bluuebunny/crossref_metadata_2025 huggingface dataset:
Hitting times of decreasing sets for regenerative processes
Authors: Athreya K. B.
Published: April 2006

Let {X(t): t ≧ 0} be a regenerative process with excursion process {Y(u): 0 ≦ u < T, T} and state space . Let An be a sequence of sets in such that pn=P(Y(u)∊An for some 0≦ u < T) → 0. Let Vn be the hitting time of An for the process X. This paper gives a variety of conditions on the excursion process Y to obtain limit theorems for Vn. Apart from obtaining an exponential limit in the positive recurrent case, i.e. ET < x, some non-exponential limits are obtained in the null recurrent case. The results are illustrated via the age process.
A generalization of the classical secretary problem: dependent arrival sequences
Authors: Reeves J. H. , Flack V. F.
Published: April 2006

This paper presents new methods for selecting the best population value under the classical secretary problem sequential selection rules. Unlike the classical case, the possible population arrival sequences considered here are not necessarily equally likely. When arrival orders in the sequence are rank dependent, the performance of the optimal solution to the standard problem (totally random arrival order assumed) is very poor, compared with procedures designed to guard against non-uniform arrival orders.
Stochastic scrabble: large deviations for sequences with scores
Authors: Arratia Richard , Morris Pricilla , Waterman Michael S.
Published: April 2006

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q=(q(i))i∊S letters from a finite alphabet S and v=(v(i))i∊S be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i1 · ·· im is defined to be V(w)=v(i1) + · ·· + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2 · ·· Xn and Y1Y2· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q,v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.

Implicit in these results is a large-deviation result for the additive functional, H ≡ Σn < τv(Xn), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.
Multivariate conditional hazard rates and the MIFRA and MIFR properties
Authors: Shaked Moshe , Shanthikumar J. George
Published: April 2006

IfT= (T1, · ··,Tn) is a vector of random lifetimes then its distribution can be determined by a set λof multivariate conditional hazard rates. In this paper, sufficient conditions on λare found which imply thatTis Block–Savits MIFRA (multivariate increasing failure rate average) or Savits MIFR (multivariate increasing failure rate). Applications for a multivariate reliability model of Ross and for load-sharing models are given. The relationship between Shaked and Shanthikumar model of multivariate imperfect repair and the MIFRA property is also discussed.
On the optimality of static priority policies in stochastic scheduling on parallel machines
Authors: Kämpke Thomas
Published: April 2006

n jobs are to be preemptively scheduled for processing on n machines. The machines may have differing speeds and the jobs have processing requirements which are distributed as independent exponential random variables with different means. Holding cost g(U) is incurred per unit time that the set of uncompleted jobs is U and it is desired to minimize the total expected holding cost which is incurred until all jobs are complete. We show that if g satisfies certain simple conditions then the optimal policy is one which takes the jobs in the order 1, 2, ···, n and assigns each uncompleted job in turn to the fastest available machine. In the special case in which the objective is to minimize the expected weighted flowtime, where there is a holding cost of wi while job i is incomplete, the sufficient condition is simply w1 ≧ … ≧ wn and λ1 w1 ≧ … ≧ λn wn.
The limit distribution of the largest nearest-neighbour link in the unit d-cube
Authors: Dette Holger , Henze Norbert
Published: April 2006

Let·p denote the p-norm on . For n points drawn independently and uniformly from the unit d-cube, we obtain the limit distribution of the largest nearest-neighbour link (with respect to | · |p) in the cases ; (2) d = 3, p = 1, 2, ∞and (3) 4, p =∞.
On a non-Markovian time-dependent queueing system
Authors: Sharma S. D.
Published: April 2006

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.
Asymptotic expansions for time series statistics
Authors: Tanaka Katsuto
Published: April 2006

Asymptotic expansions for the distributions of estimators and test statistics are derived in connection with time series models. The expansions relate to marginal and joint distributions together with the percentiles of marginal distributions. We also consider transforming a statistic so that the transformed statistic has a distribution that coincides with its asymptotic distribution up to a higher order.
A comparison of the ordinary and a varying parameter exponential smoothing
Authors: Bonsdorff Heikki
Published: April 2006

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the processZn=ß(Zn –1)Xn+(1 –ß(Zn–1))Zn–1, whereXnare i.i.d. taking values in the interval [0,M],M≦ ∞ andßis a monotonically increasing function [0,M] → [c,d], 0 <c<d< 1.

