bluuebunny/crossref_metadata_2025 huggingface dataset:
Disproof of the conjectured subexponentiality of certain functions in percolation theory
Authors: Van den berg J.
Published: April 2006

Consider bond-percolation on a graph G with sites S(G). We disprove the conjecture of Hammersley (1957) that the function n → sups ϵ S(G)E [the number of sites s′ at distance n from s which can be reached from s by an open path which, except for s′, only passes through sites at distance smaller than n from s] is always subexponential.
A comment on star-ordering and tail-ordering
Authors: Sathe Y. S.
Published: April 2006

In a recent letter, Deshpande and Kochar (1983) gave the following theorem.
On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel
Authors: Fayolle Guy , Flajolet Philippe , Hofri Micha
Published: April 2006

We analyse a stack protocol of the Capetanakis–Tsybakov–Mikhailov type for resolving collisions in a random multiple-access channel. We obtain a functional equation for the generating function of the expected collision resolution interval (CRI) durations, which is non-local with a non-commutative iteration semigroup. Using Mellin transform techniques and geometric properties of the iteration semigroup we show that for arrival rates smaller than a fixed threshold, the mean CRI duration for n initial colliders is asymptotically proportional to n. Ergodicity conditions are also demonstrated.
Probabilistic analysis of a learning matrix
Authors: Faris William G. , Maier Robert S.
Published: April 2006

A learning matrix is defined by a set of input and output pattern vectors. The entries in these vectors are zeros and ones. The matrix is the maximum of the outer products of the input and output pattern vectors. The entries in the matrix are also zeros and ones. The product of this matrix with a selected input pattern vector defines an activity vector. It is shown that when the patterns are taken to be random, then there are central limit and large deviation theorems for the activity vector. They give conditions for when the activity vector may be used to reconstruct the output pattern vector corresponding to the selected input pattern vector.
Model fields in crossing theory: a weak convergence perspective
Authors: Wilson Richard J.
Published: April 2006

In this paper, the behavior of a Gaussian random field near an ‘upcrossing' of a fixed level is investigated by strengthening the results of Wilson and Adler (1982) to full weak convergence in the space of functions which have continuous derivatives up to order 2. In Section 1, weak convergence and model processes are briefly discussed. The model field of Wilson and Adler (1982) is constructed in Section 2 using full weak convergence. Some of its properties are also investigated. Section 3 contains asymptotic results for the model field, including the asymptotic distributions of the Lebesgue measure of a particular excursion set and the maximum of the model field as the level becomes arbitrarily high.
Maxima and exceedances of stationary Markov chains
Authors: Rootzén Holger
Published: April 2006

Recent work by Athreya and Ney and by Nummelin on the limit theory for Markov chains shows that the close connection with regeneration theory holds also for chains on a general state space. Here this is used to study extremal behaviour of stationary (or asymptotically stationary) Markov chains. Many of the results center on the ‘clustering’ of extremes of adjacent values of the chains. In addition one criterion for convergence of extremes of general stationary sequences is derived. The results are applied to waiting times in the GI/G/1 queue and to autoregressive processes.
A note on the optimal replacement problem
Authors: Yeh Lam
Published: April 2006

In this note, we study a new repair replacement model for a deteriorating system, in which the successive survival times of the system form a geometric process and are stochastically non-increasing, whereas the consecutive repair times after failure also constitute a geometric process but are stochastically non-decreasing. Two kinds of replacement policy are considered, one based on the working age of the system and the other one determined by the number of failures. The explicit expressions of the long-run average costs per unit time under these two kinds of policy are calculated.
Markov renewal processes, counters and repeated sequences in Markov chains
Authors: Biggins J. D. , Cannings C.
Published: April 2006

The theory of Markov renewal processes is applied to study the occurrence of specific sequences of states in a Markov chain. Çinlar&s (1969) results are used to study both the basic process, and that obtained when the overlap of sequences is not permitted, as in the theory of counters. These results are applied to the fragments formed when DNA is digested using one, or more, restriction enzymes.
Classes of probability density functions having Laplace transforms with negative zeros and poles
Authors: Sumita Ushio , Masuda Yasushi
Published: April 2006

