bluuebunny/crossref_metadata_2025 huggingface dataset:
Stochastic ordering of random processes with an imbedded point process
Authors: Baccelli François
Published: April 2006

We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.
Asymptotic properties of intensity estimators for Poisson shot-noise processes
Authors: Liese Friedrich , Schmidt Volker
Published: April 2006

Stochastic processes {X(t)} of the form X(t) = Σ n f(t – Tn) are considered, where {Tn} is a stationary Poisson point process with intensity λ and f: R → R is an unknown response function. Conditions are obtained for weak consistency and asymptotic normality of estimators of λ based on long-run observations of {X(t)}.
On the first zero of an empirical characteristic function
Authors: Bräker H. U. , Hüsler J.
Published: April 2006

We deal with the distribution of the first zero Rn of the real part of the empirical characteristic process related to a random variable X. Depending on the behaviour of the theoretical real part of the underlying characteristic function, cases with a slow exponential decrease to zero are considered. We derive the limit distribution of Rn in this case, which clarifies some recent results on Rn in relation to the behaviour of the characteristic function.
Dynamic multivariate mean residual life functions
Authors: Shaked Moshe , Shanthikumar J. George
Published: April 2006

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied.

A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given.

Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.
Hitting probabilities for random ellipses and ellipsoids
Authors: Duma Andrei , Stoka Marius
Published: April 2006

Let denote a rectangular lattice in the Euclidean plane E2, generated by (a × b) rectangles. In this paper we consider the probability that a random ellipse having main axes of length 2α and 2ß, with intersects . We regard the lattice as the union of two orthogonal sets and of equidistant lines and evaluate the probability that the random ellipse intersects or . Moreover, we consider the dependence structure of the events that the ellipse intersects or . We study further the case when the main axes of the ellipse are parallel to the lines of the lattice and satisfy 2ß = min (a, b) < 2α = max (a, b). In this case, the probability of intersection is 1, and there exist almost surely two perpendicular segments in within the ellipse. We evaluate the distribution function, density, mean and variance of the length of these segments. We conclude with a generalization of this problem in three dimensions.
Closure of the NBUC class under formation of parallel systems
Authors: Hendi M. I. , Mashhour A. F. , Montasser M. A.
Published: April 2006

The class of new better than used in convex ordering (NBUC) is shown to be closed under formation of parallel systems with independent and identically distributed components.
A linear random growth model
Authors: Quine M. P. , Robinson J.
Published: April 2006

Points start to form on an ‘uncovered' unit interval according to a Poisson process with parameter λ. From newly formed points a covering region grows in both directions at velocity v, while new points continue to form on uncovered parts of the interval. Eventually the whole interval will be covered. Let N ≧ 1 denote the total number of points formed. We derive integral expressions for E(N) and Var(N) and give precise asymptotic expressions for these moments as ρ = λ/v →∞. Asymptotic normality of N is also established.
On certain aspects of non-homogeneous Markov systems in continuous time
Authors: Gerontidis Ioannis I.
Published: April 2006

In the present paper we study three aspects in the theory of non-homogeneous Markov systems under the continuous-time formulation. Firstly, the relationship between stability and quasi-stationarity is investigated and conditions are provided for a quasi-stationary structure to be stable. Secondly, the concept of asymptotic attainability is studied and the possible regions of asymptotically attainable structures are determined. Finally, the cyclic case is considered, where it is shown that for a system in a periodic environment, the relative structure converges to a periodic vector, independently of the initial distribution. Two numerical examples illustrate the above theoretical results.
A unification of some approaches to Poisson approximation
Authors: Witte Hans-Jürgen
Published: April 2006

Let Sn be a sum of independent random variables. For the approximation of Sn by a Poisson random variable Y with the same mean, the complex analysis approaches based on generating functions and the semigroup approach are presented in a unified setting which permits us to refine Kerstan's complex analysis approach obtaining considerably sharper upper bounds for some metric distances of Sn and Y. These results are applied to some special Sn counting the records of an i.i.d. sequence of random variables which is important to various applied problems, for instance the secretary problem.
Maintenance comparisons: block policies
Authors: Block Henry W. , Langberg Naftali A. , Savits Thomas H.
Published: April 2006