Together with (Zn), we consider the ordinary exponential smoothingYn=αXn+ (1 –α)Yn –1whereαis a constant, 0 <α< 1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and thatEYn, EZnconverge to limitsEY, EZ,respectively, with a geometric convergence rate. Moreover, we show thatEzis strictly less thanEY=EXn.
Interchange arguments in stochastic scheduling
Authors: Nain Philippe , Tsoucas Pantelis , Walrand Jean
Published: April 2006

Interchange arguments are applied to establish the optimality of priority list policies in three problems. First, we prove that in a multiclass tandem of two ·/M/1 queues it is always optimal in the second node to serve according to the cµ rule. The result holds more generally if the first node is replaced by a multiclass network consisting of ·/M/1 queues with Bernoulli routing. Next, for scheduling a single server in a multiclass node with feedback, a simplified proof of Klimov's result is given. From it follows the optimality of the index rule among idling policies for general service time distributions, and among pre-emptive policies when the service time distributions are exponential. Lastly, we consider the problem of minimizing the blocking in a communication link with lossy channels and exponential holding times.
Invariance principles in queueing theory
Authors: Alex Michael , Steinebach Josef
Published: April 2006

Several stochastic processes in queueing theory are based upon compound renewal processes. For queues in light traffic, however, the summands {Xk}and the renewal counting process {N(t)} are typically dependent on each other. Making use of recent invariance principles for such situations, we present some weak and strong approximations for theGI/G/1 queues in light and heavy traffic. Some applications are discussed including convergence rate statements or Darling–Erdös-type extreme value theorems for the processes under consideration.
Autoregressive logistic processes
Authors: Arnold Barry C. , Robertson C. A.
Published: April 2006

A stochastic model is presented which yields a stationary Markov process whose invariant distribution is logistic. The model is autoregressive in character and is closely related to the autoregressive Pareto processes introduced earlier by Yeh et al. (1988). The model may be constructed to have absolutely continuous joint distributions. Analogous higher-order autoregressive and moving average processes may be constructed.
Random sequential packing simulations in three dimensions for aligned cubes
Authors: Cooper Douglas W.
Published: April 2006

This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L3/NS3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L3/NS)⅓. The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.
A model for responding to promotion blockages
Authors: Kalamatianou A. G.
Published: April 2006

A model is proposed for a manpower system in which promotion probabilities are functions of the seniority structure within the grades. The aim is to investigate the adequacy of a strategy which appeared to be being used by the management of the nurses in an Athens hospital. It is shown that the strategy is capable of restoring the system to normal state but that the problem will usually recur.
Divided differences and the joint distribution of linear combinations of spacings
Authors: Huffer Fred
Published: April 2006

We present an identity which can sometimes express the joint distribution of several linear combinations of uniform spacings as a sum of simpler distributions. Examples are given to show that this identity is useful in the exact computation of probabilities and expectations which arise in testing for uniformity. The identity is also used to provide a new derivation of the distribution of a linear combination of spacings.
On the dependence structure of hitting times of univariate processes
Authors: Ebrahimi Nader , Ramalingam T.
Published: April 2006

Some concepts of dependence have recently been introduced by Ebrahimi (1987) to explore the structural properties of the hitting times of bivariate processes. In this framework, the special case of univariate processes has curious features. New properties are derived for this case. Some applications to sequential inference and inequalities for Brownian motion and new better than used (NBU) processes are also provided.
Sample-path derivations of the excess, age, and spread distributions
Authors: Wolff Ronald W.
Published: April 2006

The familiar limiting distributions of excess, age, and spread are derived on sample paths, where these distributions are fractions of time. Similar results are obtained for the distribution of remaining service at a queue. For stochastic processes, where specified limits exist as finite constants with probability 1, the derived sample-path results are shown to imply that the same limiting distributions hold, where in a stochastic setting, these distributions may be defined in a more conventional way.
Discrimination between monotonic trends and long-range dependence
Authors: Künsch H.
Published: April 2006

