bluuebunny/crossref_metadata_2025 huggingface dataset:
The vectorial representation of the frequency of encounters of freely flowing vehicles
Authors: Garwood F.
Published: April 2006

It is known that if q vehicles at speed v pass a fixed point per unit time, randdomly and independently, and mix with another set q', v', then the expected number of overtakings per unit length of road per unit time is qq' (v' – v)/vv'. If k, k' are the concentrations, i.e., number per unit length of road, this rate can also be expressed as kk' (v'– v) and as q'k – qk'. It is also the expected number of intersections per unit area on the space-time diagram, where the vehicle paths are represented by sets of random parallel lines. If each set is represented by a vector with magnitude equal to the density (i.e., inverse of the spacing), the rate is equal to the magnitude of the vector product. This is extended to the general case of a stream of vehicles with distributed speeds. This has an equivalent vector, whose components are the total concentration (in the direction of the time axis) and total flow (distance direction). This leads to the concept of the cumulative vector curve in problems of geometrical probability. The extension to the case of variable flows and speeds is indicated. The vector representing the distribution of speeds is then variable, but satisfies a condition of continuity which makes its divergence vanish.
Another definition of genealogical distance
Authors: Mallows C. L.
Published: April 2006

Mycielski and Ulam have introduced three metrics on the set of all members of a species that reproduces by pairwise mating; their discussion concerns only a very orderly society in which the whole population is divided into disjoint generations, with random mating, universal monogamy and exactly two offspring of each mating. Here we discuss another metric, derived from certain probabilistic considerations, that is free of all such restrictions. It does not derive from Mendelian laws.
A correction to “The area under the infectives trajectory of the general stochastic epidemic”
Authors: Downton F.
Published: April 2006

In a recent paper (Downton (1972)) an attempt was made to use the combinatorial approach of Downton (1967) to obtain an expression for the Laplace transform (and hence the moments) of the area under the infectives trajectory of the general stochastic epidemic. The basic property of that area, which makes this possible, is that it is composed of a random number of rectangles, each of whose area has a distribution depending on the infection rate and the number of susceptibles present at the appropriate stage in the development of the epidemic, but which is independent of the number of infectives. However, Mr A. Abakuks has pointed out that in one respect the argument used was faulty. It was argued that the contribution to the area under the infectives curve of an interval ending in a reduction in the number of infectives from i to i – 1 had an exponential distribution with parameter p (the infection rate); and if the interval ended in a reduction in the number of susceptibles from s to s – 1 the distribution was exponential with parameter s. While this is true unconditionally, the combinatorial situation described was essentially a conditional one, in which each of these contributions to the total area had an exponential distribution, independent of i, but with parameter (ρ + s) regardless of whether the interval ended in a reduction in the number of infectives or of susceptibles.
Geometric and exponential decay in derived Markov chains
Authors: Callaert Herman
Published: April 2006

It is known [9] that geometric or exponential decay of a Markov chain is preserved under derivation (as defined by Cohen [2]). In this paper we consider the inverse problem, i.e., does a derived Markov chain with a geometrical or exponential decay necessarily arise from a Markov chain having the same property. For a large class of Markov chains (including time reversible chains) a complete solution is found. The method used in this paper proved accurate for obtaining exact results on the value of the decay parameter. An example extending results of Miller [8] and Teugels [9] illustrates the procedure.
Order preserving measures of information
Authors: Shiva S. G. S. , Ahmed N. U. , Georganas N. D.
Published: April 2006

If the information content of a complete finite probability scheme is greater than that of another such scheme in one measure of information, it seems reasonable to expect that this relation remains true in any other valid measure. In this paper Shannon and Rényi measures are discussed, regarding this aspect.
The reverse Galton-Watson process
Authors: Esty Warren W.
Published: April 2006

Consider the following path,Zn(w), of a Galton-Watson process in reverse. The probabilities thatZN–n= jgivenZN=iconverge, asN→ ∞ to a probability function of a Markov process,Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not onlyZN= 0 but alsoZN–1> 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives then-step transition probabilities for the reverse process, a stationary distribution ifm ≠1, and a limit law forXn/nifm= 1 andσ2<∞. Two related results aboutZcn,0 <c< 1, for Galton-Watson processes conclude the paper.
Inspection and replacement policies
Authors: Zuckerman Dror
Published: April 2006

