bluuebunny/crossref_metadata_2025 huggingface dataset:
Asymptotic bounds on the time to fatigue failure of bundles of fibers under local load sharing
Authors: Tierney Luke
Published: April 2006

A fiber bundle is a parallel arrangement of fibers. Under a steady tensile load, fibers fail randomly in time in a manner that depends on how they share the applied load. The bundle fails when all its fibers have failed in a specified region.

In this paper we consider the fatigue failure of such a bundle in a fiber load-sharing setting appropriate for composite materials, that is, to bundles impregnated with a flexible matrix. The bundle is actually modelled as a chain of short bundles, and local load sharing is assumed for the fibers within each short bundle. The chain of bundles fails once all the fibers in one of the short bundles have failed.

Reasonable assumptions are made on the stochastic failure of individual fibers. A general framework for describing fiber bundles is developed and is used to derive the limiting distribution of the time to the first appearance of a set ofkor more adjacent failed fibers as the number of fibers in the bundle grows large. These results provide useful bounds on the distribution of the time to total bundle failure. Some implications and extensions of these results are discussed.
Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and theGI/G/1 queue
Authors: Asmussen Søren
Published: April 2006

LetSn=X1+ · · · +Xnbe a random walk with negative drift μ < 0, letF(x) =P(Xk≦x),v(u) =inf{n:Sn>u} and assume that for some γ > 0is a proper distribution with finite meanVarious limit theorems for functionals ofX1,· · ·,Xv(u)are derived subject to conditioning upon {v(u)< ∞} withularge, showing similar behaviour as if theXiwere i.i.d. with distributionFor example, the deviation of the empirical distribution function fromproperly normalised, is shown to have a limit inD, and an approximation forby means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and theGI/G/1 queue considered either within a busy cycle or in the steady state. The methods produce an alternate approach to known asymptotic formulae for ruin probabilities as well as related waiting-time approximations for theGI/G/1 queue. For exampleuniformly inN, withWNthe waiting time of the Nth customer.
Sojourn times and the overtaking condition in Jacksonian networks
Authors: Walrand J. , Varaiya P.
Published: April 2006

Consider an open multiclass Jacksonian network in equilibrium and a path such that a customer travelling along it cannot be overtaken directly by a subsequent arrival or by the effects of subsequent arrivals. Then the sojourn times of this customer in the nodes constituting the path are all mutually independent and so the total sojourn time is easily calculated. Two examples are given to suggest that the non-overtaking condition may be necessary to ensure independence when there is a single customer class.
Prediction of level crossings for normal processes containing deterministic components
Authors: Lindgren Georg
Published: April 2006

A level crossing predictor is a predictor process Y(t) which can be used to predict whether a specific process X(t) will cross a predetermined level or not. A natural criterion of how good a predictor is can be the probability that a crossing is detected a sufficient time ahead, and the number of times the predictor makes a false alarm.

Suppose the process X(t) consists of a deterministic part A(t) which can be calculated with sufficient accuracy, and a stochastic part Xe(t) which can be predicted by some statistically based predictor An example of this is the prediction of water level near a coast, when A(t) is a sum of known tide components.

The paper develops a tool to handle the detection properties of such predictor processes when used to predict level crossings for the case when A(t) is periodic or is a sum of such functions.
Forwards and backwards models for finite-state Markov processes
Authors: Anderson B. D. O. , Kailath T.
Published: April 2006

The construction and properties of reversible and dynamically reversible models for finite-state Markov processes are studied. Certain results on approximating processes with rational power spectra with dynamically reversible finite-state models are also obtained.
A birth–death model of advertising and pricing
Authors: Albright S. Christian , Winston Wayne
Published: April 2006

This paper employs the methods currently used to solve many queuing control models in order to investigate the behavior of a firm's optimal advertising and pricing strategies over time. Given that a firm's market position expands or deteriorates in a probabilistic way which depends upon the current position, the rate of advertising, and the price the firm charges, we present conditions which ensure that the optimal level of advertising is a monotonic function of the firm's market position, and we discuss the economic meaning of these conditions. Furthermore, although the primary focus is upon a non-competitive environment, we develop the above model as a non-zero sum, two-person stochastic game and show that an equilibrium strategy exists which is simple to compute.
On the steady-state solution of the M/G/2 queue
Authors: Hokstad Per
Published: April 2006

The asymptotic behaviour of the M/G/2 queue is studied. The difference-differential equations for the joint distribution of the number of customers present and of the remaining holding times for services in progress were obtained in Hokstad (1978a) (for M/G/m). In the present paper it is found that the general solution of these equations involves an arbitrary function.