Complete repair and minimal repair models with a block maintenance policy are considered. Each of these models gives rise to a counting process, and these processes are compared stochastically. This contrasts with most previous work on maintenance policies where only univariate marginal comparisons were made. Also a more general block schedule is considered than is customary.
Wicksell's corpuscle problem in spherical spaces
Authors: Zähle M.
Published: April 2006

Wicksell's problem of unfolding ball size distributions is solved for the case of spherical spaces of curvatureK> 0. The well-known formulas for the euclidean case result as limits ifK →0. The paper is completed by a brief discussion of some computer-aided statistical procedures.
Characterization of the exponential distribution by the relevation transform
Authors: Lau Ka Sing , Rao B. L. S. Prakasa
Published: April 2006

A characterization of the exponential distribution was obtained by Grosswald et al. (1980) using the relevation transform introduced by Krakowski (1973). Here we obtain an improved version of the result in Grosswald et al. (1980).
Analysis of the Luria–Delbrück distribution using discrete convolution powers
Authors: Ma W. T. , Sandri G. vH. , Sarkar S.
Published: April 2006

The Luria–Delbrück distribution arises in birth-and-mutation processes in population genetics that have been systematically studied for the last fifty years. The central result reported in this paper is a new recursion relation for computing this distribution which supersedes all past results in simplicity and computational efficiency:p0=e–m;wheremis the expected number of mutations. A new relation for the asymptotic behavior ofpn(≈c/n2) is also derived. This corresponds to the probability of finding a very large number of mutants. A formula for thez-transform of the distribution is also reported.
Applications of the key renewal theorem: crudely regenerative processes
Authors: Serfozo Richard F.
Published: April 2006

Limit Statements obtainable by the key renewal theorem are of the form EXt = v(t) + o(1), as t →∞. We show how to delineate the limit function v for processes X associated with crudely regenerative phenomena. Included are refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EXt.
Stationary distributions for piecewise-deterministic Markov processes
Authors: Costa O. L. V.
Published: April 2006

In this paper we show that the problem of existence and uniqueness of stationary distributions for piecewise-deterministic Markov processes (PDPs) is equivalent to the same problem for the associated Markov chain, so long as some mild conditions on the parameters of the PDP are satisfied. Our main result is the construction of an invertible mapping from the set of stationary distributions for the PDP to the set of stationary distributions for the Markov chain. Some sufficient conditions for existence are presented and an application to capacity expansion is given.
A renewal look at switching rules in the MIL-STD-105D sampling system
Authors: Yang Grace L.
Published: April 2006

A sampling system, MIL-STD-105D, used in quality control consists of three sampling plans with different acceptance probabilities used in turn for lot inspection. The decision to switch plan is based on the history of the lot acceptance records and a set of stopping rules. We derive the performance measure, average outgoing quality (AOQ), of this sampling system from a renewal process in which AOQ is expressed in terms of the moments of the stopping times. The renewal approach is simpler than that of the Markov chain generally used in computing AOQ for an infinite sequence of lots; it also provides a formula for AOQ for a finite sequence of lots.
A representation of a discrete distribution by its binomial moments
Authors: Brandt Andreas , Brandt Manfred , Sulanke Hannelore
Published: April 2006

Letpk, k≧ 0, be a probability distribution having finite binomial momentsBr, r≧ 0, and the probability generating functionU(z) with a radius of convergenceα(≧ 1). In this note explicit and recursive formulae are derived allowing computation of thepkin terms of theBrifα> 1. Ch. Jordan's formula, which holds ifα> 2, turns out to be a special case.
L'intégrale du mouvement brownien
Authors: Lachal Aimé
Published: April 2006

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process .

Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb).
A Swiss Army formula of Palm calculus
Authors: Brémaud P.
Published: April 2006

We obtain a single formula which, when its components are adequately chosen, transforms itself into the main formulas of the Palm theory of point processes: Little's L = λW formula [10], Brumelle's H = λG formula [5], Neveu's exchange formula [14], Palm inversion formula and Miyazawa's rate conservation law [12]. It also contains various extensions of the above formulas and some new ones.
Joint exceedances of high levels under a local dependence condition
Authors: Ferreira Helena
Published: April 2006

Under appropriate long-range dependence conditions, it is well known that the joint distribution of the number of exceedances of several high levels is asymptotically compound Poisson. Here we investigate the structure of a cluster of exceedances for stationary sequences satisfying a suitable local dependence condition, under which it is only necessary to get certain limiting probabilities, easy to compute, in order to obtain limiting results for the highest order statistics, exceedance counts and upcrossing counts.
On a new formula for the transient state probabilities forM/M/1 queues and computational implications
Authors: Conolly B. W. , Langaris Christos
Published: April 2006

