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Both paper and author (D. R. McNeil, (1968)) will be referred to below as DRM. The said paper deals with the following situation: an intersection is controlled by a traffic light with a fixed cycle time, T; the possibility of other delays, e.g., due to turning vehicles, is ignored; arrivals at the light form a compound Poisson process; if vehicles arrive to find the light green and the queue empty they are not delayed, while in the contrary case they depart when they reach the head of the queue, providing the light is green, each vehicle taking a constant time to move off. The length of the effective red period is R. For further details and discussion, DRM may be consulted.

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.

The purpose of this note is to point out some difficulties which arise in connection with the stochastic calculus used in the previous paper and to suggest some ways in which these difficulties may be overcome. It turns out that if one wishes to restrict the application of the stochastic calculus to stochastic processes whose sample functions are step-functions with probability one, then difficulties of the following sort arise. Suppose X(ω, t) is a stochastic process whose sample functions are, with probability one, right-continuous step-functions in t on the interval [a, b). Then for every ω outside a null set N and every t ɛ [a, b), there is a δ(ω, t) such that < 5(w, t) implies X(ω, t + h) = X(ω, t) which in turn implies (X(ω, t + h) — X(ω, t))/h → 0 with probability one as Since convergence with probability one implies convergence in probability, it follows that the right-hand derivative of X(ω, t) in probability is zero at every t ɛ [a, b). Consequently, if the left-hand t ɛ [a, b), then it is also zero, but in general left-hand derivatives in probability on [a, b) may not even exist as finite-valued random functions. It is also worthwhile to observe that if a process is not differentiable in probability, then it is not differentiable in quadratic mean. From these observations it follows that the claims in Section 6 of the previous paper regarding the existence of non-zero derivatives in quadratic mean are in error, for if a derivative in quadratic mean exists, then it is a derivative in probability and is therefore zero. Incidentally, it turns out that the Poisson process has a zero derivative in probability but is not differentiable in quadratic mean. The fact that the Poisson process has a derivative in probability seems to depend rather crucially on the fact that it is also a process with independent increments.

This paper provides simple proofs of two stardard .esults for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argum-nt is applicable to more general single server systems than the queue GI/G/1. STABLE GI/G/I1 QUEUE; STATIONARY VIRTUAL WAITING TIME; STATIONARY QUEUELENGTH; STATIONARY ACTUAL WAITING TIME The objective of this paper is to provide simple proofs of two well-known results for the stable GI/G/1 queue, which results give the distribution function of W* (stationary virtual waiting time) and the distribution of Q (stationary queue-length) in terms of the distribution function of W(stationary actual waiting time). Our approach does not require transform methods and is applicable to more general single server systems than the GI/G/1 queue. Let {V,, n 2 0} and {U,?1, n > 0} denote independent sequences of independent and identically distributed, real, non-negative random variables defined on the probability space (Q, Y, P). The variable VI, represents the service time of customer n, and U,+1 is the time between the arrivals of customers n and n + 1, n 2 0. Denote by F and B the distribution functions of Vo and U, , respectively, and by f and b the first moments of F and B. We assume B is non-lattice, that each off and b is strictly positive and finite, and that f W* and Q(t) Q, as n -+ oo and t oo, where ">" denotes weak convergence. If J is a proper distribution on the real line with J(O-) = 0 and finite first moment j > 0, denote by j the proper distribution with J(0-) = 0 and Received in revised form 4 December 1973. Research supported by N.S.F. Grant No. GK-39073. 849 This content downloaded from 157.55.39.244 on Fri, 17 Jun 2016 04:45:48 UTC All use subject to http://about.jstor.org/terms 850 AUSTIN J. LEMOINE J(x) = p-1f: [ -J(y)]dy. Then, if Y is a random variable having distribution J, E{Y A x} = pf(x) for x = 0, where u A v = min(u, v) for real numbers u and v. Let H denote the distribution function of W, K = H * P, where "*" denotes convolution, and p = f/b. Let Bo denote the distribution function assigning unit mass to the origin, and Bi the j-fold convolution of B with itself, j _ 1. With the above notation, there are the following basic results from queueing theory; see Takics (1963) or Cohen (1969). Theorem 1. P{W* 1, d = Vo 0+ + V,_I, t, = U1 + -+ U,, and S, = d,t,. Next, set n = inf{n: n > 1 and S, 0, so that the queueing system is aperiodic in the sense of (6) from [2]. Since U, has a non-lattice distribution so does the variable ?. And, if m = E{7t}, then E{jt} = mf and E{r} = mb on account of Wald's Lemma. Hence, by (9) and (22) from [2], for x > 0, (1) mH(x) = E I1wx) }X n=O and (2) mbP{W* < xj} = E 1(W(s)xdS We have S1(w(S)x)ds I 1 1 Fd,-xds o n=1 . , = I I, I_-t_ I _yldy = 1 lis,-i+v.,-sy)dylls-_,+v._,_dy n=1 0 (3) = [(x+ U -S_1V._)+ -_(x-S--1V,,-W)] n=l1 This content downloaded from 157.55.39.244 on Fri, 17 Jun 2016 04:45:48 UTC All use subject to http://about.jstor.org/terms On two stationary distributions for the stable GI/G/1 queue 851 n-I = [(-S,)+ (x-S,V,)+] +x-S,-(x-SoVo)+

