Showing papers in "Journal of Applied Probability in 1969"

A new family of life distributions

Abstract: : A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.

Estimation for a family of life distributions with applications to fatigue

Abstract: : The estimation problem is studied for a new two-parameter family of life length distributions which has been previously derived from a model of fatigue crack growth. Maximum likelihood estimates of both parameters are obtained and iterative computing procedures are given and examined. A simple estimate of the median life is exhibited, shown to be consistent and then compared, favorably, with the maximum likelihood estimate. More importantly the asymptotic distribution of this estimate is shown to be within the same class of distributions as the observations themselves. This model, and these estimation procedures, are tried by fitting this distribution to several extensive sets of fatigue data and then some comparisons of practical significance are made.

The total progeny in a branching process and a related random walk

Abstract: This paper is a continuation of [1]. The techniques of [1] are used to get specific information about the distribution of the total progeny in a branching process. This distribution is also related to one which arises in a random walk problem.

Markov population processes

Abstract: The processes of the title have frequently been used to represent situations involving numbers of individuals in different categories or colonies. In such processes the state at any time is represented by the vector n = (n 1, n 2, …, nk ), where nt is the number of individuals in the ith colony, and the random evolution of n is supposed to be that of a continuous-time Markov chain. The jumps of the chain may be of three types, corresponding to the arrival of a new individual, the departure of an existing one, or the transfer of an individual from one colony to another.

Diffusion approximations in collective risk theory

Abstract: Collective risk theory concerned with random fluctuations of total assets of insurance company

Random secants of a convex body

Abstract: Let two points be taken at random in an n-dimensional convex body K, and let σ be the line joining them. The distribution of σ is found and compared with other distributions for random secants of K. More generally, if r + 1 ≦ npoints are taken in K, the distribution of the r-dimensional affine subspace containing them is computed. The results find application to the n-dimensional case of a problem of Sylvester.

Random paths through convex bodies

Abstract: Several mechanisms under which the randomness of straight line paths through convex bodies can arise are described. Some general results are given relating four of these mechanisms, and the corresponding distributions of the lengths of the straight line paths are found for the circle, the rectangle, and the cube.

The minimum of a stationary Markov process superimposed on a U-shaped trend

Abstract: 1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).) There is a dearth of particular examples of processes and trends for which the distributions of interest are known exactly. Such examples could give useful experience of the form of distribution to be expected in typical cases, and could serve as material on which to test out approximate methods. The object of the present paper is to provide an example of this kind. One process for which exact results are available in the trend-free case is the Ornstein-Uhlenbeck process, i.e., the stationary Gaussian Markov process X(t) generated by

A recursion theorem on solving differential-difference equations and applications to some stochastic processes

Abstract: A recursive procedure leading to the solution of triangular systems of differential-difference equations is presented. It is shown that the equations for special families of multi-dimensional pure birth processes and pure death processes can be represented as such a system. The theory is used to resolve a difficult problem of stochastic cross-infection among groups. Additional applications to problems in epidemic theory are discussed.

Used item replacement policy

Abstract: The block replacement policy, wherein items are replaced at regular intervals of time and on failure, is rather wasteful because sometimes almost new items are also removed. As an alternative a policy of replacement by new items at regular intervals of time and by used items on failure, is suggested. The consequences of this policy, called used item replacement policy, are studied for Erlangian and sub-exponential life-time distributions. The latter distribution which is the difference of two negative exponential distributions, does not seem to have received much attention in the literature so far.

Some results for infinite server poisson queues

Abstract: In the first section we show that both W(t), the number of customers being served at time t, and S(t) the number of customers who have completed service by time t, are distributed as compound Poisson laws. In the second section we derive, in the time homogeneous case (Gt,( ) = G(* ), Pt,( ) = P( ), M(t)= 2t) the limiting proportion of the time that the system is non-empty. In the third section we derive the distribution of the occupation time O(t), where O(t) is defined as the amount of time past t until the system becomes empty, under the assumption that no new customers are served after time t. In the fourth section we derive the distribution of the traffic time average T- j'W(t)dt. In the fifth section asymptotic properties of the traffic time average are derived in the time homogeneous case. These include almost sure and quadratic mean convergence as well as asymptotic normality. In [7] Shanbhag considered a special case of the above model, G, = G, P,( " ) = P( - ), and by solving a differential equation, derived the joint generating function of W(t) and S(t). His method, however, does not seem applicable to the present model unless some conditions (such as t-continuity) are placed on