Past work relating to the computation of time-dependent state probabilities inM/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at timetlater.
An integer-valuedpth-order autoregressive structure (INAR(p)) process
Authors: Alzaid A. A. , Al-osh M.
Published: April 2006

An extension of the INAR(1) process which is useful for modelling discrete-time dependent counting processes is considered. The model investigated here has a form similar to that of the Gaussian AR(p) process, and is called the integer-valuedpth-order autoregressive structure (INAR(p)) process. Despite the similarity in form, the two processes differ in many aspects such as the behaviour of the correlation, Markovian property and regression. Among other aspects of the INAR(p) process investigated here are the limiting as well as the joint distributions of the process. Also, some detailed discussion is given for the case in which the marginal distribution of the process is Poisson.
A new continuous-time search model
Authors: Stadje Wolfgang
Published: April 2006

We study a search model in which offers of random size are received randomly over time. The arrival times form a point process of a certain type, and the offer size distribution may depend on the corresponding arrival time. The search costs have a time-dependent cost rate. The objective is to stop the search process such that the expected discounted net reward (associated with the maximal offer received so far) is maximized. A stopping time σ is suggested, and conditions are specified under which σ turns out to be optimal.
Sequential limits in Markov set-chains
Authors: Hartfiel D. J.
Published: April 2006

Let T be a non-empty subset of η X n stochastic matrices. Define

The sequence T1, T2, · ·· is called a Markov set-chain. An important problem in this area is to determine when such a set-chain converges. This paper gives a notion of a sequential limiting set and shows how it can be used to obtain a result on set-chain convergence.
Convex hulls of selected subsets of a Poisson process
Authors: Blackwell Paul
Published: April 2006

This paper considers sets of points from a Poisson process in the plane, chosen to be close together, and their properties. In particular, the perimeter of the convex hull of such a point set is investigated. A number of different models for the selection of such points are considered, including a simple nearest-neighbour model. Extensions to marked processes and applications to modelling animal territories are discussed.
On closed ring queueing networks
Authors: Bambos Nicholas
Published: April 2006

In this paper we first study ring structured closed queueing networks with distinguishable jobs. Under assumptions of periodicity and ergodicity of the service times, essentially the most general, it is shown that the limits defining the average flows of the jobs exist almost surely, and methods for their estimation by simulation are given. However, it turns out that the values of the flows depend on the initial positions of the jobs, due to the emergence of distinct persistent blocking modes. The effect of these modes on the behavior of general networks with queueing loops is examined.

For independent and identically distributed service times, conditions are specified for the network to asymptotically approach a steady state at large times.

Finally, we study the special case of ring networks with indistinguishable items and stationary and ergodic service times. It is shown that as the number of jobs in the network increases towards infinity, the average circulation time converges to the maximum of the expectations of the service times.
Finite capacity vacation models with non-renewal input
Authors: Blondia C.
Published: April 2006

This paper studies a single server queue with finite waiting room where the server takes vacations according to two different strategies: (i) an exhaustive service discipline, where the server takes a vacation whenever the system becomes empty and these vacations are repeated as long as there are no customers in the system upon return from a vacation, i.e. a repeated vacation strategy; (ii) a limited service discipline, where the server begins a vacation either if K customers have been served in the same busy period or if the system is empty and then a repeated vacation strategy is followed. The input process is a general Markovian arrival process introduced by Lucantoni, Meier-Hellstern and Neuts, which as special cases includes the Markov modulated Poisson process and the phase-type renewal process. The service times and vacation times each are generally distributed random variables. For both models, we obtain the queue length distribution at departures, at an arbitrary time instant and at arrival time. We also derive the loss probability of an arriving customer. We obtain formulae for the LST of the virtual waiting time distribution and for the LST of the waiting time distribution at arrival epochs.
Lumpability and marginalisability for continuous-time Markov chains
Authors: Ball Frank , Yeo Geoffrey F.
Published: April 2006