In order to decide which of the possible solutions is the answer to the queueing problem one has to consider the singularities of the Laplace transforms involved. When the service time has a rational Laplace transform, a method of obtaining the queue length distribution is outlined. For a couple of examples the explicit form of the generating function of the queue length is obtained.
Uniform approximation to distributions of extreme order statistics
Authors: Reiss R.-D.
Published: April 2006

This paper deals with asymptotic expansions of the distribution of the kth-largest order statistic Zn–k+1:n for the sample size n. These expansions establish higher-order approximations which hold uniformly over all Borel sets. In the particular case of the distribution of Zn–k+1:n under the uniform distribution and the exponential distribution, the approximating measures are linear combinations of ‘negative’ gamma distributions and linear combinations of extreme-value distributions. These results can be extended to the case of the joint distribution of the k largest order statistics. A numerical comparison to a different asymptotic expansion is given where the normal distribution is the leading term.
Glivenko–Cantelli theorems for classes of convex sets
Authors: Elker J. , Pollard D. , Stute W.
Published: April 2006

Let denote the empirical distribution obtained from a sequence of i.i.d. -valued random vectors with common distribution P. If is a class of Borel subsets of then we say that it forms a Glivenko–Cantelli class for P if

In this paper we describe a simple technique for identifying such classes, based on the idea of uniformity classes for setwise convergence. Classes for which the method proves successful include the closed half-spaces, closed balls, and the class of all convex subsets of .
A rate-conservative principle for stationary piecewise Markov processes
Authors: Rolski Tomasz
Published: April 2006

A stationary processyt,t∈R1is considered which is Markov between points of changeover from a stationary point process, and at these points it changes over according to a distribution dependent only on the value ofytjust before the change is analysed. An explicit form of a rate-conservative principle is stated, and its relationship with formulae relating the distribution of the process at an instanttand the distribution at a point of changeover is shown. The theory is applied to discrete state processes and to processes which are generalizations of the Takács processes, and is also applied in the theory ofG/M/landG/G/kqueues to obtain relations between distributions of a process and some process imbedded in it.
Invariant problems in discounted dynamic programming
Authors: Assaf David
Published: April 2006

Discounted dynamic programming problems whose transition mechanism depends only on the action taken and does not depend on the current state are considered. A value determination operation and method of obtaining optimal policies for the case of finite action space (and arbitrary state space) are presented.

The solution of other problems is reduced to this special case by a suitable transformation. Results are illustrated by examples.
The time to failure of fiber bundles subjected to random loads
Authors: Taylor Howard M.
Published: April 2006

The effect on cable reliability of random cyclic loading such as that generated by the wave-induced rocking of ocean vessels deploying these cables is examined. A simple model yielding exact formulas is first explored. In this model, the failure time of a single fiber under a constant load is assumed to be exponentially distributed, and the random loadings are a two-state stationary Markov process. The effect of load on failure time is assumed to follow a power law breakdown rule. In this setting, exact results concerning the distribution of bundle or cable failure time, and especially the mean failure time, are obtained. Where the fluctuations in load are frequent relative to bundle life, such as may occur in long-lived cables, it is shown that randomness in load tends to decrease mean bundle life, but it is suggested that the reduction in mean life often can be restored by modestly reducing the base load on the structure or by modestly increasing the number of elements in the bundle.

In later pages this simple model is extended to cover a broader range of materials and random loadings. Asymptotic distributions and mean failure times are given where fibers follow a Weibull distribution of failure time under constant load, and loads that are general non-negative stationary processes subject only to some mild condition of asymptotic independence. When the power law breakdown exponent is large, the mean time to bundle failure depends heavily on the exact form of the marginal probability distribution for the random load process and cannot be summarized by the first two moments of this distribution alone.
Energy of Markov chains
Authors: Syski R.
Published: April 2006

After preliminaries on Markov chains, supermartingales and potential theory (Section 1), the energy of a potential supermartingale generated by an increasing process is defined. The paper examines some properties of the energy of potentials of the formUt=p(Xt) wherepis a purely excessive function (which is also a potential of a charge) for a Markov chain (Xt). Also, the mutual energy of two potentials associated with the same Markov chain is discussed. Finally, several applications and examples are worked out in detail (Section 3).
On a tandem queueing model with identical service times at both counters, II
Authors: Boxma O. J.
Published: April 2006

