bluuebunny/crossref_metadata_2025 huggingface dataset:
The derivation of invariance relations in complex queueing systems with stationary inputs
Authors: Miyazawa Masakiyo
Published: April 2006

We discuss a method of obtaining invariance relations in complex systems by using the theory of point processes. New formulae are given for obtaining them generally, and in particular in many-stage models such as tandem and network queues. The formulae are shown to be useful by applications to a many-server queue and a tandem queue. Stochastic inequalities in a tandem queue are also discussed using the invariance relations obtained.
Dispersive ordering results
Authors: Lynch James , Mimmack Gillian , Proschan Frank
Published: April 2006

A distribution F is less dispersed than a distribution G if for all .

We generalize a characterization of dispersive ordering of Shaked (1982) concerning sign changes of Fc – G, where Fc is a translate of F. We then use this generalization plus total positivity to develop a simple proof of a characterization of dispersive distributions due to Lewis and Thompson (1981); a distribution H is dispersive if
Conditional expected sojourn times in insensitive queueing systems and networks
Authors: Jansen Uwe
Published: April 2006

We consider queueing systems where the stationary state probabilities are insensitive with respect to the distribution of certain basic random variables such as service requirements, interarrival times, repair times, etc. The conditional expected sojourn times are stated as Radon–Nikodym densities of the stationary distribution at jump points of the queueing system. The conditions are the given values of such basic random variables for which the insensitivity is valid. We use stationary point processes as our main tool. This means that dependences between certain basic random variables are permitted. Conditional expected real service times, conditional mean response times in closed queueing networks, and similar conditional expected values, are dealt with as special cases.
An age-dependent counting process generated from a renewal process
Authors: Sumita Ushio , Shanthikumar J. George
Published: April 2006

Let {N(t)} be a renewal process having the associated age process {X(t)}. Of interest is the counting process {M(t)} characterized by a non-homogeneous Poisson process with age-dependent intensity function λ (X(t)). The trivariate process {Y(t) = [M(t), N(t), X(t)]} is analyzed obtaining its Laplace transform generating function explicitly. Based on this result, asymptotic behavior of {S(t) = cM(t) + dN(t)} as t → ∞ is discussed. Furthermore, a sufficient condition is given under which {M(t), –N(t), X(t)} is stochastically monotone and associated. This condition also assures increasing stochastic convexity of {M(t)}. The usefulness of these results is demonstrated through an application to the age-dependent minimal repair problem.
A Markov chain of triangle shapes
Authors: Mannion David
Published: April 2006

The process of choosing a random triangle inside a compact convex region, K, may be iterated when K itself is a triangle. In this way successive generations of random triangles are created. Properties of scale, location and orientation are filtered out, leaving only the shapes of the triangles as the objects of study. Various simulation investigations indicate quite clearly that, as n increases, the nth-generation triangle shape converges to collinearity. In this paper we attempt to establish such convergence; our results fall slightly short of a complete proof.
Necessary optimality conditions for priority policies in stochastic weighted flowtime scheduling problems
Authors: Kämpke Thomas
Published: April 2006

Conditions implied by the optimality of static priority policies in a special class of stochastic scheduling problems will be derived.
ESS modelling of diploid populations I: anatomy of one-locus allelic frequency simplices
Authors: Hines W. G. S.
Published: April 2006

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, relevant features of the probability simplex of allelic frequencies for a population with genetic variability at a single locus are explored. Singularities and related properties of mappings from the space of allele frequencies to the space of strategy frequencies are examined, and related to a certain covariance measure of variability present in the population.

A companion paper builds on this characterization to establish that previous claims of stability in fact hold under slightly weaker conditions than initially indicated. The pair of papers also determines conditions under which unstable equilibria can occur, and establishes that these conditions are exceptional in practice.
The interchangeability of tandem queues with heterogeneous customers and dependent service times
Authors: Weber Richard R.
Published: April 2006

Consider m queueing stations in tandem, with infinite buffers between stations, all initially empty, and an arbitrary arrival process at the first station. The service time of customer j at station i is geometrically distributed with parameter pi, but this is conditioned on the fact that the sum of the m service times for customer j is cj. Service times of distinct customers are independent. We show that for any arrival process to the first station the departure process from the last station is statistically unaltered by interchanging any of the pi's. This remains true for two stations in tandem even if there is only a buffer of finite size between them. The well-known interchangeability of ·/M/1 queues is a special case of this result. Other special cases provide interesting new results.
On the departure processes of M/M/1/N and GI/G/1/N queues
Authors: Chao Xiuli
Published: April 2006

