bluuebunny/crossref_metadata_2025 huggingface dataset:
Networks of queues
Authors: Kelly F. P.
Published: April 2006

The behaviour in equilibrium of networks of queues is studied. Equilibrium distributions are obtained and in certain cases it is shown that the state of an individual queue is independent of the state of the rest of the network. The processes considered in this paper are irreversible; however, the method used to establish equilibrium distributions is one which has previously only been used when dealing with reversible processes. Results are obtained for models of communication networks, machine interference and birth-illness-death processes.
Branching diffusion processes in population genetics
Authors: Sawyer Stanley
Published: April 2006

A branching random field is considered as a model of either of two situations in genetics in which migration or dispersion plays a role. Specifically we consider the expected number of individualsNAin a (geographical) setAat timet, the covariance ofNAandNBfor two setsA, B, and the probabilityI(x, y, u) that two individuals found at locationsx, yat timetare of the same genetic type if the population is subject to a selectively neutral mutation rateu.The last also leads to limit laws for the average degree of relationship of individuals in various types of branching random fields. We also find the equations that the mean and bivariate densities satisfy, and explicit formulas when the underlying migration process is Brownian motion.
Homecomings of Markov processes
Authors: Kingman J. F. C.
Published: April 2006

Ifx0is a particular state for a continuous-time Markov processX, the random time setis often of both practical and theoretical interest. Ignoring trivial or pathological cases, there are four different types of structure which this random set can display. To some extent, it is possible to treat all four cases in a unified way, but they raise different questions and require different modes of description. The distributions of various random quantities associated withcan be related to one another by simple and useful formulae.
Everywhere irregularity of certain classes of random processes with stationary Gaussian increments
Authors: Davies P. L.
Published: April 2006

This paper is concerned with everywhere local behaviour of certain classes of random processes which have stationary Gaussian increments. It is shown that for two classes of processes almost all the sample functions have the following property. The supremum of the increments in the neighbourhood of a point is everywhere of larger order than the standard deviation. For a third class of processes it is shown that the supremum is at least of the same order as the standard deviation.
Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach
Authors: Scott D. J.
Published: April 2006

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.
Regular variation of the tail of a subordinated probability distribution
Authors: Stam A. J.
Published: April 2006

Let F be a probability measure on the real line and G = Σ C(k)Fk∗ the probability measure subordinate to F with subordinator C restricted to the nonnegative integers. Let V(x) vary regularly of order p for x→ ∞ and either (1) V(x) F[x, ∞)→ α ≧ 0 or (2) V(x) C[x, ∞)→ γ ≧ 0. If ρ > 1 and F(–∞, 0) = 0, necessary and sufficient in order that V(x) G[x, ∞)→b, is that both (1) and (2) hold for suitable α and γ. For 0 ≦ ρ ≦ 1 the conditions are of different type. For two-sided F a different situation arises and only sufficient conditions are found. An application to renewal moments of negative order is given.
Optimal control of batch service queues
Authors: Deb Rajat K. , Serfozo Richard F.
Published: April 2006

A batch service queue is considered where each batch size and its time of service is subject to control. Costs are incurred for serving the customers and for holding them in the system. Viewing the system as a Markov decision process (i.e., dynamic program) with unbounded costs, we show that policies which minimize the expected continuously discounted cost and the expected cost per unit time over an infinite time horizon are of the form: at a review point whenxcustomers are waiting, serve min {x, Q} customers (Qbeing the, possibly infinite, service capacity) if and only ifxexceeds a certain optimal levelM.Methods of computingMfor both the discounted and average cost contexts are presented.
A semi-Markov storage model
Authors: Senturia Jerome , Puri Prem S.
Published: April 2006

In this paper a storage model is described in which fluctuations in the content are governed by a sequence of independent identically distributed (i.i.d.) random inputs and i.i.d. random releases. This sequence proceeds according to an underlying semi-Markov process. Laplace transforms of the exact distribution of the content are given for the case of negative exponential distributions for both inputs and releases. Exact expressions for limiting (in time) content distributions are found. In the general case, the asymptotic behavior of the content is described for critical and supercritical limiting conditions.
Space-time harmonic functions and age-dependent branching processes
Authors: Savits Thomas H.
Published: April 2006