We consider lumpability for continuous-time Markov chains and provide a simple probabilistic proof of necessary and sufficient conditions for strong lumpability, valid in circumstances not covered by known theory. We also consider the following marginalisability problem. Let {X{t)} = {(X1(t), X2(t), · ··, Xm(t))} be a continuous-time Markov chain. Under what conditions are the marginal processes {X1(t)}, {X2(t)}, · ··, {Xm(t)} also continuous-time Markov chains? We show that this is related to lumpability and, if no two of the marginal processes can jump simultaneously, then they are continuous-time Markov chains if and only if they are mutually independent. Applications to ion channel modelling and birth–death processes are discussed briefly.
A general recursive discrete-time filter
Authors: Elliott Robert J.
Published: April 2006

A reference probability is explicitly constructed under which the signal and observation processes are independent. A simple, explicit recursive form is then obtained for the conditional density of the signal given the observations. Both non-linear and linear filters are considered, as well as two different information patterns.
Minimizing the expected rank with full information
Authors: Bruss F. Thomas , Ferguson Thomas S.
Published: April 2006

The full-information secretary problem in which the objective is to minimize the expected rank is seen to have a value smaller than 7/3 for all n (the number of options). This can be achieved by a simple memoryless threshold rule. The asymptotically optimal value for the class of such rules is about 2.3266. For a large finite number of options, the optimal stopping rule depends on the whole sequence of observations and seems to be intractable. This raises the question whether the influence of the history of all observations may asymptotically fade. We have not solved this problem, but we show that the values for finite n are non-decreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908.
A logarithmic reduction algorithm for quasi-birth-death processes
Authors: Latouche Guy , Ramaswami V.
Published: April 2006

Quasi-birth-death processes are commonly used Markov chain models in queueing theory, computer performance, teletraffic modeling and other areas. We provide a new, simple algorithm for the matrix-geometric rate matrix. We demonstrate that it has quadratic convergence. We show theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.
Yule process approximation for the skeleton of a branching process
Authors: O'Connell Neil
Published: April 2006

The skeleton of a (super-) critical Galton-Watson process with offspring mean 1 +r, r≧ 0, and finite offspring variance, is considered. Whenr= 0 it is trivial. Ifr> 0 is small and the time unit is taken asα /rgenerations (α> 0) then the skeleton can be approximated by a Yule (linear pure birth) process of rateα.This approximation can be used to study the evolution of genetic types over a long period of time in an exponentially growing population.
Zero-sum two-person semi-Markov games
Authors: Lal Arbind K. , Sinha Sagnik
Published: April 2006

Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.
Discrete minification processes and reversibility
Authors: Littlejohn R. P.
Published: April 2006

Discrete minification processes are introduced and it is proved that the discrete first-order autoregression of McKenzie (1986) and the discrete minification process are mutually time-reversible if and only if they have common marginal geometric distribution, corresponding to a result for continuous processes given by Chernick et al. (1988). It is also proved that a discrete minification process is time-reversible if and only if it has marginal Bernoulli distribution.
Asymptotic reliability of consecutive k-out-of-n systems
Authors: Canfield E. R. , Mccormick W. P.
Published: April 2006

By studying the minimum of moving maxima, that is the maxima taken over a sliding window of length k in an i.i.d. sequence, we obtain new results on the reliability of consecutive k-out-of-n systems. In particular, we give the reliability asymptotically with both k and n varying. The underlying method of our approach is to analyze the singularities of a generating function.
The allocation of offensive and defensive resources in a territorial game
Authors: Cowan Richard
Published: April 2006

How should a team allocate its players to achieve the optimal balance between offence and defence? By posing a simple territorial game between two teams, insight into this question is gained. The saddlepoint solution to the game is derived.
An operation which inverts Bernoulli multiplication and associated stationary reversible Markov processes
Authors: Littlejohn R. P.
Published: April 2006

A simple operation is described which inverts Bernoulli multiplication. It is used to define two classes of stationary reversible Markov processes with general marginal distribution. These are compared to the DAR(1) process of Jacobs and Lewis (1978). LJAR(1) is used to model ovulation rate time series.
A spatial Markov property for nearest-neighbour Markov point processes
Authors: Kendall W. S.
Published: April 2006

Nearest-neighbour Markov point processes were introduced by Baddeley and Møller (1989) as generalizations of the Markov point processes of Ripley and Kelly. This note formulates and discusses a spatial Markov property for these point processes.
Stochastic processes directed by randomized time
Authors: Lee Mei-Ling Ting , Whitmore G. Alex
Published: April 2006

