bluuebunny/crossref_metadata_2025 huggingface dataset:
A rendezvous-evasion game on discrete locations with joint randomization
Authors: Lim Wei Shi
Published: April 2006

We consider a problem proposed by S. Alpern (European Journal of Operational Research (1997)) of how two players can optimally rendezvous while at the same time evading an enemy searcher. This problem can be modelled as a two-person, zero-sum game between the rendezvous team R (with agents R1, R2) and the searcher S. This paper gives the first solution to such a rendezvous-evasion game by considering a version that is discrete in time and space, as in the pure rendezvous problem of Anderson and Weber (Journal of Applied Probability28, pp. 839–851). R1,R2 and S start at different locations among the n identical locations where there is no common labelling and at each integer time they may rellocate to any one of the n locations. When some location is occupied by more than one player, the game ends. If S is at this location, S (maximizer) wins and the payoff is 1; otherwise R(minimizer) wins and the payoff is 0. The value of the game is the probability that Swins under optimal play. We assume that R1 and R2 can jointly randomize their strategies. When n equals 3, the value of the game is . We also prove that the value of the game is bounded above by asymptotically. If, in addition, the players share a common notion of a directed cycle containing all the nlocations (while still able to move between any two locations), the value of the game is ((1 – 2/n)n–1 + l)/2. Finally, we prove that with this extra information, R can secure a strictly lower value for all n.
On the Survival Probability of a Branching Process in a Random Environment
Authors: D'souza J. C. , Hambly B. M.
Published: April 2006

We derive a variety of estimates for the survival probability of a branching process in a random environment. There are three cases of interest, the critical, weakly and strongly subcritical. The large deviation result, first obtained by Dekking for the class of finite state space i.i.d. environments, is shown to hold in more general environments. We also obtain some finer convergence results.
On the Existence and Application of Continuous-Time Threshold Autoregressions of Order Two
Authors: Brockwell P. J. , Williams R. J.
Published: April 2006

A continuous-time threshold autoregressive process of order two (CTAR(2)) is constructed as the first component of the unique (in law) weak solution of a stochastic differential equation. The Cameron–Martin–Girsanov formula and a random time-change are used to overcome the difficulties associated with possible discontinuities and degeneracies in the coefficients of the stochastic differential equation. A sequence of approximating processes that are well-suited to numerical calculations is shown to converge in distribution to a solution of this equation, provided the initial state vector has finite second moments. The approximating sequence is used to fit a CTAR(2) model to percentage relative daily changes in the Australian All Ordinaries Index of share prices by maximization of the ‘Gaussian likelihood'. The advantages of non-linear relative to linear time series models are briefly discussed and illustrated by means of the forecasting performance of the model fitted to the All Ordinaries Index.
The convex hull of a normal sample
Authors: Hueter Irene
Published: April 2006

Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices Nn, the perimeter Ln and the area An of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of Nn, Ln and An for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.
Clustering in a Continuum Percolation Model
Authors: Quintanilla J. , Torquato S.
Published: April 2006

We study properties of the clusters of a system of fully penetrable balls, a model formed by centering equal-sized balls on the points of a Poisson process. We develop a formal expression for the density of connected clusters ofkballs (calledk-mers) in the system, first rigorously derived by Penrose [15]. Our integral expressions are free of inherent redundancies, making them more tractable for numerical evaluation. We also derive and evaluate an integral expression for the average volume ofk-mers.
Consistency in systematic sampling for stereology
Authors: Arnau X. Gual , Cruz-Orive L. M.
Published: April 2006

In design-based stereology, fixed parameters (such as volume, surface area, curve length, feature number, connectivity) of a non-random geometrical object are estimated by intersecting the object with randomly located and oriented geometrical probes (e.g. test slabs, planes, lines, points), [4], [5], [8], [11], [12]. Estimation accuracy may in principle be increased by increasing the number of probes, which are usually laid in a systematic pattern, [1], [2], [3], [7], [9], [10]. An important prerequisite to increase accuracy, however, is that the relevant estimators are unbiased and consistent. Our purpose is therefore to give sufficient conditions for the unbiasedness and strong consistency of design-based stereological estimators obtained by systematic sampling.
The random division of faces in a planar graph
Authors: Cowan Richard , Chen Simone
Published: April 2006