We consider a class of functions on [0,∞), denoted by Ω, having Laplace transforms with only negative zeros and poles. Of special interest is the class Ω+of probability density functions in Ω. Simple and useful conditions are given for necessity and sufficiency off∊ Ω to be in Ω+. The class Ω+contains many classes of great importance such as mixtures ofnindependent exponential random variables(CMn),sums ofnindependent exponential random variables(PF∗n), sums of two independent random variables, one inCMrand the other inPF∗1(CMPFnwithn=r+l)and sums of independent random variables inCMn(SCM).Characterization theorems for these classes are given in terms of zeros and poles of Laplace transforms. The prevalence of these classes in applied probability models of practical importance is demonstrated. In particular, sufficient conditions are given for complete monotonicity and unimodality of modified renewal densities.
On the intensity of crossings by a shot noise process
Authors: Hsing Tailen
Published: April 2006

The crossing intensity of a level by a shot noise process with a monotone response is studied, and it is shown that the intensity can be naturally expressed in terms of a marginal probability.
Counts of long aligned word matches among random letter sequences
Authors: Karlin Samuel , Ost Friedemann
Published: April 2006

Asymptotic distributional properties of the maximal length aligned word (a contiguous set of letters) among multiple random Markov dependent sequences composed of letters from a finite alphabet are given. For sequences of length N, Cr,s(N) defined as the longest common aligned word found in r or more of s sequences has order growth log N/(–logλ) where λis the maximal eigenvalue of r-Schur product matrices from among the collections of Markov matrices that generate the sequences. The count Z∗r,s(N, k) of positions that initiate an aligned match of length exceeding k = log N/(–logλ) + x but fail to match at the immediately preceding position has a limiting Poisson distribution. Distributional properties of other long aligned word relationships and patterns are also discussed.
The moments of the random variable for the number of returns of a simple random walk
Authors: Kemp Adrienne W.
Published: April 2006

The number of returns to the origin for a simple random walk starting and ending at the origin is reconsidered; closed-form expressions in μ are given for μ3 and μ4.
Bonferroni-type inequalities
Authors: Recsei Elaine , Seneta E.
Published: April 2006

We derive the Sobel–Uppuluri and Galambos-type extensions of the Bonferroni bounds, and further extensions of the same nature, as consequences of a single non-probabilistic inequality. The methodology follows that of Galambos.
Shape averages and their Bias
Authors: Mardia K. V. , Dryden I. L.
Published: April 2006

The paper considers the bias of Bookstein's mean estimator for shape under the isotropic normal model. This work depends on certain distributional properties of shape variables. An alternative unbiased modified estimator is proposed and its performance is compared with various estimators, including Procrustes and the exact maximum likelihood estimator, in a simulation study.
Stability conditions for some distributed systems: buffered random access systems
Authors: Szpankowski Wojciech
Published: April 2006

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).
On the mean sojourn time of jobs in queues by general service disciplines
Authors: Ritter Gunter
Published: April 2006

Existence and finiteness of the sample-mean limit of sojourn times of jobs in a queueing system are investigated. The queueing system operates under rather general multiprocessor disciplines allowing job classes and priorities. The input stream of jobs consisting of job classes and interarrival and processing times is stationary and ergodic and may contain batch arrivals. Existence of the sample-mean limit is proved by means of the superadditive ergodic theorem, and its finiteness is controlled by uniform mixing of the input stream.
Sensitivity analysis for stationary and ergodic queues: additional results
Authors: Konstantopoulos P. , Zazanis M.
Published: April 2006

Perturbation analysis estimators for expectations of possibly discontinuous functions of the time-stationary workload were derived in [2]. The expressions obtained may, however, not be valid if the customer-stationary distribution of the workload has atoms (at points other than zero). This was pointed out by Brémaud and Lasgouttes in [1]. In this note we clearly state the additional condition required for the validity of the expressions in [2]. We furthermore show how our approximation scheme can also be used to obtain the correct expressions for the right and left derivatives given in [1].
Non-linear time series and Markov chains
Authors: Tjøstheim Dag
Published: April 2006

It is shown how Markov chain theory can be exploited to study non-linear time series, the emphasis being on the classification into stationary and non-stationary models. A generalized h-step version of the Tweedie (1975), (1976) criteria is formulated, and applications are given to a number of non-linear models. New results are obtained, and known results are shown to emerge as special cases in both the scalar and vector case. A connection to stability theory is briefly discussed, and it is indicated how the Markov property can be utilized for estimation purposes.
Resource pooling in queueing networks with dynamic routing
Authors: Laws C. N.
Published: April 2006