In this article we examine a breakdown model in which the system's status can be determined only by a test. Upon detection of failure the system must be replaced by a new identical one. The costs incurred include cost of inspection, operating costs, failure cost and a cost associated with planned replacements. Throughout the paper we restrict attention to replacement rules in which the time interval between two successive inspections is regarded as a fixed quantity. The decision variables include the inspection interval and the scheduling for preventive (planned) replacements. The problem is to specify a replacement rule which minimizes the long-run average cost per unit time. We show that under certain monotone conditions there is a natural candidate for an optimal replacement rule.
The fair charge on a car ferry
Authors: Grey D. R.
Published: April 2006

Vehicles whose lengths are independent identically distributed random variables with known distribution function are loaded onto ferries of fixed known length, each ferry departing as soon as it can no longer accommodate the next vehicle in the queue. We work out how much a vehicle of any particular length ought to pay for use of the ferry, as well as the expected number of vehicles per ferry and expected revenue per ferry in equilibrium.
High-Level exceedances of regenerative and semi-stationary processes
Authors: Serfozo Richard
Published: April 2006

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.
New results in the theory of repeated orders queueing systems
Authors: Choo Qui Hoon , Conolly Brian
Published: April 2006

The repeated orders queueing system (ROO) permits no waiting or queue in the normal sense. Instead customers who find the service (or device, to use an engineering term) busy make reapplications at random intervals and in random order until their needs are met. Thus a second demand stream supplements the basic first arrival stream. Familiar examples are provided in a telephone communication setting, in particular in the context of a multiaccess computing system. Cohen [3] and Aleksandrov [1] made the first contributions to the theory of ROO. This paper complements their work with a steady-state analysis of system time (waiting time including service of a new arrival), of service idle time, and of system busy period.
Some inequalities for moments and coefficients of variation for a large class of probability functions
Authors: Bronk Burt V.
Published: April 2006

Some inequalities for moments and coefficients of variation of probability densities over the positive real line are obtained by means of simple geometrical relationships. As an illustrative application rigorous bounds are obtained for the ratio of weight average to number average molecular weight for a large class of distributions of macromolecules, giving a more precise characterization of this empirical measure of heterogeneity.
On linearly increasing mean residual lifetimes
Authors: Morrison Donald G.
Published: April 2006

It is shown that the gamma distribution is the unique mixing distribution of exponentials that leads to a linearly increasing mean residual lifetime function. A corollary of this result is that regardless of the mixing process the Pareto distribution is the unique aggregate distribution mixture with this property.
The actual waiting time of each customer in a GI/G/1 queue
Authors: Le Minh Do
Published: April 2006

This paper studies a generalization of the GI/G/1 queueing system in which the inter-arrival times are not necessarily identically distributed and there is a random set-up time for customers who arrive when the server is idle. A recursive scheme is derived to obtain the distribution of the actual waiting time of each customer in the system.
Weak moment conditions for time coordinates in first-passage percolation models
Authors: Wierman John C.
Published: April 2006

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.
On entrance—exit distributions of Markov processes
Authors: Gzyl Cristina , Gzyl Henryk
Published: April 2006

We use a result on integration by parts for stochastic integrals together with a technique developed by Getoor in [6], to express entrance—exit distributions for a standard processX, and a set Φ which is the support of a continuous additive functionalC, in terms of the infinitesimal generators of semigroups associated with the time-changed process (Xτt), where (τt) is the right-continuous inverse ofC.
A note on networks of infinite-server queues
Authors: Harrison J. Michael , Lemoine Austin J.
Published: April 2006

The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady-state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network. We consider closed systems in which a fixed and finite number of customers circulate through the network and no external arrivals or departures are permitted, and open systems in which customers originate from an external source according to a Poisson process, possibly non-homogeneous, and each customer eventually leaves the system.
On the distribution of random lines
Authors: Hall Peter
Published: April 2006

Draw two lines, L1 and L2, in a plane. Along L1 place the points of a Poisson process, and through each point draw a line, the angles of intersection with L1 being distributed independently and uniformly on (0, π) . The intercepts of these random lines with L2 form a new process, which is Poisson if and only if L1 and L2 are parallel. Curiously, if L1 and L2 are inclined then, with probability 1, the new process forms a dense subset of L2. It is not really even a point process. In the present paper we shall investigate this anomaly and some of its generalizations.
Waveforms and velocities for models of spatial infection
Authors: Renshaw Eric
Published: April 2006

