bluuebunny/crossref_metadata_2025 huggingface dataset:
Poisson flats in Euclidean spaces Part II: Homogeneous Poisson flats and the complementary theorem
Authors: Miles R. E.
Published: April 2006

Part I [21] treated the case of a finite number of independent random uniforms-flats in an ‘admissible’ subset ofEd(s= 0, · · ·,d− 1). In this second part, the natural and fruitful ‘Poisson extension’ to a ‘countable number of independent random uniforms-flats inEditself” is considered. It is worth mentioning at the outset that to have read Part I is not a prerequisite for reading the present paper. Although results of that part are often applied here, they serve only in an auxiliary capacity, thereby allowing the main thread of the theory to be developed without interruption.
The geometry of random drift I. Stochastic distance and diffusion
Authors: Antonelli Peter L. , Strobeck Curtis
Published: April 2006

A stochastic distance measure is defined for a general diffusion process on a parameter spaceX.This distance is defined bywhere (gij) is the inverse of the covariance matrix of the diffusion equation. This permits the study of the geometry associated with a diffusion equation, since the matrix (gij) is the fundamental tensor of the Riemannian space (X, gij), and of a diffusion process in terms of Brownian motion. For the diffusion equation approximation to random drift withnalleles the covariance matrix is that of a multinomial distribution. The resulting stochastic distance is equal to twice the genetic distance as defined by Cavalli-Sforza and Edwards and is a generalization of the angular transformation of Fisher tonalleles. The geometry associated with the diffusion equation for random drift withnalleles is that of apart of an (n− 1)-sphere of radius two. We also show that the diffusion equation for random drift isnotspherical Brownian motion, although it is approximated by it near the centroid of frequency space.
Asymptotic properties of the equilibrium probability of identity in a geographically structured population
Authors: Sawyer Stanley
Published: April 2006

LetI(x, u) be the probability that two genes found a vector distancexapart are the same type in an infinite-allele selectively-neutral migration model with mutation rateu.The creatures involved inhabit an infinite of colonies, are diploid and are held atNper colony. Setin one dimension andin higher dimensions, where σ2is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Thenin one dimension,in two dimensions, andin three dimensions uniformly forHereC0is a constant depending on the migration law,K0(y) is the Bessel function of the second kind of order zero, andare the eigenvalues of σ2. For symmetric nearest-neighbor migrations,in one dimension andlogmiin two. Foris known in one dimension andC0does not appear. In two dimensions,These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.
A general markov decision method I: Model and techniques
Authors: De Leve G. , Federgruen A. , Tijms H. C.
Published: April 2006

This paper provides a new approach for solving a wide class of Markov decision problems including problems in which the space is general and the system can be continuously controlled. The optimality criterion is the long-run average cost per unit time. We decompose the decision processes into a common underlying stochastic process and a sequence of interventions so that the decision processes can be embedded upon a reduced set of states. Consequently, in the policy-iteration algorithm resulting from this approach the number of equations to be solved in any iteration step can be substantially reduced. Further, by its flexibility, this algorithm allows us to exploit any structure of the particular problem to be solved.
The supremum distribution of a Lévy process with no negative jumps
Authors: Harrison J. Michael
Published: April 2006

Let be a process with stationary, independent increments and no negative jumps. Let be this same process modified by a reflecting barrier at zero (a storage process). Assuming that – and denote by ψ(s) the exponent function of X. A simple formula is derived for the Laplace transform of as a function of W(0). Using the fact that the distribution of M is the unique stationary distribution of the Markov process W, this yields an elementary proof that the Laplace transform of M is µs/ψ(s). If it follows that These surprisingly simple formulas were originally obtained by Zolotarev using analytical methods.
Estimated factorisation of the spectral density of a stationary point process
Authors: Rice John
Published: April 2006

A statistical method for estimating the factorisation of the spectral density of a stationary point process is presented and asymptotic properties of the resulting estimates are derived. The estimated functions are of interest in the analysis of a self-exciting process and more generally in the problem of linear prediction, and can be viewed as an alternative second-order analysis of the process. Some examples are discussed.
The asymptotic distribution of random molecules
Authors: Rehder Wulf
Published: April 2006

If n solid spheres Kn of some volume V(Kn) are scattered randomly in the unit cube of euclidean d-space, some of them will overlap to form Ln(s) molecules with exactly s atoms Kn. The random variable Ln(s) has a limit distribution if V(Kn) tends to zero but nV(Kn) tends to infinity at a certain rate: it is shown that for

Ln(s) is asymptotically Poisson.
Two-armed bandits with a goal, I. One arm known
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 the current part of the paper, the distribution of X is unknown and that of Y is known. We characterize the situations in which optimal strategies exist and, for certain kinds of information concerning X and Y, we characterize optimal sequential strategies for choosing to observe X and Y.