The purpose of this note is to point out the connection between the invariance property of M/M/1 and GI/G/1 queues (which has been reported in several papers) and the interchangeability and reversibility properties of tandem queues. This enables us to gain new insights for both problems and obtain stronger invariance results for M/M/1, GI/G/1, as well as loss systems M/M/1/N, GI/G/1/N and tandem systems.
On conditional intensities and on interparticle correlation in non-linear death processes
Authors: Brown Timothy C. , Donnelly Peter
Published: April 2006

Ball and Donnelly (1987) announced a result giving circumstances in which there is positive or negative correlation between the death times in a non-linear, Markovian death process. A proof is provided here, based on results concerning the distribution of optional random variables in terms of their conditional intensities.
A constant regression characterization of the gamma law
Authors: Wesołowski Jacek
Published: April 2006

A constant regression of the quotient on the sum of two i.i.d. non-degenerate positive random variables is a characteristic property of the gamma distribution.
Extremal properties of shot noise processes
Authors: Hsing Tailen , Teugels J. L.
Published: April 2006

Consider the shot noise process X(t):= Σih(t – τi), , where h is a bounded positive non-increasing function supported on a finite interval, and the are the points of a renewal process η on [0, ). In this paper, the extremal properties of {X(t)} are studied. It is shown that these properties can be investigated in a natural way through a discrete-time process which records the states of {X(t)} at the points of η. The important special case where η is Poisson is treated in detail, and a domain-of-attraction result for the compound Poisson distribution is obtained as a by-product.
On hazard rate ordering of dependent variables
Authors: Aly Emad-Eldin A. A. , Kochar Subhash C.
Published: April 2006

Shanthikumar and Yao (1991) introduced some new stochastic order relations to compare the components of a bivariate random vector (X1, X2). As they point out in their paper, even if according to their hazard rate (or likelihood ratio) ordering, the marginal distributions may not be ordered accordingly. We introduce some new concepts where the marginal distributions preserve the corresponding stochastic orders. Also a relation between the bivariate scale model and the introduced bivariate hazard rate ordering is established.
Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes
Authors: Van Doorn Erik A.
Published: April 2006

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.
A reliability model based on the gamma process and its analytic theory
Authors: Wenocur Michael L.
Published: April 2006

This paper presents a variation of a state-dependent reliability model first proposed in Lemoine and Wenocur [4], [5], and develops some of its corresponding analytical theory. In particular, we develop a reliability model in which system state is a random process satisfying a stochastic differential equation where the driving process is gamma distributed.
The Horizon of the Random Cone Field Under a Trend
Authors: Nosko Vladimir P.
Published: April 2006

The horizon ξ T(x) of a random field ζ (x, y) of right circular cones on a plane is investigated. It is supposed that bases of cones are centered at points sn = (xn, yn), n = 1, 2, ···, on the (X, Y)-plane, constituting a Poisson point process S with intensity λ0 > 0 in a strip ΠT = {(x, y): – ∞< x <∞, 0 ≦ y ≦ T}, while altitudes of the cones h1, h2, · ·· are of the form hn = hn* + f(yn), n = 1, 2, ···, where f(y) is an increasing continuous function on [0,∞), f(0) = 0, and h1*, h2*, · ·· is a sequence of i.i.d. positive random variables which are independent of the Poisson process S and have a distribution function F(h) with density p(h).

Let denote the expected mean number of local maxima of the process ξ T(x) per unit length of the X-axis. We obtain an exact formula for under an arbitrary trend function f(y). Conditions sufficient for the limit being infinite are obtained in two cases: (a) h1* has the uniform distribution in [0, H], f(y) = kyγ; (b) h1* has the Rayleigh distribution, f(y) = k[log(y + 1)]γ. (In both cases γ 0 and 0 < k∞.) The corresponding sufficient conditions are: 0 < γ< 1 in case (a), 0 < γ< 1/2 in case (b).
Stochastic fm models and non-linear time series analysis
Authors: Huang D.
Published: April 2006