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times of an arbitrary customer at both queues are identical. For this purpose some tables and graphs are included. A comparison is made—mainly by numerical and asymptotic techniques—between the following two phenomena: (i) the queueing behaviour at the second counter of the two-stage tandem queue and (ii) the queueing behaviour at a single-server queue with the same offered (Poisson) traffic as the first counter and the same service-time distribution as the second counter. This comparison makes it possible to assess the influence of the first counter on the queueing behaviour at the second counter. In particular we note that placing the first counter in front of the second counter in heavy traffic significantly reduces both the mean and variance of the total time spent in the second system.
Optimal replacement under a minimal repair strategy—a general failure model
Authors: Aven Terje
Published: April 2006

In this paper we generalize the minimal repair replacement model introduced by Barlow and Hunter (1960). We assume that there is available information about the underlying condition of the system, for instance through measurements of wear characteristics and damage inflicted on the system. We assume furthermore that the system failure rate and the expected cost of a repair/replacement at any point of time are adapted to this information. At time t = 0 a new system is installed. At a stopping time T, based on the information about the condition of the system, the system is replaced by a new and identical one, and the process is repeated. Failures that occur before replacement are rectified through minimal repair. We assume that a minimal repair changes neither the age of the system nor the information about the condition of the system. The problem is to find a T which minimizes the total expected discounted cost. Under appropriate conditions an optimal T is found. Some generalizations and special cases are given.
On Arnold's treatment of Moran's bounds
Authors: Buckley M. J. , Seneta E.
Published: April 2006

We prove a conjecture of Arnold (1968) which simplifies the determination of an optimal bound on absorption probability originally due to Moran (1960).
On simulation from infinitely divisible distributions
Authors: Bondesson Lennart
Published: April 2006

A general method based on a limit theorem for generation of random numbers from infinitely divisible distributions with essentially given Lévy measure is studied. Some classes of infinitely divisible distributions that appear in a natural way in this context are paid particular attention.
The behavior of the renewal sequence in case the tail of the waiting-time distribution is regularly varying with index −1
Authors: Frenk J. B. G.
Published: April 2006

A second-order asymptotic result for the probability of occurrence of a persistent and aperiodic recurrent event is given if the tail of the distribution of the waiting time for this event is regularly varying with index −1.
Reversibility, invariance and μ-invariance
Authors: Pollett P. K.
Published: April 2006

In this paper we consider a number of questions relating to the problem of determining quasi-stationary distributions for transient Markov processes. First we find conditions under which a measure or vector that is µ-invariant for a matrix of transition rates is also μ-invariant for the family of transition matrices of the minimal process it generates. These provide a means for determining whether or not the so-called stationary conditional quasi-stationary distribution exists in the λ-transient case. The process is not assumed to be regular, nor is it assumed to be uniform or irreducible. In deriving the invariance conditions we reveal a relationship between μ-invariance and the invariance of measures for related processes called the μ-reverse and the μ-dual processes. They play a role analogous to the time-reverse process which arises in the discussion of stationary distributions. Secondly we bring the related notions of detail-balance and reversibility into the realm of quasi-stationary processes. For example, if a process can be identified as being μ-reversible, the problem of determining quasi-stationary distributions is made much simpler. Finally, we consider some practical problems that emerge when calculating quasi-stationary distributions directly from the transition rates of the process. Our results are illustrated with reference to a variety of processes including examples of birth and death processes and the birth, death and catastrophe process.
Diffusion models of temporally varying selection in population genetics
Authors: Seno Shoichiro , Shiga Tokuzo
Published: April 2006

We consider a diploid model of random selection. Assuming that the selective fitness is of the SAS–CFF type introduced by Gillespie, we obtain a diffusion approximation. We also discuss the existence of stationary distributions and recurrent properties.
Strong unimodality
Authors: Klaassen Chris A. J.
Published: April 2006

It is proved that a distribution F is strongly unimodal iff any two quantiles of the convolution of F with any other distribution are further apart than the corresponding quantiles of F itself. Some related characterizations of strong unimodality are also given.
On the fraction of random points by specified nearest-neighbour interrelations and degree of attraction
Authors: Henze Norbert
Published: April 2006