In Part II (Berry and Fristedt (1980)), 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.
Average optimal policies in Markov decision drift processes with applications to a queueing and a replacement model
Authors: Hordijk Arie , Van Der Duyn Schouten Frank A.
Published: April 2006

Recently the authors introduced the concept of Markov decision drift processes. A Markov decision drift process can be seen as a straightforward generalization of a Markov decision process with continuous time parameter. In this paper we investigate the existence of stationary average optimal policies for Markov decision drift processes. Using a well-known Abelian theorem we derive sufficient conditions, which guarantee that a ‘limit point' of a sequence of discounted optimal policies with the discounting factor approaching 1 is an average optimal policy. An alternative set of sufficient conditions is obtained for the case in which the discounted optimal policies generate regenerative stochastic processes. The latter set of conditions is easier to verify in several applications. The results of this paper are also applicable to Markov decision processes with discrete or continuous time parameter and to semi-Markov decision processes. In this sense they generalize some well-known results for Markov decision processes with finite or compact action space. Applications to an M/M/1 queueing model and a maintenance replacement model are given. It is shown that under certain conditions on the model parameters the average optimal policy for the M/M/1 queueing model is monotone non-decreasing (as a function of the number of waiting customers) with respect to the service intensity and monotone non-increasing with respect to the arrival intensity. For the maintenance replacement model we prove the average optimality of a bang-bang type policy. Special attention is paid to the computation of the optimal control parameters.
A note on the processor-sharing queue in a quasi-reversible network
Authors: Tsoucas Pantelis , Walrand Jean
Published: April 2006

Consider a processor-sharing queue placed in a quasi-reversible network in equilibrium. This note explains why the expected sojourn time of a customer in such a queue is proportional to his service time.
On the spectral decomposition of stationary time series using walsh functions. I
Authors: Kohn R.
Published: April 2006

The paper looks at the asymptotic properties of the finite Walsh–Fourier transform applied to a discrete-time stationary time series, and shows that in many ways we have analogous results to those obtained when using the finite trigonometric Fourier transform.
Improved bounds for the availability and unavailability in a fixed time interval for systems of maintained, interdependent components
Authors: Natvig B.
Published: April 2006

In this paper we arrive at a series of bounds for the availability and unavailability in the time interval I = [tA, tB] ⊂ [0, ∞), for a coherent system of maintained, interdependent components. These generalize the minimal cut lower bound for the availability in [0, t] given in Esary and Proschan (1970) and also most bounds for the reliability at time t given in Bodin (1970) and Barlow and Proschan (1975). In the latter special case also some new improved bounds are given. The bounds arrived at are of great interest when trying to predict the performance process of the system. In particular, Lewis et al. (1978) have revealed the great need for adequate tools to treat the dependence between the random variables of interest when considering the safety of nuclear reactors.

Satyanarayana and Prabhakar (1978) give a rapid algorithm for computing exact system reliability at time t. This can also be used in cases where some simpler assumptions on the dependence between the components are made. It seems, however, impossible to extend their approach to obtain exact results for the cases treated in the present paper.
Estimating probabilities for normal extremes
Authors: Hall Peter
Published: April 2006

Let Xnn denote the largest of n independent N(0, 1) variables. Several methods of estimating P(Xnn ≦ x) are considered. It is shown that X2nn, when normalized in an optimal way, converges to the extreme value distribution at a rate of only 1/(log n)2, and that if 0 < t ≠ 2 then |Xnn|t converges at a rate of 1/log n. Therefore it is not feasible to use the extreme value distribution to estimate probabilities for normal extremes unless the sample size is extremely large. An alternative approach is presented, which gives very good estimates of P(Xnn ≦ x) for n ≧ 10. The case of rth extremes is also considered.
Application of Stieltjes theory forS-fractions to birth and death processes
Authors: Bordes G. , Roehner B.
Published: April 2006

We are interested in obtaining bounds for the spectrum of the infinite Jacobi matrix of a birth and death process or of any process (with nearest-neighbour interactions) defined by a similar Jacobi matrix.