It is known [9] that geometric or exponential decay of a Markov chain is preserved under derivation (as defined by Cohen [2]). In this paper we consider the inverse problem, i.e., does a derived Markov chain with a geometrical or exponential decay necessarily arise from a Markov chain having the same property. For a large class of Markov chains (including time reversible chains) a complete solution is found. The method used in this paper proved accurate for obtaining exact results on the value of the decay parameter. An example extending results of Miller [8] and Teugels [9] illustrates the procedure.

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.

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β > 0 and a γ > 0 such that for t∈ R +. The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

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

Summary The present paper generalizes the results of the author's paper [2] on “Discontinuous Markov processes” to processes that are incompletely specified. These results are then used to develop a step-by-step method of solution for processes involving several kinds of “jumps”, where at the first step one solves for the transition probabilities involving jumps of the first kind, but not of the second, third, etc., at the second step for transition probabilities involving jumps of the first and second, but not the third, fourth, etc. kinds, and so on. Some examples are given.

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.

The random walk between an absorbing and a reflecting or semi-reflecting barrier has been discussed in the literature. Indeed, a biological example of such a random walk, arising from a system of countercurrent dialysis, was described by this author. In this earlier paper a technique was suggested for obtaining the probability of being in a specific state after a given number of transitions, if the probability is unity of being in the reflecting state at time zero. The present paper extends this technique to yield the full matrix of higher transition probabilities. Introduction The random walk between an absorbing and a reflecting or semi-reflecting barrier has been discussed in several recent papers in the literature [6], [7], [8]. A biological example of such a random walk, arising from a system of countercurrent dialysis, was described by Kastenbaum [4]. In his paper, Kastenbaum presented a technique for obtaining the probability of being in a certain state after a given number of transitions, if the probability is 1 of being in the reflecting state at time zero. In a subsequent report, Kemperman [5] showed that these probabilities could be achieved using the concept of path lengths. The present paper demonstrates how Kastenbaum's procedure may be extended to yield the full matrix of higher transition probabilities. Knowledge of these k-step transition probabilities is of interest to the experimenter who may wish to stop the random walk after a small and finite number of steps, k. The procedure does not require that the eigenvalues of the stochastic matrix be known explicitly. It is rather the elementary symmetric functions of these eigenvalues which play the important role in the mathematical reduction. These symmetric functions are coefficients of polynomials described in Section 2. Because the matrix of transition probabilities is a special case of a Jacobi-matrix, the polynomials associated with it are of a special and interesting form. As a Received in revised form 27 April 1966. Sponsored by Mathematics Research Center, U. S. Army, Madison, Wisconsin, under contract No. DA-11-022-ORD-2059. 365 This content downloaded from 157.55.39.175 on Sun, 02 Oct 2016 05:28:15 UTC All use subject to http://about.jstor.org/terms 366 MARVIN A. KASTENBAUM result the matrix of eigenvectors is a slightly modified orthogonal matrix, whose inverse may be easily written. The application of all this information results in explicit expressions for the k-step transition probabilities. 