Let X be an age-dependent branching process with lifetime distribution G and age-dependent generating function π(y,s) = σk = 0∞pk(y) sk. We assume that G is right-continuous and G(0+) = G(0) = 0. The base state space S is [0,T) where T = inf{t : G(t) = 1}. Set m(y) = σk = 0∞k pk(y) and Then extinction occurs with probability one iff m ≤ 1. In the case where m > 1, define the Malthusian parameter λ to be the unique (positive) root of and set

on S. is a -space-time harmonic function of the process X and the corresponding non-negative martingale converges w.p.l to a random variable W; furthermore, under a regularity assumption, W is non-trivial iff where and If 0 < a ≤ Φ ≤ β < ∞, for some constants a, β, then w.p.l, where Zt is the number of particles at time t.
Bayesian dynamic programming
Authors: Rieder Ulrich
Published: April 2006

We consider a non-stationary Bayesian dynamic decision model with general state, action and parameter spaces. It is shown that this model can be reduced to a non-Markovian (resp. Markovian) decision model with completely known transition probabilities. Under rather weak convergence assumptions on the expected total rewards some general results are presented concerning the restriction on deterministic generalized Markov policies, the criteria of optimality and the existence of Bayes policies. These facts are based on the above transformations and on results of Hindererand Schäl.
Ergodicity of age structure in populations with Markovian vital rates, III: Finite-state moments and growth rate; an illustration
Authors: Cohen Joel E.
Published: April 2006

Leslie (1945) models the evolution in discrete time of a closed, single-sex population with discrete age groups by multiplying a vector describing the age structure by a matrix containing the birth and death rates. We suppose that successive matrices are chosen according to a Markov chain from a finite set of matrices. We find exactly the long-run rate of growth and expected age structure. We give two approximations to the variance in age structure and total population size. A numerical example illustrates the ergodic features of the model using Monte Carlo simulation, finds the invariant distribution of age structure from a linear integral equation, and calculates the moments derived here.
Renewal theory in two dimensions: bounds on the renewal function
Authors: Hunter Jeffrey J.
Published: April 2006

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.
Countable non-homogeneous Markov chains: asymptotic behaviour
Authors: Cohn Harry
Published: April 2006

The paper deals with asymptotic properties of the transition probabilities of a countable non-homogeneous Markov chain, the main concept used in the proofs being that of the tail σ-field of the chain. A state classification similar to that existing in the homogeneous case is given and a strong ratio limit property is shown to parallel the basic limit theorem for positive homogeneous chains. Some global asymptotic properties for null chains are also derived.
One-dimensional Brownian motion and the three-dimensional Bessel process
Authors: Pitman James W.
Published: April 2006

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.
Limit theorems for stationary distributions
Authors: Norman M. Frank
Published: April 2006

For everyN≧ 1, letbe a real-valued stationary process, and let. Suppose thatand var, where∊NandτNare positive null sequences. Limiting distributions ofasN→∞are obtained for the casesτN=∊NandτN=o(∊N). These results are established by an extension of a method due to Moran. The theory is illustrated by a variety of applications to genetic models.
The theory of periodic screening II: doubly bounded recurrence times and mean lead time and detection probability estimation
Authors: Prorok Philip C.
Published: April 2006

Additional theory important in the design and evaluation of a periodic screening program is presented. The distributions of doubly bounded preclinical state forward and backward recurrence times of the stochastic screening model are derived. Properties of these distributions are used to obtain an expression for estimating the mean lead time gained by screening, an alternative theoretical expression for the mean lead time and an equation for calculating an estimate of the probability of detecting preclinical disease.
The control of a multipurpose reservoir
Authors: Haslett John
Published: April 2006

A technique known as potential cost, used by Faddy [3] for assessing the operation of a dam is seen to be capable of extension to allow for

(i) a very general cost function, as is required for a multipurpose reservoir (the norm nowadays) and

(ii) the use of discounting of future costs, a very widespread accounting procedure.

Numerical results are obtained for an optimal policy based on such an assessment, and demonstrate the need for an accurate specification of the costs associated with the operation of a reservoir. As a by-product a very full description of the steady-state stochastic behaviour of the dam is obtained.
Mathematical models in the interpretation of the interaction between infective agents
Authors: Gart John J.
Published: April 2006

The effect of an epidemic of chickenpox on the course of an epidemic of yaws in a village of New Guinea was studied by Gart and deVries (1966) and Gart (1968). A deterministic continuous time model was used to confirm that the yaws epidemic accelerated dramatically in the month following the chickenpox epidemic. However, by the nature of the model, no statistical tests are possible.
Hyperplasia: The spread of abnormal cells through a plane lattice
Authors: Williams Trevor , Bjerknes Rolf
Published: April 2006