In this paper we investigate dynamic routing in queueing networks. We show that there is a heavy traffic limiting regime in which a network model based on Brownian motion can be used to approximate and solve an optimal control problem for a queueing network with multiple customer types. Under the solution of this approximating problem the network behaves as if the service-stations of the original system are combined to form a single pooled resource. This resource pooling is a result of dynamic routing, it can be achieved by a form of shortest expected delay routing, and we find that dynamic routing can offer substantial improvements in comparison with less responsive routing strategies.
Sensitivity analysis for stationary and ergodic queues
Authors: Konstantopoulos P. , Zazanis Michael A.
Published: April 2006

Starting with some mild assumptions on the parametrization of the service process, perturbation analysis (PA) estimates are obtained for stationary and ergodic single-server queues. Besides relaxing the stochastic assumptions, our approach solves some problems associated with the traditional regenerative approach taken in most of the previous work in this area. First, it avoids problems caused by perturbations interfering with the regenerative structure of the system. Second, given that the major interest is in steady-state performance measures, it examines directly the stationary version of the system, instead of considering performance measures expressed as Cesaro limits. Finally, it provides new estimators for general (possibly discontinuous) functions of the workload and other steady-state quantities.
A modification of the general stochastic epidemic motivated by AIDS modelling
Authors: Ball Frank , O'neill Philip
Published: April 2006

This paper considers a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rateβxy/(x + y), wherexandyare the numbers of susceptible and infectious individuals, respectively, andβis an infection parameter. This contrasts with the standard general epidemic in which new infections occur at rateβxy.Both the deterministic and stochastic versions of the modified epidemic are analysed. The deterministic model is completely soluble. The time-dependent solution of the stochastic model is derived and the total size distribution is considered. Threshold theorems, analogous to those of Whittle (1955) and Williams (1971) for the general stochastic epidemic, are proved for the stochastic model. Comparisons are made between the modified and general epidemics. The effect of introducing variability in susceptibility into the modified epidemic is studied.
Worth of perfect information in bernoulli bandits
Authors: Berry Donald A. , Kertz Robert P.
Published: April 2006

For k-armed Bernoulli bandits with discounting, sharp comparisons are given between average optimal rewards for a gambler and for a ‘perfectly informed' gambler, over natural collections of prior distributions. Some of these comparisons are proved under general discounting, and others under non-increasing discount sequences. Connections are made between these comparisons and the concept of ‘regret' in the minimax approach to bandit processes. Identification of extremal cases in the sharp comparisons is emphasized.
Stochastic non-linear oscillators
Authors: Markus Lawrence , Weerasinghe Ananda
Published: April 2006

Non-linear stochastic systems driven by white noise are analysed from the viewpoint of non-linear oscillation theory. Under various familiar hypotheses concerning dissipative and restorative dynamical forces, the existence and uniqueness, asymptotic growth, and oscillatory behavior of the solutions are demonstrated.
The policy which maximises long-term survival of an animal faced with the risks of starvation and predation
Authors: McNamara J. M.
Published: April 2006

This paper presents a simple model of the decision problems which face many animals in the wild. Animals have an upper limit to the amount of energy reserves which can be stored as body fat. They constantly use energy and find food as a stochastic process. Death occurs if energy reserves fall to zero or if the animal is taken by a predator. Typically animals have a range of behavioural options which differ in the distribution of food gained and in the associated predation risk. There is often a trade-off between starvation and predation in that animals have to expose themselves to higher predation risks in order to gain more food.

The above situation is modelled as a finite-state, finite-time-horizon Markov decision problem. The policy which maximises long-term survival probability is characterised. As special cases two trade-offs are analysed. It is shown that an animal should take fewer risks in terms of predation as its reserves increase, and that an animal should reduce the variability of its food supply at the expense of its mean gain as its reserves increase.
On a generalization of a model by Lindley and Singpurwalla
Authors: Bandyopadhyay Dipankar , Basu Asit P.
Published: April 2006

The flexibility of a notion considered by Lindley and Singpurwalla is pointed out. It is shown that their set-up can be generalized by looking at systems whose component life lengths are a priori dependent.
A Note on Electrical Networks and the Inverse Gaussian Distribution
Authors: Barndorff-Nielsen O. E.
Published: April 2006