Let Z1, …, Zn be i.i.d. random vectors (‘points') defined in having common density f(x) that is assumed to be continuous almost everywhere. For a fixed but otherwise arbitrary norm |.| on , consider the fraction Vn of those points Z1, …, Zn that are the lth nearest neighbour (with respect to |.|) to their own kth nearest neighbour, and write Sn for the fraction of points that are the nearest neighbour of exactly k other points. We derive the stochastic limits of Vn and Sn, as n tends to∞, and show how the results may be applied to the multivariate non-parametric two-sample problem.
New exact non-linear filters by large Lie algebras
Authors: Beneš V. E.
Published: April 2006

The ideas previously used (Beneš (1981)) to construct some finite-dimensional non-linear filters also yield related new filters of finite dimension with arbitrarily large bases; this is because the finite dimensionality is not destroyed by insertion of noiseless linear differential operations on the observations.
A stochastic model of particle shattering
Authors: Griffith William S. , Kryscio Richard J. , Purdue Peter
Published: April 2006

Motivated by the problem of solid catalytic particle attrition during chemical reactions we formulate a continuous-time Markov model to describe the shattering of a particle when it is assumed that particles can be classified into a small number of types by size. We then obtain a recursive expression for the joint probability generating function of the count of the different types of particle at time t and derive a system of differential equations for the mean number of particle counts and a system of matrix differential equations for the covariance matrix of the particle counts. Solutions to these differential equations are presented in an important special case.
Obtaining remainder term estimates by an inversion technique
Authors: Englund Gunnar
Published: April 2006

Let be a double sequence of random variables such that there exists a ‘dual' sequence satisfying , and that can be expressed as a sum of independent random variables. If (suitably centered and rescaled) is approximately normally distributed as k and N → ∞ in some fashion, we can use this fact to obtain a remainder-term estimate for the asymptotic normality of as n and N → ∞ in some prescribed manner. The result in the general theorem is used in two specific situations: (i) classical occupancy where the balls can fall through the boxes, (ii) a capture-recapture problem where tagging affects catchability.
Optimal selection based on relative ranks of a sequence by ties
Authors: Campbell Gregory
Published: April 2006

The optimal selection of a maximum of a sequence with the possibility of ties is considered. The object is to examine each observation in the sequence of known length n and, based only on the relative rank among predecessors, either to stop and select it as a maximum or to continue without recall. Rules which maximize the probability of correctly selecting a maximum from a sequence with ties are investigated. These include rules which randomly break ties, rules which discard tied observations, and minimax rules based on the atoms of a discrete distribution function. If the sequence is random from F, a random distribution function from a Dirichlet process prior with non-atomic parameter, optimal rules are developed. The limiting behavior of these rules is studied and compared with other rules. The selection of the parameter of the Dirichlet process regulates the ties.
Asymptotic methods in reliability theory: a review
Authors: Gertsbakh I. B.
Published: April 2006

Section 1 of this paper reviews some works related to reliability evaluation of systems without repair. The assumption that element failure rates are low enables one to obtain an expression for the main term in the asymptotic representation of system reliability function. Section 2 is devoted to repairable systems. The main index of interest in reliability is the time to the first system failure. A typical situation in reliability is that the repair time is much smaller than the element lifetime. This ‘fast repair' property leads to an interesting phenomenon, that for many renewable systems the time to system failure converges in distribution, under appropriate norming, to an exponential random variable. Some basic theorems explaining this fact are presented and a series of typical examples is considered. Special attention is paid to reviewing the works describing the exponentiality phenomenon in birth-and-death processes. Some important aspects of computing the normalizing constants are considered, among them the role played by the so-called ‘main event'. Section 2 also reviews various bounds on the deviation from exponentiality. Section 3 gives brief comments on some works and techniques related to asymptotic reliability analysis. In particular, attention is paid to the works presenting upper and lower bounds on the reliability function.

A considerable part of this review is based on sources originally published in Russian.
Perturbation theory for unbounded Markov reward processes with applications to queueing
Authors: Van Dijk Nico M.
Published: April 2006

Consider a perturbation in the one-step transition probabilities and rewards of a discrete-time Markov reward process with an unbounded one-step reward function. A perturbation estimate is derived for the finite horizon and average reward function. Results from [3] are hereby extended to the unbounded case. The analysis is illustrated for one- and two-dimensional queueing processes by an M/M/1-queue and an overflow queueing model with an error bound in the arrival rate.
Asymptotic performance of a multistate coherent system
Authors: Iyer Srinivas
Published: April 2006