When a basal cell divides, both daughter cells remain in the basal layer of the epithelium, with one of the neighbouring cells being pushed out to make room. This fact opens the possibility that a cell with a heritable advantage over the normal cells may gradually produce a clone covering more and more of the basal layer. The advantage in question may consist in a faster rate of division than normal, or a more tenacious hold on the basement membrane; we shall limit consideration to the former situation.
An algebraic proof of the threshold theorem for the general stochastic epidemic
Authors: Williams Trevor
Published: April 2006

Bailey (1957) has considered the general stochastic epidemic with relative removal rate ρ, one initial infective and n susceptibles. His Equation (5.51) may be re-cast as

where the fk satisfy the recurrence relationship
Computer modelling of influenza epidemics for the whole country (USSR)
Authors: Baroyan O. V. , Rvachev L. A. , Basilevsky U. V. , Ermakov V. V. , Frank K. D. , Rvachev M. A. , Shashkov V. A.
Published: April 2006

Influenza epidemics at a city or country level can be observed only through a daily officially registered morbidity, or briefly DORM. The influenza model for any country is defined as some set of equations containing the DORM for cities in that country as unknown functions of time, these being designed to approximate the cities' real DORM.
A branching process with a state dependent immigration component
Authors: Pakes A. G.
Published: April 2006

Consider the well known Galton-Watson branching process (Harris (1963)) in which individuals reproduce independently of each other and have probabilityaj(j= 0, 1, · · ·) of giving rise tojindividuals in the next generation. In recent years some attention has been given to the branching process in which there is an independent immigration component at each generation,bj(j= 0, 1, · · ·) being the probability ofjimmigrants entering each generation. For a review of this work see Seneta (1969), and Pakes (1971 a, b and c) for further results.
On asymptotic independence of the partial sums of positive and negative parts of independent random variables
Authors: Tucker Howard G.
Published: April 2006

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution function F, assume F is in the domain of attraction of a stable distribution with characteristic exponent α, 0 < α ≦ 2, and let {Bn} be normalizing coefficients for F. Let us denote Xn+ = XnI[Xn > 0] and Xn− = − XnI[Xn<0], so that Xn = Xn+ - Xn−. Let F+ and F− denote the distribution functions of X1+ and X1− respectively, and let Sn(+) = X1+ + · · · + Xn+, Sn(-) = X1− + · · · + Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+) + an, Bn-1Sn(-) + bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α < 2 and when α = 2. When α < 2, there always exist such sequences so that the above is true, and in this case both U and V are stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 < ∫ x2dF(x) < ∞, then there always exist such sequences such that the above convergence takes place; both U and V are normal (possibly one is a constant), and if neither is a constant, then U and V are not independent. If α = 2 and ∫ x2dF(x) = ∞, then at least one of F+, F− is in the domain of partial attraction of the normal distribution, and a modified statement on the independence of U and V holds. Various specialized results are obtained for α = 2.
A broad analysis of single server priority queues with two independent input streams, one of them Poisson
Authors: Schassberger R.
Published: April 2006

A single server faces a renewal stream of low and another one of high priority customers, one of them being a Poisson stream. For the preemptive resume and head-of-the line disciplines the transient behaviour of the queue is studied. Various limiting distributions are also obtained.
Asymptotic rate of a Markov process
Authors: Craven B. D.
Published: April 2006

For a Markov process in discrete time, both (a) the time-dependent behaviour of the process at time t, and (b) the behaviour of moments (if finite) of the process, whether or not stationary, as a function of time-lag t, are determined by solutions to a single equation, the key equation. The state space of the process is required only to be both a measure space, and a locally convex topological vector space, which may have infinite dimension. The asymptotic behaviour of both (a) and (b), for large times t, is shown to be (nearly) the same, being proportional to tξe−βt, where the exponent β is calculable from an eigenvalue of a linear operator, describing the Markov process. For a large class of Markov processes, the spectrum of this operator is shown to consist of discrete eigenvalues only. Queueing applications are discussed.
Sojourn times, exit times and jitter in multivariate Markov processes
Authors: Keilson J.
Published: April 2006

To treat the transient behavior of a system modeled by a stationary Markov process in continuous time, the state space is partitioned into good and bad states. The distribution of sojourn times on the good set and that of exit times from this set have a simple renewal theoretic relationship. The latter permits useful bounds on the exit time survival function obtainable from the ergodic distribution of the process. Applications to reliability theory and communication nets are given.
Some remarks on a paper of Kingman
Authors: Getoor R. K.
Published: April 2006

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b1, …, bn}.
Regression and autoregression with infinite variance
Authors: Kanter Marek , Steiger W. L.
Published: April 2006

