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This is an extension of our previous paper [4] on the queue GI/Hm/s. In that paper we have derived results for the actual waiting time, the number of customers in the system at arrival epochs and the number of customers during a busy cycle. Here we obtain results for the virtual waiting time and the number of customers in the system at arbitrary times.

In a previous paper, the author has described a method for obtaining envelope processes for oscillatory stochastic processes. These are processes which can be represented as the output of a time-varying linear filter whose input is a stationary process. It is shown in this paper that the proposed definition of the envelope process may not be unique, but may depend on the particular representation of the oscillatory process chosen. It is then shown that for a class of oscillatory processes which is of particular interest, the class of transient processes, there is a class of natural representations which all lead to a unique envelope process.

This paper discusses infinite server queues whose input is a Phase Type Renewal Process. The problems of obtaining the transient and steady-state distributions and moments of the queue length are reduced to the solution of certain well-behaved systems of linear differential equations. Sample computations are provided with as many as ten phases. The paper contains some useful explicit formulas and also discusses the interesting special case where the service time is also of phase type. The Phase Type Distributions include a wide variety of models such as generalized Erlang, hyperexponential (mixtures of a finite number of exponentials) as very special cases and possess great versatality in modeling a number of interesting qualitative features such as bimodality.

Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such as ϕ -irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measure π for general space chains, and for π to have a finite f-moment ∫π (dy)f(y), where f is a general function. In the case f ≡ 1 these cover the question of finiteness of π itself.In this paper we show that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain.The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible.

The matrix-geometric results of M. Neuts are extended to ergodic row-continuous bivariate Markov processes [J(t), N(t)] on state space B = {(j, n)} for which: (a) there is a boundary level N for N(t) associated with finite buffer capacity; (b) transition rates to adjacent rows and columns are independent of row level n in the interior of B. Such processes are of interest in the modelling of queue-length for voice-data transmission in communication systems. One finds that the ergodic distribution consists of two decaying components of matrix-geometric form, the second induced by the finite buffer capacity. The results are obtained via Green's function methods and compensation. Passage-time distributions for the two boundary problems are also made available algorithmically. MATRIX-GEOMETRIC; GREEN'S FUNCTION; COMPENSATION In a previous paper [10] a theoretical treatment of finite row-continuous bivariate Markov chains B(t)= [J(t), N(t)] was developed, providing an algorithmic basis for finding their ergodic distributions and associated passagetime moments. The continuous-time chain B(t) with state space = {(j, n):'0 j J, 0 n N} was described as row-continuous in the sense that the marginal process N(t), indexed by row coordinate n, changed at transition epochs by at most 1. In the present paper we restrict our discussion to those row-continuous chains for which the transition rate matrices, v°, vn, v, describing rates local to row n, are independent of n for each 1 _ n 5 N 1. For n = 0, one has vo = 0, and for n = N, v = 0. Such processes may be described as row-homogeneous, row-continuous processes modified by two retaining boundaries, as for earlier similar univariate contexts [4]. This research was conducted at the MIT Laboratory for Information and Decision Systems with partial support provided by the USAF OSR Grant Number AFOSR-79-0043, U.S. Air Force Geophysics Laboratory Grant Number F19628-80-C-0003, and the National Science Foundation Grant Number NSF/ECS 79-19880. © Applied Probability Trust 1988

In previous papers (1976), (1977a), (1979) limit theorems were obtained for the classical Ising model, and these provided the basis for asymptotic inference. The present paper extends these results to more general Ising models. In two and more dimensions, likelihood inference for the thermodynamic parameters (i.e. the interaction energies) is effectively impossible. The problem is that the error in locating critical and/or confidence regions is as large as their diameters. To remedy this requires more accurate characterizations of the partition functions, but these seem unlikely to be forthcoming. Besag's coding estimators for these parameters are inverse hyperbolic tangents of the roots of simultaneous polynomial equations and hence avoid such location errors. However, little is yet known about their sampling characteristics. Finally, likelihood inference for lattice averages (an alternative parametrization) is straightforward from the limit theorems.