A planar graph contains faces which can be classified into types depending on the number of edges on the face boundaries. Under various natural rules for randomly dividing faces by the addition of new edges, we investigate the limiting distribution of face type as the number of divisions increases.
A Stochastic Model of Carcinogenesis and Tumor Size at Detection
Authors: Hanin L. G. , Rachev S. T. , Tsodikov A. D. , Yakovlev A. Yu.
Published: April 2006

This paper discusses the distribution of tumor size at detection derived within the framework of a new stochastic model of carcinogenesis. This distribution assumes a simple limiting form, with age at detection tending to infinity which is found to be a generalization of the distribution that arises in the length-biased sampling. Two versions of the model are considered with reference to spontaneous and induced carcinogenesis; both of them show similar asymptotic behavior. When the limiting distribution is applied to real data analysis its adequacy can be tested through testing the conditional independence of the size, V, and the age, A, at detection given A > t*, where the value of t* is to be estimated from the given sample. This is illustrated with an application to data on premenopausal breast cancer. The proposed distribution offers the prospect of the estimation of some biologically meaningful parameters descriptive of the temporal organization of tumor latency. An estimate of the model stability to the prior distribution of tumor size and some other stability results for the Bayes formula are given.
Linear Programming Estimators and Bootstrapping for Heavy Tailed Phenomena
Authors: Feigin Paul D. , Resnick Sidney I.
Published: April 2006

For autoregressive time series with positive innovations which either have heavy right or left tails, linear programming parameter estimates of the autoregressive coefficients have good rates of convergence. However, the asymptotic distribution of the estimators depends heavily on the distribution of the process and thus cannot be used for inference. A bootstrap procedure circumvents this difficulty. We verify the validity of the bootstrap and also give some general comments on the bootstrapping of heavy tailed phenomena.
Mean size-and-shapes and mean shapes: a geometric point of view
Authors: Le Hulling
Published: April 2006

Unlike the means of distributions on a euclidean space, it is not entirely clear how one should define the means of distributions on the size-and-shape or shape spaces ofklabelled points in ℝmsince these spaces are all curved. In this paper, we discuss, from a shape-theoretic point of view, some questions which arise in practice while using procrustean methods to define mean size-and-shapes or shapes. We obtain sufficient conditions for such means to be unique and for the corresponding generalized procrustean algorithms to converge to them. These conditions involve the curvature of the size-and-shape or shape spaces and are much less restrictive than asking for the data to be concentrated.
Finite-dimensional models for hidden Markov chains
Authors: Aggoun Lakhdar , Elliott Robert J.
Published: April 2006

A continuous-time, non-linear filtering problem is considered in which both signal and observation processes are Markov chains. New finite-dimensional filters and smoothers are obtained for the state of the signal, for the number of jumps from one state to another, for the occupation time in any state of the signal, and for joint occupation times of the two processes. These estimates are then used in the expectation maximization algorithm to improve the parameters in the model. Consequently, our filters and model are adaptive, or self-tuning.
Probability metrics and recursive algorithms
Authors: Rachev S. T. , Rüschendorf L.
Published: April 2006

It is shown by means of several examples that probability metrics are a useful tool to study the asymptotic behaviour of (stochastic) recursive algorithms. The basic idea of this approach is to find a ‘suitable' probability metric which yields contraction properties of the transformations describing the limits of the algorithm. In order to demonstrate the wide range of applicability of this contraction method we investigate examples from various fields, some of which have already been analysed in the literature.
Transient characteristics of an M/M/∞ system
Authors: Guillemin Fabrice , Simonian Alain
Published: April 2006