The theory of the linear model is incomplete in that it fails to deal with variables possessing infinite variance. To fill an important part of this gap, we give an unbiased estimate, the “screened ratio estimate”, for λ in the regression E(X|Z) = λX; X and Z are linear combinations of independent, identically distributed symmetric random variables that are either stable or asymptotically Pareto distributed of index α ≤ 2. By way of comparison, the usual least squares estimate of λ is shown not to converge in general to any constant when α < 2. However, in the autoregression Xn = a1Xn-1 + … + akXn-k + Un, the least squares estimates are shown to be consistent as long as the roots of 1 - a1x2 - a2x2 - … - akxk = 0 are outside the complex unit circle, Xn is independent of Un+j,j ≥ 1, and the Un are independent and identically distributed and in the domain of attraction of a stable law of index a ≤ 2. Finally, the consistency of least squares estimates for finite moving averages is established.
Linkage disequilibrium between neutral genes in finite populations
Authors: Hill William G.
Published: April 2006

There is now a large literature on linkage disequilibrium between pairs of loci, both for selection in infinite populations and for neutral genes in finite populations, but there have been few studies with more loci. Bennett (1954) showed how the frequencies of chromosomes with any number of neutral genes would change in an infinite population, and the author (unpublished) has extended Bennett's results to find expected changes in chromosome frequencies with up to six loci in finite populations. For two linked neutral genes in finite populations the expected disequilibrium is zero, but the variance of the disequilibrium or the correlation of gene frequencies in segregating populations has been found. This has been done by Monte Carlo simulation (Hill and Robertson (1968)), but an approximation can be obtained by diffusion methods (Ohta and Kimura (1969)) and the asymptotic values using inbreeding theory (Sved (1971)). In this note we discuss the case of disequilibrium between three neutral loci and show how it relates to that between two loci.
On discrete time extremal processes
Authors: Shorrock R. W.
Published: April 2006

Upper record values and times and inter-record times are studied in their rôles as embedded structures in discrete time extremal processes. Various continuous time approximations to the discrete-time processes are analysed, especially as processes over their state spaces. Discrete time processes, suitably normalized after crossing a threshold T, are shown to converge to limiting continuous time processes as T → ∞ under suitable assumptions on the underlying CDF F, for example, when 1 — F varies regularly at ∞, and more generally. Discrete time extremal processes viewed as processes over their state spaces are noted to have an interesting interpretation in terms of processes of population growth.
Limit theorems for some functionals of certain Galton-Watson branching processes
Authors: Lindvall Torgny
Published: April 2006

This paper extends the Feller-Jiřina theorem on the diffusion approximation of Galton-Watson branching processes with reproduction mean close to one, and limit theorems are obtained for several functionals of such processes.
The statistical analysis of spatial pattern
Authors: Bartlett M. S.
Published: April 2006

A brief survey is made of the range of models and statistical techniques relevant to the analysis of spatial pattern, and more detailed discussion given to three specific classes of stochastic process in two dimensionsviz.(i) continuous processesX(r), (ii) point processesN(r), (iii) lattice processesXi. The appropriate theory for these classes of processes is indicated in Part I; and in Part II some examples of data are classified under these same three headings, and their statistical analysis discussed.
Inverses of extremal processes
Authors: Resnick Sidney I.
Published: April 2006

The inverse of an extremal process {Y(t),t≧ 0} is an additive process whose Lévy measure can be computed. This measure controls among other things the Poisson number of jumps ofYwhileYis in the vertical window (c, d]. A simple transformation of the inverse of the extremal process governed by Λ (x) = exp{–e–x} is also extremal-Λ (x) and this fact enables one to relate behavior ofY-Λ att= ∞ to behavior neart= 0. Some extensions of these ideas to sample sequences of maxima of i.i.d. random variables are carried out.
Optimal stopping problems for Brownian motion
Authors: Bather John
Published: April 2006

This paper is concerned with the general problem of choosing an optimal stopping time for a Brownian motion process, where the cost associated with any trajectory depends only on its final time and position.
A queue subject to extraneous phase changes
Authors: Neuts Marcel F.
Published: April 2006

Many service systems exhibit variations of a random nature in the intensity of the arrival process or of the speed of service or of both. Changes in work shifts, rush hours, interruptions in the arrival process, server breakdowns, etc. all fall into this category.

