Showing papers in "Journal of Applied Probability in 1967"
TL;DR: A summary of the various stochastic approaches and applications to chemical reaction kinetics can be found in this paper, but before discussing these we first briefly introduce the basic ideas and definitions of classical or deterministic chemical kinetics.
Abstract: In this article we shall present a summary of the various stochastic approaches and applications to chemical reaction kinetics, but before discussing these we first briefly introduce the basic ideas and definitions of classical or deterministic chemical kinetics. One of the basic questions to which chemists address themselves is the rate of chemical reactions, or in other words, how long it takes for a chemical reaction to attain completion, or equilibrium. Apparently the first significant quantitative investigation was made in 1850 by L. Wilhelmy [93]. He studied the inversion of sucrose (cane sugar) in aqueous solutions of acids, whose reaction is He found empirically that the rate of decrease of concentration of sucrose was simply proportional to the concentration remaining unconverted, i.e., if S(t) is the concentration of sucrose, then The constant of proportionality is called the rate constant of the reaction. If S o is the initial concentration of sucrose, then Since then an enormous number of reactions has been studied and the field of chemical kinetics is now one of the largest areas of chemical research. The importance of the field lies in the fact that it yields concise expressions for the time dependence of reactions, predicts yields, optimum economic conditions, and gives one much insight into the actual molecular processes involved. The detailed molecular picture of a reaction process is called the mechanism of the reaction.
925 citations
TL;DR: In this article, the authors discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time, and summarized the analogous results when the time parameter is continuous.
Abstract: In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.
253 citations
TL;DR: In this paper, the authors derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution and defined joint waiting times in a bivariate Poisson process.
Abstract: In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on “shock models” and one is based on the requirement that residual life is independent of age. The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define “joint waiting time”. Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.
182 citations
TL;DR: In this paper, it was shown that if max u |au | is small, then the process Xw is close to Gaussian in the sense that ∫∞ −∞(F(y) − Φ(y))2 dy ≦ g max u|a u |, where Φ (·) is the standard Gaussian d.f. of X0.
Abstract: Let U denote the set of all integers, and suppose that Y = {Yu ; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au ; u ∈ U} be a sequence of real numbers with Σ u a u 2 = 1. Then Xu = Σw aw Y u –w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu ) = 0, E(Xu 2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if max u |au | is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞ −∞(F(y) − Φ(y))2 dy ≦ g max u |a u | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w 1, … wn ), (X w 1 , … X wn ) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of max u |au | to the bandwidth of the filter a is studied.
76 citations
TL;DR: In this paper, a queueing system with a single server and a finite number of types of customers, and the types of successive arrivals form a Markov chain, is considered.
Abstract: A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the n th interarrival time has a distribution function which may depend on the types of the nth and the n– 1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.
74 citations
TL;DR: In this article, the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1] were defined on convex hull of the space curve {(t,t2, …, tn ): 0 ≦ t ≦ 1.
Abstract: Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn ): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn ) of Mn , define Note that v −, v + depend only on C1, C2, …, Cn− 1; Vm only on cn ; We shall as notational convenience dictates and as will be apparent from the context regard v ± n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.
67 citations
TL;DR: In this article, the authors considered a more realistic formulation of the problem of finding an optimal policy for controlling the path of a space-ship as it moves towards its target and derived bounds on the form of the optimal policy.
Abstract: This paper is a sequel to [1] and considers a more realistic formulation of the same question: that of finding an optimal policy for controlling the path of a space-ship as it moves towards its target. The difference here is that we no longer suppose there is an infinite quantity of fuel, always available at a fixed price, for modifying the current direction of motion. This complicates the problem of reducing the final miss distance, by introducing an extra variable. As before, we shall be particularly concerned to find a control procedure which always minimizes the mean square terminal miss. From the theoretical point of view we are also interested to see whether the techniques used to approximate the optimal policy can be extended, and how far we shall be forced to adopt a new approach. Results are derived which provide bounds on the form of the optimal policy. The derivation depends on a comparison technique whose validity is intuitively obvious, but which is still only a conjecture. However, further confirmation is obtained in the quadratic case from asymptotic expansions giving the form of the solution both when the space-ship is far away from its target and during its final approach. The sequential decision problem can be specified in terms of three variables, two of which represent both the time and the information obtained so far about the final miss p. The assumptions concerning this accumulation of information and its conversion to a convenient form were discussed in ([1], Section 2). As a result, the available information always leads to a posterior distribution for p of the normal form A' (y, s), where the variance s is a known, strictly decreasing function of time which vanishes at the terminal instant and the mean y is determined by the corresponding observed state IY(s) of a well-defined stochastic process. Then roughly speaking, s can be regarded as the time
59 citations
TL;DR: In this paper, a single server queueing system with Poisson input is considered and the time dependence of the queue size process is studied, and limiting as well as transient results are given.
