bluuebunny/crossref_metadata_2025 huggingface dataset:
Multiple channel queues in heavy traffic. II: sequences, networks, and batches
Authors: Iglehart Donald L. , Whitt Ward
Published: April 2006

This paper is a sequel to [7], in which heavy traffic limit theorems were proved for various stochastic processes arising in a single queueing facility with r arrival channels and s service channels. Here we prove similar theorems for sequences of such queueing facilities. The same heavy traffic behavior prevails in many cases in this more general setting, but new heavy traffic behavior is observed when the sequence of traffic intensities associated with the sequence of queueing facilities approaches the critical value (ρ = 1) at appropriate rates.
Stability theorems
Authors: Lukacs Eugene
Published: April 2006

A stability theorem determines the extent to which the conclusions of a given theorem are affected if the assumptions of the theorem are not exactly but only approximately satisfied. The meaning of the word ‘approximately’ has to be defined exactly. The stability of decomposition theorems, of characterizations by independence and by regression properties are the primary object of the paper.
On an extension of a theorem of Linnik and Zinger
Authors: Gyires Béla
Published: April 2006

In their paper [2] Linnik and Zinger proved a theorem which has an important role in both theoretical and applied statistics. For the proof of their theorem the authors used a condition which greatly narrows down its applicability. In this paper we prove the theorem of Linnik and Zinger without this condition, and also give other extensions of it.
Unilateral Markov fields
Authors: Pickard David K.
Published: April 2006

Recently, there has been considerable interest in some specialized binary lattice processes. However, this exciting work has been rather fragmentary and heuristic. Rigorous proofs are provided for the existence of some classes of stationary unilateral Markov fields on infinite square lattices. (Neighbours are sites which are horizontally, vertically, or diagonally adjacent.) Many of the difficulties are avoided by characterizing stationary unilateral Markov fields on finite lattices first. Detailed analyses are given for both binary and Gaussian variables.
Approximations for the steady-state probabilities in theM/G/cqueue
Authors: Tijms H. C. , Van Hoorn M. H. , Federgruen A.
Published: April 2006

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.
A dynamic programming approach to evolutionarily stable strategy theory
Authors: Cannings C. , Gardiner D.
Published: April 2006

In the war of attrition (wa), introduced by Maynard Smith (1974), two contestants play values from [0, ∞), the individual playing the longer value winning a fixed prize V, and both incurring a loss equal to the lesser of the two values. Thus the payoff, E(x, y) to an animal playing x against one playing y, is

A more general form (Bishop and Cannings (1978)) has

and it was demonstrated that with and there exists a unique evolutionarily stable strategy (ess), which is to choose a random value from a specified density function on [0, ∞). Results were also obtained for strategy spaces [0, s] and [0, s).
Detection of regulation of family size after births of affected children
Authors: Stene Jon
Published: April 2006

For family data where some of the children may have a certain inherited disorder, the number of affected children has usually been assumed to be binomially distributed given the total number of children in the family. In this assumption is included the assumption that the distribution of the total number of children does not depend on the probability that a child is affected. For many disorders this assumption is unlikely to hold because the birth of an affected child may lead either to some sort of family limitation or to some sort of overcompensation. In such cases models other than the binomial distribution have to be used.
The multitype continuous-time Markov branching process in a periodic environment
Authors: Klein B. , MacDonald P. D. M.
Published: April 2006

The multitype continuous-time Markov branching process has many biological applications where the environmental factors vary in a periodic manner. Circadian or diurnal rhythms in cell kinetics are an important example. It is shown that in the supercritical positively regular case the proportions of individuals of various types converge in probability to a non-random periodic vector, independent of the initial conditions, while the absolute numbers of individuals of various types converge in probability to that vector multiplied by a random variable whose distribution depends on the initial conditions. It is noted that the proofs are straightforward extensions of the well-known results for a constant environment.
Randomised allocation of treatments in sequential trials
Authors: Bather John
Published: April 2006

Given a finite number of different experiments with unknown probabilitiesp1,p2, ···,pkof success, the multi-armed bandit problem is concerned with maximising the expected number of successes in a sequence of trials. There are many policies which ensure that the proportion of successes converges top= max (p1,p2, ···,pk), in the long run. This property is established for a class of decision procedures which rely on randomisation, at each stage, in selecting the experiment for the next trial. Further, it is suggested that some of these procedures might perform well over any finite sequence of trials.
The N/G/1 queue and its detailed analysis
Authors: Ramaswami V.
Published: April 2006