The paper investigates stochastic processes directed by a randomized time process. A new family of directing processes called Hougaard processes is introduced. Monotonicity properties preserved under subordination, and dependence among processes directed by a common randomized time are studied. Results for processes subordinated to Poisson and stable processes are presented. Potential applications to shock models and threshold models are also discussed. Only Markov processes are considered.
Some characterizations of exponential or geometric distributions in a non-stationary record value model
Authors: Witte Hans-Jürgen
Published: April 2006

Let {Fn}n ≧ 0 be a sequence of c.d.f. and let {Rn}n ≧ 1 be the sequence of record values in a non-stationary record model where after the (n − 1)th record the population is distributed according to Fn. Then the equidistribution of the nth population and the record increment Rn – Rn– 1 (i.e. Rn – Rn– 1~ Fn) characterizes Fn to have an exponentially decreasing hazard function. To be more precise Fn is the exponential distribution if the support of Rn– 1 generates a dense subgroup in and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chosen arbitrarily. Setting λ = – log(1 – F(1)), Fn(x) = 1 – F(x – [x])exp(–λ [x]) is an admissible solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that Fn is either exponential or geometric are given. Similar results for exponential or geometric tail distributions based on the independence of Rn– 1 and Rn – Rn– 1 are proved.
Analysis of an M/G/1/K/N queue
Authors: Takine T. , Takagi H. , Hasegawa T.
Published: April 2006

A steady-state analysis of an M/G/1 queue with a finite capacity (K) and a finite population (N) of customers is given. The queue size distribution in this M/G/1/K/N system can be derived from the known queue size distribution in the corresponding M/G/1//N system. The system throughput, the mean response time, and the blocking probability are then calculated. The joint distributions of the queue size and the remaining service times are used to obtain the distributions of the unfinished work in the service facility and the waiting time of an accepted customer.
The first-order autoregressive Mittag–Leffler process
Authors: Jayakumar K. , Pillai R. N.
Published: April 2006

The first-order autoregressive semi-Mittag-Leffler (SMLAR(1)) process is introduced and its properties are studied. As an illustration, we discuss the special case of the first-order autoregressive Mittag-Leffler (MLAR(1)) process.
First-order autoregressive logistic processes
Authors: Sim C. H.
Published: April 2006

We propose an AR(1) model that can be used to generate logistic processes. The proposed model has simple probability and correlation structure that can accommodate the full range of attainable correlation. The correlation structure and the joint distribution of the proposed model are given, as well as their conditional mean and variance.
A new look at the Moran dam
Authors: Stadje W.
Published: April 2006

For the original Moran dam with independent and identically distributed inputs a representation of the stationary distribution is given which readily provides a geometric rate of convergence to this distribution. For the integer-valued case the stationary distribution can be expressed in terms of simple boundary crossing probabilities for the underlying random walk.
On three classical problems for Markov chains with continuous time parameters
Authors: Chen Mu-Fa
Published: April 2006

For a given transition rate, i.e., a Q-matrix Q = (qij) on a countable state space, the uniqueness of the Q-semigroup P(t) = (Pij(t)), the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and classical problems, treated in many textbooks. As an addition, this paper introduces some practical results motivated from the study of a type of interacting particle systems, reaction diffusion processes. The main results are theorems (1.11), (1.17) and (1.18). Their proofs are quite straightforward.
On the matrix occurring in a linear search problem
Authors: Phatarfod R. M.
Published: April 2006

In this paper we consider the Markov chain formed by the operation of the move-to-front scheme. We show that the eigenvalues of the transition probability matrix are of the form pi, pi + pj, ···, where pi is the probability of selecting the ith item and N is the number of items; further, that the multiplicity of the eigenvalues of the form Σpi where the summation is over m items is equal to the number of permutations of N – m objects, ordered in some way, such that no object is in its natural position. Finally, we show that the Markov chain is lumpable – many times over.
Limit distributions for compounded sums of extreme order statistics
Authors: Beirlant Jan , Teugels Jozef L.
Published: April 2006

LetX(1)≦X(2)≦ ·· ·≦X(N(t))be the order statistics of the firstN(t) elements from a sequence of independent identically distributed random variables, where {N(t);t≧ 0} is a renewal counting process independent of the sequence ofX's. We give a complete description of the asymptotic distribution of sums made from the topktextreme values, for any sequencektsuch thatkt→ ∞,kt/t→ 0 ast→ ∞. We discuss applications to reinsurance policies based on large claims.
Search models
Authors: Bather J. A.
Published: April 2006