1. Background and notation Countercurrent dialysis is a diffusion system in which a linear series of cells are used. Each cell is a container into which is suspended a membranous dialysis sac. During dialysis, material flows from the inside to the outside volume of the sac. After a fixed interval of time the dialysis is stopped, and the material is manually transferred among the cells to the inside volumes, according to a prescribed set of rules. Dialysis is then continued through another cycle at which time it is stopped so that the material may again be transferred manually. The operation is continued in this manner for prescribed number of cycles, at which time the experimenter may wish to know the proportion of isolate at a specified stage in the system. For additional details on countercurrent dialysis, see [1], [2], [3]. Let m be the number of cells in the system, p the probability that, in any cell, a particle inside the dialysis sac will not penetrate to the outside volume, and q = (1p), the probability that a particle will penetrate to the outside volume. Also specify that, once in the outside volume, a particle cannot return inside, except by manual transfer. Then the diffusion probabilities for a system with m cells may be represented by the square matrix of order 2m given by D, 0 D = L. , where D = 0 1for all j = 1,2,..., m. O Dm I The odd rows and columns of D correspond to events taking place in the inside volume and the even rows and columns correspond to events taking place in the outside volume in successive cells during dialysis. In the same manner, the manual transfer of material among the cells at the end of each cycle of dialysis is defined in terms of transfer matrix T. The matrix T is a square matrix of order 2m whose elements are probabilities equal to zero or unity. Its structure is determined by the following rules: At the end of each cycle (i) a particle in the inside volume of cell 1 will remain in the inside volume of cell 1 with probability 1; (ii) a particle in the inside volume of cell j will be transferred to the inside volume of cell (j 1) with probability 1, for j = 2, 3, ..., nm; (iii) a particle in the outside volume of cell j will be transferred to the inside volume of cell (j + 1) with probability 1, for j = 1,2, .-, m1; This content downloaded from 157.55.39.175 on Sun, 02 Oct 2016 05:28:15 UTC All use subject to http://about.jstor.org/terms A dialysis system with one absorbing and one semi-reflecting state 367 (iv) a particle in the outside volume of cell m will remain in the outside volume of cell m with probability 1. This transfer mechanism is described pictorially in Figure 1. In the nomenclature of Markov chains, the inside volume of the first cell is a semi-reflecting barrier, and the outside volume of the mth cell is an absorbing barrier. If D and T are two square matrices as defined, their product DT is a square matrix of order 2m with non-negative elements and unit row sums. The elements of DT are the conditional probabilities that a particle will make a transition from Cell 1 Cell 2 Cell 3 Cell (m--l) Cell m Figure 1 Manual transfer after each dialysis cycle one of the 2m states to another after both dialysis and transfer have taken place. Moreover, (DT)k is the matrix of transition probabilities before the (k + 1)th cycle of dialysis has begun, and (DT)kD is the matrix of transition probabilities at the end of the (k + 1)th cycle of dialysis before the manual transfer has been effected. We shall be concerned with the algebraic reduction of the matrix (DT)k for a dialysis system with m cells. 2. The matrix reduction By the rules for its construction, the matrix T will have (m-1) null columns, namely all those columns, except the 2mth, involving manual transfers to the outside volume. Hence the matrix DT as well as its power (DT)kwill have (m-l) null columns whose positions are known. This property suggests that an initial simplification would be to reduce the matrix DTof order 2m to a matrix P of order (m + 1), by eliminating the (m-1) null columns and their respective rows. The resulting (m + 1) x (m + 1) matrix is of full rank, and is given by -p q 0 0 *.. 0 00p 0 q 0 ..** 0 0 0 0 p 0 q ... 0 0 0 (2.1) P= 0 0 00 *.. p0 q 0 0 0 0 ... 0 0 1_ This content downloaded from 157.55.39.175 on Sun, 02 Oct 2016 05:28:15 UTC All use subject to http://about.jstor.org/terms 368 MARVIN A. KASTENBAUM Let A be the Jordan normal form of P. Then (2.2) pk = M-1AkM, where M is a square matrix of order (m + 1) of eigenvectors resulting from the solution of the equations (2.3) x[P AI] = 0, and where x is the (m + 1)-dimensional invariant row vector (xl,x29E,,x m+ *). A is a diagonal matrix of order (m + 1) whose elements, the eigenvalues of P, are the roots of the determinantal equation (2.4) P AI = 0. Equation (2.4) is a polynomial of degree (m + 1) with (m + 1) real and distinct roots (see [4]) one of which is known and equal to unity. Thus we may rewrite Equation (2.4) in the form (2.5) (A 1)/im(A) = 0. It has been shown [4] that for any m, the mth degree polynomial '(3A) may be written directly as