An important model in communications is the stochastic FM signal st = A cos , where the message process {mt} is a stochastic process. In this paper, we investigate the linear models and limit distributions of FM signals. Firstly, we show that this non-linear model in the frequency domain can be converted to an ARMA (2, q + 1) model in the time domain when {mt} is a Gaussian MA (q) sequence. The spectral density of {St} can then be solved easily for MA message processes. Also, an error bound is given for an ARMA approximation for more general message processes. Secondly, we show that {St} is asymptotically strictly stationary if {mt} is a Markov chain satisfying a certain condition on its transition kernel. Also, we find the limit distribution of st for some message processes {mt}. These results show that a joint method of probability theory, linear and non-linear time series analysis can yield fruitful results. They also have significance for FM modulation and demodulation in communications.
On the Distributions of Scan Statistics of a Two-Dimensional Poisson Process
Authors: Alm Sven Erick
Published: April 2006

Given a two-dimensional Poisson process, X, with intensity λ, we are interested in the largest number of points, L, contained in a translate of a fixed scanning set, C, restricted to lie inside a rectangular area.

The distribution of L is accurately approximated for rectangular scanning sets, using a technique that can be extended to higher dimensions. Reasonable approximations for non-rectangular scanning sets are also obtained using a simple correction of the rectangular result.
Clustering of Bursts of Openings in Markov and Semi-Markov Models of Single Channel Gating
Authors: Ball Frank , Davies Sue
Published: April 2006

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean [N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝ Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.
Integral Probability Metrics and Their Generating Classes of Functions
Authors: Müller Alfred
Published: April 2006

We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.
Strong Approximations of Irreducible Closed Queueing Networks
Authors: Zhang Hanqin
Published: April 2006

A sequence of irreducible closed queueing networks is considered in this paper. We obtain that the queue length processes can be approximated by reflected Brownian motions. Using these approximations, we get rates of convergence of the distributions of queue lengths.
Smoothed Perturbation Analysis for Single-Server Queues by Some General Service Disciplines
Authors: Miyoshi Naoto , Hasegawa Toshiharu
Published: April 2006

We consider some single-server queues with general service disciplines, where the family of the queueing processes are parameterized by the service time distributions. Through the smoothed perturbation analysis (SPA) technique, we present under some mild conditions a unified approach to give the strongly consistent estimator for the gradient of the steady-state mean sojourn time with respect to the parameter of service time distributions, provided that it exists. Although the implementation of the SPA requires the additional sub-paths in general, the derived estimator is given as suitable for single-run computation. Simulation results are presented for queues with non-preemptive and preemptive-resume priority disciplines which demonstrate the performance of our estimators.
Stereological estimation of the size distribution for systems of random convex polyhedrons
Authors: Ohser Joachim , Nagel Werner
Published: April 2006

Consider homogeneous spatial particle systems consisting of homothetic particles. Important examples are systems of balls or systems of homothetic convex polyhedrons. For such models the distribution of the typical particle can be expressed by the distribution of one size parameter. We are interested in the stereological estimation of this size parameter distribution from observations on two-dimensional section planes.
Loss networks under diverse routing: the symmetric star network
Authors: Hunt P. J.
Published: April 2006

A highly symmetric loss network is considered, the symmetric star network previously considered by Whitt and Ziedins and Kelly. As the number of links, K, becomes large, the state space for this process also grows, so we consider a functional of the network, one which contains all information relevant to blocking probabilities within the network but which is easier to analyse. We show that this reduced process obeys a functional law of large numbers and a functional central limit theorem, the limit in this latter case being an Ornstein-Uhlenbeck diffusion process. Finally, by considering the network in equilibrium, we are able to prove that the well-known Erlang fixed point approximation for blocking probabilities is correct to within o(K-1/2) as K → ∞.
Abelian-type expansions and non-linear death processes(II)
Authors: Lefèvre Claude , Picard Philippe
Published: April 2006

This paper is concerned with the study of death processes with time-homogeneous non-linear death rates. An explicit formula is obtained for the joint distribution of the state XT and the variable ∫T0g(Xt), where g is any given real function and T corresponds to some appropriate stopping time. This is achieved by constructing a family of martingales and then by using a particular family of Abel–Gontcharoff pseudopolynomials (the theory of which has been introduced in a companion paper) and related Abelian-type expansions. Moreover, the distribution of the first crossing level of such a death process through a general upper boundary is also evaluated in terms of pseudopolynomials of that kind. The flexibility of the methods developed makes easy the extension to multidimensional death processes.
Typification of Linnaean plant names in Brassicaceae (Cruciferae)
Authors: Cafferty Steve , Jarvis Charles E.
Published: April 2006