This paper aims to demonstrate that the general Crump–Mode–Jagers branching process may be used in a natural way to model the asymmetric growth of budding yeast cells. The models obtained are generalisations of the deterministic model proposed by Hartwell and Unger (1977): all the results that are derived in that paper may be obtained using branching-process methods, but such methods also apply when account is taken of the biologically obvious fact that the various phases of the cell growth are of random rather than fixed duration. In their full generality, branching processes involve more parameters than can be estimated by experiment, but we present below a special case in which this problem is not likely to arise. A recent paper, Lord and Wheals (1980), discusses more of the biological background than is appropriate here. In the present paper, we show how certain statistical procedures for our model may be developed.

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in the M/GI/s queue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived for M/GI/s/k with finite and infinite k. Similar approximations are possible for M/GI/s/k (k < +∞).

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index. Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions. The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index.Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions.The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4]. In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γ n (p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n (s) the number of vertices in the union of all those components of Γ n (p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n (s) and the limit distribution of ρ n (s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.

This paper is concerned with the characterization of the cumulative pensionable service over an individual's working life that is made up of random lengths of service in different employments in a given industry, under partial coverage, transferability, and a uniform vesting rule. This characterization uses some results that are developed in the paper involving a functional and cumulative constrained sojourn times (constrained in the sense that if a sojourn time is less than a given constant it is not counted) in semi-Markov processes.

In a previous paper, Natvig [4], we suggested a new measure of the importance of a component in a coherent system and derived some of its properties. The measure is for the case of components not undergoing repair proportional to the expected reduction in remaining system lifetime due to the failure of the component. In the present paper, we arrive at the whole distribution of this reduction in remaining system lifetime. Furthermore, for the case where components have proportional hazards, and are not repaired, a speculation of another measure is given. This measure is proportional to the derivative of the expected total lifetime of a new system with respect to the inverse of the component's proportional hazard rate.

The purpose of this paper is to give some very simple applications of martingales to epidemics. The results are all connected with stopping times T (for instance the classical end of epidemic) and include the expression of the joint generating function Laplace transform of and and simple relations between moments of these three variables. (Here Xt and Yt respectively denote the numbers of susceptibles and carriers.) We also give several relations between different types of epidemics. Although this paper only deals with Downton's model, some of the methods are still valid for more general models.

In a previous paper a branching Poisson process model was derived to explain deviations from a Poisson process in computer failure patterns. Physically the deviations arise because an attempt to repair a computer is not always successful and the failure recurs a relatively short time later. In this paper we discuss the implications of this model for the use and maintenance of computers.

This paper treats the well known model of sib mating of individuals with autosomal genes; a single locus is studied. The general derivations are valid for any number of alleles. At the end of the paper, some simple examples are included; here several familiar results concerning models for two alleles are obtained as special cases of the general theory.

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

This paper deals with the simple Galton-Watson process with immigration, {Xn} with offspring probability generating function (p.g.f.)F(s) and immigration p.g.f.B(s), under the basic assumption that the process is subcritical (0

The purpose of this paper is to construct a theory of the amount of information provided by an experiment which does not rely on what Good (1962) has termed the modern Bayesian principle that it is legitimate to use the axioms of probability even when this involves the use of probabilities of hypotheses. In this respect the theory of this paper differs from the Lindley (1956), Mallows (1959) and Good (1960) each of which is written from a Bayesian viewpoint. Lindley (1956) expresses the opinion that Bayesian ideas would seem to be necessary to the development of a theory of the amount of information provided by an experiment and it is of interest therefore to determine how far such a theory may be developed without Bayesian ideas. There is further a need to use such a theory to examine how prior knowledge can be expressed quantitatively, and used in accordance with that theory.

This paper applies the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials. The model characterizes the patient as being in one of a finite number of states at any given time with an arbitrary probability distribution to describe the length of stay in a state. Transitions between states are assumed to be chosen according to a stationary finite Markov chain.Other attempts have been made to develop stochastic models of clinical trials. However, these have all been essentially Markovian with constant transition probabilities which implies that the distribution of time spent during a visit to a state is exponential (or geometric for discrete Markov chains). Markov models need also to assume that the transitions in the state of a patient depend only on absolute time whereas the semi-Markov model assumes that transitions depend on time relative to a patient. Thus the models are applicable to degenerative diseases (cancer, acute leukemia), while Markov models with time dependent transition probabilities are applicable to colds and epidemic diseases. In this paper the Laplace transforms are obtained for (i) probability of being in a state at timet, (ii) probability distribution to reach absorption state and (iii) the probability distribution of the first passage times to go from initial states to transient or absorbing states, transient to transient, and transient to absorbing. The model is applied to a clinical study of acute leukemia in which patients have been treated with methotrexate and 6-mercaptopurine. The agreement between the data and the model is very good.