Mathematical models have been proposed for oil exploration and other kinds of search. They can be used to estimate the amount of undiscovered resources or to investigate optimal stopping times for the search. Here we consider a continuous search for hidden objects using a model which represents the number and values of the objects by mixtures of Poisson processes. The flexibility of the model and its complexity depend on the number of components in the mixture. In simple cases, optimal stopping rules can be found explicitly and more general qualitative results can sometimes be obtained.
On the optimality of LEPT and cµ rules for machines in parallel
Authors: Chang Cheng-Shang , Chao Xiuli , Pinedo Michael , Weber Richard
Published: April 2006

We consider scheduling problems with m machines in parallel and n jobs. The machines are subject to breakdown and repair. Jobs have exponentially distributed processing times and possibly random release dates. For cost functions that only depend on the set of uncompleted jobs at time t we provide necessary and sufficient conditions for the LEPT rule to minimize the expected cost at all t within the class of preemptive policies. This encompasses results that are known for makespan, and provides new results for the work remaining at time t. An application is that if the cµ rule has the same priority assignment as the LEPT rule then it minimizes the expected weighted number of jobs in the system for all t. Given appropriate conditions, we also show that the cµ rule minimizes the expected value of other objective functions, such as weighted sum of job completion times, weighted number of late jobs, or weighted sum of job tardinesses, when jobs have a common random due date.
Mixture models for time series
Authors: Jalali Assad , Pemberton John
Published: April 2006

In this paper we extend the class of zero-order threshold autoregressive models to a much richer class of mixture models. The new class has the important property of duality which, as we show, corresponds to time reversal. We are then able to obtain the time reversals of the zero-order threshold models and to characterise the time-reversible members of this subclass. These turn out to be quite trivial. The time-reversible models of the more general class do not suffer in this way. The complete stationary distributional structure is given, as are various moments, in particular the autocovariance function. This is shown to be of ARMA type. Finally we give two examples, the second of which extends from the finite to the countable mixture case. The general theory for this extension will be given elsewhere.
New partial ordering of survival functions based on the notion of uncertainty
Authors: Ebrahimi Nader , Pellerey Franco
Published: April 2006

A new partial ordering among life distributions in terms of their uncertainties is introduced. Our measure of uncertainty is Shannon information applied to the residual lifetime. The relationship between this ordering and various existing orderings of life distributions are discussed. Various properties of our proposed concept are examined. Based on our proposed ordering and various existing orderings, the notion of a ‘better system' is introduced.
Characterizing the idle-period distribution of GI/G/1 queues
Authors: Li Jingwen , Ou Jihong
Published: April 2006

A variety of performance measures of a GI/G/1 queue are explicitly related to the idle-period distribution of the queue, suggesting that the system analysis can be accomplished by the analysis of the idle period. However, the ‘stand-alone' relationship for the idle-period distribution of the GI/G/1 queue (i.e. the counterpart of Lindley's equation) has not been found in the literature. In this paper we develop a non-linear integral equation for the idle period distribution of the GI/G/1 queue. We also show that this non-linear system defines a unique solution. This development makes possible the analysis of the GI/G/1 queue in a different perspective.
On constant tail behaviour for the limiting random variable in a supercritical branching process
Authors: Hambly B. M.
Published: April 2006

We examine a family of supercritical branching processes and compute the density of the limiting random variable, W, for their normalized population size. In this example the left tail of W decays exponentially and there is no oscillation in this tail as typically observed. The branching process is embedded in the n-adic rational random walk approximation to Brownian motion on [0, 1]. This connection allows the explicit computation of the density of W.
On large deviations of empirical measures in the τ-topology
Authors: De Acosta A.
Published: April 2006

We prove a generalization of Sanov's theorem in which the state space S is arbitrary and the set of probability measures on S is endowed with the τ -topology.
Sample quantiles of additive renewal reward processes
Authors: Dassios Angelos
Published: April 2006

The distribution of the sample quantiles of random processes is important for the pricing of some of the so-called financial ‘look-back' options. In this paper a representation of the distribution of the α-quantile of an additive renewal reward process is obtained as the sum of the supremum and the infimum of two rescaled independent copies of the process. This representation has already been proved for processes with stationary and independent increments. As an example, the distribution of the α-quantile of a randomly observed Brownian motion is obtained.
On the generating functions of a random walk on the non-negative integers
Authors: Dette Holger
Published: April 2006