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

The optimal strategies of any finite matrix game can be characterized by means of the Snow-Shapley Theorem [1]. However, in order to use this theorem to compute the optimal strategies, it may be necessary to invert a large number of matrices, most of which are not related to the solutions of the game. The present paper will show that when the columns of the pay-off matrix satisfy some special relations, it is possible to enumerate a much smaller class of matrices from which the optimal strategies may be obtained. Furthermore, the maximizing strategies that are determined by these matrices can be written down by inspection as soon as the matrices are specified. The method used in this paper is based on a simple combinatorial lemma. The games considered here arose in the study of some strategic models [2]. These models constitute extensions of some games considered by Gross (unpublished) and Dresher [3] and include examples considered in the books by Dresher [4] and Karlin [5]. These models, in the most general form considered in [2], describe a situation where an attack force of A units may be used against one or several cities whose total defense force is D units, and where, if the attacking and defensive forces assigned to the ith city are x, and yj units, respectively, the attacker gains there the amount

A previous paper [2] was concerned with the determination of optimal policies for restocking an inventory which is continuously depleted by a random process of demands. The purpose of the present paper is to develop a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process. This reversal of the roles of input and output does not, by itself, change the character of the problem. But the consideration of set-up costs for ordering replacements, which leads to inventory policies of the (s, S) type, has no counterpart here. It is natural to regard the dam as a device for smoothing out random fluctuations in a flow of water and, under utility assumptions which reflect this attitude, it follows that the optimal output rate is a continuous function of the level of water in the reservoir. Our main object is to determine this function.

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.

A general age-dependent branching process is studied in which the population is distributed in N colonies. Migration from one colony to another is possible and no particular geometrical configuration is assumed for the colonies. Asymptotic expressions for the first two moments are given in several different cases. In this paper we consider a stochastic model for a population whichis distributed over N colonies. The model was motivated in part by a similar stochastic population process discussed in a paper by Bailey (1968). Bailey's process is a birth and death process with the additional feature that members of the population may migrate from one colony to another. These colonies are infinite in number and are located at equally spaced nodes on a one-, two- or three-dimensional lattice. Bailey assumes essentially that an individual migrates at times which occur in accordance with a Poisson process and can migrate only to one of the nearest adjacent colonies. The process we shall study differs from Bailey's in several respects. First of all the underlying process governing population growth is a general-type branching process which includes the birth and death process as well as the age-dependent Bellman-Harris branching process as special cases. Secondly, and perhaps most important in view of possible applications, there are only a finite number of colonies and we need not assume they have any particular geometric configuration. Thirdly, we make only very minimal assumptions governing the migration of individuals from colony to colony. This paper represents only a first attempt at elucidating the process since we have restricted ourselves to an investigation of the first two moments. These moments are studied by means of systems of renewal-type integral equations and we shall rely heavily on two recent papers by Crump (1970a,b) on these kinds of integral equations.

Questions of the decomposability of distribution functions into real-valued components of bounded variation were discussed by P. Lévy (1964) in relation to the nature of the components, whether non-decreasing (distribution functions in particular) or absolutely continuous (a.c.) or both. Hanson (1965), in a review of Lévy's paper, raised the question of whether or not a rectangular distribution could be decomposed into two a.c. distributions. In fact, D. G. Kendall had conjectured earlier (Kendall (1960)) that no such decomposition is possible. The object of this paper is to state and prove the truth of Kendall's conjecture. “Decomposition” or “factorisation” will be understood throughout the paper to mean decomposition into distributions. Decompositions of the rectangular distribution into one a.c. and one discrete factor are well known (see, e.g., Lukacs (1960) pp. 128–9), and decompositions in which both factors are singular continuous (s.c.) have been discovered by Kendall and by P. M. Lee; it is shown here that no other combinations of factor-type can exist. References to other work on related decomposability properties are given in the papers by Lévy and Kendall cited above.