To this aim we use some results of Stieltjes theory forS-fractions, after reviewing them. We prove a general theorem giving a lower bound of the spectrum. The theorem also gives sufficient conditions for the spectrum to be discrete.

The expression for the lower bound is then worked out explicitly for several, fairly general, classes of birth and death processes. A conjecture about the asymptotic behavior of a special class of birth and death processes is presented.
On an explanation for the characteristic clutch size of some bird species
Authors: Heyde C. C.
Published: April 2006

Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. For example, the ground-living pigeons lay two eggs, the white stork four and the sulphur-crested cockatoo two and rarely three. The spotted nightjar lays one egg and the white-tailed nightjar lays two. The (European) swift lays two or three eggs, the (Australian) wood swallows lay three or four eggs and a number of (Australian) finches lay five or six eggs. These clutch sizes should be the result of evolutionary forces and it could be expected that the clutch size, for a particular species and environment, is the one which results in maximization of the probability of survival of the species to each time point t. It is consequently of interest to see whether it is possible to prescribe a population process which provides a reasonable description of a bird population and which exhibits this maximization behaviour when the clutch size is either k or is k or k + 1. Such a model has been suggested by Heyde and Schuh (1978) and it is the object of the present note to indicate how this model may be extended while still retaining the desired characteristic features.
An infinite particle system with continual input and random death
Authors: McDonald J. N. , Weiss N. A.
Published: April 2006

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.
The branching property in generalized information theory
Authors: Ebanks Bruce
Published: April 2006

It is shown that every measure of expected information which has the branching property is of the form where J is a given information measure which is compositive under a regular binary operation and the Ψn are antisymmetric, bi-additive functions. In a probability space, such measures (entropies) take the form
Heavy traffic limit theorems in fluctuation theory
Authors: Stam A. J.
Published: April 2006

Letbe a family of random walks withFor ε↓0 under certain conditions the random walkU(∊)nconverges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. LetM∊= max {U(∊)n,n≧ 0},v0= min {n:U(∊)n=M∊},v1= max {n:U(∊)n=M∊}. The joint limiting distribution of ∊2σ∊–2v0and ∊σ∊–2M∊is determined. It is the same as for ∊2σ∊–2v1and ∊σ–2∊M∊. The marginal ∊σ–2∊M∊gives Kingman's heavy traffic theorem. Also lim ∊–1P(M∊= 0) and lim ∊–1P(M∊<x) are determined. Proofs are by direct comparison of corresponding probabilities forU(∊)nand for a special family of random walks related toMI/M/1 queues, using the central limit theorem.
The insensitivity of stationary probabilities in networks of queues
Authors: Schassberger R.
Published: April 2006

The stationary probabilities for certain networks of queues as defined by Kelly [4] were recently shown by Barbour [1] to depend on the service-time distributions involved only through their means. This type of insensitivity has been studied by König and Jansen [5] for a general class of stochastic processes. Kelly's networks yield special cases of such processes. We point this out in the present paper, thus shedding new light on the insensitivity phenomenon observed in these networks and its connection with the phenomenon of local balance. As a consequence of our recent study [8] we also obtain a new insensitivity result for these networks.
On the parity of individuals in birth and death processes
Authors: Srinivasan S. K. , Ranganathan C. R.
Published: April 2006

This paper deals with the parity of individuals in an age-dependent birth and death process. A more general model with parity and age-dependent birth rates is also considered. The mean number of individuals with parity 0, 1, 2, ·· ·is obtained for the two models. The first moments of the total number of births in the population up to timetand the sum of the parities of the individuals existing at timetare obtained. A brief discussion on the parity of individuals in a population including ‘twins' is also given.
Passage times for overtake-free paths in Gordon–Newell networks
Authors: Daduna Hans
Published: April 2006

Consider a path in a multiclass Gordon–Newell network such that a customer present in a node of this path cannot be overtaken by any other customer behind him in a node of this path or by probabilistic influences created by such customers. The passage time through such a path is a mixture of Erlangian distributions, where the mixing distribution is given by the steady state of the network.
Entropy inequalities for classes of probability distributions II. The multivariate case
Authors: Karlin Samuel , Rinott Yosef
Published: April 2006

In this paper we continue our investigation of entropy comparisons with emphasis on multivariate distributions. For multiparameter cases of the multinomial and negative multinomial distributions we consider various higher-order forms of multivariate convexity. For the multinormal, Wishart, and t-distributions we define a partial ordering on the set of covariance matrices and determine monotonicity of the entropy functional. We further indicate some entropy inequalities for different sampling schemes. Because of the complex nature of multivariate partial ordering relations several problems remain open.
Well-timed diffusion approximations
Authors: Walsh John B.
Published: April 2006

Let X be a Markov process on the line. Under certain conditions it is possible to find a diffusion process which is an approximation to X in the following sense:

(1) X can be embedded in that is there are stopping times (Tt) such that {Xt, t ≥ 0} and have the same distribution;

(2) for each t, E{Tt} = t.