The present study deals with a generalization of the classical M/G/1 queue by considering an extraneous process of phases which can be in one of the states {1, …,m}. During any interval spent in phasei, the arrivals are according to a homogeneous Poisson process of rate λiand any service initiated during such an interval has a duration distributed according toHi(·). The process of phases is assumed to be an irreducible Markov chain in continuous time and is fully characterized by its initial conditions, by an irreducible stochastic matrixPand by the mean sojourn times σ1-1, …, σm-1in each phase.

Independently of the queueing aspects, this arrival process is a generalization of the classical Poisson process which can be of interest in modelling simple point processes with randomly fluctuating “arrival” rate.

Two approaches to the time dependent study of this queue are presented; one generalizes the imbedded semi-Markov process obtained by considering the queue immediately following departure points; the other approach exploits the relationship between this queue and branching processes. The latter is more elegant from a purely theoretical viewpoint and involves iterates of a general type of matrix function introduced by the author. By making extensive use of the Perron-Frobenius theory of positive matrices the equilibrium condition of the queue is obtained. While retaining a similar intuitive interpretation the equilibrium condition is substantially more complicated than for the M/G/1 model.

The recurrence relations which yield the joint distribution of the phase state at timet, the queue length, the total number served and the virtual waiting time attare exhibited in detail. Via transform techniques a number of limiting and marginal distributions are discussed. The discussion relies heavily on the theory of Markov renewal processes.

Throughout the paper and in a final section the author advocates the use of the structural properties of the queue and the resulting recurrence relations to organize the numerical analysis of complex queueing models such as the present one.

More explicit results for the case of two phases are given and are compared to results obtained by Yechiali and Naor for a closely related two-phase generalization of the M/M/1 queue.
The analysis of queues by state-dependent parameters by Markov renewal processes
Authors: Schäl Manfred
Published: April 2006

In this paper, some results on the asymptotic behavior of Markov renewal processes with auxiliary paths (MRPAP's) proved in other papers ([28], [29]) are applied to queueing theory. This approach to queueing problems may be regarded as an improvement of the method of Fabens [7] based on the theory of semi-Markov processes. The method of Fabens was also illustrated by Lambotte in [18], [32]. In the present paper the ordinaryM/G/1 queue is generalized to allow service times to depend on the queue length immediately after the previous departure. Such models preserve the MRPAP-structure of the ordinaryM/G/1 system. Recently, the asymptotic behaviour of the embedded Markov chain (MC) of this queueing model was studied by several authors. One aim of this paper is to answer the question of the relationship between the limiting distribution of the embedded MC and the limiting distribution of the original process with continuous time parameter. It turns out that these two limiting distributions coincide. Moreover some properties of the embedded MC and the embedded semi-Markov process are established. The discussion of theM/G/1 queue closes with a study of the rate-of-convergence at which the queueing process attains equilibrium.
The geometry of random drift III. Recombination and diffusion
Authors: Antonelli Peter L. , Morgan Kenneth , Lathrop G. Mark
Published: April 2006

A new diffusion model for random genetic drift of a two-locus di-allelic system is proposed. The Christoffel velocity field and the intrinsic geometry of the diffusion is computed for the equilibrium surface. It is seen to be radically non-spherical and to depend explicitly on the recombination fraction. The model has not been shown to be a limit of discrete Markov chains. For large values of the recombination, the present model is radically different from that of Ohta and Kimura, which is an approximation to the discrete process of random mating in the limit as the value of the recombination fraction goes to zero.
A model for the parasitic disease bilharziasis
Authors: Lewis Trevor
Published: April 2006

In this paper we develop a model for describing the spread of the parasitic disease bilharziasis. The detailed model first developed (Model A) is made more tractable by replacing infecting population sizes by their expected values in the transition probabilities, giving Model B. Using this approximating model it is possible to examine the nature of the spread of the disease and the effects of control measures. The effects of a particular control measure are examined in order to compare Models A and B.
Fluctuation theory in continuous time
Authors: Bingham N. H.
Published: April 2006

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.
Wiener-Hopf methods, decompositions, and factorisation identities for maxima and minima of homogeneous random processes
Authors: Greenwood Priscilla
Published: April 2006

Some Wiener-Hopf type results are collected, related and given more direct proofs. Spitzer's random-walk method for the Wiener-Hopf integral equation also produces his factorisation relating functionals of maxima and minima. Transform equations are interpreted as decompositions of time-changed processes. Discrete- and continuous-time versions are related. Prabhu's factorisation for generators is equivalent to Fristedt's Lévy measure factorisation and to process decomposition.
Combinatorial properties of a queueing system by limited availability
Authors: Pollard J. M.
Published: April 2006