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

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

This paper applies the invariance principle to the problem of testing a distribution on a compact homogeneous space for uniformity. The notion of using a reduction by invariance in such a situation is due to Ajne[1], who considers tests invariant under rotation on a circle. In his paper, he derives the distribution of the maximal invariant and gives the general form of the most powerful invariant test for uniformity on the circle.

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]-[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry. GEOMETRICAL PROBABILITY; RANDOM LINES; RANDOM POLYGONS; PLANE OF CONSTANT CURVATURE; HYPERBOLIC PLANE; CONVEX DOMAINS; POISSON LINE PROCESS

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.

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

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.

Summary In this paper a stochastic calculus, based on limits in quadratic mean and in probability, is introduced and applied to some fundamental theorems of natural selection. The paper is divided into five principal sections. In Section 2 some elements of stochastic calculus are given, and in Section 3 the stochastic calculus is applied to the haploid case. Section 4 is devoted to the development of an analysis of variance structure applicable to a haploid population, and in Section 5 the results of Sections 3 and 4 are generalized to cover the diploid case. Finally, Section 6 is devoted to two specific examples in which the results of previous sections apply.

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.

A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full. The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results. 1. Description of the model The system of queues, discussed in this paper, consists of two units. Customers arrive at a first unit (I) according to a homogeneous Poisson process of rate 2. Their service times in unit I are independent, identically distributed random variables with common distribution function H(.). We assume that H(.) has a positive, finite mean a and we will denote the Laplace-Stieltjes transform of H( ) by h(s), Re s 0. Upon completion of service in unit I, all customers go on to a second unit (II) via a finite waitingroom. We assume that there can be not more than k customers in unit II and in the waitingroom at any time. If upon completion of service in unit I a customer finds the waitingroom full, then the unit I blocks until a service in unit II is completed. At that time he is allowed to enter the waitingroom. We assume that the service times in unit II are independent, identically distributed random variables with a negative-exponential distribution with mean 1/a. The service times in unit II are also stochastically independent of those in unit I and of the arrival process. The case k = 1, i.e., when no customers can wait between the two units, was studied by B. Avi-Itzhak and M. Yadin[1], T. Suzuki [11] and N. U. Prabhu [7]. These authors allow the service times in the second unit to have a general distribution. In the case of general k, we impose the requirement that these service Received 3 July 1967. Research supported in part by Office of Naval Research Contract NONR 1100(26). 123 This content downloaded from 157.55.39.217 on Mon, 18 Apr 2016 07:16:32 UTC All use subject to http://about.jstor.org/terms

This paper applies the invariance principle to the problem of testing a distribution on a compact homogeneous space for uniformity. The notion of using a reduction by invariance in such a situation is due to Ajne[1], who considers tests invariant under rotation on a circle. In his paper, he derives the distribution of the maximal invariant and gives the general form of the most powerful invariant test for uniformity on the circle.

Summary In a previous paper [4] the author studied the stochastic process {wn , n = 1,2, …}, recursively defined by with K a positive constant, τ1, τ2, … σ1, σ2, …, independent, nonnegative stochastic variables. τ1,τ2…, are identically distributed, and σ1,σ2,…, are also identically distributed variables. For this process the generating function of the Laplace-Stieltjes transforms of the joint distribution of Wn , σ2 + … + σ n and τ1 + … + τ n−1 has been obtained. Closely related to the process {wn , n = 1, 2,…} is the process {un , n = 1, 2,…} with {un = K + [wn + τ n − K]−, n = 1,2,…; these are dual processes. In the present paper we study the stationary distributions of the processes {wn , n= 1,2, …} and {un , n = 1,2, …}, and the distributions ot the entrance times and return times of the events “wn , n = 0” and “un = K” for some n, for discrete as well as for continuous time. For these events various taboo probabilities are also investigated. The mathematical descri ption of the processes {wn , n = 1,2, …} and {un , n= 1,2, …} gives all the necessary information about the time-dependent behaviour for the general dam model with finite capacity K, since the process {wn , n= 1,2, …} is the basic process for such dam models. In Sections 5, 6 and 7 the general theory is applied to the models M/G/1 and G/M/1. Complete explicit solutions are obtained for these models. The present theory also leads to new and important results for the queueing system or dam model G/G/1 with infinite capacity. For instance the joint distribution of the busy period (or wet period) and of the supremum of the dam content dunng this period is obtained.