We call the well-timed diffusion approximation to X, and suggest that it is useful for approximating quantities like the first-exit probabilities and expected first-exit times of X.

We determine the well-timed approximation in several special cases and give an asymptotic approximation for use in cases in which cannot be exactly determined.
On the heavy-traffic limit theorem forGI/G/∞ queues
Authors: Whitt Ward
Published: April 2006

A revealing alternate proof is provided for the Iglehart (1965), (1973)–Borovkov (1967) heavy-traffic limit theorem forGI/G/squeues. This kind of heavy traffic is obtained by considering a sequence ofGI/G/ssystems with the numbers of servers and the arrival rates going to ∞ while the service-time distributions are held fixed. The theorem establishes convergence to a Gaussian process, which in general is not Markov, for an appropriate normalization of the sequence of stochastic processes representing the number of customers in service at arbitrary times. The key idea in the new proof is to consider service-time distributions that are randomly stopped sums of exponential phases, and then work with the discrete-time vector-valued Markov chain representing the number of customers in each phase of service at arrival epochs. It is then easy to show that this sequence of Markov chains converges to a multivariate O–U (Ornstein–Uhlenbeck) diffusion process by applying simple criteria in Stroock and Varadhan (1979). The Iglehart–Borovkov limit for these special service-time distributions is the sum of the components of this multivariate O–U process. Heavy-traffic convergence is also established for the steady-state distributions ofGI/M/squeues under the same conditions by exploiting stochastic-order properties.
The branching random field
Authors: Ivanoff Gail
Published: April 2006

The branching random field is studied under general branching and diffusion laws. Under a renormalization transformation it is shown that at finite fixed time the branching random field converges in law to a generalized Gaussian random field with independent increments. Very mild moment conditions are imposed on the branching process. Under more restrictive conditions on the branching and diffusion processes, the existence of a steady state distribution is proven in the critical case. A central limit theorem is proven for the renormalized steady state, but the limiting Gaussian random field no longer has independent increments. The covariance kernel is now a multiple of the potential kernel of the diffusion process.
Some new approaches to probability distributions
Authors: Kotz S. , Shanbhag D. N.
Published: April 2006

We develop some approaches to the characterization of distributions of real-valued random variables, useful in practical applications, in terms of conditional expectations and hazard measures. We prove several representation theorems generalizing earlier results, and establish stability theorems for two general characteristics introduced in this paper.
Asymptotic properties of a certain class of Bush–Mosteller learning models
Authors: Findeisen Peter
Published: April 2006

One general and three specialized models of the Bush–Mosteller type are presented to describe the kind of learning experiment where the response of the learner is always reinforced. Inhomogeneity is admitted. The random sequences of response probabilities and of responses associated with the different models are considered. Information about the existence and the distribution of asymptotic response probabilities is provided. The stress is on sufficient and necessary conditions for convergence (a.s. or with positive probability) of the response sequence, which is what ‘learning' means.
The asymptotic distribution for the time to failure of a fiber bundle
Authors: Phoenix S. Leigh
Published: April 2006

A model is developed for the failure time of a bundle of fibers subjected to a constant load. At any time, all surviving fibers share the bundle load equally while all failed fibers support no load. The bundle may collapse immediately or fibers may fail randomly in time, possibly more than one at a time. The failure time of the bundle is the failure time of the last surviving fiber. For a single fiber, the c.d.f. for the failure time is assumed to be a specific functional of an arbitrary load history. The model is developed using a quantile process approach. In the most important case the failure time of the bundle is shown to be asymptotically normal with known parameters. The bundle failure model has the features of both static strength and fatigue failure of earlier analyses, and thus is more realistic than earlier models.
On a general storage problem and its approximating solution
Authors: Smith N. M. H. , Yeo G. F.
Published: April 2006

