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In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles 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.

We assume that the number of claims occur according to a renewal process and treat the number of claims that occur and are reported in a certain time interval as a renewal process with random displacements. We obtain a renewal equation for the mean value function and an integral equation for the Laplace transform of the distribution of the claims that are reported. We also give asymptotic expressions for the mean value function and calculate the generating function in the case where the renewal process is a Poisson process. This matter is a part of the IBNR-problem in insurance mathematics. INCURRED BUT NOT REPORTED CLAIMS; RENEWAL PROCESS WITH RANDOM DISPLACEMENTS ntroduction At a given point of time one has only partial knowledge of an insurance business. For instance, there is always a time lag between the occurrence of a road accident and the reporting of and full knowledge of the claim amount due to that accident. This lack of information generates many questions. One, which this paper touches, is: if we observe M(t) occurred and reported claims during the interval (0, t), how many claims occurred in that period? On the accounting date one has to answer this question and make an estimate of the number of claims which occurred during the last year. The starting point will be assumptions of the number of claims that occur in (0, t). Since the days of Filip Lundberg one has assumed that this is described by a Poisson process. Several authors have tried to generalize Lundberg's model. Without intending to make a complete list we mention Ove Lundberg, Sparre Andersen and Jan Grandell. We will pursue the course of Sparre Andersen who in 1957 presented a paper [1] where the epochs of claims form a renewal process. As to the distribution of the time lag between the occurrence of a claim and the report of that claim there are several possible ways. We will make the assumption that the lags are independent and equally distributed, which seems to be a reasonable assumption as a first approximation at least when the claim amounts are of a fairly equal size. Received in revised form 21 March 1973. 382 This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms A stochastic modelfor time lag in reporting of claims 383 A model We assume that Feller [3], ch. XI is known. Assume that claims occur according to a delayed renewal process on [0, oo). Number the claims 0, 1, 2, -.. in the order in which they occur. So is a stochastic variable representing the time between the starting point zero and the occurrence of the first claim and So has the distribution function Fo. T, is a stochastic variable representing the time between the occurrence of claim number n 1 and claim number n, n = 1,2, ---. The variables TI, T2, -" have the same non-arithmetic distribution function F, with F(O) = 0. Put P, = E T, and aF = var T. and assume that ao 1, This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms 384 JAN-ERIK KARLSSON A fundamental property of the renewal process is that N(t) > n + 1 if and only if S, n if and only if S, + X, . We omit the index Fo, if P(So = 0) = 1. Thus we have MFo(t) = I e,(t) 0 and VFo(t) = P((t) P((t) = 1) = (G*Fo*F*)(t) 0 0 or putting Go = G,*Fo VF,(t) = (Go * U) (t). Thus VFo(t) is the solution of the renewal equation VFo(t) = Go(t) + VFr(tx)F {dx}. Conditioning by So, we see that VFo(t) = (V* Fo)(t). We can also get the renewal equation for VFo(t) by the usual renewal reasoning: EMFo(t) = f E(MFo(t) lSo + X0 = S, So + T1 = y)Pso+xo{ds} Pso+r,{dy} where (0 if s > t, y > t, E(MF(t) So + Xo = s, S + T1 = ) = 1 if s t, y> t, EM(t y) if s > t, y t, 1 +EM(t y) if s ?t, y t. This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms A stochastic modelfor time lag in reporting of claims 385 This technique will be used later. According to the renewal theorem for a delayed renewal process, we have for h > 0 lim (VFo(t + h) VFo(t)) = h I/,. If we put z(t) = VFo(t) t /ip it can be shown that Z(t) satisfies Z(t) = z(t) + f (t x) F{dx} with z(t) = (1 F(y))dy (1 Go(t)). Since z(t) is directly integrable (we assume that Po = ESo All use subject to http://about.jstor.org/terms A stochastic model for time lag in reporting of claims 387 References [1] ANDERSEN, E. SPARRE (1957) On the collective theory risk in the case of contagion between claims. Transactions XVth International Congress of Actuaries, New York. Vol. II. [2] DALEY, D. J. AND VERE-JONES, D. (1972) A summary of the theory of point processes. Stochastic Point Processes: Statistical Analysis, Theory and Applications (P. A. W. Lewis, ed.), Wiley, New York. 299-383. [3] FELLER, W. (1970) An Introduction to Probability Theory and its Applications. Vol. 2, 2nd edition. Wiley, New York. [4] PRABHU, N. U. (1965) Queues and Inventories. Wiley, New York. [51 THEDtEN, T. (1967) On point .v.vtoms in R, under random motion. Doctor's thesis, Royal Institute of Technology, Stockholm. This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:40:22 UTC All use subject to http://about.jstor.org/terms