When a random electrical network has the structure of a rooted tree and the edge resistances are either inverse Gaussian or reciprocal inverse Gaussian random variables then, subject to some restrictions, the overall resistance of the network is shown to follow a reciprocal inverse Gaussian distribution.
A non-standard family of polynomials and the final size distribution of Reed-Frost epidemic processes
Authors: Lefevre Claude , Picard Philippe
Published: April 2006

This paper provides a global treatment of the final size distribution of Reed–Frost epidemic processes. Exact and asymptotic results are derived for both single and multipopulation situations. The key tool is a non-standard family of polynomials, introduced initially by Gontcharoff (1937) for one variable, revisited and extended here for several variables. The attractiveness of these polynomials will be enhanced in forthcoming works in the epidemic context as well as in other fields.
On the solution of the Fokker–Planck equation for a Feller process
Authors: Sacerdote L.
Published: April 2006

Use of one-parameter group transformations is made to obtain the transition p.d.f. of a Feller process confined between the origin and a hyperbolic-type boundary. Such a procedure, previously used by Bluman and Cole (cf., for instance, [4]), although useful for dealing with one-dimensional diffusion processes restricted between time-varying boundaries, does not appear to have been sufficiently exploited to obtain solutions to the diffusion equations associated to continuous Markov processes.
Existence of moments in a stationary stochastic difference equation
Authors: Karlsen Hans Arnfinn
Published: April 2006

The stationary stochastic difference equationXt=YtXt–1+Wtis analyzed with emphasis on conditions ensuring that ||Xt||p<∞. Some general results are obtained and then applied to different classes of input processes {(Yt, Wt)}. Especially both necessary and sufficient conditions are given in the Gaussian case. We also obtain results concerning moments of products of dependent variables.
A geometrical derivation of the shape density
Authors: Goodall Colin R. , Mardia Kanti V.
Published: April 2006

The density for the shapes of random configurations ofNindependent Gaussian-distributed landmarks in the plane with unequal means was first derived by Mardia and Dryden (1989a). Kendall (1984), (1989) describes a hierarchy of spaces for landmarks, including Euclidean figure space containing the original configuration, preform space (with location removed), preshape space (with location and scale removed), and shape space. We derive the joint density of the landmark points in each of these intermediate spaces, culminating in confirmation of the Mardia–Dryden result in shape space. This three-step derivation is an appealing alternative to the single-step original derivation, and also provides strong geometrical motivation and insight into Kendall's hierarchy. Preform space and preshape space are respectively Euclidean space with dimension 2(N–1) and the sphere in that space, and thus the first two steps are reasonably familiar. The third step, from preshape space to shape space, is more interesting. The quotient by the rotation group partitions the preshape sphere into equivalence classes of preshapes with the same shape. We introduce a canonical system of preshape coordinates that include 2(N–2)polar coordinatesfor shape and one coordinate for rotation. Integration over the rotation coordinate gives the Mardia–Dryden result. However, the usual geometrical intuition fails because the set of preshapes keeping the rotation coordinate (however chosen) fixed is not an integrable manifold. We characterize the geometry of the quotient operation through the relationships between distances in preshape space and distances among the corresponding shapes.
A bernoulli excursion and its various applications
Authors: Takács Lajos
Published: April 2006

This paper is concerned with a random walk process in which and for i = 1, 2, ···, 2n. This process is called a Bernoulli excursion. The main object is to find the distribution, the moments, and the asymptotic distribution of the random variable ω n defined by . The results derived have various applications in the theory of probability, including random graphs, tournaments and order statistics.
Distributions of ballot problem random variables
Authors: Chao Chern-Ching , Severo Norman C.
Published: April 2006

Suppose that in a ballot candidateAscoresavotes and candidateBscoresbvotes, and that all the possible voting records are equally probable. Corresponding to the firstrvotes, letαrandβrbe the numbers of votes registered forAandB, respectively. Let p be an arbitrary positive real number. Denote byδ(a, b, p)[δ*(a,b,ρ)] the number of values ofrfor which the inequality,r =1, ···,a+b, holds. Heretofore the probability distributions of δand δ* have been derived for only a restricted set of values ofa, b, andρ, although, as pointed out here, they are obtainable for all values of (a,b,ρ) by using a result of Takács (1964). In this paper we present a derivation of the distribution ofδ[δ*] whose development, for any (a, b, ρ), leads to both necessary and sufficient conditions forδ[δ*] to have a discrete uniform distribution.
Random spherical triangles I: Geometrical background
Authors: Le Huiling
Published: April 2006