A new technique is developed for deriving velocities of propagation and shapes of travelling wave profiles in non-linear situations. Two models are considered for the spread of infection through a linear community. In one the infected individuals move, in the other the phenomenon of infection.
A note on Gaussian processes and zero-memory non-linear transformations
Authors: Masry Elias
Published: April 2006

We consider the problem of the distinguishability between Gaussian processes observed through non-linear systems. For a broad class 𝒜 of zero-memory time-dependent non-linear transformations, the following uniqueness relationship is obtained. If Xi, i = 1, 2 are two normalized jointly Gaussian processes, then they are indistinguishable if and only if their transformed processes Yi = A[Xi], i = 1, 2, A ∈ 𝒜, are indistinguishable. Some interesting examples of functions in 𝒜 are presented.
Distortion of certain Voronoi tesselations when one particle moves
Authors: Orive Luis-M. Cruz
Published: April 2006

Some properties of the Voronoi polygon associated with a moving particle in a regular tessellation are studied. In particular, the area of the variable polygon has played a relevant role in previous studies on plant competition, animal predation, adjustment of the sex ratio in bisexual populations and the gregarious behaviour of animal species.
An enumerative approach to a problem in probability
Authors: Devitt John S.
Published: April 2006

Sequences over the finite alphabet {1, …, m} are used to model the various job queues of a machine, classifying each job according to which one of a set of m resources is used by the job. Known results concerning the enumeration of such sequences according to the number of rises, levels and falls are then used to obtain the various moments of the distribution of the random variable for the number of changes of resources occurring in a sequence of n jobs given the conditional probabilities of changing resources between independent jobs. The techniques involved form a useful approach to problems where such enumerative information is available.
Return times in nearly-completely decomposable stochastic processes
Authors: Latouche G. , Louchard G.
Published: April 2006

Consider a finite irreducible aperiodic Markov chain with nearly-completely decomposable stochastic matrix: i.e. a Markov chain for which the states can be grouped into disjoint aggregates, in such a way that the probabilities of transition between states of the same aggregate are large compared to the probabilities of transition between states belonging to different aggregates. Let Ω be a subset of one of the aggregates. Second-order approximations are determined for the first and second moments of the time to reach Ω and the return time to Ω.
The age distribution of Markov processes
Authors: Levikson Benny
Published: April 2006

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.
Multivariate Poisson flows on Markov step processes
Authors: Beutler Frederick J. , Melamed Benjamin
Published: April 2006

A Markov step process Z equipped with a possibly non-denumerable state space X can model a variety of queueing, communication and computer networks. The analysis of such networks can be facilitated if certain traffic flows consist of mutually independent Poisson processes with respective deterministic intensities λi (t). Accordingly, we define the multivariate counting process N = (N1, N2, · ·· Nc) induced by Z; a count in Ni occurs whenever Z jumps from x (χ into a (possibly empty) target set . We study N through the infinitesimal operator Ã of the augmented Markov process W = (Z, N), and the integral relationship connecting Ã with the transition operator Tt of W. It is then shown that Ni depends on a non-negative function ri defined on χ; ri (x) may be interpreted as the expected rate of increase in Ni, given that Z is in state x.

A multivariate N is Poisson (i.e., composed of mutually independent Poisson streams Ni) if and only if simultaneous jumps are impossible in a certain sense, and if the conditional expectation E[ri (Z(t) | 𝒩i] = E[ri (Z(t))] for i = 1, 2, ···, c and each t ≧ 0, where 𝒩i is the σ-algebra σ{N(s), s ≦ t}. Necessary and sufficient conditions are also specified that, for each s ≦ t, the variates [Ni(t)-Ni(s)] are mutually independent Poisson distributed; this involves a weakened version of E[ri(Z(v)) | N(v) – N(u)] = E [ri(Z(v))] for i = 1, 2, ···, c and all 0 ≦ u ≦ v.