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 problem discussed in this paper arose from the study of the effects of unresolved resonances on neutron cross-sections, but it is considered here in more general terms. Events in a modified renewal process occur at successive intervals x 1, x 2, …, and the ith event has associated with it a parameter Γ i . The random variables xi and Γ i are all independent, and their probability distribution functions are known. Each event contributes to two quantities F(u) and G(v) measured at u and v respectively. The value of the total contribution of all events to G(v) is assumed to be known from observation: this paper gives formulae for the mean value of F(u) conditional on this known value of G(v).

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.

In a previous paper (Moran (1956)) the theory of a dam with a continuous release was developed for a situation in which the input was defined by an additive homogeneous random process of gamma distribution type, and the release was defined to occur continuously at a fixed rate when the content of the dam was non-zero. In the present paper we modify these conditions by assuming first that the input is defined by a general additive homogeneous process with non-negative increments with finite second moment, and secondly that the rate of release is proportional to the content of the dam. This modification to the release rule ensures that the content of the dam is never zero and the theory is then so simplified that the distributional properties of the contents of a sequence of dams in series can be easily found.

Summary In this paper a stochastic model is set up for a certain type of reversible chemical reaction and a solution given for the equilibrium distribution; this solution is then extended to deal with any system of chemical reactions. Three different types of reaction are considered: (1) Several substances Ai react together and give a set of substances Bj. The reaction is reversible, with the substances Ai appearing only on one side of the reaction equation and the substances Bj only on the other. (2) Several different reactions involving the substances Ai and Bj take place simultaneously, but in each reaction equation the substances Ai can appear only on one side and the Bj only on the other. (3) The restriction of the sets Ai and Bj to different sides of the reaction equations is removed: any reaction involving any of the substances Ai and Bj on either side of the equation is permissible. The paper concludes with some applications of the results to problems of molecular adsorption.

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.

This paper arises from the observation by Dirac [1] that an exact measurement of the velocity of an electron should give one of the instantaneous values ± c. The velocity v < c with which the electron moves bodily is in some statistical sense an average. This aspect of the quantum states of motion of a relativistic particle has frequently been mentioned [2], [3] and, as is wellknown, is connected with the self energy problem. Such deductions from quantum mechanics have led the author to consider the possibility of constructing from purely statistical ideas an entity having the physical properties of a relativistic elementary particle. One might reasonably hope to build up characteristics such as rest mass and spin in such a scheme. Properties such as energy and momentum will arise as averages over fluctuating random quantities. In this paper we shall concentrate on rest mass, energy and momentum. We shall show that a zero rest mass object which is regarded as being imbedded in a statistical structure emulates the characteristic of rest mass. The average energy and momentum of this statistical system are then shown to satisfy the usual classical relativistic relation between energy and momentum. The statistical structure chosen is that of an object with a randomly orientated velocity vector, c, which has a fixed magnitude c … |c| where c is the velocity of light.