A Correlated Queue

Abstract: A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

On the Maximum of Sums of Random Variables and the Supremum Functional for Stable Processes

Abstract: Let X i i =1,2,3,... be a sequence of independent and identically distributed random variable which belong to the domain of attraction of a stable law of index α. Wrie \(S_0=0,\quad S_n\sum olimits^{n}_{i}=1 \,{\mathbf{X}}_i, n\geqq 1,\) and \(M_n={\rm max}_{0\leqq k \leqq n} S_k.\) In the case where the X i are such that \(\sum olimits^{\infty}_{1}\, n^{-1}\mathbf{P}{\rm r}\left(S_n > 0\right) 0\right)=\infty\) (the case of oscillation of the random walk generated bu the S n) and the only result available with case α=2 (Erdos and Kac [5]) and the case where the X i themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit therome for M n in the case of oscillation. Specifically, if {B n,n=1,2,3,...} is a monotone sequence of constants such that B -1 n S n converges in distribution to the stable with charactistic function $$\begin{array}{ccc} {\rm exp}\left\{-\lambda \mid t\mid^{\infty}\left(1+\,i\beta \,{\rm sgn}\, t\, {\rm tan}\frac{\pi \alpha}{2}\right)\right\},\\ \lambda > 0,0 < \alpha \leqq 2, \beta = 0\, {\rm if} \,\alpha =1, \mid \beta \mid < 1\,{\rm if}\, \alpha < 1 \,{\rm if}\, 1 < \alpha \leqq 2,{\rm we\, shall\, find}\\ H\left(x\right)={\rm \lim \limits_{n\rightarrow \infty}}\, \mathbf{P}{\rm r}\left(\mathbf{B}^{-1}_{n} M_n\leqq x\right)\end{array}.$$ (1) In connection with the parameter restrictions, we note that the stable law with characteristic function (1) is one-sided if \(\alpha < 1, | \beta | = 1\) (e.g., Lukacs [12], page 106) so that the random walk generated by the S n does not oscillate ([9], Lemma). The case \(\alpha = 1, \beta eq 0\) introduces a normalization complication and is not amenable to tratment by the methods of this paper.

A theory of dams with continuous input and a general release rule

Abstract: Consider an infinite capacity dam in which the input and release occur continuously in time. Write X(t) for the total input up to time t starting from X(O) = 0 at t = 0. Let Z(t) be the content of the dam at time t and R(u) (0 ? u < oo) a release function such that in any interval of time (t, t + dt), the amount of water released is R(Z(t))dt + o(t) for any bounded realisation of the process {Z(t)}. Thus R(u) can be regarded as a "rate of release". A heuristic description of the resulting process is given by the equation

Negative binomial processes

Abstract: This paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk , and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.

Asymptotic properties and equilibrium conditions for branching poisson processes

Abstract: Some previously obtained asymptotic results for branching Poisson processes are extended and sharpened. It is shown that under rather general conditions the number of events in both the transient and the equilibrium processes, suitably normalized, have a unit normal distribution. Finally, unique initial conditions are derived for the equilibrium process.

An Age-Dependent Branching Process with Correlations Among Sister Cells

Abstract: In this note we consider an age-dependent branching process in which the life-spans of sister cells are correlated as well as the numbers of offspring of sister cells, but otherwise cells live and reproduce independently. One might surmise that in many populations there would be some positive correlation among siblings due to similar characteristics inherited from parents. Data of Powell (1955) seem to corroborate this conjecture. Powell's data indicate that in certain bacterial populations the life-spans of sister cells have significant positive correlations while the correlation between the life-spans of mother and daughter cells is negligible. More recently positive correlations have been observed in cell populations among the life-spans of first cousins and also among second cousins (Kubitschek (1967)). However, the model presented in this note permits correlations only among siblings.