Convergence results are given for transient characteristics of an M/M/∞ system such as the period of time the occupation process remains above a given state, the area swept by this process above this state and the number of customers arriving during this period. These results are precise in contrast to approximations derived in the framework of the Poisson clumping heuristic introduced by Aldous.
Homogeneous rectangular tessellations
Authors: Mackisack Margaret S. , Miles Roger E.
Published: April 2006

A rectangular tessellation is a covering of the plane by non-overlapping rectangles. A basic theory for general homogeneous random rectangular tessellations is developed, and it is shown that many first-order mean values may be expressed in terms of just three basic quantities. Corresponding values for independent superpositions of two or more such tessellations are derived. The most interesting homogeneous rectangular tessellations are those with only T-vertices (i.e. no X-vertices). Gilbert's (1967) isotropic model adapted to this two-orthogonal-orientations case, although simply specified, appears theoretically intractable, due to a complex ‘blocking' effect. However, the approximating penetration model, also introduced by Gilbert, is found to be both tractable and informative about the true model. A multi-stage method for simulating the model is developed, and the distributions of important characteristics estimated.
Anticipative portfolio optimization
Authors: Pikovsky Igor , Karatzas Ioannis
Published: April 2006

We study a classical stochastic control problem arising in financial economics: to maximize expected logarithmic utility from terminal wealth and/or consumption. The novel feature of our work is that the portfolio is allowed to anticipate the future, i.e. the terminal values of the prices, or of the driving Brownian motion, are known to the investor, either exactly or with some uncertainty. Results on the finiteness of the value of the control problem are obtained in various setups, using techniques from the so-called enlargement of filtrations. When the value of the problem is finite, we compute it explicitly and exhibit an optimal portfolio in closed form.
Gamma-type results and other related properties of Poisson processes
Authors: Møller Jesper , Zuyev Sergei
Published: April 2006

Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometric characteristics determined by finite subprocesses. In particular, we deduce Miles' complementary theorem. Applications of the results within stochastic geometry and particularly for random tessellations are discussed.
A stopping time concerning sphere data and its applications
Authors: Zhu Li-Xing , Cheng Ping , Wei Gang , Shi Pei-De
Published: April 2006

Denote byA(x) = {a: |aτx| ≦h} a circle zone on the three-dimensional sphere surface for each givenh> 0. For a given integerm, we investigate how many zones chosen randomly are needed to contain at least one of the points on the sphere surfacemtimes. As an application, the lifetime of a sphere roller is investigated. We present empirical formulas for the mean, standard deviation and distribution of the lifetime of the sphere roller. Furthermore, some limit behaviors of the above stopping time are obtained, such as the limit distribution, the law of the iterated logarithm, and the upper and lower bounds of the tail probability with the same convergent order.
The allocation of resources in a multiple-trial war of attrition conflict
Authors: Whittaker J. C.
Published: April 2006

We consider a model in which players must divide a fixed amount of resource between a number of trials of an underlying contest, with the underlying contest based on the War of Attrition. We are able to find the unique ES set (a simple generalisation of the idea of an evolutionarily stable strategy) in certain circumstances: in particular we find the conditions under which the ES set may contain a pure strategy.
Multitype branching processes in random environments
Authors: Weissner Edward W.
Published: April 2006

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.
Weak convergence theorems for priority queues: preemptive-resume discipline
Authors: Whitt Ward
Published: April 2006

We shall consider a single-server queue with r priority classes of customers and a preemptive-resume discipline. In this system customers are served in order of their priority while customers of the same priority are served in order of their arrival. Higher priority customers, immediately upon arrival, replace lower priority customers at the server, while customers displaced in this way return to the server before any other customers of the same priority receive service. When a displaced customer returns to the server, his remaining service time is the uncompleted portion of his original service time (cf. Jaiswal (1968)).
Diffusion models in population genetics
Authors: Kimura Motoo
Published: April 2006