It is shown that the above criteria are automatically met by the more stringent classical requirement that N(t) and Z(t) be independent for each t ≧ 0.
Dispersive ordering of distributions
Authors: Shaked Moshe
Published: April 2006

Two distributions, F and G, are said be ordered in dispersion if F-1(β)-F-1(α)≦G-1(β)-G-1(α) whenever 0<α <β <1. This relation has been studied by Saunders and Moran (1978). The purpose of this paper is to study this partial ordering in detail. Few characterizations of this concept are given. These characterizations are then used to show that some particular pairs of distributions are ordered by dispersion. In addition to it some proofs of results of Saunders and Moran (1978) are simplified. Furthermore, the characterizations of this paper can be used to throw a new light on the meaning of the underlying partial ordering and also to derive various inequalities. Several examples illustrate the methods of this paper.
Optimal stopping rules in proofreading
Authors: Yang Mark C. K. , Wackerly Dennis D. , Rosalsky Andrew
Published: April 2006

Optimal stopping rules under various conditions are obtained for a proofreader who has a probability p (known or unknown) of detecting a misprint in proofsheets which contain an unknown but Poisson-distributed number of misprints.
Aspects of population genetics theory
Authors: Ewens W. J.
Published: April 2006

Population genetics and evolution have long supplied a stimulus to mathematical and statistical research. In this note a brief survey of current theory is given, with special reference to the work of P. A. P. Moran. Finally, an analysis of various biometrical questions is presented in the spirit of Moran's contributions to mathematical genetics.
Comparison results for diffusions conditioned on positivity
Authors: Day Martin V.
Published: April 2006

We consider a diffusion process on the reals subject to the conditional probability that the process is positive from t = 0 to the present. We establish comparison results between the conditioned diffusion and a second unconditioned Markov diffusion. One result allows the initial process to be non-Markov before conditioning. A stronger comparison theorem is shown to hold in the Markov case.
On the asymptotic distribution of multivariate renewal processes
Authors: Niculescu Ştefan P.
Published: April 2006

Using a result of Bikjalis (1971) concerning the rate of convergence in the multidimensional central limit theorem we obtain informations about some limit distributions in multivariate renewal theory.
On weak convergence within the hnbue family of life distributions
Authors: Basu Sujit K. , Bhattacharjee Manish C.
Published: April 2006

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.
A correlated random walk model for two-dimensional diffusion
Authors: Henderson Robin , Renshaw Eric , Ford David
Published: April 2006

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.
An operator-analytic approach to the Jackson network
Authors: Massey William A.
Published: April 2006

Operator methods are used in this paper to systematically analyze the behavior of the Jackson network. Here, we consider rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system. By deriving the equations that govern an arbitrary subnetwork, we can see how the mean and variance for the queue length of one node as well as the covariance for two nodes vary in time.

We can estimate the transient behavior by deriving a stochastic upper bound for the joint distribution of the network in terms of a judicious choice of independent M/M/1 queue-length processes. The bound we derive is one that cannot be derived by a sample-path ordering of the two processes. Moreover, we can stochastically bound from below the process for the total number of customers in the network by an M/M/1 system also. These results allow us to approximate the network by the known transient distribution of the M/M/1 queue. The bounds are tight asymptotically for large-time behavior when every node exceeds heavy-traffic conditions.
Continuum structures I
Authors: Baxter Laurence A.
Published: April 2006

A generalisation of multistate coherent structures is proposed where the state of each component in a binary coherent structure can take any value in the unit interval, as can the structure function. The notions of duality, critical elements and strong coherency for such a structure are discussed and the functional form of the structure function is analysed. An expression is derived for the distribution function of the state of the system, given the distributions of the states of the components, and generalisations of the Moore–Shannon and IFRA and NBU closure theorems are proved. The states of the components are then permitted to vary with time and a first-passage-time distribution is discussed. A simple model for such a process, based on the concept of partial availability, is then proposed. Lastly, an alternative continuum structure function is introduced and discussed.
A central limit theorem for a security of supply model
Authors: Platz Ole
Published: April 2006

A security of supply model for energy is described. The energy demand is a non-random periodic function of time and the supply model is a continuous-time Markov chain. Asymptotic normality is proved for the two important reliability quantities: the amount of energy not supplied and the time the demand is not met within a time interval.
On some classes of population-size-dependent Galton–Watson processes
Authors: Höpfner Reinhard
Published: April 2006

Some classes of population-size-dependent Galton–Watson processes {Z(t)}t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.
Lower bounds for the probability of overload in certain queueing networks
Authors: Kanter Marek
Published: April 2006