We discuss a single-server queue whose input is the versatile Markovian point process recently introduced by Neuts [22] herein to be called the N-process. Special cases of the N-process discussed earlier in the literature include a number of complex models such as the Markov-modulated Poisson process, the superposition of a Poisson process and a phase-type renewal process, etc. This queueing model has great appeal in its applicability to real world situations especially such as those involving inhibition or stimulation of arrivals by certain renewals. The paper presents formulas in forms which are computationally tractable and provides a unified treatment of many models which were discussed earlier by several authors and which turn out to be special cases. Among the topics discussed are busy-period characteristics, queue-length distributions, moments of the queue length and virtual waiting time. We draw particular attention to our generalization of the Pollaczek–Khinchin formula for the Laplace–Stieltjes transform of the virtual waiting time of the M/G/1 queue to the present model and the resulting Volterra system of integral equations. The analysis presented here serves as an example of the power of Markov renewal theory.
Storage processes with general release rule and additive inputs
Authors: Brockwell P. J. , Resnick S. I. , Tweedie R. L.
Published: April 2006

A construction is given of a process in which X(t) represents the content at time t of a dam whose cumulative input process is a Lévy process with measure v and whose release rate at time t is r(X(t)). It is assumed only that r(0) = 0 and that r is strictly positive and left-continuous with strictly positive finite right limits on (0,∞). The sample-paths of X are shown to satisfy the storage equation

The process X is analyzed using renewal theory and stochastic comparison techniques, and necessary and sufficient conditions are found in terms of v and r for X to have a stationary distributionπ. These generalize previous results which were obtained under the assumption that v is finite. Conditions for Πto have an atom at 0 are considered in some detail, and related results on the positivity of the expected occupation time of level 0 are given.

Necessary and sufficient conditions for the existence of Πare expressed in terms of the existence of non-negative integrable solutions of certain integral equations and conditions are given under which such solutions are necessarily stationary densities for X. A simple sufficient condition for X to have a stationary distribution is found in terms of and in the case when r is non-decreasing the condition is shown to be also necessary. Finally some examples are considered; these show that the results described above unify various known conditions in special cases, and confirm several conjectures in the related literature.
Alignments in two-dimensional random sets of points
Authors: Kendall David G. , Kendall Wilfrid S.
Published: April 2006

Letnpoints in the plane be generated by some specified random mechanism and suppose thatN(∊) of theresulting triads form triangles with largest angle ≧ π – ∊. The main object of the paper is to obtain asymptotic formulae forand Var (N(∊)) when ∊ ↓ 0, and to solve the associated data-analytic problem of testing whether an empirical set ofnpoints should be considered to contain too many such ∊-blunt triads in the situation where the generating mechanism is unknown and where all that can be said about the tolerance ∊ is that it must be allowed to take values anywhere in a given interval (T0,T1) (0 <T0<T1). This problem is solved by the introduction of a plot to be called thepontogramand by the introduction of simulation-based significance tests constructed by random lateral perturbations of the data.
Heavy traffic results for single-server queues with dependent (EARMA) service and interarrival times
Authors: Jacobs Patricia A.
Published: April 2006

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.
Poisson approximation for some statistics based on exchangeable trials
Authors: Barbour A. D. , Eagleson G. K.
Published: April 2006

Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables.

Let be exchangeable random elements of a space and, for I a k-subset of , let XI be a 0–1 function. The statistics studied here are of the form

where N is some collection of k -subsets of .