An expression for the asymptotic or steady-state performance function is derived for a multistate coherent system when each component changes states in time according to a semi-Markov process, the stochastic processes being mutually independent. This generalizes the expression for system availability of a binary coherent system when the components are governed by mutually independent alternating renewal processes.
L'application des representations au calcul explicite sur certaines chaînes de Markov finies
Authors: Ycart Bernard
Published: April 2006

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.
Dispersive ordering of distributions
Authors: Saunders D. J.
Published: April 2006

Shaked (1982) gives a characterisation of dispersive ordering for distributions which are absolutely continuous with interval supports. However there is a technical error in his proof of the second part of Theorem 2.1, although the result is still true. The difficulty arises because zero terms are discarded in the definition of the sign change operator S. Therefore the equality of and x′ does not imply that
Blocking probabilities in large circuit-switched networks
Authors: Kelly F. P.
Published: April 2006

This paper is concerned with blocking and loss probabilities in circuit-switched networks. We show that when the capacity of links and the offered traffic are increased together, a limiting regime emerges in which loss probabilities are as if links block independently, with blocking probabilities given by the solution of a simple convex programming problem. We then show that an approximate procedure, based on solving Erlang&s formula under the assumption of independent blocking, produces a unique solution when routes are fixed, and that under the limiting regime the estimates of loss probabilities obtained from the procedure converge to the correct values.
Nearest neighbors and Voronoi regions in certain point processes
Authors: Newman C. M. , Rinott Y. , Tversky A.
Published: April 2006

We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that Poisson (λ= 1). An asymptotic Poisson (λ= 1) distribution also holds for many of the other models. On the other hand, we find that . Related results concern the (random) volume, , of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that while
Relaxed Markov processes
Authors: Whittle P.
Published: April 2006

The concept of relaxing a Markov process is introduced; this is the creation of additional transitions between ergodic classes of the process in such a way as to conserve the existing equilibrium distribution within ergodic classes. The ‘open' version of a ‘closed' model of migration, polymerisation etc. often has this character. As further examples, generalized versions of Jackson networks and networks with clustering nodes are given.
Arret optimal avec contrainte
Authors: Pontier Monique , Szpirglas Jacques
Published: April 2006

Given two optional positive bounded processesYandY′, defined on a probability space, and a non-negative reala,the problem is to maximize the average rewardE(YT) among all the stopping timesTverifying the following constraint:The problem is solved by Lagrangian saddlepoint techniques in the set of randomized stopping times including the set of stopping times.
Evaluation of the total time in system in a preempt/resume priority queue via a modified Lindley process
Authors: Keilson J. , Sumita U.
Published: April 2006

A Poisson stream of arrival rate λIand service-time distributionAI(x) has preempt/resume priority over a second stream of rate λIIand distributionAII(x). Abundant theoretical results exist for this system, but severe numerical difficulties have made many descriptive distributions unavailable. Moreover, the distribution of total time in system of low-priority customers has not been discussed theoretically. It is shown that the waiting-time sequences of such customers before first entry into service is a Lindley process modified by replacement. This leads to the total time distribution needed. A variety of descriptive distributions, transient and stationary, is obtained numerically via the Laguerre transform method.
Continuous-time branching processes with decreasing state-dependent immigration
Authors: Mitov K. V. , Vatutin V. A. , Yanev N. M.
Published: April 2006

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.
Exponential expansion for the tail of the waiting-time probability in the single-server queue with batch arrivals
Authors: Van Ommeren J. C. W.
Published: April 2006

This paper deals with the single-server queue with batch arrivals. We show that under suitable conditions the waiting-time distribution of an individual customer has an asymptotically exponential expansion. Computationally useful characterizations of the amplitude factor and the decay parameter are given for the practically important case in which the interarrival time and the service time have phase-type distributions.
Probabilistic profiles of continuously replenished spherically symmetric clouds
Authors: Sturgeon M. L.
Published: April 2006

The evolution of a spherically symmetric cloud, continuously replenished from a Gaussian source with particle speed parameter σ, is characterized at time τ by a folded normal radial density with parameter στ and constant growth rates for p-percentile envelopes. Density functions for the speed of particles seen in profile by an optical sensor with a 4π steradian field of view at various times τ>0 and distances L ≦ ∞ are derived and related to the Rayleigh and Maxwell laws associated with i.i.d. N(0, σ) velocity components.
Transient behavior of regulated Brownian motion, I: Starting at the origin
Authors: Abate Joseph , Whitt Ward
Published: April 2006