Conservative estimates for the probability that open queueing networks are not overloaded during any finite set of time points are derived. The queueing networks have infinitely many servers at each queue and general service times. A key preliminary result is that for multidimensional Markov population processes, the correspondence between initial distributions and their temporal development preserves stochastic order.
Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution
Authors: Nobile A. G. , Ricciardi L. M. , Sacerdote L.
Published: April 2006

The asymptotic behavior of the first-passage-time p.d.f. through a constant boundary is studied when the boundary approaches the endpoints of the diffusion interval. We show that for a class of diffusion processes possessing a steady-state distribution this p.d.f. is approximately exponential, the mean being the average first-passage time to the boundary. The proof is based on suitable recursive expressions for the moments of the first-passage time.
Tail behaviour of ladder-height distributions in random walks
Authors: Grübel Rudolf
Published: April 2006

We give necessary and sufficient conditions for various results connecting the tail behaviour of a distribution with that of its right Wiener–Hopf factor.
The Hurst effect under trends
Authors: Bhattacharya R. N. , Gupta Vijay K. , Waymire Ed
Published: April 2006

Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the formf(n) =c(m+n)ß, where m is an arbitrary non-negative parameter andcis not 0. For – ½ <ß< 0 the Hurst exponent is shown to be precisely given by 1 +ß.Forß≦ – ½ and forß= 0 the Hurst exponent is 0.5, while forß > 0it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.
On a class of continuous-time Markov processes
Authors: Gani J. , Tin Pyke
Published: April 2006

This paper considers a certain class of continuous-time Markov processes, whose time-dependent and stationary distributions are studied. In the stationary case, the analogy with Whittle's relaxed Markov process is pointed out. The derivation of the probability generating functions of the general process provides useful results for the analysis of some population and queueing processes.
On recurrence and transience of growth models
Authors: Kersting G.
Published: April 2006

Let Xn be non-negative random variables, possessing the Markov property. We given criteria for deciding whether Pr(Xn →∞) is positive or 0. It turns out that essentially this depends on the magnitude of E(Xn+1 | Xn = x) compared to that of E(X2n+1 | Xn = x) for large x. The assumptions are chosen such that for example population-dependent branching processes can be treated by our results.
Multichannel queueing systems with infinite waiting room and stochastic control
Authors: Dshalalow Jewgeni
Published: April 2006

A wide class of multichannel queueing models appears to be useful in practice where the input stream of customers can be controlled at the moments preceding the customers' departures from the source (e.g. airports, transportation systems, inventories, tandem queues). In addition, the servicing facility can govern the intensity of the servicing process that further improves flexibility of the system. In such a multichannel queue with infinite waiting room the queueing process {Zt; t ≧ 0} is under investigation. The author obtains explicit formulas for the limiting distribution of (Zt) partly using an approach developed in previous work and based on the theory of semi-regenerative processes. Among other results the limiting distributions of the actual and virtual waiting time are derived. The input stream (which is not recurrent) is investigated, and distribution of the residual time from t to the next arrival is obtained. The author also treats a Markov chain embedded in (Zt) and gives a necessary and sufficient condition for its existence. Under this condition the invariant probability measure is derived.
Approximations of the optimal stopping problem in partial observation
Authors: Mazziotto G.
Published: April 2006

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.
The Wright–Fisher model with varying selection
Authors: Weber N. C.
Published: April 2006

The Wright–Fisher model with varying population size is examined in the case where the selective advantage varies from generation to generation. Models are considered where the selective advantage is not always in favour of a particular genotype. Sufficient conditions in terms of the selection coefficients and the population growth are given to ensure ultimate homozygosity.
The discrete uhlenbeck–ornstein process
Authors: Renshaw Eric
Published: April 2006

A correlated random walk is studied in which, at each stage, the velocity changes according to a first-order process. Motion is considered both with and without friction, the former situation being the discrete analogy of the Uhlenbeck–Ornstein process. Exact and limiting expressions are developed for the cumulant structures.
Subexponential distributions and integrated tails
Authors: Klüppelberg Claudia
Published: April 2006

Let F be a distribution function on [0,∞) with finite expectation. In terms of the hazard rate of F several conditions are given which simultaneously imply subexponentiality of F and of its integrated tail distribution F1. These conditions apply to a wide class of longtailed distributions, and they can also be used in connection with certain random walks which occur in risk theory and queueing theory.
Sums and weighted sums of a gamma Markov sequence
Authors: Phatarfod Ravindra M.
Published: April 2006