We show that the periodogram behaves differently for a weakly dependent process with a small monotonic trend and a stationary strongly dependent process. In the former case it has a non-central-distribution with noncentrality parameter tending to 0 uniformly outside intervals. In the latter case it has λα−1times a central-distribution, 0 <α< 1.
Some results for quasi-stationary distributions of birth-death processes
Authors: Kijima Masaaki , Seneta E.
Published: April 2006

Quasi-stationary distributions are considered in their own right, and from the standpoint of finite approximations, for absorbing birth-death processes. Results on convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stationary distribution are obtained. These results are akin to those of Keilson and Ramaswamy (1984). The methodology is a synthesis of Good (1968) and Cavender (1978).
Asymptotic behavior of near-critical multitype branching processes
Authors: Klebaner Fima C.
Published: April 2006

Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.
Nested Monte Carlo study of random packing on the sphere
Authors: Rodgers R. P. C. , Baddeley A. J.
Published: April 2006

We consider two random sequential packing processes in which spheres of unit radius are randomly attached to the surface of a fixed unit sphere. Independent random spheres are generated and added successively, provided there is no overlap with previous spheres. In model 1, the process stops when a trial sphere intersects one of the previously-accepted spheres. In model 2, random sequential packing, any such overlapping trial sphere is discarded and the next random sphere is tried, until it is impossible to add any further spheres.

Previous workers have conjectured convincingly that no exact analytical solution is possible for this type of problem. We use Monte Carlo simulation methods to estimate transition probabilities for the two models. Because some probabilities are extremely small, a simulation using independent repetitions of the model would be inefficient. We designed a branching process of conditionally binomial trials, and performed over 108 trials on a supercomputer.
On the cut-off phenomenon in some queueing systems
Authors: Konstantopoulos P. , Baccelli F.
Published: April 2006

A number of stochastic queueing systems exhibit an interesting phenomenon known as the cut-off phenomenon. A properly scaled version of the distance between the transient process and the stationary one converges to a step function as the initial load converges to infinity. The purpose of this paper is to promote the idea that this phenomenon is a direct consequence of the coupling between the two processes, being thus generalizable to systems lacking any kind of Markovian structure.
Random walks on discrete and continuous circles
Authors: Rosenthal Jeffrey S.
Published: April 2006

We consider a large class of random walks on the discrete circleZ/(n), defined in terms of a piecewise Lipschitz function, and motivated by the ‘generation gap' process of Diaconis. For such walks, we show that the time until convergence to stationarity is bounded independently ofn.Our techniques involve Fourier analysis and a comparison of the random walks onZ/(n) with a random walk on the continuous circleS1.
Mean passage times for tridiagonal transition matrices and a two-parameter ehrenfest urn model
Authors: Krafft Olaf , Schaefer Martin
Published: April 2006

A two-parameter Ehrenfest urn model is derived according to the approach taken by Karlin and McGregor [7] where Krawtchouk polynomials are used. Furthermore, formulas for the mean passage times of finite homogeneous Markov chains with general tridiagonal transition matrices are given. In the special case of the Ehrenfest model they have quite a different structure as compared with those of Blom [2] or Kemperman [9].
Equilibrium distribution of block-structured Markov chains with repeating rows
Authors: Grassmann Winfried K. , Heyman Daniel P.
Published: April 2006

In this paper we consider two-dimensional Markov chains with the property that, except for some boundary conditions, when the transition matrix is written in block form, the rows are identical except for a shift to the right. We provide a general theory for finding the equilibrium distribution for this class of chains. We illustrate theory by showing how our results unify the analysis of the M/G/1 and GI/M/1 paradigms introduced by M. F. Neuts.
A convergence rate in extreme-value theory
Authors: Balkema A. A. , De Haan L.
Published: April 2006