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 discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found. RENEWAL PROCESS; POISSON PROCESS; INTERACTION; QUEUING THEORY; FINITE WAITING ROOM 1. The model and its connection with queuing theory Ten Hoopen and Reuver [2] have suggested the following kind of interaction between two random point processes as a mathematical model for neuron firing. Points in a renewal process, the excitatory process, are eliminated by points in a Poisson process, the inhibitory process, according to the following rule: whenever one or more points of the inhibitory process arrive, the next point of the excitatory process is eliminated. In this paper this model is generalized by assuming that one or more points of the inhibitory process eliminate a random number of points of the excitatory process. Ten Hoopen and Reuver's model has been generalized by Coleman and Gastwirth [1] in a different way: the points of the inhibitory process are effective during a time period of constant or random length, during which they eliminate all arriving points of the excitatory process. We assume in the following that the excitatory process is a renewal process and denote by F the c.d.f. of the time intervals between points in this process. Furthermore, we assume that the inhibitory process is a Poisson process with intensity A and that one or more points of the inhibitory process eliminate the next k points of the excitatory process with probability Pk, 0 Pk = 1. We also assume that points of the inhibitory process arriving during a time period when points are eliminated by previous points of the inhibitory process, do not affect the elimination process. The case studied by Ten Hoopen and Reuver is characterized by pl = 1. Received in revised form 28 July 1971. 451 This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:41:49 UTC All use subject to http://about.jstor.org/terms 452 LENNART RADE Under these assumptions the random process of remaining points of the excitatory process, the response process, is a renewal process. We denote by F, the c.d.f. of the time intervals between points of this process. There is a close connection between the problem of determining the c.d.f. F, of the response process for a given c.d.f. F of the renewal excitatory process and a given elimination rule and queuing theory. As a matter of fact, the first problem can be interpreted as a problem of determining the busy period of a suitably chosen queuing situation. Consider for instance the case originally studied by Ten Hoopen and Reuver, when one or more points of the inhibitory process eliminate exactly one point of the excitatory process. In this case F, is the c.d.f. of the length of the busy period for the M/G/1 queuing system, where the arrival process is a Poisson process with intensity ), the service times have c.d.f. F and there is a finite waiting room with space for one waiting customer. For the case studied in [1] by Coleman and Gastwirth, when the inhibitors are effective during a random time period, one has to consider a similar queuing situation but with the waiting time of the customers limited by a random variable. For the case studied in this paper the appropriate queuing situation is an M/G/1 system with the following characteristics. The arrival process is a Poisson process with intensity A^ and there is room for only one waiting customer. The busy period is initiated by a virtual (imaginary) customer whose service time has c.d.f. F, while the subsequent customers have a service time with c.d.f. ,km=0 PkF*, where F[ is the convolution of k functions F; the latter service time can be considered as consisting of k phases with probability Pk. Arrivals are permitted only during the service time of the virtual customer, and during the last phase of the service of the actual customer, but at no other time. Put P,(x) = Probability that the length of the busy period is at most x, and the number of customers (including the virtual customer) served during the busy period is n. We now get PA(x) = fe-tdF(t) + Po (1 e-t) dF(t), P(x) = (1 e-'t')P,,1(xtl t2) dF(tl) d Pk_ l(t2) , n> 1. 1t=0 2=0 k=1 Taking Laplace-Stieltjes transforms yields (we denote by f the Laplace-Stieltjes transform off) P1(s) = P(s + o) + po(i(s)[(s + 2)), 00 P(s) = P~1(s)((s)P(s + )) pk p,(s)"k, ,1 > I k=l This content downloaded from 157.55.39.17 on Wed, 31 Aug 2016 04:41:49 UTC All use subject to http://about.jstor.org/terms A modelfor interaction of a Poisson and a renewalprocess 453 But Fi(x) = ~,P= 1P(x) and thus Pl(s) = ~~, 1P(s). Summation of the formulas above then gives P(s) = Poo(s) + (1 po)(s + ) PA(s) (+(s) f(s +)) pkF(S)k-1 k=

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.