Lectotypes, neotypes and epitypes are designated by 23 specialists for 86 previously untypified Linnaean plant names belonging to the family Brassicaceae (Cruciferae), and one epitype is designated for a previously lectotypified name. These newly proposed types support the current usage of the names concerned. Earlier but ineffective or supersedable type statements are discussed.
The status of South American Grimmia herzogii Broth. (Musci)
Authors: Delgadillo M. Claudio , Villaseñor R. José Luis
Published: April 2006

Grimmia herzogii Broth. recently became the centre of a controversy among specialists who consider it separate from or synonymous with G. longirostris Hook. Cluster and ordination analyses have shown that by detailed morphological study, specimens with small ovate to ovate­lanceolate leaves and sinuous setae belong in G. herzogii. Those with comparatively large leaves and straight setae correspond to the concept of G. longirostris and include G. nigella Herz., a species described from heterogeneous materials.
A reassessment of the nomenclature of Matricaria L. and Tripleurospermum Sch. Bip. (Asteraceae)
Authors: Applequist Wendy L.
Published: April 2006

The nomenclature of the taxa often called Matricaria recutita L. (common chamomile) and Tripleurospermum perforatum (Mérat) Laínz (scentless mayweed) is discussed with reference to nomenclatural history, recent typifications, and the provisions of the International Code of Botanical Nomenclature. Justification is presented for the use of the name Matricaria chamomilla L. for common chamomile and the epithet inodorum rather than perforatum for scentless mayweed, both of which have been avoided in much recent literature. Varietal nomenclature of M. chamomilla is addressed, and the combination Tripleurospermum maritimum subsp. inodorum is validated.
Some applications of the theory of infinite capacity service systems to a single server system with linearly state dependent service
Authors: Conolly B. W.
Published: April 2006

A certain single server queueing system with negative exponential service with mean rate nμ, when the system contains n customers, and Poisson arrivals, is formally equivalent to the infinite capacity system M/M/∞. This equivalence is exploited to yield in a very simple manner results for the single server system which were previously obtained by difficult analysis (see Hadidi (1969)).
Mean free paths in a convex reflecting region
Authors: Kingman J. F. C.
Published: April 2006

Kendall and Moran ([4], §3.41) consider the following problem, which first arose in connexion with the acoustical design of auditoria. Imagine a convex room with perfectly reflecting walls, and suppose that a particle is projected from a point inside the room. Then it will bounce around the room, and its trajectory will consist of a large number of straight segments. The average length of these segments is the mean free path of the particle.
Selective interaction between two independent stationary recurrent point processes
Authors: Srinivasan S. K. , Rajamannar G.
Published: April 2006

In an earlier contribution to this Journal, Ten Hoopen and Reuver [5] have studied selective interaction of two independent recurrent processes in connection with the unitary discharges of neuronal spikes. They have assumed that the primary process called excitatory is a stationary renewal point process characterised by the interval distribution ϕ(t). The secondary process called the inhibitory process also consists of a series of events governed by a stationary renewal point process characterised by the interval distribution Ψ(t). Each secondary event annihilates the next primary event. If there are two or more secondary events without a primary event, only one subsequent primary event is deleted. Every undeleted event gives rise to a response. For this reason, undeleted events may be called registered events. Ten Hoopen and Reuver have studied the interval distribution between two successive registered events. As is well-known, the interval distribution does not fully characterise a point process in general and in this case it would be interesting to obtain other statistical features like the moments of the number of undeleted events in a given interval as well as correlations of these events. The object of this short note is to point out that the point process consisting of the undeleted events can be studied directly by the recent techniques of renewal point processes ([1], [3]).
On the optimal control of stationary diffusion processes with inaccessible boundaries and no discounting
Authors: Morton R.
Published: April 2006

Because there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.
Some perturbation results for the single-server queue with Poisson input
Authors: Hasofer A. M.
Published: April 2006

In a previous paper [2] the author has studied the single-server queue with non-homogeneous Poisson input and general service time, with particular emphasis on the case when the parameter of the Poisson input is of the form
A persistency problem connected with a point process
Authors: Elfving G.
Published: April 2006