A uniform convergence rate is determined for maxima of i.i.d. random variables from a distribution in the domain of attraction of the double-exponential distribution. The result is proved under a second-order condition on the underlying distribution parallelling the one given in Smith (1982) for the domain of attraction of the bounded-below and bounded-above families of limit distributions.
Complementary generating functions for theMX/GI/1/kandGI/My/1/Kqueues and their application to the comparison of loss probabilities
Authors: Miyazawa Masakiyo
Published: April 2006

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in theMx/GI/1/kqueue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensityρ< 1 and for the virtual waiting time process inM/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality ofGI/MY/1/ktoMx/GI/1/k +1, we get a similar result for the system queue length distribution just before the arrival of a customer inGI/MY/1/k.We apply those results to prove that the loss probabilities ofMx/GI/1/kandGI/MY/1/kare increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.
Throughput maximization in a loss queueing system with heterogeneous servers
Authors: Sobel Matthew J.
Published: April 2006

Assigning each arriving customer to the fastest idle server is shown to maximize throughput (equivalently, minimize blocking probability) in a queueing model with Poisson arrivals, heterogeneous exponential servers, and no waiting room. If a cost structure is imposed on this model, under specified conditions the same policy minimizes the expected discounted cost and the long-run average cost per unit time.
The abscissa of convergence of the Laplace transform
Authors: Hall Peter , Teugels Jozef L. , Vanmarcke Ann
Published: April 2006

Assume that we want to estimate –σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator ofσcan converge at a faster rate than (logn)–1, wherenis the sample size. An optimal convergence rate is achieved by an estimator of the formwherexn=O(logn) andis the mean of the sample values overshootingxn.Under further parametric restrictions this (logn)–1phenomenon is also illustrated by a weak convergence result.
The maximum and mean of a random length sequence
Authors: Haas Peter J.
Published: April 2006

We obtain a limit theorem for the joint distribution of the maximum value and sample mean of a random length sequence of independent and identically distributed random variables. This extends a previous bivariate convergence result for fixed length sequences and incidentally yields a new proof of Berman's classical limit theorem for the maximum value of a random number of random variables. Our approach uses a property of record time sequences and leads to probabilistically intuitive proofs. We also consider the partition of a finite interval into a random number of subintervals by the points of a non-delayed renewal process. Using the bivariate convergence result for random length sequences, we establish a limit theorem for the joint distribution of the number and maximum length of the subintervals as the interval length becomes large. This leads to limiting results for the ratio of the maximum to the mean subinterval length. Such results are of interest in connection with a simple model of parallel processing.
Tree-dependent extreme values: the exponential case
Authors: Gupta Vijay K. , Mesa Oscar J. , Waymire E.
Published: April 2006

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.
Optimal stationary strategies in leavable Markov decision processes
Authors: Fassbender Matthias
Published: April 2006

This paper establishes the existence of an optimal stationary strategy in a leavable Markov decision process with countable state space and undiscounted total reward criterion.

Besides assumptions of boundedness and continuity, an assumption is imposed on the model which demands the continuity of the mean recurrence times on a subset of the stationary strategies, the so-called ‘good strategies'. For practical applications it is important that this assumption is implied by an assumption about the cost structure and the transition probabilities. In the last part we point out that our results in general cannot be deduced from related works on bias-optimality by Dekker and Hordijk, Wijngaard or Mann.
On a new characterization of the exponential distribution related to a queueing system with an unreliable server
Authors: Dimitrov B. , Khalil Z.
Published: April 2006

In this paper we derive a new property of the exponential distribution, closely related to a single-server queueing system with unreliable server. We show that this new property is another characterization of the exponential law.
Boundary-crossing probabilities of some random fields related to likelihood ratio tests for epidemic alternatives
Authors: Yao Qiwei
Published: April 2006

We consider the likelihood ratio tests to detect an epidemic alternative in the following two cases of normal observations: (1) the alternative specifies a square wave drift in the mean value of an i.i.d. sequence; (2) the alternative permits a square wave drift in the intercept of a simple linear regression model. To develop the approximations for the significance levels leads us to consider boundary-crossing problems of some two-dimensional discrete-time Gaussian fields. By the method which was proposed originally by Woodroofe (1976) and adapted to study maxima of some random fields by Siegmund (1988), some large deviations for the conditional non-linear boundary-crossing probabilities are developed. Some results of Monte Carlo experiments confirm the accuracy of these approximations.
Random splittings of an interval
Authors: Mountford T. S. , Port S. C.
Published: April 2006