Population genetics is that branch of genetics, whose object is the study of the genetical make-up of natural populations. By investigating the laws which govern the genetic structure of natural populations, we intend to clarify the mechanism of evolution.
A review of transient behavior in regular diffusion and birth-death processes
Authors: Keilson J.
Published: April 2006

Our concern is with passage times and rate of approach to ergodicity for two types of temporally homogeneous processes, doubly bounded diffusion processes in one dimension, and birth-death processes on the finite lattice. The passage time problems associated with these processes are of considerable practical interest, but for many important cases, e.g., the Uhlenbeck-Ornstein process, only formal solutions such as Laplace transforms (cf. Darling and Siegert, 1953) have been given, with limited numerical potential.
Implications of a failure model for the use and maintenance of computers
Authors: Lewis P. A. W.
Published: April 2006

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.
Some birth-death and mutation models for phage reproduction
Authors: Gani J. , Yeo G. F.
Published: April 2006

Several models for phage reproduction within a bacterium have recently been discussed by Steinberg and Stahl (1961), Gani (1962) and Ohlsen (1963); all are effectively birth-death type processes, with the first of these considering a single type of phage mutation. A detailed description of the biological processes involved may be found in Adams (1959), Weidel (1959), Sanders (1961) and Smith (1961). In this paper we shall consider some general models extending that of Steinberg and Stahl in which mutation may take place from normalaphages to mutantAphages.
Equilibrium behavior of a stochastic system with secondary input
Authors: Şahin İzzet
Published: April 2006

Equilibrium behavior of a stochastic system with two types of input of different statistical nature and with linear continuous output is investigated. The results have applications in queueing theory, storage theory and insurance-risk theory.
A note on the first emptiness time of an infinite reservoir with inputs forming a Markov chain
Authors: Lehoczky J. P.
Published: April 2006

The first emptiness time of an infinite reservoir with unit release and discrete input forming a stationary. Markov chain is investigated. The exact distribution of the first emptiness time is derived without the use of moment generating functions. The first and second moments of this distribution are given explicity. The close relationship between the process with stationary independent input and Markov chain input is emphasized.

The first moment of the area beneath the sample path up to first emptiness is computed. This area is often used as a measure of total delay in traffic flow theory.
Some applications of renewal theory on the whole line
Authors: Crump Kenny S. , Hoel David G.
Published: April 2006

SupposeFis a one-dimensional distribution function, that is, a function from the real line to the real line that is right-continuous and non-decreasing. For any such functionFwe shall writeF{I} =F(b)– F(a) whereIis the half-open interval (a,b]. Denote thek-fold convolution ofFwith itself byFk*and letNow ifzis a non-negative function we may form the convolutionalthoughZmay be infinite for some (and possibly all) points.
A new family of point processes which are characterized by their second moment properties
Authors: Newman David S.
Published: April 2006

The most widely used probabilistic model of a series of “events” occurring randomly in a time or space continuum is the Poisson process. It is the proper basis for statistical analysis when general information about the rate of occurrence of events is the primary purpose of an investigation, or when the assumption of independent occurrences is reasonable and valid. A unified treatment of the statistical methods for estimating parameters in stationary and non-stationary Poisson models, and tests of the Poisson assumption against various alternatives, is contained in the lucid monograph by Cox and Lewis [6].
The multi-type Galton-Watson process with immigration
Authors: Quine M. P.
Published: April 2006

We consider the limiting behaviour of ak-type (k< ∞) Galton-Watson process which is augmented at each generation by a stochastic immigration component. In Section 2, conditions for ergodicity are found for a subclass of such processes. In Section 3, expressions are derived for the first two moments of thenth generation (by way of a recurrence relation) and for the first two asymptotic moments, in a manner which to some extent generalises previous results.
Selective interaction of a stationary point process and a renewal process
Authors: Lawrance A. J.
Published: April 2006

The selective interaction of a stationary point process and a renewal process is studied. Equilibrium conditions for the resulting point process are given, the equilibrium counting distribution is derived, and an explicit expression for the rate of the process is determined.
Demographic effects on the rate of genetic evolution I. constant size populations with two genotypes
Authors: Chia A.B. , Watterson G. A.
Published: April 2006