We derive the Laplace transforms of sums and weighted sums of random variables forming a Markov chain whose stationary distribution is gamma. Both seasonal and non-seasonal cases are considered. The results are applied to two problems in stochastic reservoir theory.
A note on the equilibrium M/G/1 queue length
Authors: Willmot Gordon E.
Published: April 2006

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.
A proof of simple insensitive bounds for a pure overflow system
Authors: Van Dijk Nico M.
Published: April 2006

In [1], [4] and [6] simple and intuitively obvious bounds were suggested for various overflow systems and claimed to be insensitive, that is to be valid also in the non-exponential case. This insensitivity was formalized in [4] and [6] when the service distributions have a monotone failure rate. This technical note provides another proof which for the case of a pure overflow system with one customer class extends the insensitivity to the general case.
How large is the support of an ESS?
Authors: Haigh John
Published: April 2006

Suppose the payoffs (aij) in then×nmatrixAare drawn independently from some continuous probability distribution. The number of tactics used in an ESS is investigated. Asymptotic results on the size of the ESS with largest support are given, using the work of Karlin and Kingman on the size of polymorphisms in one-locus multi-allele diploid selection models.
Limit theorems for jump shock models
Authors: Yamada Keigo
Published: April 2006

We consider an additive shock process where shocks occur according to a Poisson point process and they are accumulated in an appropriate way to the damage. It is shown that suitably normalized shock processes converge weakly to a process which is represented as a sum of a stable process and a deterministic process.
On the distribution of points in a poisson dirichlet process
Authors: Griffiths R. C.
Published: April 2006

A probability density function important in the Poisson Dirichlet process of population genetics is studied. An accurate computational algorithm is given for this density and for the marginal distributions of the points in the Poisson Dirichlet process. The distribution of the maximal point of the process is tabulated. Rational polynomial approximations in θ, the mutation parameter, are found for the expected values of the first three maximal points.
Some multivariate distributions derived from a non-fatal shock model
Authors: Savits Thomas H.
Published: April 2006

A non-homogeneous Poisson shock model has a continuous mean function Λ(t). The kth shock Sk causes simultaneous failure of the components j ∊ J ∊ {1, ···, n} with probability pJ(Sk). If Tj is the lifetime of component j, it is shown that (T1, · ··, Tn) belongs to various multivariate non-parametric life classes depending on the life class of .
Mixing properties of harris chains and autoregressive processes
Authors: Athreya Krishna B. , Pantula Sastry G.
Published: April 2006

Let {Yn: n ≧ 1} be a Harris-recurrent Markov chain on a general state space. It is shown that {Yn} is strong mixing, provided there exists a stationary probability distribution π (·) for {Yn}. Necessary and sufficient conditions for an autoregressive process to be uniform mixing are given.
A simulation study of random caps on a sphere
Authors: Solomon H. , Sutton C.
Published: April 2006

This paper describes the computer simulation of a coverage problem in geometric probability, that of placing random caps on the surface of a sphere. The simulation results were compared with exact values, where known, and the differences were negligible. This suggested the use of simulation results to assess several approximation formulas in the literature.
Notes on random chords in convex bodies
Authors: Enns E. G. , Ehlers P. F.
Published: April 2006

Consider a compact bodyEembedded in a convex compact bodyG.A pointPis randomly chosen inEand three different rays to the boundary∂GofGare generated byP.Rayis in a uniformly random direction and has lengthR, rayis through a second random point chosen from withinGand has lengthW, and rayis to a random point in∂Gand has lengthY.The distribution ofYis obtained and is related to previous work. WhenEis specialized toGor to∂G, other known results are retrieved. The paper ends with a discussion of a conjecture relating the means ofR, WandY.
A geometric rate of convergence to the equilibrium for Boltzmann processes with multiple particle interactions
Authors: Chauvin Brigitte , Giroux Gaston
Published: April 2006

We construct Boltzmann processes using the formalism of random trees. We are then able to extend previous results about convergence toward the equilibrium law to interactions involving random numbers of particles. We even show a geometric rate of convergence for an extended class of processes, especially for those having a scaling invariant interaction mechanism.
A convergence theorem for symmetric functionals of random partitions
Authors: Joyce Paul , Tavaré Simon
Published: April 2006