AGI/G/r(x) store is considered with independently and identically distributed inputs occurring in a renewal process, with a general release rater(·) depending on the content. The (pseudo) extinction time, or the content, just before inputs is a Markov process which can be represented by a random walk on and below a bent line; this results in an integral equation of the formgn+1(y) = ∫l(y, w)gn(w)dwwithl(y, w) a known conditional density function. An approximating solution is found using Hermite or modified Hermite polynomial expansions resulting in a Gram–Charlier or generalized Gram–Charlier representation, with the coefficients being determined by a matrix equation. Evaluation of the elements of the matrix involves two-dimensional numerical integration for which Gauss–Hermite–Laguerre integration is effective. A number of examples illustrate the quality of the approximating procedure against exact and simulated results.
Queues with non-stationary input stream: Ross's conjecture
Authors: Rolski Tomasz
Published: April 2006

Characteristics of queues with non-stationary input streams are difficult to evaluate, therefore their bounds are of importance. First we define what we understand by the stationary delay and find out the stability conditions of single-server queues with non-stationary inputs. For this purpose we introduce the notion of an ergodically stable sequence of random variables. The theory worked out is applied to single-server queues with stationary doubly stochastic Poisson arrivals. Then the interarrival times do not form a stationary sequence (‘time stationary’ does not imply ‘customer stationary’). We show that the average customer delay in the queue is greater than in a standard M/G/1 queue with the same average input rate and service times. This result is used in examples which show that the assumption of stationarity of the input point process is non-essential.
Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue
Authors: Takahashi Yukio
Published: April 2006

It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distribution W(x) has an asymptotically exponential tail, i.e., 1 – W(x) ∽ Ke–ckx. The parameter k is the unique positive number satisfying T*(ck) S*(–k) = 1, where T*(s) and S*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η = T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.

The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.

The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constant K is not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.
A transition density expansion for a multi-allele diffusion model
Authors: Griffiths R. C.
Published: April 2006

An expansion in orthogonal polynomials is found for the transition density in a neutral multi-allele diffusion model where the mutation rates of allele typesAi→Ajare assumed to beuj(≥ O). The density is found when the mutation rate is positive for all allele types, and when some or all have zero mutation. The asymptotic conditional density is found for a mixture of positive and zero mutation rates.

The infinite alleles limit with equal mutation is studied. Eigenfunctions of the process are derived and the frequency spectrum found. An important result is that the first eigenfunction depends only on the homozygosity.

A density for the time to fixation with zero mutation is found for theKallele, and infinite alleles model.
Generalizing the secretary problem
Authors: Lorenzen Thomas J.
Published: April 2006

The secretary problem refers to a certain class of optimal stopping problems based on relative ranks. To allow a more realistic formulation of the problem, this paper considers an arbitrary loss function. A finite and an infinite problem are defined and the optimal solutions are obtained. The solution for the infinite problem is given by a differential equation while the finite problem is given by a difference equation. Under general conditions, the finite problem tends to the infinite problem. An example involving the secretary problem with interview cost is considered and illustrates the usefulness of the present paper.
Comparing multi-server queues with finite waiting rooms, II: Different numbers of servers
Authors: Sonderman David
Published: April 2006

We compare two queueing systems with identical general arrival streams, but different numbers of servers, different waiting room capacities, and stochastically ordered service time distributions. Under appropriate conditions, it is possible to construct two new systems on the same probability space so that the new systems are probabilistically equivalent to the original systems and each sample path of the stochastic process representing system size in one system lies entirely below the corresponding sample path in the other system. This construction implies stochastic order for these processes and many associated quantities of interest, such as a busy period, the number of customers lost in any interval, and the virtual waiting time.
On convergence to diffusion processes of Markov chains related to population genetics
Authors: Okada Norio
Published: April 2006

A Markov chain model related to population genetics and its convergence to a diffusion process on the multi-dimensional bounded domain are treated. We discuss the case where natural selection is random and the different selection effects over successive generations are independent. Our model is a multi-allelic version of the haploid model of Karlin and Levikson. The asymptotic properties of the limiting diffusion are stated.
On a stochastic difference equation and a representation of non–negative infinitely divisible random variables
Authors: Vervaat Wim
Published: April 2006

The present paper considers the stochastic difference equation Yn = AnYn-1 + Bn with i.i.d. random pairs (An, Bn) and obtains conditions under which Yn converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A1A2 ··· An-1Bn. A second subject is the series ∑ Cnf(Tn) with (Cn) a sequence of i.i.d. random variables, (Tn) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case An and Cn are independent, Bn = AnCn, An = U1/αn with Un a uniform random variable, f(x) = e−x/α.
On monotone optimal policies in a queueing model ofM/G/1 type with controllable service time distribution
Authors: Gallisch Eckhardt
Published: April 2006