This paper continues the author's study of age-dependent branching processes allowing immigration. In this paper the multidimensional case is considered. A sufficient condition is obtained for the existence of a legitimate limiting distribution. Several corollaries are obtained, which generalize many of the results of the discrete theory and those of the one-dimensional continuous time model.

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived. MARKED BIRTH-AND-DEATH PROCESS; MULTIVARIATE LAGRANGE DIFFERENTIAL EQUATIONS; MULTITYPE BRANCHING PROCESS Introduction In a recent paper Williams (1969) has considered the following process. We have a population of elements developing as a linear birth-and-death process and each element bears a number of inanimate marks. When a marked element dies then it and its marks are removed from the population; when a marked element divides, its marks segregate independently and without loss, each with probability I into one of the two daughter elements. It is assumed that marking an element does not affect its birth and death rates (cf. Clifford and Sudbury (1972)). Williams' process was proposed as a model of an experiment conducted on bacteria (the elements) infected by phage (the inanimate marks), by Meynell (1959). (The bacteria are lysogenic with respect to the phage, which means that the phage do not interfere with the behaviour of the bacteria.) His object was to use the marking process in order to separate the birth and death rates of the bacteria; if we denote birth and death rates by A and p respectively, then the mean-value measures on the bacterial population (assumed to develop as a linear birth-and-death process) will estimate the difference (A p). As the marks are inanimate, however, they will only be affected by p and so mean-value measures on the mark population enable us to estimate p (see Williams (1969)). The papers of Meynell and Williams taken together provide a most interesting example of a virological experiment and the probability model of that experiment. Received in revised form 10 August 1973 Research sponsored by a Science Research Council grant and a World Health Organisation research traineeship. 423 This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms 424 BYRON J. T. MORGAN Williams considered the mean-value solution of his model; this was adequate for Meynell's experiment in which population sizes were large. Here we shall give a fuller probabilistic statement of the model and make some contributions to its solution. Two differences from Williams' work lie in the adoption of an upper bound, s, to the number of marks any element can have, and the assumption (except where stated) of birth and death rates independent of time. We 'shall assume general initial conditions which, in Meynell's experiment, correspond to the distribution of phage over bacteria once these have been initially exposed to a phage population. Williams took this distribution to be Poisson; the distribution of phage over bacteria has, however, been the subject of much probabilistic study (e.g., see Gani (1965) and Morgan (1971)). Finally, let us note that the value s = I, on which we shall particularly concentrate, probably gives the most realistic description of Meynell's experiment. 2. The probability model The model is a continuous-time Markov process with discrete states. Let N,(t) be a random variable denoting the number of elements with i marks at time t, i = 0, 1,2, ..., s. Let n = (n0,n1, l , n.., ), Iu = (Uo, u1, *,us), N(t) = (No(t), N(t), --,N,(t)), and suppose N,(O) = ai, i = 0, 1,2, --, s. Let P(n; t) = P(N(t) = n), then corresponding to death we have the transitions, (No, ..., Ni, ---, N,) (No, --, Ni 1, , N,) at probability rate pNi, i = 0, 1,2, ,s, while corresponding to birth we have the following transitions (No, N,) ,(No + 1, .., N,) at probability rate, 2 =1 N 2-' +,ANo, (No, , Ni ,N2i,..., X (No, Ni + 2, , N2iI, at probability rate, This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms On the distribution of inanimate marks over a linear birth-and-death process 425 where we use [is] to denote the integral part of is, and (No, " , Ni, * * *, Nk i Nk, "', Ns) -*(No, ..., Nj + 1, ..., Nk-i + 1, ..., Nj 1, ..., Nl) at probability rate 2N 2-k' i = 1,2,...,C[(k 1)], k = 3,4,...,s (smaller k give transitions already considered above). It is readily verified that the sum of the transition rates is (A + yp) X = oN,, as required in the overall birth-and-death process. Introducing the operator Di, which increases the argument ni of a function of n by 1, we then deduce that the Kolmogorov forward equations (for all possible n) are, d S S (1) +(n; t) P(n; t)(A+ t)1 (n2 + D,P(n;t)(n +1) dt ==0 1=0