A model is constructed to describe the evolution of a monoecious haploid population in which the population of adults remains of constant size, but for which generations may be overlapping, family size distributions are to a considerable degree arbitrary, and differential survival rates may apply with respect to adult and young individuals. The eigenvalues of the transition probability matrix are obtained, and those which determine the rate of evolution are found as simple functions of mutation rates, the mean lifetime of an individual, and the variance of his viable offspring number. The model builds on, and includes as special cases, certain non-overlapping generation models of Wright (1931), Karlin and McGregor (1965), and an overlapping generation model of Moran (1958).

One of the secondary conclusions of the study is that for some models, when evolutionary rates are measured on a mean-lifetime scale, the more offspring that are produced at a mating season may slow down evolution when mutation is absent, but may speed it up if mutation is present.
A note on passage times and infinitely divisible distributions
Authors: Miller H. D.
Published: April 2006

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.
A stochastic population process and its application to bubble chamber measurements
Authors: Brockwell P. J. , Moyal J. E.
Published: April 2006

A particle travels along the half-line [0,∞) in such a way that it has probability λδu + o(δu) of generating an event in any small element (u,u + δu) of its track. The particle is observed only in the line segment 0 ≦ u ≦ x and successive events occur at X1, X2, …, Xn (0 ≦ X1 ≦ X2 ≦ … ≦ Xn ≦ x) where X1, …,Xn are random variables and the number n of events in [0, x] is also random. The distances

constitute a finite univariate population process as defined by Moyal [1], the individuals being the distances Yi with state space [0, x].
The total waiting time in a busy period of a stable single-server queue, II
Authors: Daley D. J. , Jacobs D. R.
Published: April 2006

This paper is a continuation of Daley (1969), referred to as (I), whose notation and numbering is continued here. We shall indicate various approaches to the study of the total waiting time in a busy period2 of a stable single-server queue with a Poisson arrival process at rate λ, and service times independently distributed with common distribution function (d.f.) B(·). Let X'i denote3 the total waiting time in a busy period which starts at an epoch when there are i (≧ 1) customers in the system (to be precise, the service of one customer is just starting and the remaining i − 1 customers are waiting for service). We shall find the first two moments of X'i, prove its asymptotic normality for i → ∞ when B(·) has finite second moment, and exhibit the Laplace-Stieltjes transform of X'i in M/M/1 as the ratio of two Bessel functions.
On the service time distribution and the waiting time process of a potentially infinite capacity queueing system
Authors: Hadidi Nasser
Published: April 2006

In [1] the authors dealt with a particular queueing system in which arrivals occurred in a Poisson stream and the probability differential of a service completion was μσn when the queue contained n customers. Much of the theory could not be carried out further analytically for a general σn, which is a purely n-dependent quantity. To carry the analysis further to the extent of finding the “effective” service time and the waiting time distribution when σn is a linear function of n, (which is considered to be rather general and sufficient for practical purposes), constitutes the subject matter of this paper.
Random secants of a convex body
Authors: Kingman J. F. C.
Published: April 2006

Let two points be taken at random in an n-dimensional convex body K, and let σ be the line joining them. The distribution of σ is found and compared with other distributions for random secants of K. More generally, if r + 1 ≦ npoints are taken in K, the distribution of the r-dimensional affine subspace containing them is computed. The results find application to the n-dimensional case of a problem of Sylvester.
The total progeny in a branching process and a related random walk
Authors: Dwass Meyer
Published: April 2006

This paper is a continuation of [1]. The techniques of [1] are used to get specific information about the distribution of the total progeny in a branching process. This distribution is also related to one which arises in a random walk problem.
A birth, death and migration process
Authors: Adke S. R.
Published: April 2006