A queueing model ofM/G/1 type with controllable service time distribution is considered in this paper. Conditions on costs and service time distribution only are established such that there exists a monotone policy optimal relative to the average cost criterion. To prove this, it is shown that it suffices to search for a monotone optimal policy in a Markov decision model associated with the queueing model.
Optimal replacement under a general failure model
Authors: Bergman Bo
Published: April 2006

Replacement policies based on measurements of some increasing state variable, e.g. wear, accumulated damage or accumulated stress, are studied in this paper. It is assumed that the state measurements may be regarded as realizations of some stochastic process and that the proneness to failure of an active unit may be described by an increasing state-dependent failure rate function. Average long-run cost per unit time is considered. The optimal replacement rule is shown to be a control limit rule, i.e. it is optimal to replace either at failure or when the state variable has reached some threshold value, whichever occurs first. The optimal rule is determined. Some generalizations and special cases are given.
A sufficient criterion for particles performing a diffusive motion with state-dependent death rate to die with probability one
Authors: Mergenthaler Wolfgang
Published: April 2006

We consider an individual which performs a diffusive motion in a certain state space and dies according to a state-dependent death rate. An integral equation for the survival probability is derived, and finally a sufficient criterion for the existence of an initial state is given, for which the corresponding individual dies with probability one.
Random packing of an interval. II
Authors: Mannion David
Published: April 2006

We showed in [2] that if an object of initial size x (x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z(x, B) denotes the number of such cells whose sizes are points in the set B, then there exists c, (0 < ≦ 1), such that Z(x, B)x−c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y1, y2, ···, yk and if for each k x - y1 - y2 - ··· - yk (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the nth cumulant, Ψn (x, B), of Z(x, B) satisfies Ψn (x, B)x-c → κn (B), as x → ∞, for some κn(B). Thus, writing N = xc, Z(x, B) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.
The control of a grade structure in a stochastic environment using promotion control
Authors: Bartholomew D. J.
Published: April 2006

In an earlier paper (Bartholomew (1977)) the problem of maintaining a grade (or age) structure by controlling the recruitment flows was discussed. Here we carry out a parallel analysis for the case where control is by promotion. We show how to calculate the probability that a structure can be maintained and investigate the performance of several strategies for keeping a structure as near as possible to that desired. As before, the results demonstrate the limitations of a deterministic analysis of the problem.
A new autoregressive time series model in exponential variables (NEAR(1))
Authors: Lawrance A. J. , Lewis P. A. W.
Published: April 2006

A new time series model for exponential variables having first-order autoregressive structure is presented. Unlike the recently studied standard autoregressive model in exponential variables (ear(1)), runs of constantly scaled values are avoidable, and the two parameter structure allows some adjustment of directional effects in sample path behaviour. The model is further developed by the use of cross-coupling and antithetic ideas to allow negative dependency. Joint distributions and autocorrelations are investigated. A transformed version of the model has a uniform marginal distribution and its correlation and regression structures are also obtained. Estimation aspects of the models are briefly considered.
Simple approximations to the expected waiting time for a cluster of any given size, for point processes
Authors: Samuel-Cahn Ester
Published: April 2006

For point processes, such that the interarrival times of points are independently and identically distributed, let T(L, m) denote the time until at least points cluster within an interval of length at most L. Let τ (L, m) + 1 be the total number of points observed until the above happens. Simple approximations of Eτ (L, m) and ET(L, m) are derived, as well as lower and upper bounds for their value. Approximations to the variances are also given. In particular the Poisson, Bernoulli and compound Poisson processes are discussed in detail. Some numerical tables are included.
Statistical efficiency of regenerative simulation methods for networks of queues
Authors: Iglehart Donald L. , Shedler Gerald S.
Published: April 2006

This paper is concerned with the assessment of the statistical efficiency of proposed regenerative simulation methods. We compare the efficiency of the ‘marked job' and ‘labelled jobs' methods for estimation of passage times in multiclass networks of queues with general service times. Using central limit theorem arguments, we show that the confidence intervals constructed for the expected value of a general function of the limiting passage time using the labelled jobs method are shorter than those obtained from the marked job method. This is consistent with intuition since the labelled jobs method extracts more passage-time information from a fixed-length simulation run.
Convergence rates for the ultimate and pentultimate approximations in extreme-value theory
Authors: Cohen Jonathan P.
Published: April 2006