Points are independently and uniformly distributed onto the unit interval. The firstn—1 points subdivide the interval intonsubintervals. For 1we find a necessary and sufficient condition on {ln} for the events [Xnbelongs to thelnth largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of thelnth largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by timenthe events [Xrbelongs tolrth largest subinterval] occur when these events occur infinitely often.
On permutation policies for the scheduling of deteriorating stochastic jobs on a single machine
Authors: Glazebrook K. D.
Published: April 2006

A single machine is available to process a collection of stochastic jobs. Processing is preemptive and so (for example) the machine is allowed to switch away from a job before completion, should that prove advantageous. The jobs are deteriorating in the sense that their processing requirements grow (at job-specific rates) as they await processing. This phenomenon might be expected to enhance the status of non-preemptive policies. The primary objective of the paper is to find conditions which are sufficient to ensure the existence of a permutation policy to minimise the expected makespan. We also derive results for a weighted flowtime criterion. Applications of such models to the control of queues and to communication systems have been cited by other authors.
Extremal properties of the shortest/longest non-full queue policies in finite-capacity systems with state-dependent service rates
Authors: Sparaggis P. D. , Towsley D. , Cassandras C. G.
Published: April 2006

We consider the problem of routing jobs to parallel queues with identical exponential servers and unequal finite buffer capacities. Service rates are state-dependent and non-decreasing with respect to queue lengths. We establish the extremal properties of the shortest non-full queue (SNQ) and the longest non-full queue (LNQ) policies, in systems with concave/convex service rates. Our analysis is based on the weak majorization of joint queue lengths which leads to stochastic orderings of critical performance indices. Moreover, we solve the buffer allocation problem, i.e. the problem of how to distribute a number of buffers among the queues. The two optimal allocation schemes are also ‘extreme', in the sense of capacity balancing. Some extensions are also discussed.
Mixing rates for Brownian motion in a convex polyhedron
Authors: Matthews Peter
Published: April 2006

For Brownian motion on a convex polyhedral subset of a sphere or torus, the rate of convergence in distribution to uniformity is studied. The main result is a method to take a Markov coupling on the full sphere or torus and create a faster coupling on the convex polyhedral subset. Upper bounds on variation distance are computed, and applications are discussed.
Distribution of the busy period in a controllable M/M/2 queue operating under the triadic (0, K, N, M) policy
Authors: Rhee Hahn-Kyou , Sivazlian B. D.
Published: April 2006

We consider an M/M/2 queueing system with removable service stations operating under steady-state conditions. We assume that the number of operating service stations can be adjusted at customers' arrival or service completion epochs depending on the number of customers in the system. The objective of this paper is to obtain the distribution of the busy period using the theory of the gambler's ruin problem. As special cases, the distributions of the busy periods for the ordinary M/M/2 queueing system, the M/M/1 queueing system operating under the N policy and the ordinary M/M/1 queueing system are obtained.
Winding angle and maximum winding angle of the two-dimensional random walk
Authors: Bélisle Claude , Faraway Julian
Published: April 2006

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10msteps, form= 2, 3, 4, 5, 6, 7.
Transient and busy period analysis of theGIG/1 Queue as a Hilbert factorization problem
Authors: Bertsimas Dimitris J. , Keilson Julian , Nakazato Daisuke , Zhang Hongtao
Published: April 2006

In this paper we find the waiting time distribution in the transient domain and the busy period distribution of theGI G/1 queue. We formulate the problem as a two-dimensional Lindley process and then transform it to a Hilbert factorization problem. We achieve the solution of the factorization problem for theGI/R/1,R/G/1 queues, whereRis the class of distributions with rational Laplace transforms. We obtain simple closed-form expressions for the Laplace transforms of the waiting time distribution and the busy period distribution. Furthermore, we find closed-form formulae for the first two moments of the distributions involved.
Coefficients of ergodicity for stochastically monotone Markov chains
Authors: Pflug G. Ch. , Schachermayer W.
Published: April 2006