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]–[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry.

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem. Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior. 1. Concepts and definitions We consider a queueing process with two service units, unit 1 and unit 2, and a single server. The server attends to the two units alternately according to some switching rule. A switching rule (Neuts and Yadin (1968)) is a rule describing how the server changes from one unit to the other. The server may change from one unit to the other either by a non-zero switching rule or by a zero switching rule. By a non-zero switching rule the server continues to serve in a unit until some upper number of consecutive services has been completed and then he switches to the other unit. By a zero switching rule the server stays in a unit until the queue in it becomes empty and then he switches to the other unit. In this paper we discuss a non-zero switching rule for unit 1 and zero switching rule for unit 2. Since the analysis of the non-zero switching rules for both units needs heavy notations, we omit the discussion of it here. The zero switching rules for both units is dealt with separately.

This paper is concerned with the distribution of "types" of individuals in an infinite population after indefinitely many nonoverlapping generations of random mating. The absence of selection and mutation is assumed. The probabilistic law which governs the production of an offspring may be asymmetrical with respect to the "sexes" of the two parents, but the law is assumed to apply independently of the "sex" of the offspring. The question of the existence of a limit distribution of types, the rate at which a limit distribution is approached, and properties of limit distributions are treated. Such questions have already received much attention in the genetic literature. Most of the limit results which have been obtained previously appear in the series of papers [5] [12] by Geiringer (1944-1949). These papers include results of earlier workers such as those of Jennings (1917), Robbins (1918) and Haldane (1930). With the exception of Geiringer's 1948 paper [8], all papers referred to in this paper assume that differences due to sex are absent. All of Geiringer's papers admit an arbitrary number of alleles at each locus. In [5], Geiringer (1944) obtains a limit theorem for haploid gametes, an arbitrary number of loci and linkage. Additional details in the case of three loci are given by Geiringer (1945) in [6]. A limit theorem is proved in Geiringer's 1947 paper, [7], for an arbitrary degree of autopolyploidy, one locus and chromosomal segregetion. In [8], Geiringer proves a limit theorem for haploid gametes, an arbitrary number of loci, and linkage with different recombination values for males and females. In [9], Geiringer (1949a) obtains the limit distribution and rate of convergence for diploid and triploid gametes, one locus and a general chromatid segregation; tetraploid gametes are considered briefly. In the second part of the two-part paper [10]"-[11], Geiringer (1949b, c) proves a limit theorem for an arbitrary degree of autopolyploidy, an arbitrary number of loci and chromosomal segregation with linkage. Paper [12] is a review paper in which Geiringer (1949d) summarizes the results then available, and indicates what remains to be done.

It has been demonstrated by Takaics in a series of papers and in his book [6] that combinatorial methods can be successfully applied to derive certain probability distributions in queueing processes. In this paper, we further illustrate the usefulness of combinatorial techniques and determine the stochastic law of the busy period in two queueing systems particularly involving batches. It may be of interest to note that queues involving batches have been dealt with in [3] and [7]. COMBINATORIAL THEORY; QUEUEING PROCESSES 1. Queueing model I Before describing the model, we give a combinatorial result which is required later. Consider a vector x = (x, -...,x,) with xi's integers and 0? X1 _ x2 ?-_ ...

In this paper we establish the central limit theorem for a class of stochastic partial differential equations and as an application derive this theorem for two widely studied population models: super-Brownian motion and the Fleming-Viot process.

Consider an absolutely continuous distribution on [0, ∞) with finite mean μ and hazard rate function h(t) ≤ b for all t. For bμ close to 1, we would expect F to be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance between F and an exponential distribution with mean μ, as well as between F and an exponential distribution with failure rate b. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

In this paper we use the Siegert formula to derive alternative expressions for the moments of the first passage time of the Ornstein-Uhlenbeck process through a constant threshold. The expression for the nth moment is recursively linked to the lower-order moments and consists of only n terms. These compact expressions can substantially facilitate (numerical) applications also for higher-order moments.