In this paper we identify the shape space Σ(S2,k) forklabelled points on the sphereS2that gives a mathematical model applicable to data sets whose elements are, or can be represented by, configurations of labelled sequences of points onS2and for which the fundamental properties of interest are the shapes of these configurations, and we examine the geometric structures on the space, especially the riemannian structure on Σ(S2, 3). In a companion paper (pp. 581–594) we investigate the statistical properties of such shapes when thekpoints are generated by a random mechanism.
Preservation of some partial orderings under Poisson shock models
Authors: Singh Harshinder , Jain Kanchan
Published: April 2006

Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity λ. Let Pk denote the probability that the first device will survive the first k shocks and let denote such a probability for the second device. Let and denote the survival functions of the first and the second device respectively. In this note we show that some partial orderings, namely likelihood ratio ordering, failure rate ordering, stochastic ordering, variable ordering and mean residual-life ordering between the shock survival probabilities and are preserved by the corresponding survival functions and .
The job-search problem with competition: an evolutionarily stable dynamic strategy
Authors: Collins E. J. , Mcnamara J. M.
Published: April 2006

We consider a game-theoretical solution for an optimal stopping problem which we describe in terms of a job-search problem with an infinite population of candidates and an infinite population of posts of varying value. Each candidate finds posts from the post population at unit rate. If a post found is still vacant, a candidate can either accept it or reject it. The reward to a candidate is the value of the post if one is accepted and zero if he never accepts a post. There are no costs for searching, no discounting of future rewards and no recall of previously found posts. The only pressure on a candidate to accept a post comes from the changing rate at which he finds vacant posts (and the values associated with them) as a result of the actions of the other candidates. It is not possible to define optimality for a single candidate without reference to the policies followed by the other candidates. We say a policy π is an evolutionarily stable strategy (ESS) if it has the following property: if all candidates used π and an individual candidate was given the option of changing his policy, then it would not be to his advantage to do so. We first find the optimal value function and optimal policy for the case of a single candidate operating in an environment where the distribution of posts on offer and the chance of finding one both vary with time in a known way. We then show that for the infinite population there is a unique ESS given by a control-limit policy π c, where the control-limit function c is the solution of a given differential equation with a given initial condition. This function c also gives the expected future reward function for any single candidate when all candidates use π c.
The exact bispectra for bilinear realizable processes with hermite degree 2
Authors: Terdik György , Meaux Laurie
Published: April 2006

This paper deals with the stationary bilinear model with Hermite degree 2 in discrete time which is built up by the first- and second-order Hermite polynomial of a Gaussian white noise process. The exact spectrum and bispectrum is constructed in terms of the transfer functions of the model.
Calculating the extremal index for a class of stationary sequences
Authors: Chernick Michael R. , Hsing Tailen , McCormick William P.
Published: April 2006

A local mixing conditionD(k)is introduced for stationary sequences satisfying Leadbetter's conditionD.Under the local mixing condition, the asymptotic distribution of the sample maximum can be calculated with the knowledge of the joint distribution ofkconsecutive terms. Some examples are given to illustrate the notion.
Poisson approximations for runs and patterns of rare events
Authors: Godbole Anant P.
Published: April 2006

Consider a sequence of Bernoulli trials with success probabilityp, and letNn,kdenote the number of success runs of lengthamong the firstntrials. The Stein–Chen method is employed to obtain a total variation upper bound for the rate of convergence ofNn,kto a Poisson random variable under the standard conditionnpk→λ. This bound is of the same order,O(p), as the best known for the casek =1, i.e. for the classical binomial-Poisson approximation. Analogous results are obtained for occurrences of word patterns, where, depending on the nature of the word, the corresponding rate is at mostO(pk–m) for somem= 0, 2, ···,k –1. The technique is adapted for use with two-state Markov chains. Applications to reliability systems and tests for randomness are discussed.
Simulated annealing process in general state space
Authors: Haario Heikki , Saksman Eero
Published: April 2006

The stochastic process corresponding to the simulated annealing optimization algorithm is generalized to the case of an arbitrary state space. Conditions for the strong and weak convergence of the process are established. In addition the relation between the size of the generating distributions and the possible rate of cooling is studied.
Giant components in three-parameter random directed graphs
Authors: Łuczak Tomasz , Cohen Joel E.
Published: April 2006

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertexito vertexjwherei < j, the probability of ‘down arrows' fromitojwherei ≥ j,and the probability of bidirectional arrows betweeniandj.In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.
Stochastic comparisons for queueing models via random sums and intervals
Authors: Jean-Marie Alain , Liu Zhen
Published: April 2006

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered.