A model proposed by Bailey (1968) for migratory individuals which reproduce according to a simple birth-death process is generalized to include time dependent birth and death rates.
A note on the waiting times between record observations
Authors: Holmes Paul T. , Strawderman William E.
Published: April 2006

Let X1,X2,X3,··· be independent, identically distributed random variables with a continuous distribution function and let the sequence of indices {Vr} be defined as follows:

and for r ≧ 1,
Vr is the trial on which the rth (upper) record observation occurs. {Vr} will be an infinite sequence of random variables since the underlying distribution function of the X's is continuous. It is well known that the expected value of Vr. is infinite for every r (see, for example, Feller [1], page 15). Also define

and for r > 1

δr is the number of trials between the (r - l)th and the rth record. The distributions of the random variables Vr and δr do not depend on the distribution of the original random variables. It can be shown (see Neuts [2], page 206 or Tata 1[4], page 26) that

The following theorem is due to Neuts [2].
Optimal procedures for stochastically failing equipment
Authors: Folkman Jon , Port Sidney
Published: April 2006

When a complex system such as the Apollo vehicle arrives on the Moon, certain parts may no longer function. To restore the system to a working state, these parts must be replaced by new ones; but the vehicle can carry only a limited number of spare parts and must also be able to leave at a specified time after arrival. This paper investigates an abstract (and simplified) version of the problem of the optimal maintenance procedure for such a system. As a bonus of the abstract formulation, the results obtained are applicable to a variety of specific systems.
Random displacements of regularly spaced events
Authors: Nelsen Roger B. , Williams Trevor
Published: April 2006

We consider the point processi+ τi(i= 0, ± 1, ± 2, . . .), where the τi(assumed for convenience to be positive) are independent random samples from the same distribution. We define an inter-arrival interval as the stretch between neighbouring events, which need not belong to successive values ofi, because of the jumbling of position occasioned by the variability of the τ. We obtain explicit expressions for the distribution of inter-arrival intervals in general, as well as the correlation coefficient between successive inter-arrival intervals in the case where the τ are exponentially distributed.
Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test
Authors: Durbin J.
Published: April 2006

Letw(t), 0 ≦t≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1,E{w(t1)w(t2)}= min (t1,t2), and leta,bdenote the boundaries defined byy=a(t),y=b(t), whereb(0) < 0 <a(0) andb(t) <a(t), 0 ≦t≦T≦∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path ofw(t) crossesaorbbeforet=T.This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the conditionw(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.
Stochastic inventory models with bulk demand and state-dependent leadtimes
Authors: Gross Donald , Harris Carl M. , Lechner James A.
Published: April 2006

This paper describes two one-for-one ordering (S− 1,S) inventory models in which the time required for order replenishment is state-dependent. The demand is assumed to follow a compound Poisson distribution, and that portion of the leadtime corresponding to the actual filling of orders is assumed to depend on the number of outstanding orders. Since the orders placed are assumed to go into a single-server queue, queuing results are used to obtain the expected inventory cost as a function ofSin order to obtain an optimal value ofS.
On the optimal control of stationary diffusion processes with natural boundaries and discounted cost
Authors: Morton R.
Published: April 2006

Results similar to those in [3] are obtained for one-dimensional diffusion processes with discounted cost. The stronger assumption that at least one of the inaccessible boundaries is natural enables us to identify the solution of a differential equation corresponding to the future expected cost function.
A relation between stationary queue and waiting time distributions
Authors: Haji Rasoul , Newell Gordon F.
Published: April 2006

A theorem is proved which, in essence, says the following. If, for any queueing system, (i) the arrival process is stationary, (ii) the queue discipline is first-in-first-out (FIFO), and (iii) the waiting time of each customer is statistically independent of the number of arrivals during any time interval after his arrival, then the stationary random queue size has the same distribution as the number of customers who arrive during a random time interval distributed as the stationary waiting time.
Diffusion with “collisions” between particles
Authors: Harris T. E.
Published: April 2006