A type of service system of particular interest in teletraffic theory has the property that calls (customers) originating from different sources have access to different subsets of a single group of devices (servers). One may or may not allow the formation of queues. Thierer [11], [12] has studied such a system with queues assuming it to be reversible. We investigate the conditions under which this assumption is exact.
Equilibrium distributions Markov population processes
Authors: Barbour A. D.
Published: April 2006

Distributional limit theorems, together with rates of convergence, are obtained for the equilibrium distributions of a wide variety of one-dimensional Markov population processes. Three separate cases are considered. First, in the standard setting, the convergence as N→∞ of √N(xN-c) to a normal distribution is established, together with a rate of convergence of O(N−1/2), under weaker conditions than those previously imposed: here, c represents the unique equilibrium of the deterministic equations ẋ = F(x), and xN denotes the population process under its equilibrium distribution. This convergence holds if F′(c)<0: the next section shows that, if F′(c) = 0, both the normalization and the limit distribution are different. Finally, sequences of processes xN suitable for approximating genetical models are considered. In these circumstances, xN itself converges in distribution as N→∞, and the convergence rate is essentially O(N-1), though modification is sometimes needed near natural boundaries.
The steady-state solution of the M/K2/m queue
Authors: Hokstad Per
Published: April 2006

The many-server queue with service time having rational Laplace transform of order 2 is considered. An expression for the asymptotic queue-length distribution is obtained. A relatively simple formula for the mean queue length is also found. A few numerical results on the mean queue length and on the probability of having to wait are given for the case of three servers. Some approximations for these quantities are also considered.
A class of singular stochastic control problems
Authors: Karatzas Ioannis
Published: April 2006

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in a singular manner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.
Design problems for the pure birth process
Authors: Becker Gerhard , Kersting Götz
Published: April 2006

LetY(t) be a pure birth process. If a maximum likelihood estimator of the birth intensity is desired and the numbernof observational points and the last observation T are given in advance, it is shown that equidistant sampling is not an optimal procedure. Properties of ‘optimal' designs and the corresponding maximum likelihood estimators are investigated and compared with equidistant and continuous sampling.
Filtering of Markov renewal queues, I: Feedback queues
Authors: Hunter Jeffrey J.
Published: April 2006

Queueing systems which can be formulated as Markov renewal processes with basic transitions of three types, ‘arrivals', ‘departures' and ‘feedbacks' are examined. The filtering procedure developed for Markov renewal processes by Çinlar (1969) is applied to such queueing models to show that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also Markov renewal. In this part we focus attention on the derivation of stationary and limiting distributions (when they exist) for each of the embedded discrete-time processes, the embedded Markov chains. These results are applied to birth–death queues with instantaneous state-dependent feedback including the special cases of M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback.
A class of branching processes on a lattice with interactions
Authors: Schürger Klaus
Published: April 2006

In this paper, a very general class of branching processes on thed-dimensional square lattice is studied. It is assumed that the division rates as well as the spatial distribution of offspring are configuration-dependent. The main interest of this paper is in the asymptotic geometrical behaviour of such processes. Utilizing techniques mainly due to Richardson [28], we derive conditions which are necessary and sufficient for such branching processes to have the following property: there exists a norm N(·) onRdsuch that, for all 0 <∊< 1, we have that almost surely for all sufficiently larget, all sites in theN-ball of radius (1 –∊)tare contained in(the set of sites occupied at timet) andis contained in the set of all sites in theN-ball of radius (1 +∊)t(given that the process starts with finitely many particles).
Entropy inequalities for classes of probability distributions I. The univariate case
Authors: Karlin Samuel , Rinott Yosef
Published: April 2006

Entropy functionals of probability densities feature importantly in classifying certain finite-state stationary stochastic processes, in discriminating among competing hypotheses, in characterizing Gaussian, Poisson, and other densities, in describing information processes, and in other contexts. Two general types of problems are considered. For a given parametric family of densities the member of maximal (or sometimes minimal) entropy is ascertained. Secondly, we determine a natural (partial) ordering over for which the entropy functional is monotone. The examples include the multiparameter binomial, multiparameter negative binomial, some classes of log concave densities, and others.
Analysis of two parallel queues by common exit service and infinite queue size
Authors: Atkinson C. , Thompson M. E.
Published: April 2006