These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.
General shape distributions in a plane
Authors: Dryden I. L. , Mardia K. V.
Published: April 2006

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendall's or Bookstein's shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.
On the transient behavior of the repairman problem
Authors: Knessl Charles
Published: April 2006

We consider the repairman problem which corresponds to the finite population M/M/1 queue. Asymptotic approximations for the transient probability distribution of the number of broken machines constructed when the number M of machines is large and the service (repair) rate is also large, specifically, O(M). The approximations are constructed by using singular perturbation techniques such as the ray method, boundary layer theory, and the method of matched asymptotic expansions. Extensive numerical comparisons show the quality of our approximations.
Analysis of finite-capacity polling systems
Authors: Takagi Hideaki
Published: April 2006

We consider a system of N finite-capacity queues attended by a single server in cyclic order. For each visit by the server to a queue, the service is given continuously until that queue becomes empty (exhaustive service), given continuously only to those customers present at the visiting instant (gated service), or given to only a single customer (limited service). The server then switches to the next queue with a random switchover time, and administers the same type of service there similarly. For such a system where each queue has a Poisson arrival process, general service time distribution, and finite capacity, we find the distribution of the waiting time at each queue by utilizing the known results for a single M/G/1/K queue with multiple vacations.
Shape distributions for landmark data
Authors: Mardia K. V. , Dryden I. L.
Published: April 2006

The paper obtains the exact distribution of Bookstein's shape variables under his plausible model for landmark data. We consider its properties including invariances, marginal distributions and the relationship with Kendall's uniform measure. Particular cases for triangles and quadrilaterals are highlighted. A normal approximation to the distribution is obtained, extending Bookstein's result for three landmarks. The adequacy of these approximations is also studied.
Produits semi-directs de diffusions reelles et lois asymptotiques
Authors: Franchi J.
Published: April 2006

A general study of the asymptotic law of a skew-product of two real diffusions is proposed; a complete and elementary form of the Rosenkrantz theorem is given; the results are applied to three examples: the Brownian motions onSO3, onS3, and onSH3.
On the comparison of the distribution of the supremum of random fields represented by stochastic integrals
Authors: Orsingher Enzo , Bassan Bruno
Published: April 2006

In this paper we compare the distribution of the supremum of the Gaussian random fields Z(P) = ∫CpG(P, P′) dW(P′) and U(P) = ∫CpdW(P'), where CP are circles of fixed radius, dW is a white noise field and G are special deterministic response functions.

The results obtained permit us to establish upper bounds for the distribution of the supremum of Z(P) by applying some well-known inequalities on U(P).

The comparison of the suprema is carried out also, when CP = ℝ2, between fields with different response functions.
The probability of attaining a structure in a partially stochastic model
Authors: Guerry M. A.
Published: April 2006

A manpower system of constant size, controlled by recruitment, is described by a partially stochastic model in which there are fixed promotion rates, no demotions and stochastic wastage. The geometric-probabilistic relationship is examined for the attainability after one step and after two steps.
Blocking probabilities for large multirate erlang loss systems
Authors: Gazdzicki Pawel , Lambadaris Ioannis , Mazumdar Ravi R.
Published: April 2006

This paper is concerned with the computation of asymptotic blocking probabilities for a generalized Erlangian system which results whenMindependent Poisson streams of traffic with ratesaccess a trunk group ofCcircuits with traffic from streamkrequiringAkcircuits which are simultaneously held and released after a time which is randomly distributed with unit mean and independent of earlier arrivals and holding times. A call from streamkis lost if on arrival less thanAkcircuits are available. Although exact expressions for the blocking probabilities are known, their computation is unwieldy for even moderate-sized switches. It is shown that as the size of the switch increases in that both the traffic rates and trunk capacity are scaled together, simple asymptotic expressions for the blocking probabilities are obtained. In particular the expression is different for light, moderate and heavy loads. The approach is via exponential centering and large deviations and provides a unified framework for the analysis.
Markov Chains in Many Dimensions
Authors: Politis Dimitris N.
Published: April 2006