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

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

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.

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 ?-_ ...

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t. CENTRAL LIMIT THEOREM; ITERATED LOGARITHM LAW; BRANCHING PROCESS; TEMPORALLY HOMOGENEOUS MARKOV PROCESS; SPLITTING PROCESS

This paper is concerned with a class of dynamic network flow problems in which the amount of flow leaving node i in one time period for node j is the fraction pij of the total amount of flow which arrived at node i during the previous time period. The fraction pij whose sum over j equals unity may be interpreted as the transition probability of a finite Markov chain in that the unit flow in state i will move to state j with probability pij during the next period of time. The conservation equations for this class of flows are derived, and the limiting behavior of the flows in the network as related to the properties of the fractions Pij are discussed.

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/I queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for singleserver queues to obtain characterizations of steady-state distributions and heavy-traffic approximations. QUEUE; SERVICE SYSTEM; CONGESTION THEORY; MULTI-SERVER QUEUE; HETEROGENEOUS-SERVER QUEUE; QUEUE WITH ERLANG SERVICE TIMES; STOCHASTIC ORDERING OF QUEUES; STOCHASTIC BOUNDING RELATIONS FOR QUEUES; EXISTENCE THEOREMS FOR QUEUES; LIMIT THEOREMS FOR QUEUES; APPROXIMATIONS FOR QUEUES; QUEUES IN HEAVY TRAFFIC

The paper considers an extension to the reversible counters system of Lampard [1]. In Lampard's model the input processes are two independent Poisson processes; this results in a gamma Markov sequence for the time-interval between successive output pulses and a negative binomial Markov sequence for the counts at the times of out-put pulses. We consider the input process to be a bivariate Poisson process and show that the first out-put process given above is not affected, while the second out-put-process becomes of a type studied in the theory of branching processes.

In this paper, the cost of the carrier-borne epidemic is considered. The definition of duration, as used by Weiss (1965) and subsequent authors, is generalised and the probability distribution for the number of located carriers is obtained. One component of cost, namely the area generated by the trajectory of carriers, is examined and its probability density function derived. The expected area generated is then shown to be proportional to the expected number of carriers located during the epidemic, a result which has an analogue in the general stochastic epidemic.

Nonzero-sum N-person stochastic games are a generalization of Shapley's two-person zero-sum terminating stochastic game. Rogers and Sobel showed that an equilibrium point exists when the sets of states, actions, and players are finite. The present paper treats discounted stochastic games when the sets of states and actions are given by metric spaces and the set of players is arbitrary. The existence of an equilibrium point is proven under assumptions of continuity and compactness. NONCOOPERATIVE STOCHASTIC GAME; DISCOUNTED MARKOVIAN DECISION PROCESS; EQUILIBRIUM POINT; DYNAMIC PROGRAMMING

The paper discusses models for genetic drift in haploid models; non-overlapping models following Wright, and overlapping models following Moran. It is shown that these models, and their extensions by Chia and Watterson, can all be restated as non-overlapping models. This equivalence between the two sets of models greatly facilitates the specification of latent roots and vectors.

This paper contains a detailed study of the Poisson cluster process on the real line, concentrating on two aspects; first, the asymptotic distribution of the number of points in [0,t) as t→ ∞ for both transient and equilibrium cluster processes and, secondly, a general formula for the probability generating function of the equilibrium process. Asymptotic formulae for cumulants of the process are also derived. The results obtained generalize those of previous writers. The approach is analytical, in contrast to the probabilistic treatment of P. A. W. Lewis.