In this paper we show that to each distance d defined on the finite state space S of a strongly ergodic Markov chain there corresponds a coefficient ρd of ergodicity based on the Wasserstein metric. For a class of stochastically monotone transition matrices P, the infimum over all such coefficients is given by the spectral radius of P – R, where R = limkPk and is attained. This result has a probabilistic interpretation of a control of the speed of convergence of by the metric d and is linked to the second eigenvalue of P.
Minification processes and their transformations
Authors: Lewis Peter A. W. , McKenzie Ed
Published: April 2006

It is shown that the stationary, autoregressive, Markovian minification processes introduced by Tavares and Sim can be extended to give processes with marginal distributions other than the exponential and Weibull distributions. Necessary and sufficient conditions on the hazard rate of the marginal distributions are given for a minification process to exist. Results are given for the derivation of the autocorrelation function; these correct the expression for the Weibull given by Sim. Monotonic transformations of the minification processes are also discussed and generate a whole new class of autoregressive processes with fixed marginal distributions. Stationary processes generated by a maximum operation are also introduced and a comparison of three different Markovian processes with uniform marginal distributions is given.
Periodic strong ergodicity in non-homogeneous Markov systems
Authors: Gerontidis Ioannis I.
Published: April 2006

This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.
Stochastic growth models: bounds on critical values
Authors: Durrett R.
Published: April 2006

We give upper bounds on the critical values for oriented percolation and some interacting particle systems by computing their behavior on small finite sets.
A closure characterisation of phase-type distributions
Authors: Maier Robert S. , O'Cinneide Colm Art
Published: April 2006

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class.

We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.
Linnik distributions and processes
Authors: Anderson Dale N. , Arnold Barry C.
Published: April 2006

Using a simple characterization of the Linnik distribution, discrete-time processes having a stationary Linnik distribution are constructed. The processes are structurally related to exponential processes introduced by Arnold (1989), Lawrance and Lewis (1981) and Gaver and Lewis (1980). Multivariate versions of the processes are also described. These Linnik models appear to be viable alternatives to stable processes as models for temporal changes in stock prices.
Discretizations for the average impulse control of piecewise deterministic processes
Authors: Costa O. L. V.
Published: April 2006

This paper presents a state space and time discretization for the general average impulse control of piecewise deterministic Markov processes (PDPs). By combining several previous results we show that under some continuity, boundedness and compactness conditions on the parameters of the process, boundedness of the discretizations, and compactness of the state space, the discretized problem will converge uniformly to the original one. An application to optimal capacity expansion under uncertainty is given.
A result on the infinitely many neutral alleles diffusion model
Authors: Schmuland B.
Published: April 2006

The stationary version of the infinitely many neutral alleles diffusion model, as studied by Ethier and Kurtz, is constructed and analyzed using the theory of Dirichlet forms. We prove that the ‘boundary' of the state space is hit by the process iffθ< 1, whereθis the mutation parameter.
Stochastic ordering of order statistics
Authors: Barbour A. D. , Lindvall T. , Rogers L. C. G.
Published: April 2006

If Xi, i = 1, ···, n are independent exponential random variables with parameters λ1, · ··, λ n, and if Yi, i = 1, ···, n are independent exponential random variables with common parameter equal to (λ1 + · ·· + λ n)/n, then there is a monotone coupling of the order statistics X(1), · ··, X(n) and Y(1), · ··, Y(n); that is, it is possible to construct on a common probability space random variables X′i, Y′i, i = 1, ···, n, such that for each i, Y′(i)≦X′(i) a.s., where the law of the X′i (respectively, the Y′i) is the same as the law of the Xi (respectively, the Yi.) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ···, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.
The heaps process, libraries, and size-biased permutations
Authors: Donnelly Peter
Published: April 2006