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.
Stereology for some classes of polyhedrons
Authors: Ohser J. , Mücklich F.
Published: April 2006

A general method for solving stereological problems for particle systems is applied to polyhedron structures. We suggested computing the kernel function of the respective stereological integral equation by means of computer simulation. Two models of random polyhedrons are investigated. First, regular prisms are considered which are described by their size and shape. The size-shape distribution of a stationary and isotropic spatial ensemble of regular prisms can be estimated from the size-shape distribution of the polygons observed in a section plane. Secondly, random polyhedrons are constructed as the convex hull of points which are uniformly distributed on surfaces of spheres. It is assumed that the size of the polyhedrons and the number of points (i.e. the number of vertices) are random variables. Then the distribution of a spatially distributed ensemble of polyhedrons is determined by its size-number distribution. The corresponding numerical density of this bivariate size-number distribution can be stereologically determined from the estimated numerical density of the bivariate size-number distribution of the intersection profiles.
A stochastic model for the movement of a white blood cell
Authors: Heubach Silvia , Watkins Joseph
Published: April 2006

We present a stochastic model for the movement of a white blood cell both in uniform concentration of chemoattractant and in the presence of a chemoattractant gradient. It is assumed that the rotational velocity is proportional to the weighted difference of the occupied receptors in the two halves of the cell and that each of the receptors stays free or occupied for an exponential length of time. We define processes corresponding to a cell with 2nß + 1 receptors (receptor sites). In the case of constant concentration, we show that the limiting process for the rotational velocity is an Ornstein–Uhlenbeck process. Its drift coefficient depends on the parameters of the exponential waiting times and its diffusion coefficient depends in addition also on the weight function. In the inhomogeneous case, the velocity process has a diffusion limit with drift coefficient depending on the concentration gradient and diffusion coefficient depending on the concentration and the weight function.
Ruin theory with stochastic return on investments
Authors: Paulsen Jostein , Gjessing Håkon K.
Published: April 2006

We consider a risk process with stochastic interest rate, and show that the probability of eventual ruin and the Laplace transform of the time of ruin can be found by solving certain boundary value problems involving integro-differential equations. These equations are then solved for a number of special cases. We also show that a sequence of such processes converges weakly towards a diffusion process, and analyze the above-mentioned ruin quantities for the limit process in some detail.
A Remark on the Van Lieshout and Baddeley J-Function for Point Processes
Authors: Bedford T. , Van Den Berg J.
Published: April 2006

The empty space function of a stationary point process in ℝd is the function that assigns to each r, r > 0, the probability that there is no point within distance r of O. In a recent paper Van Lieshout and Baddeley study the so-called J-function, which is defined as the ratio of the empty space function of a stationary point process and that of its corresponding reduced Palm process. They advocate the use of the J-function as a characterization of the type of spatial interaction.

Therefore it is natural to ask whether J ≡ 1 implies that the point process is Poisson. We restrict our analysis to the one-dimensional case and show that a classical construction by Szász provides an immediate counterexample. In this example the interpoint distances are still exponentially distributed. This raises the question whether it is possible to have J ≡ 1 but non-exponentially distributed interpoint distances. We construct a point process with J ≡ 1 but where the interpoint distances are bounded.
Smoothing the Hill Estimator
Authors: Resnick Sidney , Stărică Cătălin
Published: April 2006

For sequences of i.i.d. random variables whose common tail 1 –Fis regularly varying at infinity wtih an unknown index –α< 0, it is well known that the Hill estimator is consistent for α–1and usually asymptotically normally distributed. However, because the Hill estimator is a function ofk = k(n), the number of upper order statistics used and which is only subject to the conditionsk→∞,k/n →0, its use in practice is problematic since there are few reliable guidelines about how to choosek.The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function ofkdecreases and the successful use of the esimator is made less dependent on the choice ofk.A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.
A limit theorem for certain disordered random systems
Authors: Ding Xinhong
Published: April 2006

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in :

where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝp-valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation

where P(t, dx dy) is the joint probability law of ξ and X(t).
Bisexual branching diffusions
Authors: Mytnik Leonid , Adler Robert J.
Published: April 2006

We study the limiting behaviour of large systems of two types of Brownian particles undergoing bisexual branching. Particles of each type generate individuals of both types, and the respective branching law is asymptotically critical for the two-dimensional system, while being subcritical for each individual population.