Abstract: Summary A single server queueing system with Poisson input is considered. There are a finite number of types of customers and the service time of the nth customers depends on the types of the nth and the (n - 1)th customers. The time dependence of the queue size process will be studied, (it will be clear how the methods of the paper can be applied to other processes of interest,) and limiting as well as transient results will be given.
58 citations
TL;DR: In this article, a man owns a commodity, e.g., a house, which is for sale. Offers at varying amounts are coming in every now and then. At each offer he must decide whether to accept it or wait for a better one.
Abstract: Imagine a man owning a commodity, e.g., a house, which is for sale. Offers at varying amounts are coming in every now and then. The longer he postpones selling the more he loses because of deterioration, interest losses, or the like. At each offer he must decide whether to accept it or wait for a better one. (A more picturesque example would be that of a girl scrutinizing successive suitors.)
47 citations
TL;DR: In this article, an asymptotically unbiased, and to that extent plausible, estimate of 4r f (λ ) is the periodogram of a weakly stationary stochastic process with spectral density f(λ).
Abstract: Suppose x 1 , …, x N are indefinitely many observations on a stochastic process which is weakly stationary with spectral density f(λ), – π ≦ λ ≦ π. An asymptotically unbiased, and to that extent plausible, estimate of 4r f ( λ )is the periodogram
Yet the periodograms of processes which possess spectral densities are notoriously subject to erratic behavior.
46 citations
TL;DR: In this paper, it was shown that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. But the central limit theorem for the index V of the rth upper record is not satisfied.
Abstract: If A, denotes the waitingtime between the (r - 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that A,. satisfies the weak law of large numbers and a central limit theorem. This theorem supplements those of Foster and Stuart [3] and R6nyi [5], who investigated the index V, of the rth upper record. Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of log V, is very inadequate for approximation purposes--Barton and Mallows [1].
TL;DR: In this article, the authors considered the asymptotic behavior of the mean duration of the epidemic and the limits of the transition probabilities p ij ( t ) as t → ∞.
Abstract: The epidemic model considered here, first given by Bartlett (see for example [2]), provides for the immigration of new susceptibles and infectives, as well as describing the spread of infection to susceptibles already present and the removal of infectives. The epidemic curve, relating the numbers of susceptibles and infectives, has been studied for certain cases by Bartlett [1], Kendall [6] and others, and provides a motivation for the results given here. With the aid of criteria given by Reuter [8], [9], the main question considered is the asymptotic behaviour of the mean duration of the epidemic. The behaviour of the limits of the transition probabilities p ij ( t ) as t → ∞ is also investigated.
TL;DR: For the simple stochastic epidemic with arbitrary initial distribution, this article provided simple iterative solutions of special systems of differential-difference equations, in which the initial distribution of uninfected susceptibles and the number of infectives was assumed to be arbitrary but the total population size bounded.
Abstract: We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additionalsteps that would sum proportionally-weighted conditional results. Let x(t) and x'(t) denote n by 1 column matrices whose ith row elements are the real-valued differentiable function xi(t) and its derivative xi(t), respectively,
TL;DR: In this paper, the authors defined the behavior of the epidemic by the number r of susceptibles and s of carriers at any given time t and the probabilities of transitions during the subsequent short interval of length st are given by.
Abstract: into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease. Mathematically the behaviour of the epidemic is defined by the number r of susceptibles and s of carriers at any given time t and the probabilities of transitions during the subsequent short interval of length st are given by
TL;DR: Theorem 2 of as mentioned in this paper is an extension of Theorem 4 of [8] for the case of nonnegative summands of various limit results of classical renewal theory, and it can be seen as a generalization of the result of Feller [6], 359-360.