The proportions of individuals of different kinds in two-type populations. a branching process problem arising in biology

Abstract: Summary Consider an age-dependent branching process with two types of individuals. Suppose that individuals of one type beget children of both types, whereas those of the other type can only give birth to individuals of their own kind. This paper is a study of the relation between two random functions occurring in such processes starting from an ancestor of the first type, the two functions being the numbers of individuals of the two kinds. Under weak assumptions it is shown that the random proportion of individuals of one type converges as time passes, in mean square as well as almost surely to a non-random limit, easily determined in terms of the reproduction laws and life-length distributions of the process. 1. The biological background The reproduction of biological populations consisting of two types of individuals often displays an irreversibility property in the sense that individuals of one type might give birth to descendants of both kinds, whereas those of the other type can beget individuals only of their own kind. For example, if diploid cells in, say, a tumour, are looked upon as individuals of a first type and the cells of higher ploidy are called individuals of a second type, then, due to endomitosis, the population of cells in the tumour has this irreversibility property. The development of the relative proportions of individuals during the evolution of such a population has naturally become the object of investigations, mostly experimental but also mathematical. Thus, Birger Jansson (1967) has constructed a deterministic model analyzing very successfully certain experiments with polyploidy in an Ehrlich mouse ascites tumour. In the study of such tumours it has been observed that irrespective of the initial

On large sample sequential analysis with applications to survivorship data

Abstract: Although his work on the application of invariance concepts to the sequential testing of composite hypotheses is better known, Cox (1963) has also outlined a large sample approach to the same problem. His method is based on Bartlett's (1946) recognition that the sequence of maximum likelihood estimates (MLE) of the parameter of interest, calculated from an increasing number of observations, resembles asymptotically a random walk of normally distributed variables. However, the large sample theory needed to justify this approach rigorously is left largely implicit. At the end of his paper, Cox suggests that these msthods may be extended to yield a sequential comparison of survival curves (Armitage (1959)), a suggestion which has been reiterated as a research problem in the monograph of Wetherill (1966). In this paper we first present a general theoretical framework in which the asymptotic validity of a wide class of large sample sequential tests may be examined, thus making explicit the justification for Cox's approach. The results of this section are fairly straightforward consequences of the increasingly well known theory of convergence in distribution for random variables which take values in separable metric spaces. Next we illustrate the theory by re-examining Cox's results on the comparison of two binomial parameters. Finally, and of greater consequence from the practical point of view, we present a large sample solution to the problem of the sequential comparison of exponential survival

Genetic correlation in the stepping stone model with non-symmetrical migration rates

Abstract: Kimura (1953) proposed a “stepping stone” model for the study of the genetic structure of a subdivided population. In this model, it was assumed that a population consists of infinitely many colonies located at grid points of an n -dimensional lattice and that each colony exchanges individuals with neighboring colonies in each generation, and also receives immigrants as random samples from the whole population. The former type of migration has been called “short range migration” and the latter type “long range migration”. Later, Kimura and Weiss (1964), and Weiss and Kimura (1965) have obtained formulas for the genetic correlation and variance between colonies for general cases of the model, assuming that the short range migration is symmetrical in each fixed direction. The purpose of the present report is to extend the results of Kimura and Weiss to cover situations where the restriction of symmetry of short range migration is removed, so that migration rates in opposite directions need not be equal. I believe that in nature there are cases which require the model presented in this report. For example, consider a plant population distributed along a river, or on a plain where the wind at the time when the seeds are scattered is stronger in one direction than in others. For animals and plants it is often true that the centre of habitat is more densely populated than the marginal regions where the environment is less suitable for the species. In such a case the migration rate toward the outside from the centre is larger than that in the opposite direction. As in other theories of population genetics, we will assume that the size of each colony is determined by the carrying capacity of the environment and is not affected by the migration rate. Thus we assume that in each generation each colony produces many more gametes than those which contribute to the next generation, and, among those many gametes, a certain number, say 2 N, are chosen from various colonies to form the individuals of a particular colony.