In the present paper we demonstrate that the concept of a list code is from a mathematical point of view a more canonical notion than the classical code concept (list size one) in that it allows a unified treatment of various coding problems. In particular we determine for small list sizes the capacities of arbitrarily varying channels.

It is common to represent taxonomic hierarchies of related objects (such as similar plant or animal species or languages of the same family) by rooted trees with labelled terminal vertices which represent the objects. The multivariate data comparing numerous characteristics of the objects is first reduced to indices of similarity (or more often of dissimilarity) between each pair of objects. These are used to classify the objects into groups which are then depicted on a tree. This paper shows that an unrooted tree with labelled terminal vertices may provide a better representation of the relationships between the objects because the similarity indices are required to conform to fewer restrictions. Also for a given number of terminal vertices, there are fewer unrooted than rooted trees so that studies using probability distributions of trees or seeking the most suitable tree to represent the data are more practicable.

In this paper we derive a successive method of solution to a general class of models pertaining to the investigation of the migration process. A specific application of the results to a problem arising in a study of the epidemiology of leprosy has been considered. The application to explorative studies in biological modelling using an interactive computing mode has been discussed.

In this paper, convergence theorems for heavy traffic queues are extended to multi-channel systems under general assumptions. Whitt (1968), and Iglehart and Whitt (1970) have proved weak convergence of the queue length process and the wait process for one-channel queues. Extensions to multi-server queues were established for the queue length process but only partly for the wait process (ρ = 1 only). We give here convergence theorems for the wait process when ρ > 1. Our approach uses weak convergence theory, but is different from previous ones in that we use the virtual delay as an intermediate result. The class of queues considered is more general than GI/G/m.

This paper discusses general bounds for coverage probabilities and moments of stopping rules for sequential coverage problems in geometrical probability. An approach to the study of the asymptotic behaviour of these moments is also presented. STOPPING RULE; SEQUENTIAL COVERAGE; STIRLING NUMBERS; ASYMPTOTIC BEHAVIOUR

In this paper, we establish that if an interarrival time exceeds a service time with a positive probability then the queueing system GI/G/s with a finite waiting room always has proper limiting distributions for its characteristics such as queue length, waiting time and the remaining service times of the customers being served. The result remains valid if we consider a GI/G/s system with bounded waiting times. A technique is also given to establish that for a system with Poisson arrivals the limiting distributions of the queueing characteristics at an epoch of arrival and at an arbitrary epoch are identical.

The paper employs methods of multitype branching processes to evaluate the probability of survival of mutable clones under environmental conditions which are unfavorable to the original parent of the clone. When other factors are taken to be constant, the long-term survival probability of a clone is implicitly demonstrated as a function of the intrinsic rate of mutation carried by this clone. The existence of a mutation rate which maximizes clone survival probability is shown and the effects of environmental deterioration on this optimal rate are studied. Finally, rigorous quantitative results are obtained for the classical situation of a Poisson distribution of offspring numbers. These results are then applied to the biological problem of indirect selection (Eshel (1972)).

This paper considers a dam of infinite capacity with a discrete-valued stationary input process and a unit release whenever possible. It is shown how, by suitable manipulations of the equation governing the dam content process, the stationary distribution of the dam being empty can be obtained, as also can (with a few additional assumptions) the expected value of the dam content in the stationary case. Results obtained are applied to particular cases of input independent and identical, Markov, Bivariate Markov and moving-average. INFINITE DAMS; STATIONARY INPUT; UNIT RELEASE; EMPTINESS PROBABILITY; EXPECTED DAM CONTENT

Some recent work on discrete time dam models has been concerned with special cases in which the input process is a Markov chain whose transition probabilities, p ij , are given by where A(·) and B(·) are probability generating functions (p.g.f.'s). In this paper we obtain some results for the general situation. The convergence norm of the matrix [p ij xj] is found and the results are used to obtain the p.g.f. of the first emptiness time. Distributions of the dam content are obtained and conditions are found for the existence of their limits. The p.g.f. of this distribution is so complicated that its identification in any special case is extremely difficult, or even impossible. Thus useful approximations are needed; we obtain a ‘heavy traffic’ limit theorem which suggests that under certain circumstances the limiting distribution can be approximated by an exponential distribution.