Abstract: M(x) = max[kj Mk _ x]. M(x) + 1 is then the first passage time out of the interval ( oo, x] for the random walk process S,. In this paper we shall concern ourselves with just those cases in which M(x) is a proper random variable with EM(x) < co and, when suitably normed, possesses a limit distribution as x -) co. It will be shown that M(x) can possess such a limit if and only if the random variables Xi belong to the domain of attraction of one of a certain group of stable laws and the limit law will be obtained under these circumstances. This result (Theorem 2) constitutes a generalization from the case of non-negative summands of various limit results of classical renewal theory (see for example Feller [6], 359-360). The significance of this type of generalization has previously been explored in Heyde [8]; Theorem 2 of this paper is in fact an extension of Theorem 4 of [8]. In order to obtain the limit distribution mentioned above we need a version of
TL;DR: In this article, it was shown that the best linear predictor of non-stationary multivariate time series from noise distorted observations is a limit of the generating function of the stationary analogue, where the difference function has elementary divisors.
Abstract: : The paper represents a generalization of one of the main theoretical results of my Ph.D. thesis. The work is an outgrowth of work first begun by E. J. Hannan and a correct 'conjecture' by P. Whittle. The main theorem of this paper proves the existence of, and gives an explicit formula for, the asymptotic best linear predictor of a certain type of non-stationary multivariate time series from noise distorted observations. The non-stationarity arises from the fact that the signal satisfies a difference equation, which when considered as a polynomial, has only elementary divisors. The proof is accomplished by showing, through Hilbert space and harmonic analysis methods, that the generating function is a limit of the generating functions of the stationary analogue; that is, where the difference function has elementary divisors. Finally, it is shown that this asymptotic generating function exactly predicts null solutions to the difference equation. The proof is direct and due to E. J. Hannan.
TL;DR: In this paper, it was shown that a biharmonic function on a bounded open subset of N-dimensional Euclidean space EN, N ≧ 1, and a separable Brownian motion starting at a point x ǫ R is polyharmonic of order k + 1 on R (i.e., Δk + 1 u = Δ(Δk u) = 0 on R).
Abstract: Let R be a bounded open subset of N-dimensional Euclidean space EN, N ≧ 1, let {xt : t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex [g(xτ )] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δk g(X Δ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1 u = Δ(Δk u) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].
TL;DR: In this article, the authors consider the problem of extending queueing theory to more complex systems of servers and propose a model suitable for the analysis of assembly lines and similar structures, and show that a reasonable model would assume:
Abstract: One of the troublesome problems in the extension of queueing theory to more complex systems of servers is the development of models suitable for the analysis of assembly lines and similar structures. Results in special cases are given in [2], [5], and [6]. A reasonable model would assume:
TL;DR: In this article, the problem of the propagation of a virile mutant in a population was discussed, and a general class of partial differential equations which describe simultaneous growth and diffusion processes was investigated.
Abstract: A number of years ago, R. A. Fisher discussed the problem of the propagation of a virile mutant in a population. At about the same time, Kolmogorov, Petrovsky, and Piscounoff, whom we shall refer to as KPP, investigated a general class of partial differential equations which describe simultaneous growth and diffusion processes.
TL;DR: In this article, the service time of the customer arriving at a given location was assumed to be independent and identically distributed with distribution function B(t) with the assumption that all interarrival times and service times are independent.
Abstract: on = t, t,1, n = 2, 3, ..., are assumed to be identically distributed and independent random variables with distribution function A(t). By rn, n = 1,2, ..., we denote the service time of the customer arriving at t,; the variables rl,,2,"". will be assumed to be independent and identically distributed with distribution function B(t). Moreover it is assumed that all interarrival times and service times are independent. Concerning A(t) and B(t) we assume that
TL;DR: In this paper, the authors derived conditions under which a unimodal incidence curve can result, and expressions for the expected total epidemic size, under the assumption of a finite population of susceptibles.