An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.
Optimal service-rate control of M/G/1 queueing systems using phase methods
Authors: Jo Kyung Y. , Stidham Shaler
Published: April 2006

A new approach to the optimal control of the service rate in M/G/1 queues is introduced using the method of phases. Each customer's work requirement is approximated by a random number of exponential phases with (possibly) different parameters (a generalized hyper-Erlang distribution). Using a semi-Markov decision-process formulation, we establish monotonicity properties of optimal policies for the finite-horizon problem, by induction on the horizon length. The analysis is then extended to the discounted infinite-horizon and the long-run average-return problems. In contrast to the models in previous papers, our model is appropriate for situations where the system controller has partial information about the work requirement of a customer, specifically the number of phases (tasks) to be performed. Because it requires a multidimensional state description, the analysis of the phase-type control model may be viewed as a first step toward the solution of control models for networks of queues.
Convergence to a diffusion of a multi-allelic model in population genetics
Authors: Sato Ken-Iti
Published: April 2006

We consider a Markov chain on thed-dimensional (d-allelî) non-negative lattice points with the sum of components beingN,for which one-step transition consists of two stages—independent reproduction and random sampling. Convergence to a degenerate diffusion process whenN→ ∞ is proved. We show how difference among alleles in means and variances of offspring numbers affects the limit diffusion, giving a rigorous multi-allelic version of a result of Gillespie.
On a class of random motions of point processes involving interaction
Authors: Pierson Harry M.
Published: April 2006

Starting with a stationary point process on the line with points one unit apart, simultaneously replace each point by a point located uniformly between the original point and its right-hand neighbor. Iterating this transformation, we obtain convergence to a limiting point process, which we are able to identify. The example of the uniform distribution is for purposes of illustration only; in fact, convergence is obtained for almost any distribution on [0, 1]. In the more general setting, we prove the limiting distribution is invariant under the above transformation, and that for each such transformation, a large class of initial processes leads to the same invariant distribution. We also examine the covariance of the limiting sequence of interval lengths. Finally, we identify those invariant distributions with independent interval lengths, and the transformations from which they arise.
Sums of random variables having the modified geometric distribution by application to two-person games
Authors: Phillips M. J.
Published: April 2006

The distribution of a fixed sum of independent and identically distributed random variables with the modified geometric distribution is the same as the distribution obtained by the compounding by a binomial distribution of either a negative binomial distribution or a Pascal distribution. This result can be used to obtain three summations for the game score probabilities of a two-person game, and leads to the consideration of various ways of dividing up the trials of the game. The game score probabilities are then used to consider the ‘fairness’ of four games and to analyse various methods of ‘setting’ (or ‘tie-breaking’) the games.
A note on random walks at constant speed
Authors: Bartlett M. S.
Published: April 2006

While the general principles involved in the formulation of random walk and Brownian motion equations (whether the random changes are directly on the position of a particle or individual, or on the velocity) are well-known, there are various situations considered in the literature involving the assumption of a constant speed (in magnitude). Thus the derivation by Goldstein (1951) of a one-dimensional wave-like equation involved the tacit assumption Ut = ± a, where Ut is the vector velocity dRt/dt,Rt being the (column) position vector (Bartlett (1957)). Biological models may involve the assumption of individuals moving at constant speed (cf. Kendall (1974)). Finally, the derivation of Schrodinger-type equations from Brownian motion models has sometimes involved the assumption U′tUt = c2, where c is the velocity of light (Cane (1967), (1975)).
Simulating samples from a genetic population
Authors: Watterson G. A.
Published: April 2006

Consider a genetic population evolving in time. In particular, suppose that the population is very large, that there is a large number of genetic types (alleles) possible for the genes making up the population, and that the numbers of the various alleles change stochastically in time, subject to Mendel's law of inheritance. Assume that there are no selective differences among the genotypes, but that mutation may occur among the alleles.
On a paper by Doeblin on non-homogeneous Markov chains
Authors: Cohn Harry
Published: April 2006

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail σ-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail σ-field structure of the chain, can be used to solve such cases.
A fourth note on recent research in geometrical probability
Authors: Baddeley Adrian
Published: April 2006

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, line-segments and flats in Euclidean spaces, the random division of space, coverage, packing, random sets, stereology and probabilistic aspects of integral geometry.
Probability functions on complex pedigrees
Authors: Cannings C. , Thompson E. A. , Skolnick M. H.
Published: April 2006

The calculation of probabilities on pedigrees of arbitrary complexity is discussed for a basic model of transmission and penetrance (encompassing Mendelian inheritance, and certain environmental influences).