In a previous paper devoted to an application of dynamic programming to pattern recognition [1], we pointed out that some identification problems could be regarded as generalized trajectory processes. The functional equation technique [2] could then be employed to obtain an analytic formulation of the determination of optimal search techniques. In many cases, however, (for example, in chess or checkers), a straightforward use of the functional equation is impossible because of dimensionality difficulties. In circumventing these obstacles to effective computational solution, we employed a decomposition technique which we called "stratification" [1, 3]. In this paper, we present a different way of avoiding the dimensionality problem, based upon the concept of "extended state variable". To indicate the utility of the concept, we shall apply it to the problem of finding a fault in a complex system. 1. Generalized trajectory process Let a system be described by a state vector p, and suppose that our objective is to transform p into a state which is immediately identifiable. Let T(p,q) be a family of transformations dependent on the control, or decision variable q, with the property that p, = T(p, q) is a state of S for all p and all admissible q. Let I denote the set of all identifiable states, and let (1) f(p) = the minimum number of transformations required to recognize p. In other words, f(p) is the minimum number of decisions required to transform p into an element of I. The principle of optimality then yields the functional equation (2) f(p) = 1 + min f[T(p, q)]. q In many cases [4], the equation can be used to determine optimal policies. In many more cases, the dimension of p is so great (or infinite) that current and even projected computers cannot be used in a routine fashion. In these cases, we must employ further structural properties of S in order to replace (2) by a more manageable equation. One technique for doing this is given in [1] and [3]; other techniques may be found in [2] and [4]. Received 2 June 1965. * Present address: University of Southern California, Los Angeles. 268 This content downloaded from 207.46.13.64 on Sat, 03 Sep 2016 04:23:52 UTC All use subject to http://about.jstor.org/terms Dynamic programming, pattern recognition and location of faults in complex systems 269 2. Extended state The computational difficulty results from our describing the system by means of a vector of high dimension, and therefore it is natural to look for ways to describe it which will require less information at any given time. With this in mind, let us extend the concept of state in the following manner: Definition. An extended state is an algorithm for determining the state. This is, of course, an imprecise definition since we have not been precise as to what we mean by "algorithm," "determine," or even "state." It does guide the intuition, however. To illustrate this concept, consider the vector differential equation: (3) dx/dt = g(x), x(O) = c, where x is of dimension 1000. Suppose that we wish to use the values x(tj) x(t), ..., x(tk), a total of 1000 k numbers. This is a strain on rapid access storage capacity. Once we have stored the algorithm embodied in (3) in the computer, however, each state x(ti) can be replaced by the extended state t,. This is a considerable reduction in dimension. In return for this extraordinary diminution in dimensionality, we face the task of computing x(t) anew for every t-value required. Without a digital computer, this would not be feasible; with a computer, it is merely a matter of trading time, of which we have a reasonable amount, for rapid access storage, which is in quite short supply. 3. Pattern recognition To illustrate the application of the foregoing ideas, consider a pattern recognition process in which we can apply either the transformation T, or the transformation T2 for a total of N stages. At the end of this transformation or experimentation period, we must decide the nature of the system. We suppose that there is a fixed procedure for making a decision on the basis of the final state PN and that the cost associated with this decision is g(PN). To simplify initially, let us suppose that T, and T2 commute. Then the extended state at any time is (m, n) where m and n are the numbers of times the first and second transformations, respectively, have been used. Introduce the function f(m, n) = the minimum cost of decision-making at the end of the remaining transformations. Then the principle of optimality yields the functional equation (5) f(m,n) = min[f(m + 1, n), f(m,n + 1)], min + n < N, with f (m, n) = g(Pmn), m + n = N. This content downloaded from 207.46.13.64 on Sat, 03 Sep 2016 04:23:52 UTC All use subject to http://about.jstor.org/terms

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 ?-_ ...

Replacement theory for equipment has been investigated by several authors. This paper introduces the 'delay' time preparing for replacement, derives the expected cost per unit time, and discusses the optimum replacement policies under several conditions. Three special cases are discussed and numerical examples are presented. REPLACEMENT PROBLEMS WITH DELAY; RENEWAL PROCESSES; COSTS; SCHEDULED REPLACEMENT; OPTIMIZATION

Sharma and Autar [6] have formed a functional equation in two variables containing two parameters which, as a particular case, includes Dar6czy's [1] generalized functional equation used in information theory. The solutions of our functional equation, under suitable boundary conditions, lead to an inaccuracy function of a new type. A characterization of this inaccuracy function and the convexity of Dar6czy's entropy of type-/ have been given in this paper. INFORMATION THEORY; FUNCTIONAL EQUATION; ENTROPY FUNCTION; INACCURACY FUNCTION; GENERALIZED INACCURACY FUNCTION; CONVEXITY OF GENERALIZED ENTROPY

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 optimal control problem of system described by stochastic McShane differential equations is considered. It is shown that this problem can be reduced to an equivalent optimal control problem of distributed parameter systems of parabolic type with controls appearing in the coefficients of the differential operator. Further, to this reduced problem, necessary conditions for optimality and an existence theorem for optimal controls are given.

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.

In a series of papers Whittle has proposed locating the optimal boundary between the continuation region and the stopping region, in the no-overshoot approximation, for a class of sequential analysis problems by using the criterion that the gradient of the objective function must be continuous at the boundary. It is shown here that the result originally proposed by Whittle is not true in the generality which he supposed, and an indication is given of some of the circumstances in which it does hold.