The total waiting time in a busy period of a stable single-server queue, II

Abstract: This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period 2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B (·). Let X' i denote 3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X' i , prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X' i in M / M /1 as the ratio of two Bessel functions.

Embedding submartingales in wiener processes with drift, with applications to sequential analysis

Abstract: Skorokhod (1961) demonstrated how the study of martingale sequences (and zero-mean random walks) can be reduced to the study of the Wiener process (without drift) at a sequence of random stopping times. We show how the study of certain submartingale sequences, including certain random walks with drift and log likelihood ratio sequences, can be reduced to the study of the Wiener process with drift at a sequence of stopping times (Theorem 4.1). Applications to absorption problems are given. Specifically, we present new derivations of a number of the basic approximations and inequalities of classical sequential analysis, and some variations on them including an improvement on Wald's lower bound for the expected sample size function (Corollary 7.5).

Random selective advantages of a gene in a finite population

Abstract: A problem of interest to many population geneticists is the process of change in a gene frequency. A popular model used to describe the change in a gene frequency involves the assumption that the gene frequency is Markovian. The probabilities in a Markov process can be approximated by the solution of a partial differential equation known as the Fokker-Planck equation or the forward Kolmogorov equation. Mathematically this equation is where subscripts indicate partial differentiation. In this equation, f(p, x; t) is the probability density that the frequency of a gene is x at time t, given that the frequency was p at time t = o. The expressions MΔX and VΔx are, respectively, the first and second moments of the change in the gene frequency during one generation. A rigorous derivation of this equation was given by Kolmogorov (1931).

The identifiability of mixtures of distributions

Abstract: This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)), where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).

Highway Delays Resulting From Flow-Stopping Incidents

Abstract: Modern highways, particularly the freeways of large cities, carry a considerable volume of traffic during certain times of day. Thus if any interruption or retardation of flow occurs, a large reaction in the shape of a monumental and timeconsuming traffic jam soon appears. For example, when an accident or mechanical breakdown gives rise to a severe flow restriction or stoppage, many other vehicles may be quickly halted, and remain stopped until the impediment is cleared away. In addition, the flow of traffic may be slowed considerably even after the original stoppage is removed owing to the existence of a queue. Consequently, vehicles that arrive long after the original restriction is removed experience prolonged, and sometimes seemingly inexplicable, delay. Our purpose is to develop a probability model for the situation described, with the aim of estimating the consequence of a temporary flow restriction. Various measures of (in)effectiveness are worth considering. We consider primarily the total vehicle-hours waited while the jam dissipates; the latter may roughly measure total social cost. The total number of vehicles involved in the jam is also of interest, as are other figures of merit. Since accidents, as well as other service stoppages, often seem to occur when demand is high, a simple model for rush-hour and other over-saturated situations is suggested and examined. The model permits the consideration of a changing flow rate into the congested area. The latter change may be either the natural course of the rush hour development, or possibly the effect of a detection system intended to relieve the congestion by re-routing flow to another part of the traffic network.

A birth, death and migration process

Abstract: A model proposed by Bailey (1968) for migratory individuals which reproduce according to a simple birth-death process is generalized to include time dependent birth and death rates. 1. The model Let I denote the set of all integers. We assume that an individual of the population can be at any point x e I. The individuals develop in numbers according to the generalized birth-death process of Kendall (1948), with birth rate A(t) and death rate #I(t), 0 < t < oo. They also undergo changes in position according to a simple random walk in continuous time, i.e., if an individual is at a position x e I at time t, then in (t, t + At) it may move to (x + 1) or to (x - 1), the probability of each shift being oAt + o(At). Here A(t), g(t) are functions of time only and a is a positive constant. The position of the newly born individual is, at birth, the same as that of its parent. The individuals act independently of each other and the probability of a multiple event in (t, t + At) is o(At). The above assumptions determine the stochastic process which is a multiplicative point process, (cf. Moyal (1962)). The individuals of the population are assumed to be indistinguishable. The development of the process can be described, as in Adke and Moyal (1963) or Adke (1964), in terms of the generating function: 00oo