We introduce a notion of $k$th order stochastic monotonicity and duality that allows one to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put - call symmetries in option pricing. Our general $k$th order duality can be financially interpreted as put - call symmetry for powered options. The main objective of the present paper is to develop an effective analytic approach to the analysis of duality leading to the full characterization of $k$th order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put -call symmetries., Comment: To appear in Journal of Applied Probability 52:1 (March 2015)

The asymptotic behaviour of many locally branching epidemic models can, at least to first order, be deduced from the limit theory of two branching processes. The first is Whittle's (1955) branching approximation to the early stages of the epidemic, the phase in which approximately exponential growth takes place. The second is the susceptibility approximation; the backward branching process that approximates the history of the contacts that would lead to an individual becoming infected. The simplest coupling arguments for demonstrating the closeness of these branching process approximations do not keep the processes identical for quite long enough. Thus, arguments showing that the differences are unimportant are also needed. In this paper we show that, for some models, couplings can be constructed that are sufficiently accurate for this extra step to be dispensed with.

The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in R d . We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

In this paper we consider a one dimensional stochastic system described by an elliptic equation. A spatially varying random coefficient is introduced to account for uncertainty or imprecise measurements. We model the logarithm of this coefficient by a Gaussian process and provide asymptotic approximations of the tail probabilities of the derivative of the solution.

In this paper we work in the framework of a k-dimensional vector of log-linear risks. Under weak conditions on the marginal tails and the dependence structure of a vector of positive risks, we derive the asymptotic tail behaviour of the aggregated risk and present an application concerning log-normal risks with stochastic volatility.

In this paper we discuss an exponential integrator scheme, based on spatial discretization and time discretization, for a class of stochastic partial differential equations. We show that the scheme has a unique stationary distribution whenever the step size is sufficiently small, and that the weak limit of the stationary distribution of the scheme as the step size tends to 0 is in fact the stationary distribution of the corresponding stochastic partial differential equations.

In this paper we consider some nonstandard renewal risk models with some dependent claim sizes and stochastic return, where an insurance company is allowed to invest her/his wealth in financial assets, and the price process of the investment portfolio is described as a geometric Lévy process. When the claim size distribution belongs to some classes of heavy-tailed distributions and a constraint is imposed on the Lévy process in terms of its Laplace exponent, we obtain some asymptotic formulae for the tail probability of discounted aggregate claims and ruin probabilities holding uniformly for some finite or infinite time horizons.

Let X, B, and Y be the Dirichlet, Bernoulli, and beta-independent random variables such that X ~ D (a 0, …, a d ), Pr(B = (0, …, 0, 1, 0, …, 0)) = a i / a with a = ∑ i=0 d a i , and Y ~ β(1, a). Then, as proved by Sethuraman (1994), X ~ X(1 - Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. In this paper we introduce a new distribution on the tetrahedron called a quasi-Bernoulli distribution B k (a 0, …, a d ) with k an integer such that the above result holds when B follows B k (a 0, …, a d ) and when Y ~ β(k, a). We extend it even more generally to the case where X and B are random probabilities such that X is Dirichlet and B is quasi-Bernoulli. Finally, the case where the integer k is replaced by a positive number c is considered when a 0 = · · · = a d = 1.

Although the exact expressions for the extinction probabilities of the Interacting Branching Collision Processes (IBCP) were very recently given by Chen et al. [4], some of these expressions are very complicated; hence, useful information regarding asymptotic behaviour, for example, is harder to obtain. Also, these exact expressions take very different forms for different cases and thus seem lacking in homogeneity. In this paper, we show that the asymptotic behaviour of these extremely complicated and tangled expressions for extinction probabilities of IBCP follows an elegant and homogenous power law which takes a very simple form. In fact, we are able to show that if the extinction is not certain then the extinction probabilities {a n } follow an harmonious and simple asymptotic law of a n ∼ kn -αρ c n as n → ∞, where k and α are two constants, ρ c is the unique positive zero of the C(s), and C(s) is the generating function of the infinitesimal collision rates. Moreover, the interesting and important quantity α takes a very simple and uniform form which could be interpreted as the ‘spectrum’, ranging from -∞ to +∞, of the interaction between the two components of branching and collision of the IBCP.