The structure of pedigrees, and the types of loops occurring, is discussed. Some results in graph theory are obtained and, using these, a recurrence relation derived for certain probabilities. The recursive procedure enables the successive peeling off of certain members of the pedigree, and the condensation of the information on those individuals into a function on a subset of those remaining. The underlying theory is set out, and examples given of the utilization of the resulting algorithm.
First-passage percolation on the square lattice. III
Authors: Smythe R. T. , Wierman John C.
Published: April 2006

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant,Lpand (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.
Stability and heavy traffic results for the general bulk queue
Authors: Dagsvik John
Published: April 2006

In this paper we prove that the limiting distribution of the general bulk queue exists and is independent of the initial conditions if and only if the traffic intensity is less than one. We further generalize the following heavy traffic results of theGI/G/1 model to the general bulk queue model. When ρ > 1 or ρ = 1 the waiting time is distributed approximately as a Gaussian variable and the absolute value of a Gaussian variable, respectively. The exponential approximation is derived from the Wiener–Hopf matrix equations established in a previous paper while the unstable case ρ ≧ 1 is treated by means of functional central limit theorems for mixing processes.
A poisson process whose rate is a hidden Markov process
Authors: Freed D. S. , Shepp L. A.
Published: April 2006

Let a Poisson process be observed whose output rate is one of two levels given by the state of an unseen Markov process. If one of the levels is 0, a simple formula is given for the best guess of the state at any instant based on the stream of past Poisson events. In other cases bounds are given for the likelihood ratio of the state probabilities given the event stream.
Hendricks libraries and Tsetlin piles
Authors: Dies Jacques-Edouard
Published: April 2006

In order to study the transience of Hendricks libraries, we introduce and study a special class of Markov chains, the Tsetlin d-piles, generalizing Tsetlin libraries and briefly defined as follows: a 1-pile is a Tsetlin library and a d-pile is a Tsetlin library where each book is replaced by a (d − 1)-pile. We give a stationary measure of these chains and establish the necessary and sufficient conditions for positive recurrence and transience. Finally, the study of d-piles allows us to determine a sufficient condition for transience of quite a large class of Hendricks libraries.
On the time constant and path length of first-passage percolation
Authors: Kesten Harry
Published: April 2006

Let U be the distribution function of the passage time of an individual bond of the square lattice, and let pT be the critical probability above which the expected size of the open component of the origin (in the usual bond percolation) is infinite. It is shown that if (∗)U(0–) = 0, U(0) < pT, then there exist constants 0 < a, C1 < ∞ such that a self-avoiding path of at least n steps starting at the origin and with passage time ≦ an} ≦ 2 exp (–C1n).

From this it follows that under (∗) the time constant μ (U) of first-passage percolation is strictly positive and that for each c > 0 lim sup (1/n)Nn(c) <∞, where Nn(c) is the maximal number of steps in the paths starting at the origin with passage time at most cn.
Optimal sequential selection and resource allocation under uncertainty
Authors: Chikte Shirish D.
Published: April 2006

Consider a decision-maker who is in charge of a number of activities. At each of a sequence of decision points in time, he selects—on the basis of their performance—the set of activities to be continued further and allocates his limited resources among them. Activities receiving larger allocations tend to improve their performance, while others receiving smaller allocations tend to deteriorate. We present a controlled random walk model for the progress of these activities. The problem of maximizing the net infinite horizon discounted return is formulated in the framework of Markov decision theory, and existence of optimal strategies established. It is shown that both the optimal selection and allocation strategies exhibit a ‘favoring the leaders' behavior. Finally, explicit solutions to certain special cases are obtained illustrating these results.
Two-armed bandits with a goal, II. Dependent arms
Authors: Berry Donald A. , Fristedt Bert
Published: April 2006

One of two random variables, X and Y, can be selected at each of a possibly infinite number of stages. Depending on the outcome, one's fortune is either increased or decreased by 1. The probability of increase may not be known for either X or Y. The objective is to increase one's fortune to G before it decreases to g, for some integral g and G; either may be infinite.