Let F be a distribution in the domain of attraction of the type I extreme-value distribution Λ(x). In this paper we derive uniform rates of convergence of Fn to Λfor a large class of distributions F. We also generalise the penultimate approximation of Fisher and Tippett (1928) and show that for many F a type II or type III extreme-value distribution gives a better approximation than the limiting type I distribution.
On inference in a one-dimensional mosaic and an M/G/∞ queue
Authors: Hall Peter
Published: April 2006

Suppose segments are distributed at random along a line, their locations being determined by a Poisson process. In the case where segment length is fixed, we compare efficiencies of several different estimates of Poisson intensity. The case of random segment length is also considered, and there we study estimation procedures based on empiric properties. The one-dimensional mosaic may be viewed as an M/G/∞ queue.
Bounds for the availabilities in a fixed time interval for multistate monotone systems
Authors: Funnemark Espen , Natvig Bent
Published: April 2006

In this paper upper and lower bounds for the availability and unavailability, to any level, in a fixed time interval are arrived at for multistate monotone systems based on corresponding information on the multistate components. These are assumed to be maintained and interdependent. Such bounds are of great interest when trying to predict the performance process of the system, noting that exact expressions are obtainable just for trivial systems. The bounds given generalize the existing bounds known in traditional binary theory, and represent improvements of the ones now being developed in multistate theory.
On Goldstein’s variance bound
Authors: Seaman John , Odell Pat
Published: April 2006

Goldstein (1974) derived an upper bound on the variance of certain non-negative functions when the first two moments of the underlying random variable are known. This bound is compared to a simple and fundamental variance bound which requires only that the range of the function be known. It is shown that Goldstein’s bound frequently exceeds the simpler bound. Finally, an interpretation of such bounds in the context of economic risk analysis is given.
Fluctuation theory for systems of signed and unsigned particles with interaction mechanisms based on intersection local times
Authors: Adler Robert J.
Published: April 2006

We consider two distinct models of particle systems. In the first we have an infinite collection of identical Markov processes starting at random throughout Euclidean space. In the second a random sign is associated with each process. An interaction mechanism is introduced in each case via intersection local times, and the fluctuation theory of the systems studied as the processes become dense in space. In the first case the fluctuation theory always turns out to be Gaussian, regardless of the order of the intersections taken to introduce the interaction mechanism. In the second case, an interaction mechanism based on kth order intersections leads to a fluctuation theory akin to a :φ k: Euclidean quantum field theory. We consider the consequences of these results and relate them to different models previously studied in the literature.
Some properties of a branching process with group immigration and emigration
Authors: Pakes Anthony G.
Published: April 2006

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.

A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.
Convergence and finite-time behavior of simulated annealing
Authors: Mitra Debasis , Romeo Fabio , Sangiovanni-Vincentelli Alberto
Published: April 2006

Simulated annealing is a randomized algorithm which has been proposed for finding globally optimum least-cost configurations in large NP-complete problems with cost functions which may have many local minima. A theoretical analysis of simulated annealing based on its precise model, a time-inhomogeneous Markov chain, is presented. An annealing schedule is given for which the Markov chain is strongly ergodic and the algorithm converges to a global optimum. The finite-time behavior of simulated annealing is also analyzed and a bound obtained on the departure of the probability distribution of the state at finite time from the optimum. This bound gives an estimate of the rate of convergence and insights into the conditions on the annealing schedule which gives optimum performance.
On the replacement of systems subject to shocks and wear-dependent failure
Authors: Rangan A. , Grace R. Ester
Published: April 2006

A system is discussed which is subject to a conditional Poisson stream of failures, whose intensity function representing the cumulative damage level is a Markov chain.
Point processes, regular variation and weak convergence
Authors: Resnick Sidney I.
Published: April 2006

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.
Paths in a random digital tree: limiting distributions
Authors: Pittel Boris
Published: April 2006

We study a rule of growing a sequence {tn} of finite subtrees of an infinite m-ary tree T. Independent copies {ω (n)} of a Bernoulli-type process ω on m letters are used to trace out a sequence of paths in T. The tree tn is obtained by cutting each , at the first node such that at most σ paths out of , pass through it. Denote by Hn the length of the longest path, hn the length of the shortest path, and Ln the length of the randomly chosen path in tn. It is shown that, in probability, Hn – logan = O(1), hn – logb (n/log n) = 0(1), (or hn – logb (n/log log n) = O(1)), and that is asymptotically normal. The parameters a, b, c depend on the distribution of ω and, in case of a, also on σ. These estimates describe respectively the worst, the best and the typical case behavior of a ‘trie’ search algorithm for a dictionary-type information retrieval system, with σ being the capacity of a page.
Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues
Authors: Neuts Marcel F.
Published: April 2006