A system of two parallel queues is considered, where each customer must leave after service through a common gate G. It is assumed that service times at the two stations I and II are independent and identically distributed, and that exit service takes a fixed length of time. A I-customer may be served at station I only if the previous I-customer has completed exit service. Integral equations are formulated from which the distribution of the total service time may be obtained when the two queue sizes are infinite. These equations are solved for exponential and generalized erlangian service times. Extensions to the case of k parallel queues and to the case of Poisson arrivals and finite queue sizes are discussed briefly.
Comparing counting processes and queues
Authors: Whitt Ward
Published: April 2006

Several partial orderings of counting processes are introduced and applied to compare stochastic processes in queueing models. The conditions for the queueing comparisons involve the counting processes associated with the interarrival and service times. The two queueing processes being compared are constructed on the same probability space so that each sample path of one process lies below the corresponding sample path of the other process. Stochastic comparisons between the processes and monotone functionals of the processes follow immediately from this construction. The stochastic comparisons are useful to obtain bounds for intractable systems. For example, the approach here yields bounds for queues with time-dependent arrival rates.
The distribution of heterogeneity in finite random mating populations
Authors: Stam P.
Published: April 2006

The probability distribution of the heterogenic (non-identical by descent) fraction of the genome in a finite monoecious random mating population has been derived. It was assumed that in any generation both the lengths of heterogenic and homogenic segments are exponentially distributed. An explicit expression is given for the expected number of ‘external junctions’ (sites that mark the end of a heterogenic segment) per unit map length in any generation. The latter necessitates the introduction of two higher-order identity relations between three genes, and their recurrence relations. Theoretical results were compared with the outcome of a series of simulation runs (showing a very good fit), as well as with the results predicted by Fisher's ‘theory of junctions’. In contrast to Fisher's approach, which only applies when the average heterogeneity is relatively small, the present model applies to any generation.
Sequential sampling schemes for sibships
Authors: Thompson E. A.
Published: April 2006

Many medical genetics studies are at least partially sequential, and in considering problems of ascertainment it seems desirable to consider such schemes, in addition to classical ascertainment models. Cannings and Thompson (1978) indicated the correction to the likelihood required when sampling a pedigree sequentially from some predetermined initial event, and Thompson and Cannings (1979) have extended this result and compared sequential sampling with more classical ascertainment models.
Proof of Fischer's fundamental theorem of natural selection
Authors: Vetta Atam
Published: April 2006

Fischer (1930) stated his fundamental theorem of natural selection (ftns) as ‘the rate of increase in fitness of any organism at any time is equal to its genetic (additive) variance in fitness at that time’. The theorem related to an idealised population and he thought that it held ‘the supreme position among the biological sciences’. His proof of the theorem is, however, not understood. This has given rise to questions concerning not only the validity of the proof but also concerning the importance of the theorem.
Stochastic monotonicity of birth–death processes
Authors: Van Doorn Erik A.
Published: April 2006

A birth–death process {x(t):t≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t1,t2) if Pr {x(t) >i} is increasing (decreasing) withton (t1,t2) for alli= 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.
Polymerisation processes with intrapolymer bonding. I. one type of unit
Authors: Whittle P.
Published: April 2006

We study clustering (‘polymerisation’) processes under the assumptions that new bonds may or may not form within an existing cluster. We show that both models exhibit criticality (‘gelation’) at the same point, and investigate statistics of polymer size and structure beyond criticality in the first case. Gelation may or may not be accompanied by critical effects at unit level; we examine a prototype of each case: the cosh model and the Poisson model.
Bounds and good policies in stationary finite–stage Markovian decision problems
Authors: Hübner Gerhard
Published: April 2006

A stationary Markovian decision model is considered with general state and action spaces where the transition probabilities are weakened to be bounded transition measures (this is useful for many applications). New and improved bounds are given for the optimal value of stationary problems with a large planning horizon if either only a few steps of iteration are carried out or, in addition, a solution of the infinite-stage problem is known. Similar estimates are obtained for the quality of policies which are composed of nearly optimal decisions from the first few steps or from the infinite-stage solution.
Mathematical models for the variation of air-pollutant concentrations
Authors: Grandell Jan
Published: April 2006

The purpose of this paper is to study simple mathematical models for the variation of the concentration of atmospheric pollutants. The particles are assumed to be emitted into the atmosphere with constant intensity and removed from the atmosphere with an intensity proportional to the precipitation intensity. The precipitation intensity is assumed to be a stationary stochastic process with no spatial variation. The mean, variance and covariance function of the concentration are calculated for some precipitation models. Under certain general assumptions it is shown that the concentration of ‘long-lived' particles is approximately described by an Ornstein–Uhlenbeck process. Finally, the different models are compared by means of Swedish precipitation data.
A thermal energy storage process with controlled input
Authors: Daley D. J. , Haslett J.
Published: April 2006