In Part I (Berry and Fristedt (1980)), the distribution of X is unknown and that of Y is known. In the current part, it is known that either X or Y has probability α of increasing the current fortune by 1 and the other has probability β of increasing the fortune by 1, where α and β are known, but which goes with X is not known. We show that optimal strategies exist in general and find all optimal schemes when α = 0 and when α + β = 1. In both cases myopic strategies are shown to be optimal. A counterexample is used to show that myopic strategies, while intuitively very appealing, are not optimal for general (α, β).
Estimating and testing selection: the two-alleles, genic selection diffusion model
Authors: Watterson G. A.
Published: April 2006

The estimation, and testing for the presence, of a selective advantage of one allele over another is considered. It is assumed that a population's allele frequencies are known from some initial time until fixation of one or other allele occurs. The statistics needed to perform the estimation and testing are the heterozygosity of the population summed over all generations, and the observation of which allele fixes. It is shown that certain asymptotic probability distributions arise which are similar to those proved by Brown and Hewitt for statistical inference from diffusion processes, but their results assumed that the diffusion had a stationary density which is not the case for alleles which fix.

The genetic diffusion may be transformed to Brownian motion with constant drift, and the inference questions concerning selection can be transformed to questions about the first exit of a Brownian motion from an interval. It is thus possible to construct significance tests, and to calculate the power of those tests, for detecting selection.
Transience and recurrence of state-dependent branching processes with an immigration component
Authors: Levy Joshua B.
Published: April 2006

We consider the following modification of an ordinary Galton–Watson branching process. If Zn = i, a positive integer, then each parent reproduces independently of one another according to the ith {P(i)k} of a countable collection of probability measures. If Zn = 0, then Zn + 1 is selected from a fixed immigration distribution. We present sufficient conditions on the means μi, the variances σ2i, and the (2 + γ)th central absolute moments β2+γ,i of the {P(i)k}'s which ensure transience of recurrence of {Zn}.
Geometric convergence of value-iteration in multichain Markov decision problems
Authors: Schweitzer P. J. , Federgruen A.
Published: April 2006

This paper considers undiscounted Markov decision problems. With no restriction (on either the periodicity or chain structure of the problem) we show that the value iteration method for finding maximal gain policies exhibits a geometric rate of convergence, whenever convergence occurs. In addition, we study the behaviour of the value-iteration operator; we give bounds for the number of steps needed for contraction, describe the ultimate behaviour of the convergence factor and give conditions for the existence of a uniform convergence rate.
The infinitely-many-neutral-alleles diffusion model
Authors: Ethier S. N. , Kurtz Thomas G.
Published: April 2006

A diffusion processX(·) in the infinite-dimensional ordered simplexis characterized in terms of the generatordefined on an appropriate domain. It is shown thatX(·) is the limit in distribution of several sequences of discrete stochastic models of the infinitely-many-neutral-alleles type. It is further shown thatX(·) has a unique stationary distribution and is reversible and ergodic. Kingman's limit theorem for the descending order statistics of the symmetric Dirichlet distribution is obtained as a corollary.
The asymptotic time to failure of a bundle of fibers with random slacks
Authors: Tierney L.
Published: April 2006

Consider a bundle that has been constructed by placingnfibers in parallel between two clamps. The simplest assumption on fiber load sharing is that all non-failed fibers share the load applied to the bundle equally. Phoenix (1978b) shows that, asntends to ∞, the time to failure of such a bundle is asymptotically normally distributed under certain assumptions on the stochastic nature of the fibers. In this paper we show that this asymptotic normality is preserved but with changed mean and variance if random slacks are introduced, as caused by random variations in the lengths of the fibers. A theorem on random linear combinations of order statistics, which is given in the appendix, is used to prove the result.
Exponential decay and ergodicity of general Markov processes and their discrete skeletons
Authors: Tuominen Pekka , Tweedie Richard L.
Published: April 2006

Lett≥ 0 be a Markov transition probability semigroup on a general spacesatisfying a suitableφ-irreducibility condition. We show the existence of (i) a decay parameter λ ≥ 0 which is a common abscissa of convergence of the integrals ƒestPt(x,A)dtfor almost allxand all suitableA, (ii) a natural classification into λ-positive, λ-null and λ-transient cases. Moreover this classification is completely determined by any one of theh-skeleton chains of (Pt). We study the convergence ofeλtPt(x, A) in the λ-positive case, and show that the limitf(X)π(A) (wherefand π are the unique λ-invariant function and measure, normalized so that π(f) = 1) is reached at a uniform exponential rate of convergence, i.e. ||eλtPt(x, ·)-f(x)π(·)||f=O(e−αt) for some α > 0 and almost allxif there is a π-positive set such that the convergence is exponentially fast on this set. These results are used to deduce conditions for (Pt) to have quasi-stationary distributions.
Asymptotic properties of time domain gaussian estimators
Authors: Kohn R.
Published: April 2006