In the random walk whose state space is a subset of the non-negative integers explicit representations for the generating functions of the n-step transition and the first return probabilities are obtained. These representations involve the Stieltjes transform of the spectral measure of the process and the corresponding orthogonal polynomials. Several examples are given in order to illustrate the application of the results.
Forward and reversed time prediction of autoregressive sequences
Authors: Cambanis Stamatis , Fakhre-Zakeri Issa
Published: April 2006

Prediction for autoregressive sequences with finite second moment and of general order is considered. It is shown that the best predictor with time reversed is linear if and only if the innovations are Gaussian. The connection to time reversibility is also discussed.
Properties of moments for s-order equilibrium distributions
Authors: Nanda Asok K. , Jain Kanchan , Singh Harshinder
Published: April 2006

In this article we define the s-order equilibrium distribution of a life distribution and establish some interesting properties regarding moments for s-order equilibrium distributions. Results are expected to be useful in reliability and renewal processes.
A Pollaczek–Khintchine formula forM/G/1 queues with disasters
Authors: Jain Gautam , Sigman Karl
Published: April 2006

A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula forM/G/1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFOM/G/1 queues.
Stochastic comparison of point random fields
Authors: Georgii Hans-Otto , Küneth Torsten
Published: April 2006

We give an alternative proof of a point-process version of the FKG–Holley–Preston inequality which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.
Preservation of multivariate stochastic orders under multivariate Poisson shock models
Authors: Wong Tityik
Published: April 2006

Consider two systems, labeled system 1 and system 2, each withmcomponents. Suppose componentiin systemk, k= 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γi(t),t >0},i= 1, ···,m.Assume that Γ1, · ··, Γmare independent ofMk= (Mk,1, · ··,Mk,m), the number of shocks each component in systemkcan sustain without failure. LetZk,ibe the lifetime of componentiin systemk.We find conditions on processes Γ1, · ··,Tmsuch that some stochastic orders betweenM1andM2are transformed into some stochastic orders betweenZ1and Z2. Most results are obtained under the assumption that Γ1, · ··, Γmare independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.
Reliability analysis for systems in a random environment
Authors: Petakos Kyriakos , Tsapelas Theothosios
Published: April 2006

We consider some important systems in reliability theory situated in a random environment, where shocks occur and cause component failure in a specific way. We study some appropriate coefficients, which play an important role in the reduction of our systems to a linear combination of parallel subsystems.
A secretary problem with two decision makers
Authors: Chen Robert W. , Rosenberg Burton , Shepp Larry A.
Published: April 2006

n applicants of similar qualification are on an interview list and their salary demands are from a known and continuous distribution. Two managers, I and II, will interview them one at a time. Right after each interview, manager I always has the first opportunity to decide to hire the applicant or not unless he has hired one already. If manager I decides not to hire the current applicant, then manager II can decide to hire the applicant or not unless he has hired one already. If both managers fail to hire the current applicant, they interview the next applicant, but both lose the chance of hiring the current applicant. If one of the managers does hire the current one, then they proceed with interviews until the other manager also hires an applicant. The interview process continues until both managers hire an applicant each. However, at the end of the process, each manager must have hired an applicant. In this paper, we first derive the optimal strategies for them so that the probability that the one he hired demands less salary than the one hired by the other does is maximized. Then we derive an algorithm for computing manager II's winning probability when both managers play optimally. Finally, we show that manager II's winning probability is strictly increasing in n, is always less than c, and converges to c as n →∞, where c = 0.3275624139 · ·· is a solution of the equation ln(2) + x ln(x) = x.
Dynamic search for a moving target
Authors: Assaf David , Sharlin-Bilitzky Ariela
Published: April 2006

An object is hidden in one of two boxes and occasionally moves between the boxes in accordance with some specified continuous-time Markov process. The objective is to find the object with a minimal expected cost. In this paper it is assumed that search efforts are unlimited. In addition to the search costs, the ‘real time' until the object is found is also taken into account in the cost structure. Our main results are that the optimal policy may consist of five regions and that the controls applied should be of the extreme 0 or ∞ type. The resulting expected cost compares favorably with that of the expected cost with bounded controls studied previously in the search literature.
MR/GI/1 queues by positively correlated arrival stream
Authors: Szekli R. , Disney R. L. , Hur S.
Published: April 2006