The heaps process (also known as a Tsetlin library) provides a model for a self-regulating filing system. Items are requested from time to time according to their popularity and returned to the top of the heap after use. The size-biased permutation of a collection of popularities is a particular random permutation of those popularities, which arises naturally in a number of applications and is of independent interest. For a slightly non-standard formulation of the heaps process we prove that it converges to the size-biased permutation of its initial distribution. This leads to a number of new characterizations of the property of invariance under size-biased permutation, notably what might be described as invariance under ‘partial size-biasing' of any order. Finally we consider in detail the heaps process with Poisson–Dirichlet initial distribution, exhibiting the tractable nature of its equilibrium distribution and explicitly calculating a number of quantities of interest.
A multivariate reward process defined on a semi-Markov process and its first-passage-time distributions
Authors: Masuda Yasushi , Sumita Ushio
Published: April 2006

A multivariate reward process defined on a semi-Markov process is studied. Transform results for the distributions of the multivariate reward and related processes are derived through the method of supplementary variables and the Markov renewal equations. These transform results enable the asymptotic behavior to be analyzed. A class of first-passage time distributions of the multivariate reward processes is also investigated.
Non-homogeneous semi-Markov systems and maintainability of the state sizes
Authors: Vassiliou P.-C. G. , Papadopoulou A. A.
Published: April 2006

In this paper we introduce and define for the first time the concept of a non-homogeneous semi-Markov system (NHSMS). The problem of finding the expected population stucture is studied and a method is provided in order to find it in closed analytic form with the basic parameters of the system. Moreover, the problem of the expected duration structure in the state is studied. It is also proved that all maintainable expected duration structures by recruitment control belong to a convex set the vertices of which are specified. Finally an illustration is provided of the present results in a manpower system.
An insensitivity property of ladder height distributions
Authors: Frenz Michael , Schmidt Volker
Published: April 2006

This paper considers the undershoot of a general continuous-time risk process with dependent increments under a certain initial level. The increments are given by the locations and amounts of claims which are described by a stationary marked point process. Under a certain balance condition, it is shown that the distribution of the undershoot depends only on the mark distribution and on the intensity of the underlying point process, but not on the form of its distribution. In this way an insensitivity property is extended which has been proved in Björk and Grandell [3] for the ruin probability, i.e. for the probability that after a finite time interval the initial level will be crossed from above.
On the asymptotic size and duration of a class of epidemic models
Authors: Svensson Åke
Published: April 2006

Models for epidemic spread of infections are formulated by defining intensities for relevant counting processes. It is assumed that an infected individual passes through k stages of infectivity. The times spent in the different stages are random. Many well-known models for the spread of infections can be described in this way. The models can also be applied to describe other processes of epidemic character (such as models for rumour spreading). Asymptotic results are derived both for the size and for the duration of the epidemic.
Consistency of Hill's estimator for dependent data
Authors: Resnick Sidney , Stărică Cătălin
Published: April 2006

Consider a sequence of possibly dependent random variables having the same marginal distributionF, whose tail 1−Fis regularly varying at infinity with an unknown index − α < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for α−1and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.
On transforming an index for generalised bandit problems
Authors: Glazebrook K. D. , Greatrix S.
Published: April 2006

Nash (1980) demonstrated that index policies are optimal for a class of generalised bandit problem. A transform of the index concerned has many of the attributes of the Gittins index. The transformed index is positive-valued, with maximal values yielding optimal actions. It may be characterised as the value of a restart problem and is hence computable via dynamic programming methodologies. The transformed index can also be used in procedures for policy evaluation.
Logarithmic asymptotics for steady-state tail probabilities in a single-server queue
Authors: Glynn Peter W. , Whitt Ward
Published: April 2006

We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have asymptotics of the formx–1logP(W> x) → –θ∗asx→ ∞forθ∗>0. We require only stationarity of the basic sequence of service times minus interarrival times and a Gärtner–Ellis condition for the cumulant generating function of the associated partial sums, i.e.n–1logEexp (θSn) →ψ(θ) asn→ ∞, plus regularity conditions on the decay rate functionψ. The asymptotic decay rateθis the root of the equationψ(θ)=0. This result in turn implies a corresponding asymptotic result for the steady-state workload in a queue with general non-decreasing input. This asymptotic result covers the case of multiple independent sources, so that it provides additional theoretical support for a concept of effective bandwidths for admission control in multiclass queues based on asymptotic decay rates.
Minimax one-sided Kolmogorov-type distribution-free tests
Authors: Pyke Ronald
Published: April 2006