We consider a general non-linear multivariate time series model which can be parameterized by a finite and fixed number of parameters and which can be rewritten, if necessary, in a form such that the disturbances are stationary martingale differences. Given a series of discrete, equally spaced observations we prove the strong consistency and asymptotic normality of the Gaussian estimators of the parameters, the parameters possibly being subject to non-linear constraints. Because the normal equations are usually highly non-linear it may be difficult to obtain explicit expressions for the Gaussian estimates. To overcome this problem we use a Gauss–Newton type algorithm to obtain a sequence of iterates which converge to, and have the same asymptotic properties as, the Gaussian estimates.
A conservation property for general GI/G/1 queues with an application to tandem queues
Authors: Nummelin E.
Published: April 2006

We show that, if the input process of a general GI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.
The lifetimes of configuration states in statistical physics
Authors: Ruben Harold
Published: April 2006

The moments and probability distribution of the lifetime of a configuration state relative tomdisjoint regions in ℝdfor particles under stochastic motion are expressed in terms of the derivatives at the origin of the probability after-effects for the m regions and for the union of the regions, together with the single integrated and themrandomized first-passage-time distributions, relative to the union of themregions and to the separate complements of themregions, respectively. The lifetime, suitably normed in terms of the mean lifetime, is shown to have a limiting standard exponential distribution. Finally, the distributions of lifetime when the motion of the particles is either Brownian or such as to generate a persistent generalized Smoluchowski process are discussed; in the first case, the distribution of lifetime reduces to a standard problem in heat conduction, and in the second case the distribution is expressed in terms of an exponential function and themprobability after-effects for themregions.
Risk-sensitive linear/quadratic/gaussian control
Authors: Whittle P.
Published: April 2006

The conventional linear/quadratic/Gaussian assumptions are modified in that minimisation of the expectation of cost G defined by (2) is replaced by minimisation of the criterion function (5). The scalar –θ is a measure of risk-aversion. It is shown that modified versions of certainty equivalence and the separation theorem still hold, that optimal control is still linear Markov, and state estimate generated by a version of the Kalman filter. There are also various new features, remarked upon in Sections 5 and 7. The paper generalises earlier work of Jacobson.
Characterization of stochastic processes by stochastic integrals
Authors: Rao B. L. S. Prakasa
Published: April 2006

Let be a continuous homogeneous stochastic process with independent increments. A review of the recent work on the characterization of Wiener and stable processes and connected results through stochastic integrals is presented. No proofs are given but appropriate references are mentioned.
Non-parametric covariance estimation from irregularly-spaced data
Authors: Masry Elias
Published: April 2006

The non-parametric discrete-time estimation of the covariance function R(t) of stationary continuous-time processes is considered. The characteristics of the sampling instants necessary for the consistent estimation of R(t) are explored. A class of covariance estimates is introduced and its asymptotic statistics are derived.
General first-passage percolation
Authors: McDiarmid Colin
Published: April 2006

We present some general results related to first-passage percolation. These results involve families of random variables arranged in independent subfamilies. Applications are given to the study of random networks and to first-passage percolation.
A remark on the time constant in first-passage percolation
Authors: Derriennic Yves
Published: April 2006

It is possible to deduce some information about the time constant μ without using Wierman and Reh's (1978) renewal theorem.
The mean waiting time to a repetition
Authors: Blom Gunnar
Published: April 2006

Let X1, X2, · ·· be a stationary sequence of random variables and E1, E2, · ··, EN mutually exclusive events defined on k consecutive X's such that the probabilities of the events have the sum unity. In the sequence Ej1, Ej2, · ·· generated by the X's, the mean waiting time from an event, say Ej1, to a repetition of that event is equal to N (under a mild condition of ergodicity). Applications are given.
On the distribution of the time to extinction in the stochastic logistic population model
Authors: Norden R. H.
Published: April 2006