Imagine a man owning a commodity, e.g., a house, which is for sale. Offers at varying amounts are coming in every now and then. The longer he postpones selling the more he loses because of deterioration, interest losses, or the like. At each offer he must decide whether to accept it or wait for a better one. (A more picturesque example would be that of a girl scrutinizing successive suitors.)
A theorem on Markov branching processes
Authors: Reynolds John F.
Published: April 2006

This paper derives an expression for the correlation function of a continuous parameter branching process, and obtains conditions for the convergence of this function to a zero or positive value.
A more general form of a theorem of Crofton
Authors: Ruben H. , Reed W. J.
Published: April 2006

Let Dj be a domain in nj,-dimensional Euclidean space, for j = 1, …, k. Suppose that for each j = 1,…, k, Nj points are chosen independently at random in Dj. A theorem, which is an extension of a theorem of Crofton, is proved about the expected value of functions of the points.
General stochastic epidemic with recovery
Authors: Billard L.
Published: April 2006

The general epidemic model does not provide for an infective recovering and thence being susceptible to further infection in the course of the epidemic. By considering the case in which recovery can occur once, we show how the state probabilities can be found for the stochastic model. This is readily extended to allow recovery up to a finite number of times.
The Markovian self-exciting process
Authors: Oakes David
Published: April 2006

The self-exciting point process with exponential exciting function is investigated using the immigration birth representation of the process. The counting distribution is derived explicitly and some simpler interval properties are given. In particular, it is shown that the serial covariances of the interval sequence decrease monotonically to zero.
On a quickest detection problem with costly information
Authors: Balmer D. W.
Published: April 2006

Problems of detecting the arrival of a ‘disorder' are of central importance in quality control. One such problem is formulated as an optimal stopping problem in which information about the process to be stopped may be bought at any time. A control policy will involve a determination of times at which information should be bought and of the time at which the process should be stopped. It is shown that in all circumstances there is a unique policy satisfying necessary conditions for optimality; the various circumstances and control policies are specified.
The behavior near the origin of the supremum functional in a process with stationary independent increments
Authors: Resnick Sidney I. , Rubinovitch Michael
Published: April 2006

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z1(t) = X(t), Z2(t) = sup0≦s≦tX(s) and Z3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: limt↓0t—1P[Zi(t) > x] = M+(x) at all points of continuity of M+, the Lévy measure of X.
On Gordin's central limit theorem for stationary processes
Authors: Eagleson G. K.
Published: April 2006

The central limit theorem for ergodic stationary processes obtained by Gordin is shown to hold for general stationary processes. In this case, the limit law is a mixture of normals.
A queue with poisson input and semi-Markov service times: busy period analysis
Authors: Purdue Peter
Published: April 2006

We discuss here an extension of a queueing model studied by Neuts and also by Çinlar. We obtain a matrix form of Takács' equations and exhibit the equilibrium conditions. We also show that the conditions imposed by Neuts and by Çinlar in order to obtain their results concerning the busy period are not necessary.
Buffon's problem with a long needle
Authors: Diaconis Persi
Published: April 2006

A needle of length l dropped at random on a grid of parallel lines of distance d apart can have multiple intersections if l > d. The distribution of the number of intersections and approximate moments for large l are derived. The distribution is shown to converge weakly to an arc sine law as l/d→∞.
On a functional equation for general branching processes
Authors: Doney R. A.
Published: April 2006

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.
A non-central multivariate hypergeometric distribution arising from biased sampling with application to selective predation
Authors: Chesson Jean
Published: April 2006

Selective predation is an example of biased sampling which gives rise to a non-central multivariate hypergeometric distribution. The probability distribution function of this distribution is derived.
An elementary proof for the Rao-Rubin characterization of the Poisson distribution
Authors: Shanbhag D. N.
Published: April 2006

This note gives elementary proofs for the characterizations of the Poisson distribution given by Rao and Rubin (1964) and Talwalker (1970) and disproves a conjecture of R. C. and A. B. L. Srivastava (1970).
Information flow in some classes of Markov systems
Authors: Dawson D. A.
Published: April 2006

The information flow in discrete Markov systems provides a method for determining that such a system is ergodic. Estimates are obtained for the information flow in some classes of Markov systems and using these estimates criteria for the ergodicity of the systems are established.
Results on the intersection of randomly located sets
Authors: Streit Franz
Published: April 2006