A family of distribution-free statistics along with related tests is defined and properties of its members are studied. These statistics, one of which is the Smirnov-Wald and Wolfowitz statistic, Dn+ yield tests of the one-sided hypotheses. The minimax and maximin tests in this family against a restricted class of alternatives of minimum power are obtained. The connection to a confidence bound of Stringer for the mean is also remarked.
The iterated Galton–Watson process
Authors: Gawel Boleslaw , Kimmel Marek
Published: April 2006

We consider the mathematical properties of a time-discrete stochastic process describing explosive proliferation of DNA repeats in human genetic diseases. The process is constructed using a cascade of Galton–Watson branching processes. The main results concern the probability of absorption and the supergeometric growth of the process in the supercritical case. Examples of simulations are provided.
Convergence properties of simulated annealing for continuous global optimization
Authors: Locatelli M.
Published: April 2006

In this paper conditions for the convergence of a class of simulated annealing algorithms for continuous global optimization are given. The previous literature about the subject gives results for the convergence of algorithms in which the next candidate point is generated according to a probability distribution whose support is the whole feasible set. A class of possible cooling schedules has been introduced in order to remove this restriction.
On the rate of convergence for extremes of mean square differentiable stationary normal processes
Authors: Kratz Marie F. , Rootzén Holger
Published: April 2006

Letξ (t);t≧ 0 be a normalized continuous mean square differentiable stationary normal process with covariance functionr(t).Further, letand set. We give bounds which are roughly of orderΤ–δfor the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level byξ (t)in the interval [0,T].The results assume thatr(t)andr′(t)decay polynomially at infinity and thatr″(t) is suitably bounded. For the number of upcrossings it is in addition assumed thatr(t) is non-negative.
Archimedean copulas, exchangeability, and max-stability
Authors: Ballerini Rocco
Published: April 2006

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.
Preservation of certain classes of life distributions under Poisson shock models
Authors: Fagiuoli Enrico , Pellerey Franco
Published: April 2006

Recently defined classes of life distributions are considered, and some relationships among them are proposed. The life distribution H of a device subject to shocks occurring randomly according to a Poisson process is also considered, and sufficient conditions for H to belong to these classes are discussed.
Asymptotic distributions for the Ehrenfest urn and related random walks
Authors: Voit Michael
Published: April 2006

The distributions of nearest neighbour random walks on hypercubesin continuous timet0 can be expressed in terms of binomial distributions; their limit behaviour fort, N →∞ is well-known. We study here these random walks in discrete time and derive explicit bounds for the deviation of their distribution from their counterparts in continuous time with respect to the total variation norm. Our results lead to a recent asymptotic result of Diaconis, Graham and Morrison for the deviation from uniformity forN →∞.Our proofs use Krawtchouk polynomials and a version of the Diaconis–Shahshahani upper bound lemma. We also apply our methods to certain birth-and-death random walks associated with Krawtchouk polynomials.
Poissonian approximation for the tagged particle in asymmetric simple exclusion
Authors: Ferrari P. A. , Fontes L. R. G.
Published: April 2006

We consider the position of a tagged particle in the one-dimensional asymmetric nearest-neighbor simple exclusion process. Each particle attempts to jump to the site to its right at rate p and to the site to its left at rate q. The jump is realized if the destination site is empty. We assume p > q. The initial distribution is the product measure with density λ, conditioned to have a particle at the origin. We call X, the position at time t of this particle. Using a result recently proved by the authors for a semi-infinite zero-range process, it is shown that for all t ≧ 0, Xt = Nt − Bt + B0, where {Nt} is a Poisson process of parameter (p – q)(1– λ) and {Bt} is a stationary process satisfying E exp (θ | B, |) < ∞ for some θ > 0. As a corollary we obtain that, properly centered and rescaled, the process {Xt} converges to Brownian motion. A previous result says that in the scale t1/2, the position Xt is given by the initial number of empty sites in the interval (0, λt) divided by λ. We use this to compute the asymptotic covariance at time t of two tagged particles initially at sites 0 and rt. The results also hold for the net flux between two queues in a system of infinitely many queues in series.