The aim of this paper is to investigate the distribution of the extinction times,T, of the stochastic logistic process from both the numerical and the theoretical standpoint. The problem is approached first by deriving formulae for the moments ofT; it is then shown that in most casesTis, very nearly, a gamma variate. Some simulated results are given and these agree well with the theory. Furthermore, a consideration of the process conditioned on non-extinction is shown to be an effective way of obtaining a larget(time) description of the unconditioned process. Finally, a more general form of the model in which the death-rate as well as the birth-rate is ‘density-dependent' is considered, and by comparison with the usual form of the model the effect onTof this additional factor is assessed.
Optimal sampling for pedigree analysis: tracing a single gene
Authors: Thompson E. A.
Published: April 2006

In fitting genetic models on the basis of observations on an interrelated structure, sequential procedures can enhance efficiency. In this paper we consider the case of a rare gene segregating in a single large pedigree. The sampling rule is dictated by the effect of observations on the genotypic probability distribution of unobserved relatives; this effect is investigated.
Sojourns and extremes of a diffusion process on a fixed interval
Authors: Berman Simeon M.
Published: April 2006

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.
A geometric interpretation of the relations between the exponential and generalized Erlang distributions
Authors: Dehon Michel , Latouche Guy
Published: April 2006

Linear combinations of exponential distribution functions are considered, and the class of distribution functions so obtainable is investigated. Convex combinations correspond to hyperexponential distributions, while non-convex combinations yield, among other, generalized Erlang distributions obtainable as sums of independent exponential random variables with different parameters.

For a given number n of different exponential distributions, the class investigated is an (n – 1)-dimensional convex subset of the n-dimensional real vector space generated by the n distribution functions. The geometric aspect of this subset is revealed, together with the location of hyperexponential and generalized Erlang distributions.
Number of successes in Markov trials
Authors: Bhat U. Narayan , Lal Ram
Published: April 2006

Markov trials are a sequence of dependent trials with two outcomes, success and failure, which are the states of a Markov chain. The distribution of the number of successes in n Markov trials and the first-passage time for a specified number of successes are obtained using an augmented Markov chain model.
A simple proof of a result of Iglehart and Shedler
Authors: Berman Mark
Published: April 2006

Iglehart and Shedler (1983) prove that the ‘labelled jobs’ method for estimation of passage-time characteristics in closed multiclass networks of queues with general service times provides asymptotically shorter confidence intervals than does the ‘marked job’ method. A simple alternative proof of this result, under slightly more restrictive conditions, is given here.
The structure and explicit determination of convex-polygonally generated shape-densities
Authors: Kendall David G. , Le Hui-Lin
Published: April 2006

This paper is concerned with the shape-density for a random triangle whose vertices are randomly labelled and i.i.d.-uniform in a compact convex polygon K. In earlier work we have already shown that there is a network of curves (the singular tessellation T(K)) across which suffers discontinuities of form. In two papers which will appear in parallel with this, Hui-lin Le finds explicit formulae for (i) the form of within the basic tile T0 of T(K), and (ii) the jump-functions which link the local forms of on either side of any curve separating two tiles. Here we exploit these calculations to find in the most general case. We describe the geometry of T(K), we examine the real-analytic structure of within a tile, and we establish by analytic continuation an explicit formula giving in an arbitrary tile T as the sum of the basic-tile function and the members of a finite sequence of jump-functions along a ‘stepping-stone' tile-to-tile route from T0 to T. Finally we comment on some of the problems that arise in the use of this formula in studies relating to the applications in archaeology and astronomy.
On the multiserver queue with finite waiting room and controlled input
Authors: Dshalalow Jewgeni
Published: April 2006

In this paper we study a multi-channel queueing model of type with N waiting places and a non-recurrent input flow dependent on queue length at the time of each arrival. The queue length is treated as a basic process. We first determine explicitly the limit distribution of the embedded Markov chain. Then, by introducing an auxiliary Markov process, we find a simple relationship between the limiting distribution of the Markov chain and the limiting distribution of the original process with continuous time parameter. Here we simultaneously combine two methods: solving the corresponding Kolmogorov system of the differential equations, and using an approach based on the theory of semi-regenerative processes. Among various applications of multi-channel queues with state-dependent input stream, we consider a closed single-server system with reserve replacement and state-dependent service, which turns out to be dual (in a certain sense) in relation to our model; an optimization problem is also solved, and an interpretation by means of tandem systems is discussed.
Dynamic priority rules for cyclic-type queues
Authors: Browne Sid , Yechiali Uri
Published: April 2006

A cyclic service system is composed of K channels (queues) and a single cyclically roving server who typically takes a positive amount of time to switch between channels. Research has previously focused on evaluating and computing performance measures (notably, waiting times) of fixed template routing schemes under three main service disciplines, the exhaustive, gated and limited service regimes.