This paper gives general conditions under which symmetric functionals of random partitions of the integer m converge in distribution as m → ∞. The main result is used to settle a conjecture of Donnelly et al. (1991) to the effect that the mean of the sum of the square roots of the relative sizes of the components of a random mapping of m integers converges to π/2 as m → ∞.
Some characterizations of the Poisson process and geometric renewal process
Authors: Huang Wen-Jang , Li Shun-Hwa , Su Jyh-Cherng
Published: April 2006

Let γ t and δ t denote the residual life at t and current life at t, respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as holds for a single positive integer n, then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t, we find that for some fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t.
A note on the classification of Q-processes when Q is not regular
Authors: Pollett P. K.
Published: April 2006

We shall provide a classification of states for processes other than the Feller minimal process. In particular, we shall study the regularity, recurrence, transience, etc. of these processes. Our results are based on the observation that certain honest Q-processes exhibit regenerative behaviour. We shall also comment on the problem of determining stationary distributions directly from the transition rates.
A non-exponential queueing system with independent arrivals and batch servicing
Authors: Van Dijk Nico M. , Smeitink Eric
Published: April 2006

We study a queueing system with a finite number of input sources. Jobs are individually generated by a source but wait to be served in batches, during which the input of that source is stopped. The service speed of a server depends on the mode of other sources and thus includes interdependencies. The input and service times are allowed to be generally distributed. A classical example is a machine repair system where the machines are subject to shocks causing cumulative damage. A product-form expression is obtained for the steady state joint queue length distribution and shown to be insensitive (i.e. to depend on only mean input and service times). The result is of both practical and theoretical interest as an extension of more standard batch service systems.
Reliability of a consecutive-k-out-of-n system in a random environment
Authors: Chryssaphinou Ourania , Papastavridis Stavros
Published: April 2006

We consider a consecutive-k-out-of-n system situated in a random environment where shocks occur and cause components to fail with certain probabilities. Under certain reasonable assumptions we compute the moment generating function of the time of failure of the system and the distribution of the number of shocks required to cause failure of the system.
Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes
Authors: Csenki Attila
Published: April 2006

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB(t) as t →∞.
Stochastic comparisons of density profiles for the road-hog process
Authors: Srinivasan Rengarajan
Published: April 2006

We consider the asymmetric simple exclusion process which starts from a product measure such that all the sites to the left of zero (including zero) are occupied and the right of 0 (excluding 0) are empty. We label the particle initially at 0 as the leading particle. We study the long-term behaviour of this process near large sites when the leading particle's holding time is different from that of the other particles. In particular, we assume that the leading particle moves at a slower rate than the other particles. We call this modified asymmetric simple exclusion process the road-hog process. Coupling and stochastic ordering techniques are used to derive the density profile of this process. Road-hog processes are useful in modelling series of exponential queues with Poisson and non-Poisson input process. The density profiles dramatically illustrate the flow of customers through the queues.
A gaussian process with parabolic covariances
Authors: Cabaña Enrique M.
Published: April 2006

The centred, periodic, stationary Gaussian process X(z), ≧ z ≧ 1 with covariances , appears when one studies the solutions of the vibrating string equation forced by noise, corresponding to the case of a finite string with the extremes tied together. The close relationship between this process and a Brownian bridge permits us to compute the distribution of the maximum excursion of the string at particular times.
A note on optimal inspection policy for stochastically deteriorating series systems
Authors: Qiu Yuping
Published: April 2006

A counterexample is given to demonstrate that a previously proposed characterization of optimal inspection policy for series systems is not correct in the discounted case. A set of valid necessary conditions is provided which leads to the characterization of optimality for a special case. The limiting case of these necessary conditions is also examined.
Stabilities and instabilities in population dynamics
Authors: Jagers Peter
Published: April 2006

Stability in populationsizeis illusory: populations left to themselves either grow beyond all bounds or die out. But if they do not die out theircompositionstabilizes. These problems are discussed in terms of general abstract, multitype branching processes. The life and descent of a typical individual is described.
Optimal control of an M/G/1 queue with imperfectly observed queue length when the input source is finite
Authors: Wakuta Kazuyoshi
Published: April 2006