A classical result in queueing theory states that in the stable M/G/1 queue, the stationary distribution W(x) of the waiting time of an arriving customer or of the virtual waiting time satisfies a linear Volterra integral equation of the second kind, of convolution type. For many variants of the M/G/1 queue, there are corresponding integral equations, which in most cases differ from the Pollaczek–Khinchin equation only in the form of the inhomogeneous term. This leads to interesting factorizations of the waiting-time distribution and to substantial algorithmic simplifications. In a number of priority queues, the waiting-time distributions satisfy Volterra integral equations whose kernel is a functional of the busy-period distribution in related M/G/1 queues. In other models, such as the M/G/1 queue with Bernoulli feedback or with limited admissions of customers per service, there is a more basic integral equation of Volterra type, which yields a probability distribution in terms of which the waiting-time distributions are conveniently expressed.

For several complex queueing models with an embedded Markov renewal process of M/G/1 type, one obtains matrix Volterra integral equations for the waiting-time distributions or for related vectors of mass functions. Such models include the M/SM/1 and the N/G/1 queues, as well as the M/G/1 queue with some forms of bulk service.

When the service-time distributions are of phase type, the numerical computation of waiting-time distributions may commonly be reduced to the solution of systems of linear differential equations with constant coefficients.
Dispersive and superadditive ordering
Authors: Ahmed A. N. , Alzaid A. , Bartoszewicz J. , Kochar S. C.
Published: April 2006

Recently many authors have established connections between dispersive ordering and some other partial orderings of distributions. This paper presents the connection which superadditive ordering has with dispersive ordering.
Convergence of perturbation analysis estimates for discontinuous sample functions: a general approach
Authors: Rubinstein Reuven Y. , Szidarovszky Ferenc
Published: April 2006

Generalized perturbation analysis (PA) estimates to study sensitivity of performance measures of discrete events dynamic systems for discontinuous sample functions are introduced. Their convergence conditions and rate of convergence are given. It is shown that the PA estimates based on a single sample path always converge faster to the unknown sensitivity parameter (vector of parameters) than their counterpart—crude Monte Carlo ones.
Insensitivity and reversed Markov processes
Authors: Henderson W.
Published: April 2006

This paper is concerned with the relationship between insensitivity in a certain class of Markov processes and properties of that process when time is reversed. Necessary and sufficient conditions for insensitivity are established and linked to already proved results. A number of examples of insensitive systems are given.
Bounds for the variance of the busy period of the M/G/∞ queue
Authors: Ramalhoto M. F.
Published: April 2006

Some bounds for the variance of the busy period of an M/G/∞ queue are calculated as functions of parameters of the service-time distribution function. For any type of service-time distribution function, upper and lower bounds are evaluated in terms of the intensity of traffic and the coefficient of variation of the service time. Other lower and upper bounds are derived when the service time is a NBUE, DFR or IMRL random variable. The variance of the busy period is also related to the variance of the number of busy periods that are initiated in (0, t] by renewal arguments.
Some multivariate lifetime distributions
Authors: Ghurye S. G.
Published: April 2006

The family of distributions which have the lack-of-memory property is extended by incorporating simple patterns of ageing in the model. Some relatively simple multivariate distributions, obtained in this manner, might prove to be more realistic than distributions like the multivariate exponential.
Transient behavior of regulated Brownian motion, II: Non-zero initial conditions
Authors: Abate Joseph , Whitt Ward
Published: April 2006

This paper continues an investigation of the time-dependent behavior of regulated or reflecting Brownian motion (RBM). Part I focused on RBM starting at the origin; Part II focuses on RBM starting at a fixed positive state. The first two moments of RBM as functions of time are analyzed by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied in Part I. By appropriate normalization, the two monotone components can be converted into cumulative distribution functions that can be analyzed probabilistically, e.g., their moments can be calculated. Simple approximations are then developed by fitting convenient distributions to these moments. Overall, the analysis yields a better understanding of the way RBM and related stochastic flow systems approach steady state.
Stochastic concavity of throughput in series of queues with finite buffers
Authors: Anantharam Venkat , Tsoucas Pantelis
Published: April 2006

Stochastic concavity of the output process with respect to buffer sizes is established in a series of ·/M/1/B queues with loss at the first node.