First consider two particles diffusing on the same line, with positions at time t given by y1(t) and y2(t) respectively. We suppose that they cannot pass one another, so that if initially y1(0) < y2(0), then . As long as the two particles are not in contact, we suppose that each moves, independently of the other, according to the Wiener process. We must also prescribe what happens when collisions occur. There seems to be no unique natural way to do this, and we shall adopt a method recommended by its mathematical convenience. However, we shall see that there are two possible motivations for the prescription, one in terms of random walks, the other in terms of a more mechanical model.
Sufficiency for multinomial and transition probabilities
Authors: Bofinger Eve
Published: April 2006

Gani (1955) finds the form of the multinomial probabilities which admit a sufficient estimator for the single parameter θ to be
where λ θ is a positive function of θ, the Ki, are constants and the αi are nonnegative constants.
Some passage-time generating functions for discrete-time and continuous-time finite Markov chains
Authors: Darroch J. N. , Morris K. W.
Published: April 2006

Let T denote a subset of the possible transitions between the states of a finite Markov chain and let Yk denote the time of the kth occurrence of a T-transition. Formulae are derived for the generating functions of Yk, of Yj + k — Yj and of Yj + k — Yj in the limit as j → ∞, for both discrete-time and continuoustime chains. Several particular cases are briefly discussed.
The Factorisation of the rectangular distribution
Authors: Lewis T.
Published: April 2006

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.
Distribution of the branching-process population among generations
Authors: Samuels M. L.
Published: April 2006

In a standard age-dependent branching process, let Rn(t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn(t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn(t) is asymptotically linear in t. Further, it is found that, for large n, Rn(t) has the shape of a normal density function (of t).

Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn(t)}.

For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.
Stationary distributions of open Markov processes in discrete time with application to hospital planning
Authors: Staff P. J. , Vagholkar M. K.
Published: April 2006

Suppose that a new hospital is to be planned or an existing one reorganized according to the modern Progressive Patient Care System (Hartley et al. (1968)). In this system, there are three types of wards, viz.
Interconnected birth and death processes
Authors: Puri Prem S.
Published: April 2006

Two cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f)gof the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution forg.This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends togasn → ∞in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λiandμi, i =1,2, interconnected linearly with transition rates v andδ(see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1=δ= 0, with the remaining rates time dependent and (ii) λ2=δ= 0, with the remaining rates constant, explicit solutions forgare obtained and studied.
A queue with Markov-dependent service times
Authors: Pestalozzi Gerold
Published: April 2006

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.
First-Order Conservative Processes with Multiple Latent Roots
Authors: Radcliffe J. , Staff P. J.
Published: April 2006

There are now many examples in various fields where the behaviour of ‘particles' as exhibited by their transition from one state to another is described by a multidimensional stochastic process. The linear migration model in which particles move independently of one another through a number of states has been dealt with by Bartlett (1949). This process has been used by Siegert (1949) in considering the approach to equilibrium of non-interacting gas molecules and by Krieger and Gans (1960) and Gans (1960) to examine the distribution of a multicomponent system disturbed from its equilibrium distribution and relaxing by first-order processes to another equilibrium. The correspondence between the deterministic model based on the principle of mass action and the stochastic model has been discussed by Darvey and Staff (1966) in the context of unimolecular multicomponent chemical reactions.
Sums of Lifetimes in Age Dependent Branching Processes
Authors: Weiner J. Howard
Published: April 2006

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σk=0∞pksk and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.
Some aspects of the stochastic model for the attachment of phages to bacteria
Authors: Srivastava R. C.
Published: April 2006

Recently Gani (1965) has considered a stochastic model for the attachment of phages to bacteria. In this paper we study some of the basic properties of this model. These results will be used in a subsequent paper dealing with the estimation of the parameter which occurs in this model.
A note on the black box problem
Authors: Keats R. G.
Published: April 2006