The stochastic process {Xn} satisfyingXn+1= max{Yn+1+ αβ Xn,βXn} where {Yn} is a stationary sequence of non-negative random variables and, 0<β<1, can be regarded as a simple thermal energy storage model with controlled input. Attention is mostly confined to the study ofμ=EXwhere the random variableXhas the stationary distribution for {Xn}. Even for special cases such as i.i.d.Ynorα= 0, little explicit information appears to be available on the distribution ofXorμ. Accordingly, bounding techniques that have been exploited in queueing theory are used to studyμ. The various bounds are illustrated numerically in a range of special cases.
An extended Fatou equation and continuous-time gambling
Authors: Pestien Victor C.
Published: April 2006

Let be an optional stochastic process and let be the directed set of almost-surely-finite stopping times. If lim , then

This equality is employed in studying the continuous-time gambler's problem, as formulated by Heath and Sudderth (1974). The optimal return function is shown to be upper semi-analytic and excessive whenever the utility function is Borel.
Branching random walk in varying environments
Authors: Klebaner C. F.
Published: April 2006

The branching random walk model is generalized towards generation-dependent displacement and reproduction distributions. Asymptotic theory of branching random walk in varying environments from the L2 point of view is given. If Zn(x) is the number of nth-generation particles to the left of x, then under appropriate conditions for suitably chosen xn, Zn (xn)/Zn (+∞) converges in L2 completely to a limiting distribution. Sufficient conditions for almost sure convergence are given. As a corollary an analogue of the central limit theorem for the proportion of particles of the nth generation in time interval In in the age-dependent Crump–Mode–Jagers process is obtained.
Threshold limit theorems for some epidemic processes
Authors: Von Bahr Bengt , Martin-Löf Anders
Published: April 2006

The Reed–Frost model for the spread of an infection is considered and limit theorems for the total size, T, of the epidemic are proved in the limit when n, the initial number of healthy persons, is large and the probability of an encounter between a healthy and an infected person per time unit, p, is λ/n. It is shown that there is a critical threshold λ = 1 in the following sense, when the initial number of infected persons, m, is finite: If λ ≦ 1, T remains finite and has a limit distribution which can be described. If λ > 1 this is still true with a probability σm < 1, and with probability 1 – σmT is close to n(1 – σ) and has an approximately Gaussian distribution around this value. When m → ∞ also, only the Gaussian part of the limit distribution is obtained. A randomized version of the Reed–Frost model is also considered, and this allows the same result to be proved for the Kermack–McKendrick model. It is also shown that the limit theorem can be used to study the number of connected components in a random graph, which can be considered as a crude description of a polymerization process. In this case polymerization takes place when λ > 1 and not when λ ≦ 1.
On the spectral decomposition of stationary time series using walsh functions. II
Authors: Kohn R.
Published: April 2006

The paper derives the asymptotic properties of a class of estimators of the Walsh–Fourier spectral density of a stationary time series. The spectral density is defined in Kohn (1980).
The critical branching diffusion in a varying environment
Authors: Ivanoff B. Gail
Published: April 2006

A critical branching diffusion in an environment which varies in time is defined. Conditions are given for convergence of the process to a limit, and the triviality or non-triviality of the limit is discussed. If the migration is a Gaussian process in two dimensions, the limit is trivial, and an exponential limit law is proven.
On queueing systems with renewal departure processes
Authors: Berman Mark , Westcott Mark
Published: April 2006

It is proved that, for a large class of stable stationary queueing systems with renewal arrival processes and without losses, a necessary condition for the departure process also to be a renewal process is that its interval distribution be the same as that of the arrival process. This result is then applied to the classicalGI/G/squeueing systems. In particular, alternative proofs of known characterizations of theM/G/1 andGI/M/1 systems are given, as well as a characterization of theGI/G/∞ system. In the course of the proofs, sufficient conditions for the existence of all the moments of the stationary queue-size distributions of both theGI/G/1 andGI/G/∞ systems are derived.
Optimal control of the service rate in an M/G/1 queueing system
Authors: Doshi Bharat T.
Published: April 2006

We consider an M/G/1 queue in which the service rate is subject to control. The control is exercised continuously and is based on the observations of the residual workload process. For both the discounted cost and the average cost criteria we obtain conditions which are sufficient for a stationary policy to be optimal. When the service cost rate and the holding cost rates are non-decreasing and convex it is shown that these sufficient conditions are satisfied by a monotonic policy, thus showing its optimality.