The main result of the paper is that the limiting behaviour of suitably scaled sums and differences of the two populations is given by a pair of measure and distribution valued processes which, together, determine the limit behaviours of the individual populations.

Our proofs are based on the martingale problem approach to general state space processes. The fact that our limit involves both measure and distribution valued processes requires the development of some new methodologies of independent interest.
Likelihood ratio gradient estimation for stochastic recursions
Authors: Glynn Peter W. , L'ecuyer Pierre
Published: April 2006

In this paper, we develop mathematical machinery for verifying that a broad class of general state space Markov chains reacts smoothly to certain types of perturbations in the underlying transition structure. Our main result provides conditions under which the stationary probability measure of an ergodic Harris-recurrent Markov chain is differentiable in a certain strong sense. The approach is based on likelihood ratio ‘change-of-measure' arguments, and leads directly to a ‘likelihood ratio gradient estimator' that can be computed numerically.
An algorithm to compute blocking probabilities in multi-rate multi-class multi-resource loss models
Authors: Choudhury Gagan L. , Leung Kin K. , Whitt Ward
Published: April 2006

In this paper we consider a family of product-form loss models, including loss networks (or circuit-switched communication networks) and a class of resource-sharing models. There can be multiple classes of requests for multiple resources. Requests arrive according to independent Poisson processes. The requests can be for multiple units in each resource (the multi-rate case, e.g. several circuits on a trunk). There can be upper-limit and guaranteed-minimum sharing policies as well as the standard complete-sharing policy. If all the requirements of a request cannot be met upon arrival, then the request is blocked and lost. We develop an algorithm for computing the (exact) steady-state blocking probability of each class and other steady state descriptions in these loss models. The algorithm is based on numerically inverting generating functions of the normalization constants. In a previous paper we introduced this approach to product-form models and developed a full algorithm for a class of closed queueing networks. The inversion algorithm promises to be even more useful for loss models than for closed queueing networks because fewer alternative algorithms are available for loss models. Indeed, for many loss models with sharing policies other than traditional complete sharing, our algorithm is the first effective algorithm. Unlike some recursive algorithms, our algorithm has a low storage requirement. To treat the loss models here, we derive the generating functions of the normalization constants and develop a new scaling algorithm especially tailored to the loss models. In general, the computational complexity grows exponentially in the number of resources, but the computation can often be reduced dramatically by exploiting conditional decomposition based on special structure and by appropriately truncating large finite sums. We illustrate our numerical inversion algorithm by applying it to several examples. To validate our algorithm on small models, we also develop a direct algorithm. The direct algorithm itself is of interest, because it tends to be more efficient when the number of resources is large, but the number of request classes is small. Furthermore, it also allows a form of conditional decomposition based on special structure.
Markov-modulated traffic with nearly complete decomposability characteristics and associated fluid queueing models
Authors: Kontovasilis Kimon P. , Mitrou Nikolas M.
Published: April 2006

This paper considers fluid queuing models of Markov-modulated traffic that, due to large differences in the time-scales of events, possess structural characteristics that yield a nearly completely decomposable (NCD) state-space. Extension of domain decomposition and aggregation techniques that apply to approximating the eigensystem of Markov chains permits the approximate subdivision of the full system to a number of small, independent subsystems (decomposition phase), plus an ‘aggregative' system featuring a state-space that distinguishes only one index per subsystem (aggregation phase). Perturbation analysis reveals that the error incurred by the approximation is of an order of magnitude equal to the weak coupling of the NCD Markov chain.

The study in this paper is then extended to the structure of NCD fluid models describing source superposition (multiplexing). It is shown that efficient spectral factorization techniques that arise from the Kronecker sum form of the global matrices can be applied through and combined with the decomposition and aggregation procedures. All structural characteristics and system parameters are expressible in terms of the individual sources multiplexed together, rendering the construction of the global system unnecessary.

Finally, besides providing efficient computational algorithms, the work in this paper can be recast as a conceptual framework for the better understanding of queueing systems under the presence of events happening in widely differing time-scales.