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times are shown.
On the existence of finite-dimensional filters for Markov-modulated traffic
Authors: Olivier C. , Walrand J.
Published: April 2006

A Markov-modulated Poisson process (MMPP) is a Poisson process whose rate is a finite Markov chain. The Poisson process is a simple MMPP. An MMPP/M/1 queue is a queue with MMPP arrivals, an infinite capacity, and a single exponential server. We prove that the output of an MMPP/M/1 queue is not an MMPP process unless the input is Poisson. We derive this result by analyzing the structure of the non-linear filter of the state given the departure process of the queue. The practical relevance of the result is that it rules out the existence of simple finite descriptions of queueing networks with MMPP inputs.
Non-uniqueness in probabilistic numerical identification of bacteria
Authors: Gyllenberg Mats , Koski Timo , Reilink Edwin , Verlaan Martin
Published: April 2006

In this note we point out an inherent difficulty in numerical identification of bacteria. The problem is that of uniqueness of the taxonomic structure or, in mathematical terms, the lack of statistical identifiability of finite mixtures of multivariate Bernoulli probability distributions shown here.
An application of random packing to the multiple fracture of composite materials
Authors: Kimber Alan C.
Published: April 2006

A one-dimensional packing approach is used to obtain limiting results for inter-crack distances after multiple fracture of a long brittle-matrix composite with continuous aligned fibres. The results may also be appropriate for applications of the Rényi car-parking model in which there is a reduced probability of cars parking bumper to bumper.
On maintained systems operating in a random environment
Authors: Wortman Martin A. , Klutke Georgia-Ann
Published: April 2006

This paper examines the availability of a maintained system where the rate of deterioration is governed by an exogenous random environment. We provide a qualitative result that exposes the relationship between remaining lifetime, environment, and repairs. This result leads to simple bounds that can be used to choose inspection rates that guarantee a specified level of availability. The principal result requires no specific distributional assumptions, is intuitively appealing and can be directly applied by practitioners. Our development employs techniques from stochastic calculus.
Strong approximations for some open population epidemic models
Authors: O'Neill Philip
Published: April 2006

This paper considers a class of epidemic models in which susceptibles may enter or leave the population according to a general continuous time density dependent Markov chain. A sequence of such epidemics indexed byN,the initial number of susceptibles, is constructed on the same probability space as a time-inhomogeneous birth-and-death process. A coupling argument is then used to demonstrate the strong convergence of the sequence of infectives to the birth-and-death process. This result is used to provide a threshold analysis of the epidemic model in question.
Cost comparison of a spectrum of self-organizing rules
Authors: Chong K. S. , Lam K.
Published: April 2006

Consider the following self-organizing rule called POS(i): after a book in the jth position of a shelf is borrowed, it is moved up one position if ji, and is moved to the ith position if j > i. This is a family of move-forward rules, with POS(l) being the move-to-front rule and POS(n − 1) being the transposition rule where n is the number of books to be organized. We derive explicitly the stationary distribution under the POS(i) rule and show that its search cost compares favorably with that of move-to-front rule under any book access probabilities p1, p1, ···, pn.
A power-law model and other models for long-range dependence
Authors: Martin R. J. , Walker A. M.
Published: April 2006

It is becoming increasingly recognized that some long series of data can be adequately and parsimoniously modelled by stationary processes with long-range dependence. Some new discrete-time models for long-range dependence or slow decay, defined by their correlation structures, are discussed. The exact power-law correlation structure is examined in detail.
Regularity and reversibility results for birth-death-migration processes
Authors: Renshaw Eric , Dai Yonglong
Published: April 2006

A spatial process is considered in which two general birth-death processes are linked by migration of individuals. We examine conditions for weak symmetry and regularity, and develop necessary and sufficient conditions for recurrence. The results are easily extended to thek-process case.
A note on stochastic ordering of order statistics
Authors: Ma Chunsheng
Published: April 2006

A necessary and sufficient condition is obtained for a Poisson binomial random variable to be stochastically larger (or smaller) than a binomial random variable. It is then used to deal with the stochastic comparisons of order statistics from heterogeneous populations with those from a homogeneous population. The result has obvious applications in the stochastic comparisons of lifetimes of k-out-of-n systems having independent components.