Abstract: This paper applies the constant population approximation to the study of epidemics which involve more than a single type of infective. An example of this would be a situation in which both clinically infected individuals and subclinically infected individuals or carriers are present. We derive equations for the expected numbers of clinically infected individuals and carriers at any time t for the model with zero latent period and infectious periods having negative exponential distributions. From these equations we derive conditions under which a unimodal incidence curve can result, and expressions for the expected total epidemic size. The equations describing the course of an epidemic are analytically intractable except in certain simple cases, [1]. When one examines the reason for the mathematical difficulties that arise, one finds that the principal stumbling block is the assumption of a finite population of susceptibles. It can be argued, however, that in modern societies epidemics very rarely menace an entire population, and that observed epidemic sizes are usually much smaller than the total susceptible population. This is due to a variety of reasons, among which are public health measures and good communication facilities available to a modern society. In consequence it is reasonable to try to bypass some of the mathematical difficulties inherent in the theory of epidemics by assuming a susceptible population whose size does not change through the course of the epidemic. Bailey [2], Morgan [3], and Williams [4] have discussed in some detail the theory of epidemics using the constant population approximation, although such an approximation was used by several authors earlier, [5], [6], [7]. The theory developed so far has dealt mainly with epidemic processes in which there is only a single type of infective and a single type of susceptible. Recently Gart [8] has considered a model for epidemics involving more than one type of susceptible. It is the purpose of this paper to analyze the development in time of epidemics which involve more than a single type of infective individual. This Received in revised form 3 January 1967. 257 This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms 258 HUGH M. PETTIGREW AND GEORGE H. WEISS problem, in which there may be several types of infectives, is suggested by diseases in which carriers are important, [9]. In the present paper we derive equations for the expected numbers of carriers and clinically infected individuals at any time t, in the case of a zero latent period and negative exponential distributions of infectious periods. From these equations we will give the conditions under which an epidemic arises from the introduction of a bearer of the disease, i.e., the conditions under which the reporting curve increases initially. These conditions are of some theoretical interest since Bailey has shown that in the infinite population approximation to an epidemic with only one class of susceptibles and one class of infected individuals, no initial increase in the reporting curve can occur. Finally, we derive expressions; for the expected total epidemic size. Let us consider a homogeneously mixing population which, in the present approximation, can be characterized by four parameters, yi(t), y,(t), zi(t), zc(t). These are, respectively, the number of clinically infected individuals, the number of carriers, the cumulative number of removals of infectives, and the cumulative number of removals of carriers, all evaluated at time t. The probability that a single susceptible individual will become clinically infected in (t, t + dt) is assumed to be fli(y, + yc)dt, and the probability that a susceptible will become a carrier is assumed to be Ic(yi + ye)dt, where both l's are assumed to be constant and it is assumed for simplicity that carriers and infectives are equally infectious. The constancy of these p's is the principal assumption of the present theory; in the more detailed stochastic theory discussed by Bailey [1] the f's are proportional to the number of susceptibles as well. In the simplest model, we assume that the latent period is zero, i.e., that newly infected individuals are immediately capable of infecting others, and that the infectious period is a random variable with a negative exponential distribution. The rate parameter appearing in the distribution for clinically infected will be denoted by vi and that appearing in the distribution for carriers will be denoted by v,. In order to describe the stochastic process we introduce a set of probabilities p(r,,r2, r3, r4, t) defined by p(r, t) = Pr{y,(t) = r, yc(t) = r2, z(t) = r3, zc(t) = r4}. By our assumptions the p(r, t) satisfy ap = fl (r1 + r2 1)p(r, 1,r2,r3, r4, t) + fl(r, + r2 -1)p(r1,r2 1,r3, r4, t) (1) + vA(r1 + 1)p(r + 1,r2, r 1,r4)+ vc(r2 + 1)p(r, r2 + 1, r3,r4 1) [(fl + ic + vi)rl + (fl + flPc + ve)r2] p(r, t). This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms Epidemics with carriers: the large population approximation 259 The moment generating function (2) M(01, 02, 03 , 0,t) = M(O, t) = E{eo'Y'+2yYc+03zi+042C} therefore satisfies aM M = [f#,(ee I1) + flc(e02 1) + vi(ee, +03 1)] (3) aM + [fl(ee' 1) + flc(e02 1) + vc(e-02+4 1)] The interesting features of the development of the epidemic can be determined by analyzing the properties of the mean values Lpi(t) = E{yi(t)}, pc(t) = E{yc(t)} (4) ow(t) = E{z,(t)}, wc(t) = E{zc(t)}. The equations for these mean values are easily determined by equating coefficients of the O's on both sides of Equation (3). They are Pi = (/01v-i)it?+fliltc (5) Pc = flcpi+ (fc v)Pc
TL;DR: In this article, the authors define a recurrent event e which involves the occurrence of a unique pattern of successes (S) and failures (F), the final element of which is the result of the nth trial.
Abstract: On a sequence of Bernoulli trials, the definition of a recurrent event e involves the occurrence of a unique pattern of successes (S) and failures (F), the final element of which is the result of the nth trial. Success runs are the best known of such recurrent events, but Feller (1959, §13.8) mentions more complicated patterns, among which two types may be distinguished. The simpler involves a single more complex pattern such as SSFFSS; the second type involves a set of alternative events defining e, which is said to occur when any one of the alternatives occurs at trial number n. Thus if e stands for “either a success run of length r or a failure run of length ρ”, there are two alternatives in the set; the problem is elementary because the component events are “non-overlapping”.