We consider the optimal control of an M/G/1 queue with finite input source. The queue length, however, can be only imperfectly observed through the observations at the initial time and the times of successive departures. At these times, the service rate can be chosen, based on the observable histories. A service cost and a holding cost are incurred. We show that such a control problem can be formulated as a semi-Markov decision process with imperfect state information, and present sufficient conditions for the existence of an optimal stationary I-policy.
Strong stationary times and eigenvalues
Authors: Matthews Peter
Published: April 2006

This note gives a new strong stationary time (SST) for reversible finite Markov chains. A modification of the initial distribution is represented as a mixture of distributions which have eigenvector interpretations, and for which good simple SSTs exist. This provides some insight into the relationship between SSTs and eigenvalues. Connections to duality and the threshold phenomenon are discussed.
Availability of a series system with replacement and repair
Authors: Fawzi B. B. , Hawkes A. G.
Published: April 2006

We consider two models for the reliability of a series system of components with spares, serviced by a single service facility. Replacement of failed components by a spare has pre-emptive priority over repair of the failed components; the models differ in whether interrupted repairs resume from where they left off, or restart from the beginning. We find the availability of such systems: numerical results indicate that this is insensitive to the shape of the repair-time and replacement-time distributions. We also obtain the distribution of system downtime for these models.
Dynamic conditional marginal distributions in reliability theory
Authors: Shaked Moshe , Shanthikumar J. George
Published: April 2006

Motivated by the need of studying a subset of components, ‘separate' from the other components, we introduce a new definition of ‘marginal distribution'. This is done by fixing the lives of the other components, but without the ‘knowledge' of the components of interest. Formally this is done by minimally repairing the components of no interest up to a predetermined time. Preservation properties of these ‘conditional marginal distributions', with respect to several stochastic orderings, are obtained. Also, inheritance of positive dependence properties, by the conditional marginal distributions, is shown. In addition, the preservations of dynamic multivariate aging properties, by the dynamic conditional marginal distributions, are obtained. The definitions and results are illustrated by a set of examples. Some applications for modelling ‘combinations' of sets of random lifetimes, and for bounding complex sets of random lifetimes, are described.
The geometricity of the limiting distributions in queues and dams
Authors: Phatarfod R. M.
Published: April 2006

There are a number of cases in the theories of queues and dams where the limiting distribution of the pertinent processes is geometric with a modified initial term — herein called zero-modified geometric (ZMG). The paper gives a unified treatment of the various cases considered hitherto and some others by using a duality relation between random walks with impenetrable and with absorbing barriers, and deriving the probabilities of absorption by using Waldian identities. Thus the method enables us to distinguish between those cases where the limiting distribution would be ZMG and those where it would not.
A note on a one-compartment model with clustering
Authors: Booth James G.
Published: April 2006

The classical one-compartment model with no input or pure death process is shown to be a limiting case of a ‘binomial cascade' model which has the same mean and in which particles exit the compartment in binomial clusters. The transition probabilities of the binomial cascade process are derived in closed form. The model is easily modified to allow Poisson input into the compartment. Distributional results are given for this model also. In particular, it is shown that the M/M/∞ queue is a limiting case.
Dependence structure of Poisson-paced records
Authors: Bunge J. A. , Nagaraja H. N.
Published: April 2006

Let Y0, Y1, Y2, ··· be an i.i.d. sequence of random variables with absolutely continuous distribution function F, and let {N(t), t ≧ 0} be a Poisson process with rate λ (t) and mean Λ(t), independent of the Yj's. We associate Y0 with the point t = 0, and Yj with the jth point of N(·), j ≧ 1. The first Yj (j ≧ 1) to exceed all previous ones is the first record value, and the time of its occurrence is the first record time; subsequent record values and times are defined analogously. For general Λ, we give the joint distribution of the values and times of the first n records to occur after a fixed time T, 0 ≦ T < ∞. Assuming that F satisfies Von Mises regularity conditions, and that λ (t)/Λ (t) → c ∈ (0, ∞) as t → ∞, we find the limiting joint p.d.f. of the values and times of the first n records after T, as T → ∞. In the course of this we correct a result of Gaver and Jacobs (1978). We also consider limiting marginal and conditional distributions. In addition, we extend a known result for the limit as the number of recordsK → ∞, and we compare the results for the limit as T → ∞ with those for the limit as K → ∞.