussed for G/G'/c queues. The basic results and notations used in this paper are contained in the author's previous paper (Miyazawa (1977)). FCFS MANY-SERVER QUEUES; STATIONARY INPUT; INVARIANCE RELATIONS; EXISTENCE OF MOMENTS; STEADY-STATE DISTRIBUTIONS; POINT PROCESS; PALM MEASURE
69 citations
TL;DR: In this article, it was shown that the output process has the same distribution for all series arrangements of the queues, in the sense that customers leave the first queue according to an arbitrary stochastic input process and then pass through the queues in order.
Abstract: A series of queues consists of a number of ./M/1 queues arranged in a series order. Each queue has an infinite waiting room and a single exponential server. The rates of the servers may differ. Initially the system is empty. Customers enter the first queue according to an arbitrary stochastic input process and then pass through the queues in order: a customer leaving the first queue immediately enters the second queue, and so on. We are concerned with the stochastic output process of customer departures from the final queue. We show that the queues are interchangeable, in the sense that the output process has the same distribution for all series arrangements of the queues. The 'output theorem' for the M/M/1 queue is a corollary of this result. OUTPUT PROCESSES; DEPARTURE PROCESSES; SERIES OF QUEUES; TANDEM QUEUES
58 citations
TL;DR: In this article, the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations, is investigated.
Abstract: This paper is concerned with the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations. The maximum likelihood estimate and the minimum contrast estimate are investigated. For these estimates strong consistency and asymptotic normality are proved. The unknown parameter is assumed to take its values either in an open or in a compact set of real numbers.
54 citations
TL;DR: Theorem limit distributions for occupancy problems and for dispersion statistics for the binomial, Poisson and negative-binomial distribution were derived in this paper for ZM when M, N, n - oo is proved by a general method.
Abstract: An urn contains A balls of each of N colours. At random n balls are drawn in succession without replacement, with replacement or with replacement together with S new balls of the same colour. Let X, be the number of drawn balls having colour k, k = 1, - - -, N. For a given function f the characteristic function of the random variable Z, = f(X)+ - -... + f(X,), M N, is derived. A limit theorem for ZM when M, N, n - oo is proved by a general method. The theorem covers many special cases discussed separately in the literature. As applications of the theorem limit distributions are obtained for some occupancy problems and for dispersion statistics for the binomial, Poisson and negative-binomial distribution.
TL;DR: The repeated orders queueing system (ROO) permits no waiting or queueing in the normal sense as discussed by the authors, instead customers who find the service (or device, to use an engineering term) busy make reapplications at random intervals and in random order until their needs are met.
Abstract: The repeated orders queueing system (ROO) permits no waiting or queue in the normal sense. Instead customers who find the service (or device, to use an engineering term) busy make reapplications at random intervals and in random order until their needs are met. Thus a second demand stream supplements the basic first arrival stream. Familiar examples are provided in a telephone communication setting, in particular in the context of a multiaccess computing system. Cohen [3] and Aleksandrov [1] made the first contributions to the theory of ROO. This paper complements their work with a steady-state analysis of system time (waiting time including service of a new arrival), of service idle time, and of system busy period.
TL;DR: In this article, the authors proposed two estimators of 0, say 0, 2, by minimizing two criteria D,(-), D,( ) respectively, which measure the nearness of a parametric family of spectral densities to a Gaussian stationary process with the true spectral density g(x).
Abstract: In fitting a certain parametric family of spectral densities f,(x) to a Gaussian stationary process with the true spectral density g(x), we propose two estimators of 0, say 0,, 2, by minimizing two criteria D,(-), D,( ) respectively, which measure the nearness of fo(x) to g(x). Then we investigate some asymptotic behavior of 0,, 2, with respect to efficiency and robustness. GAUSSIAN STATIONARY PROCESS; SPECTRAL DENSITY; PERIODOGRAM; ASYMPTOTIC EFFICIENCY; ROBUSTNESS
TL;DR: In this paper, the problem of finding optimal (or good suboptimal) strategies for non-stoppable families of alternative bandit processes is considered, and the performance of a simple strategy is shown to be optimal under certain conditions.
Abstract: Stoppable families of alternative bandit processes are decision processes with the property that at each decision epoch the choice is between allocating service to one of the constituent bandit processes or stopping and deciding in favour of one of them. The problem is considered of finding optimal (or good suboptimal) strategies for such processes. The theory for non-stoppable families leads us to study the performance of a simple strategy. This is shown to be optimal under certain conditions. These conditions are discussed and an example relating to research planning is given.
TL;DR: In this paper, it was shown that in two or more dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual, and thus the mortality of blocks does not make the limiting distribution of N(t) singular.
Abstract: Assume a population is distributed in an infinite lattice of colonies in a migration and random mating model in which all individuals are selectively equivalent. In one and two dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual. Let N(t) be the (random) size of a block intersecting a fixed colony at time t. Then E[N(t)] grows like Vt in one dimension, t/log t in two, and t in three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove that N(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution of N(t) singular. Results are proven for discrete time and sketched for continuous time. If a mutation rate u > 0 is imposed, the 'block structure' has an equilibrium distribution. If N(u) is the size of a block intersecting a fixed colony at equilibrium, then as u --0 N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systems u = 10-' is usually quite small. The proofs are by using multiple kinship coefficients for a stepping stone population. POPULATION GENETICS; MIGRATION; POINT PROCESSES; RATE OF CONSOLIDATION; STEPPING STONE; LIMIT LAW; GAMMA DISTRIBUTION
TL;DR: In this article, an L 1 analogue of the least square estimator for the parameters of stationary, finite-order autoregressions is considered and the Monte Carlo study provides evidence for a conjecture on the convergence rate.