The first part of the paper gives a multitude of essentially different representations of a stationary stochastic process. The second part gives a sufficient condition for the sum of two oscillatory processes to be again oscillatory.

Let {Xk : k ≧ 1} be a sequence of independent, identically distributed random variables with EX 1 = μ < 0. Form the random walk {Sn : n ≧ 0} by setting S 0 = 0, Sn = X 1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X 1) that Sn , conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam. FINITE DAMS; POISSON INPUTS; EXPONENTIAL RELEASE; FIRST PASSAGE TIMES; RECURRENCE RELATIONS, NUMERICAL INTEGRATION; SERIES SOLUTION; SIMULATION

Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range or adjusted range of partial sums of random variables. In this paper, formulas for the expected values of the range and adjusted range of partial sums of exchangeable random variables are presented. Such formulas are based on an elegant result given in Spitzer (1956). Some consequences of the aforementioned formulas are discussed. EXPECTED ADJUSTED RANGE OF PARTIAL SUMS OF EXCHANGEABLE RANDOM VARIABLES; ADJUSTED RANGE IN STORAGE THEORY; HURST PHENOMENON; STABLE DISTRIBUTIONS

Let P= {p 1,···, PN } be a finite discrete probability distribution. Then the entropy of the distribution P, introduced by Shannon [12], is defined as Throughout this paper ∑ will stand for and logarithms will be taken to the base D (D > 1).

In this paper we present a method for obtaining explicit results for some two-sided boundary problems involving sums of independent random variables with alternating distributions. We apply the method to finding the first passage time to either one of two finite barriers, and to some situations arising in queueing and dam theory. The results can be expressed in terms of a finite sum of simple repeated integrals (or sums) of known functions (cf. formulae (3.6)– (3.11)).

In the present paper we deal with two models of the traffic on a divided highway. In both models traffic is flowing in one direction only, in two lanes, of which one (the left hand lane) is used only for overtaking. We suppose that vehicles enter the highway at the same entrance so that the instants at which a vehicle enters the highway form a homogeneous Poisson process, with density λ. Thus λ is the rate at which vehicles enter the highway per unit time. We suppose that there are no junctions, or exits (i.e. the highway extends in one direction to infinity).

Yu. V. Linnik showed that certain transformations, given by Formulae (1.1), (1.6) and (1.7) transform a normal sample into itself. The transformations (1.1) and (1.7) apply to samples of size 2 while (1.6) admits an arbitrary sample size. It is also assumed that the population mean is zero. In the present paper the converse theorems are proven so that characterizations of the normal distribution are obtained. The problem leads to the functional equations (2.3) and (2.13) whose solution yields the desired results.

The paper deals with the asymptotic analysis of waiting time distribution for service systems with finite queue capacity. First an M/MIm system is considered and the rate of approximation is given. Then the case of the M/G/1 system is studied for traffic intensity p > 1. In the last section a condition is given under which an estimate can be derived for the remainder term in central limit theorems for randomly stopped sums. QUEUEING THEORY; SERVICE SYSTEM WITH FINITE CAPACITY; CENTRAL LIMIT THEOREM; RANDOMLY STOPPED SUMS; RATE OF CONVERGENCE

This paper is concerned with the cost Cis = aWis + bTis (a, b > 0) of a general stochastic epidemic starting with i infectives and s susceptibles; Tis denotes the duration of the epidemic, and Wis the area under the infective curve. The joint Laplace-Stieltjes transform of (Wis, Tis ) is studied, and a recursive equation derived for it. The duration Tis and its mean Nis are considered in some detail, as are also Wis and its mean Mis . Using the results obtained, bounds are found for the mean cost of the epidemic.