In this paper, probabilistic results are derived that allow control strategies and optimal policies to be considered for the first time. By concentrating on a new objective function, we are able to derive rules of index form amenable for direct implementation to dynamically control the system at suitably defined decision epochs. These rules utilize current system information, are of an adaptive nature, and are shown to emanate from a general physical principle.
An extremal property of FIFO discipline in G/IFR/1 queues
Authors: Hirayama Tetsuji , Kijima Masaaki
Published: April 2006

In a recent paper by Shanthikumar and Sumita (1987), it is conjectured that the ergodic sojourn time of a customer in G/IFR/1 queues is minimized by FIFO (first in, first out) discipline in the sense of increasing and convex ordering. This paper shows that their conjecture is true. In fact, FIFO discipline minimizes for any non-decreasing and convex function f, where N is the (constant) number of arrivals, θ (k) is the customer identity of the kth departing customer, and an and τ n denote the arriving and departing times of the nth customer, respectively.
Periodic queues in heavy traffic
Authors: Falin G. I.
Published: April 2006

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.
Two remarks on insensitive stochastic models
Authors: Schassberger R.
Published: April 2006

This paper contains two contributions to the insensitivity theory for generalized semi-Markov schemes, namely

(1)

a detailed account of a close relationship between insensitive schemes and partially balanced discrete-state Markov chains, and

(2)

an extension of the notion of an insensitive scheme in order to incorporate insensitivity phenomena not previously covered.
Overload performance comparisons of several processor queueing disciplines
Authors: Dishi B. T. , Heffes H.
Published: April 2006

We analyze and compare the performance, under overload, of a class of queueing and service disciplines for an M/M/l queue. The disciplines we study are (i) FIFO with finite buffer (M/M/1/N), (ii) FIFO with push out (M/M/1/N-PO), (iii) LIFO with push out, (iv) FIFO with time out (M/M1-TO) and (v) LIFO with time out. For each of these disciplines we obtain the performance measures such as the throughput, the moments and the distribution of the waiting time of the customers who get served and numerically compare the disciplines with respect to these measures.
On population-size-dependent branching processes
Authors: Klebaner F. C.
Published: April 2006

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.
A branching-process solution of the polydisperse coagulation equation
Authors: Spouge John L.
Published: April 2006

The polydisperse coagulation equation

models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when

The critical time tc when diverges corresponds to a critical branching process, while post-critical times t> tc correspond to supercritical branching processes.
Some remarks on the renewal function of the uniform distribution
Authors: Jensen U.
Published: April 2006

Russell (1983) considers the distribution properties of N(a), the minimum value of n for which the sum Sn of n random variables, which are i.i.d. uniformly on (0, 1), exceeds the constant a, .
Multivariate linear time series models
Authors: Hannan E. J. , Kavalieris L.
Published: April 2006

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.

The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.

The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.
On two classes of interacting stochastic processes arising in cancer modeling
Authors: Bartoszyński Robert , Puri Prem S.
Published: April 2006

The processes X and Y are said to interact if the laws governing the changes of either of them at time t depend on the values of the other process at times up to t. For bivariate interacting Markov processes, their limiting behavior is analysed by means of an approximation suggested by Fuhrmann, consisting of discretizing time, and assuming that in each time interval the processes develop independently, according to the laws obtained by fixing the value of the other process at its level attained at the beginning of the interval.

In this way the conditions for almost sure extinction, escape to ∞ with positive probability, etc., are obtained (by using the martingale convergence theorem) for state-dependent branching processes studied by Roi, and for bivariate processes with one component piecewise determined.
Extreme order statistics with cost of sampling
Authors: Pickands James
Published: April 2006

Let X1, X2, …, Xn, … be mutually independent with common CDF F and, for each m, n, let Xm:n be the mth largest among the first n. We consider max1≤n<∞ (X1:n – cn) and the ‘optimal stopping rule' N which maximizes where all l and In particular, we consider and All of these are considered for c ϵ (0,∞) as well as asymptotically as c → 0+.