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that the kth-moment c.d.f is the k-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k + 1)th moment approaches its steady-state limit more slowly than the kth moment. It is also easy to derive the asymptotic behavior as t →∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2) c.d.f.&s (mixtures of two exponentials). The H2 approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of the H2 approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.
A note on asymptotic inference for an asymmetric Ising model around a torus
Authors: Qian Wei
Published: April 2006

Limiting distribution results are obtained for the sufficient statistics of an asymmetric Ising model on a torus. Applications are discussed.
A new ordering for stochastic majorization: theory and applications
Authors: Chang Cheng-Shang
Published: April 2006

In this paper, we develop a unified approach for stochastic load balancing on various multiserver systems. We expand the four partial orderings defined in Marshall and Olkin, by defining a new ordering based on the set of functions that are symmetric, L-subadditive and convex in each variable. This new partial ordering is shown to be equivalent to the previous four orderings for comparing deterministic vectors but differs for random vectors. Sample-path criteria and a probability enumeration method for the new stochastic ordering are established and the ordering is applied to various fork-join queues, routing and scheduling problems. Our results generalize previous work and can be extended to multivariate stochastic majorization which includes tandem queues and queues with finite buffers.
Analysis of separable Markov-modulated rate models for information-handling systems
Authors: Stern Thomas E. , Elwalid Anwar I.
Published: April 2006

In many communication and computer systems, information arrives to a multiplexer, switch or information processor at a rate which fluctuates randomly, often with a high degree of correlation in time. The information is buffered for service (the server typically being a communication channel or processing unit) and the service rate may also vary randomly. Accurate capture of the statistical properties of these fluctuations is facilitated by modeling the arrival and service rates as superpositions of a number of independent finite state reversible Markov processes. We call such models separable Markov-modulated rate processes (MMRP).

In this work a general mathematical model for separable MMRPs is presented, focusing on Markov-modulated continuous flow models. An efficient procedure for analyzing their performance is derived. It is shown that the ‘state explosion' problem typical of systems composed of a large number of subsystems, can be circumvented because of the separability property, which permits a decomposition of the equations for the equilibrium probabilities of these systems. The decomposition technique (generalizing a method proposed by Kosten) leads to a solution of the equilibrium equations expressed as a sum of terms in Kronecker product form. A key consequence of decomposition is that the computational complexity of the problem is vastly reduced for large systems. Examples are presented to illustrate the power of the solution technique.
Synchronous service on a circle
Authors: Chen Robert W. , Shepp Larry A.
Published: April 2006

In this paper, we study a synchronous computer network model which may be described as follows. There are n service stations around a circle and there are k people to be serviced with at most one person per station. Time is slotted and in each time slot b adjacent people with an empty station ahead can move synchronously and simultaneously to the next counterclockwise station with probability pb. We show that the stationary distribution is uniform on all possible states, and using the Descartes' rule of signs we find the optimal value of k (fixed p and n) that yields maximal ‘throughput', i.e., the expected number of people served per time slot in equilibrium. We also briefly study an asynchronous model (introduced by D. Sarkar) where in each time slot a person moves with probability p to the next counterclockwise station if it is empty. We find, among other results, the new stationary distribution (Ramakrishnan et al. (1989) also find independently the stationary distribution and they also find the optimal value of k (fixed p and n) that yields maximal throughput in this model). We give tables comparing the synchronous and asynchronous models. Some applications which motivate this study are briefly presented.
A unified analysis of the final size and severity distribution in collective Reed-Frost epidemic processes
Authors: Picard Philippe , Lefevre Claude
Published: April 2006

An extended version, called collective, of the randomized Reed-Frost processes is considered where each infective during his survival time fails to transmit the infection within any given set of susceptibles with a probability depending only on the size of that set. Our purpose is to provide a unified analysis of the distribution of the final size and severity, the two main components of the cost generated by the infection process. The method developed relies on the construction of a family of martingales and the use of a family of polynomials studied recently by the authors (Lefèvre and Picard (1990)). The results generalize a number of earlier ones and are derived in a more direct and systematic way than before.
On preservation of some partial orderings under shock models
Authors: Kochar Subhash C.
Published: April 2006

Singh and Jain (1989) have proved some preservation results for partial orderings of life distributions assuming that shocks occur according to a homogeneous Poisson process. It is shown that their results hold under less restrictive conditions.
On the association of the lifelengths of components subjected to a stochastic environment
Authors: Lefevre Claude , Milhaud Xavier
Published: April 2006