It is well known (Lee 1960), especially to communications and control systems engineers, that the weighting function (unit impulse function) of a linear causal system may be obtained by cross correlating its output with its input. In general a Wiener-Hopf integral equation must be solved for the weighting function; but if the input were “white noise” of unit spectral density then, as shown by Lee (1950), the weighting function would be equal to this cross correlation function for positive values of its argument. Although a perfectly “white” input cannot be obtained in practice, a number of modifications of this technique have been used; Anderson, Buland and Cooper (1959), for example, describe the use of specially selected samples of “discrete-interval binary noise” in a problem in adaptive control.
A mathematical model for delays caused by right-turning vehicles at an uncontrolled intersection
Authors: Reid D. H.
Published: April 2006

It is apparent that much of the congestion in road traffic is due to interference between vehicles at intersections, and this has led to the appearance of several mathematical discussions of the problems of delay at intersections of various types (Cohen and Stam (1963), Cohen and de Lange (1965), Hawkes (1965), (1966), Tanner (1962), Weiss and Maradudin (1962)). Comparatively little work has appeared on the problem of delay caused by vehicles turning right* across an opposing stream of vehicles, although Hawkes (1966) and Newell (1959) discuss this problem explicitly, and the models considered by all of these authors may be regarded as having some application to the right-turning problem.
Some characterizations based on the Bhattacharya matrix
Authors: Shanbhag D. N.
Published: April 2006

Laha and Lukacs (1960) have studied distributions with the property that a quadratic statistic has quadratic regression on the sample mean. In doing this, they have arrived at some interesting characterizations for the normal, Poisson, gamma, binomial and negative binomial distributions. Starting with an exponential-type probability density function, the present paper investigates all the distributions for which the 3 × 3 Bhattacharya matrix is diagonal. It is found that the normal, Poisson, gamma, binomial and negative binomial distributions can be characterized by this property. Further, it is observed that for these distributions an s × s Bhattacharya matrix is defined for all s and is also diagonal.
On the distribution of the number of cyclically occurring random events
Authors: Podkaminer Leon
Published: April 2006

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.
On dams with Markovian inputs
Authors: Pakes A. G.
Published: April 2006

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, pij, 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 [pijxj] 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.
A note on Kendall's autoregressive series
Authors: Brown T. J. , Hardin J. C.
Published: April 2006

Realizations of the autoregressive series considered by Kendall are generated using both Gaussian and uniformly distributed random variables. Estimates of the autocorrelation and power spectrum of the process obtained from these realizations are compared with the theoretical expressions derived by Bartlett. Agreement is well within the theoretical variance.
On the optimal search for a target whose motion is a Markov process
Authors: Saretsalo Lauri
Published: April 2006

We will consider the optimal search for a target whose motion is a Markov process. The classical detection law leads to the use of multiplicative functionals and the search is equivalent to the termination of the Markov process with a termination density. A general condition for the optimality is derived and for Markov processes in n-dimensional Euclidean space with continuous transition functions we derive a simple necessary condition which generalizes the result of Hellman (1972).
On the extinction times of varying and random environment branching processes
Authors: Agresti Alan
Published: April 2006

Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.
Two dependent Poisson processes whose sum is still a Poisson process
Authors: Jacod Jean
Published: April 2006

We show that there exists a pair of Poisson processes which are not stochastically independent, and whose sum is still a Poisson process.
The exponential rate of convergence of the distribution of the maximum of a random walk
Authors: Veraverbeke N. , Teugels J. L.
Published: April 2006

Let Gn(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn(x) — G(x) is asymptotically equal to c.H(x)n−3/2γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.
Conditional limit theorems for a left-continuous random walk
Authors: Pakes A. G.
Published: April 2006

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.
Information flow in discrete Markov systems
Authors: Dawson D. A.
Published: April 2006

Infinite systems of interacting Markov chains are investigated. Some basic concepts of the ergodic theory of such systems are first presented. In particular, the question of the uniqueness and multiplicity of invariant probability measures is considered. In the case of one-dimensional systems, the question is studied in detail by investigating the flow of information throughout the system and some criteria for the uniqueness of invariant probability measures are obtained.