Abstract: We consider an L 1 analogue of the least squares estimator for the parameters of stationary, finite-order autoregressions. This estimator, the least absolute deviation (LAD), is shown to be strongly consistent via a result that may have independent interest. The striking feature is that the conditions are so mild as to include processes with infinite variance, notably the stationary, finite autoregressions driven by stable increments in Lα, α > 1. Finally, sampling properties of LAD are compared to those of least squares. Together with a known convergence rate result for least squares, the Monte Carlo study provides evidence for a conjecture on the convergence rate of LAD.
TL;DR: In this paper, the steady state distribution of the customer waiting time in the GI/G/1 queue was derived for a dam in which the instantaneous release rate is proportional to the level.
Abstract: The new concept of embedded level crossings is combined with the old principle of stationary set balance to produce an alternative approach for obtaining the steady-state distribution of the level in a dam with general release rule. The method yields the steady state distribution of the customer waiting time in the GI/G/1 queue as a special case. Results for a dam in which the instantaneous release rate is proportional to the level, and for the M/G/1, GI/M/1, E,/M/1 and DIM/1 queues are derived using the new technique.
TL;DR: In this paper, the authors consider the situation in which the decision-maker is allowed to have two choices and he must choose both the best and the second best from a group of N applicants, and derive the optimal stopping rule and the maximum probability of choosing both of them.
Abstract: We consider the situation in which the decision-maker is allowed to have two choices and he must choose both the best and the second best from a group of N applicants. The optimal stopping rule and the maximum probability of choosing both of them are derived.
TL;DR: In this paper, conditions under which the transition probabilities of various Markovian storage processes approach a stationary limiting distribution π at an exponential rate were investigated, including the waiting time of the M/G/1 queue, and models for dams with additive input and state-dependent release rule satisfying a negative mean drift condition.
Abstract: We investigate conditions under which the transition probabilities of various Markovian storage processes approach a stationary limiting distribution π at an exponential rate. The models considered include the waiting time of the M/G/1 queue, and models for dams with additive input and state-dependent release rule satisfying a ‘negative mean drift' condition. A typical result is that this exponential ergodicity holds provided the input process is ‘exponentially bounded'; for example, in the case of a compound Poisson input, a sufficient condition is an exponential bound on the tail of the input size distribution. The results are proved by comparing the discrete-time skeletons of the Markov process with the behaviour of a random walk, and then showing that the continuous process inherits the exponential ergodicity of any of its skeletons.
TL;DR: In this paper, the replacement problem of a parallel system in a random environment is extended to the case where the failure probability depends on the number of failed components and the replacement cost is a linear function of the failed components.
Abstract: We have recently discussed the replacement problem of a parallel system in a random environment. This paper extends the same replacement problem for the following two cases which are more plausible: (i) The probability that an operating component fails by the j th shock depends on the number j. (ii) The replacement cost is a linear function of failed components. The expected cost of the above model is obtained. A numerical example is finally presented when the probability of failure time of a component is a negative binomial distribution.
TL;DR: There are some martingales associated with the branching random walk which are natural generalizations of the classical martingale occurring in the Galton-Watson process as discussed by the authors, and some continuity properties of the distributions of their limits are discussed.
Abstract: There are some martingales associated with the branching random walk which are natural generalizations of the classical martingale occurring in the Galton–Watson process. Some continuity properties of the distributions of their limits are discussed.
TL;DR: In this article, the main properties of the classes and the class of Poisson mixtures are presented, including characterisations of membership, relation with cumulants, and closure properties.
Abstract: Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.
TL;DR: In this article, the probability that no common allozyme signature is found among n randomly selected cell culture lines when the s mutually exclusive and exhaustive signatures have arbitrary known probabilities is given.
Abstract: This paper gives the probability that no common allozyme signature is found among n randomly selected cell culture lines when the s mutually exclusive and exhaustive signatures have arbitrary known probabilities. This result is useful in detecting cell culture contamination by a single cell line. This problem is a generalization of the classic problem of finding the probability of no common birthday among n individuals when it is assumed that each individual has a chance 1/s = 1/365 of a birthday on a given day. Both an exact recursive solution and an approximate solution are given for the general problem.
TL;DR: The p -content of the p -parallelotope ∇ p, n determined by p independent isotropic random points z 1, …, z p in ℝ n (1 p ≦ n ) can be expressed as a product of independent variates in two ways, by successive orthogonal projection onto linear subspaces and by radial projection of the points.
Abstract: The p -content of the p -parallelotope ∇ p, n determined by p independent isotropic random points z 1 , …, z p in ℝ n (1 p ≦ n ) can be expressed as a product of independent variates in two ways, by successive orthogonal projection onto linear subspaces and by radial projection of the points, enabling calculation of the actual distribution as well as the moments of ∇ p, n . This is done explicitly in several cases. The results also have interest in multivariate statistics.
TL;DR: In this article, the authors investigate a random collision model for competition between types of individuals in a population, where dominance relations are defined for each pair of types such that if two individuals of different types collide then after the collision both are of the dominant type.
Abstract: We investigate a random collision model for competition between types of individuals in a population. There are dominance relations defined for each pair of types such that if two individuals of different types collide then after the collision both are of the dominant type. These dominance relations are represented by an oriented graph, called a tournament. It is shown that tournaments having a particular form are relatively stable, while other tournaments are relatively unstable. A measure of the stability of the stable tournaments is given in the main theorem.