This note is concerned with a system of non-renewable components in parallel subjected to a common environment which is described by a real-valued external stochastic process. Given the environment process, the components are supposed to work independently, and the corresponding failure rates are all increasing (or decreasing) functions of the observed current state. It is then proved that, under these assumptions, the association of the external process implies the association of the component lifelengths. Connection with existing results is underlined.
Unbiased Stereological Estimation of d-Dimensional Volume in ℝn from an Isotropic Random Slice Through a Fixed Point
Authors: Jensen E. B. Vedel , Kiêu K.
Published: April 2006

Unbiased stereological estimators of d-dimensional volume in ℝn are derived, based on information from an isotropic random r-slice through a specified point. The content of the slice can be subsampled by means of a spatial grid. The estimators depend only on spatial distances. As a fundamental lemma, an explicit formula for the probability that an isotropic random r-slice in ℝn through O hits a fixed point in ℝn is given.
New kinds of multidimensional IFR distribution
Authors: Wang Yuedong , Cao Jinhua
Published: April 2006

Two kinds of multidimensional IFR distribution are defined by using a partial order in Rn+, which is derived from a non-negative, strictly increasing function in Rn+. Some closure properties under operations and an application to a shock model are discussed.
Insensitivity in processes with zero speeds
Authors: Taylor P.
Published: April 2006

Many authors have discussed the equivalence of partial balance and insensitivity in in stochastic processes. When speeds are introduced into a stochastic process there arises a difficulty in proving the necessity of partial balance for insensitivity. Previous authors have overcome this difficulty by assuming that a process has the property of instantaneous attention. This property enforces the requirement that no lifetime can be created in a state in which that lifetime has zero speed.

In this paper it is shown that for processes with a finite state space it is unnecessary to make this assumption provided the notion of partial balance is slightly changed. Thus we give a criterion, analogous to partial balance, which is necessary and sufficient for insensitivity even in processes which do not possess the property of instantaneous attention. When a process does have instantaneous attention this criterion is equivalent to partial balance.
Dynamics of an epidemic in a closed population
Authors: Svensson Åke
Published: April 2006

A simple model for the intensity of infection during an epidemic in a closed population is studied. It is shown that the size of an epidemic (i.e. the number of persons infected) and the cumulative force of an epidemic (i.e. the amount of infectiousness that has to be avoided by a person that will stay uninfected during the entire epidemic) satisfy an equation of balance. Under general conditions, small deviances from this balance are, in large populations, asymptotically mixed normally distributed. For some special epidemic models the size of an asymptotically large epidemic is asymptotically normally distributed.
A note on gittins indices for pharmaceutical research
Authors: Wang You-Gan , Gittins John
Published: April 2006

The Bernoulli/exponential target process is considered. Such processes have been found useful in modelling the search for active compounds in pharmaceutical research. An inequality is presented which improves a result of Gittins (1989), thus providing a better approximation to the Gittins indices which define the optimal search policy.
Extension of the bivariate characterization for stochastic orders
Authors: Righter Rhonda , Shanthikumar J. George
Published: April 2006

The bivariate characterization of stochastic ordering relations given by Shanthikumar and Yao (1991) is based on collections of bivariate functions g(x, y), where g(x, y) and g(y, x) satisfy certain properties. We give an alternate characterization based on collections of pairs of bivariate functions, g1(x, y) and g2(x, y), satisfying certain properties. This characterization allows us to extend results for single machine scheduling of jobs that are identical except for their processing times, to jobs that may have different costs associated with them.
The joint distribution of sojourn times in finite Markov processes
Authors: Csenki Attila
Published: April 2006

Rubino and Sericola (1989c) derived expressions for the mth sojourn time distribution associated with a subset of the state space of a homogeneous irreducible Markov chain for both the discrete- and continuous-parameter cases. In the present paper, it is shown that a suitable probabilistic reasoning using absorbing Markov chains can be used to obtain respectively the probability mass function and the cumulative distribution function of the joint distribution of the first m sojourn times. A concise derivation of the continuous-time result is achieved by deducing it from the discrete-time formulation by time discretization. Generalizing some further recent results by Rubino and Sericola (1991), the joint distribution of sojourn times for absorbing Markov chains is also derived. As a numerical example, the model of a fault-tolerant multiprocessor system is considered.
Invariant measures for Q-processes when Q is not regular
Authors: Pollett P. K.
Published: April 2006

The problem of determining invariant measures for continuous-time Markov chains directly from their transition rates is considered. The major result provides necessary and sufficient conditions for the existence of a unique ‘single-exit' chain with a specified invariant measure. This generalizes a result of Hou Chen-Ting and Chen Mufa that deals with symmetrically reversible chains. A simple sufficient condition for the existence of a unique honest chain for which the specified measure is invariant is also presented.
Applications of generalized inverses to Markov chains
Authors: Rising William
Published: April 2006

First it is shown that any generalized inverse of the infinitesimal generator of an irreducible Markov chain can be used to compute the exact stationary distribution and all the expected first-passage times of the chain. In the special case of a single-server queue this allows all computations to be done with upper-triangular matrices.

Next it is shown that the effect of a perturbation of the infinitesimal generator on the stationary distribution and expected first-passage times can also be computed using generalized inverses. These results extend and generalize Schweitzer's [9] original work using fundamental matrices. It is then shown that any perturbation can be broken up into a series of perturbations each involving a single state.
Addendum to ‘On an index policy for restless bandits'
Authors: Weber Richard R. , Weiss Gideon
Published: April 2006

We show that the fluid approximation to Whittle's index policy for restless bandits has a globally asymptotically stable equilibrium point when the bandits move on just three states. It follows that in this case the index policy is asymptotic optimal.
On critically loaded loss networks
Authors: Hunt P. J. , Kelly F. P.
Published: April 2006

This paper studies the behaviour of large loss networks, paying particular attention to links at a certain critical loading where the load offered very nearly matches capacity. We correct and extend an earlier central limit theorem for the stationary distribution of a loss network with critically loaded links. We then use this result to show that acceptance probabilities have a limiting product-form decomposition, despite marked dependencies between the occupancies of critically loaded links.
L-superadditive structure functions
Authors: Block Henry W. , Griffith William S. , Savits Thomas H.
Published: April 2006

Structure functions relate the level of operations of a system as a function of the level of the operation of its components. In this paper structure functions are studied which have an intuitive property, called L-superadditive (L-subadditive). Such functions describe whether a system is more series-like or more parallel-like. L-superadditive functions are also known under the names supermodular, quasi-monotone and superadditive and have been studied by many authors. Basic properties of both discrete and continuous (i.e., taking a continuum of values) L-superadditive structure functions are studied. For binary structure functions of binary values, El-Neweihi (1980) showed that L-superadditive structure functions must be series. This continues to hold for binary-valued structure functions even if the component values are continuous (see Proposition 3.1). In the case of non-binary-valued structure functions this is no longer the case. We consider structure functions taking discrete values and obtain results in various cases. A conjecture concerning the general case is made.
Another visit to two halflines
Authors: Hooghiemstra G.
Published: April 2006

We shall use three basic properties of Brownian motion to derive in an elegant and non-computational way the probability that standard Brownian motion, starting from 0, will ever cross the halflines t → αt + β or t → γt + δ where γ, δ < 0 < α, β.
Optimal replacement policies for a deteriorating system with imperfect maintenance
Authors: Rangan A. , Grace R. Esther
Published: April 2006

A system is repaired on failure. With probability p, it is returned to the ‘good as new' state (perfect repair) and with probability 1 – p, it is returned to the functioning state, but is only as good as a system of age equal to its age at failure (imperfect repair). In this article, we develop replacement policies for a deteriorating system with imperfect maintenance. The successive survival times and consecutive repair times form a geometric process which is stochastically non-increasing or non-decreasing respectively. Explicit expressions are obtained for the long-run expected cost under two kinds of replacement policies based on the working age of the system and the number of imperfect repairs before a replacement.
A Functional Limit Theorem for the Total Cost of a Multitype Standard Epidemic
Authors: Andersson Håkan , Djehiche Boualem
Published: April 2006

By imbedding the multitype version of the standard epidemic model in a multiparameter process, we derive a functional limit theorem for the total cost of the epidemics.
Simple formulae for counting processes in reliability models
Authors: Ledoux James , Rubino Gerardo
Published: April 2006

Dependability evaluation is a basic component in the assessment of the quality of repairable systems. We develop a model taking simultaneously into account the occurrence of failures and repairs, together with the observation of user-defined success events. The model is built from a Markovian description of the behavior of the system. We obtain the distribution function of the joint number of observed failures and of delivered services on a fixed mission period of the system. In particular, the marginal distribution of the number of failures can be directly related to the distribution of the Markovian arrival process extensively used in queueing theory. We give both the analytical expressions of the considered distributions and the algorithmic solutions for their evaluation. An asymptotic analysis is also provided.