Randomly generated subsets of a point-set A0 in the k-dimensional Euclidean space Rk are investigated. Under suitable restrictions the probability is determined that a randomly located set which hits A0. is a subset of A0. Some results on the expected value of the measure and the surface area of the common intersection-set formed by n randomly located objects and A0 are generalized and derived for arbitrary dimension k.
On single- and multi-type general age-dependent branching processes
Authors: Doney R. A.
Published: April 2006

We consider a p-type general age-dependent branching process (g.a.d.b.p) in which each individual lives a random length of time and at random times during its life and possibly after its death gives birth to offspring. If this process starts with one individual of type i then it turns out that the number of individuals of type i alive at time t forms a one-type g.a.d.b.p. This fact is exploited to establish a NASC for a properly normalized, supercritical p-type g.a.d.b.p. to have a limiting distribution which is not degenerate at zero.
Two-stage output procedure of a finite dam
Authors: Zuckerman Dror
Published: April 2006

The input of water into a finite dam is a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with cost KM, (K ≧ 0), or decreased from M to 0 with zero cost. There is a reward of A monetary units for each unit of output, (A > 0). We will consider the problem of specifying an optimal control output policy under the following optimal criteria:

(a)

Minimum total long-run average cost per unit time.

(b)

Minimum expected total discounted cost.
Evaluations of absorption probabilities for the Wiener process on large intervals
Authors: Park C. , Schuurmann F. J.
Published: April 2006

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦TW(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.
Random walks on a dodecahedron
Authors: Letac G. , Takács L.
Published: April 2006

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn+1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X0 = A] and P[In = A | I0 = A]. The methods used are applicable to other solids.
Average delay in queues with non-stationary Poisson arrivals
Authors: Ross Sheldon M.
Published: April 2006

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1.

We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.
Shock models with NBUE survival
Authors: Block Henry W. , Savits Thomas H.
Published: April 2006

Conditions are given on a process so that a shock model governed by the process will have nbue survival time. A related condition is given so that survival will be NBU and it is shown that a pure birth process satisfies this condition. A result of A-Hameed and Proschan is obtained as a corollary.
Branching processes with random absorbing processes
Authors: Bruss F. Thomas
Published: April 2006

A discrete Galton–Watson process is modified by an absorbing process, which, within each generation, eliminates a subset of the living particles without leaving offspring. The absorbing process will be only roughly specified by its expected efficiency on the associated process. We give a sharp sufficient condition for the joint process to be extinguished with probability one and give, after an easy generalization, an example for possible biomedical applications.
La formule de Steiner pour les érosions
Authors: Matheron G.
Published: April 2006

If A and K are compact convex sets in ℝn, a Steiner-type formula is valid for the erosion of A by K if and only if A is open with respect to K. The if part is proved in the general case. The only if part is conjectured, and proved only in the case n = 2.
Limiting diffusion for random walks with drift conditioned to stay positive
Authors: Kao Peichuen
Published: April 2006

Let {ξk : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ1} = μ ≠ 0. Form the random walk {Sn : n ≧ 0} by setting S0, Sn = ξ1 + ξ2 + ··· + ξn, n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ1) that the finite-dimensional distributions of Xn, conditioned on n < N < ∞ converge to those of the Brownian excursion process.
On the quantiles of the gamma and F distributions
Authors: Saunders I. W. , Moran P. A. P.
Published: April 2006

We show, in the context of a neurobiological problem, that ζ a(α)−ζ a(β) increases with a, while ζ a (α)/ζ a (β) decreases with a, where 0 < β < α < 1 and ζ a (α) is the α th quantile of either the gamma distribution with shape parameter a, or a scaled F distribution with parameters 2a and 2m for any m.
Dryness of discrete dams: comments on a paper by Tin and Phatarfod
Authors: Balagopal K.
Published: April 2006

The utilisation factor for a discrete dam, defined as the stationary probability of non-emptiness of the dam just before release, is obtained for a class of models that includes the Odoom–Lloyd model, the Anis–Lloyd model, the model of Herbert, etc. The method used points to a different approach to measuring dam utilisation via the actual utilisation, which appears to be more fruitful, and this is discussed in the last section.
A Markov construction for a multidimensional point process
Authors: Isham Valerie
Published: April 2006

A sequence of finite point processes {Pn} is constructed in using a Markov sequence of points. Essentially, in the process Pn consisting of n events, the coordinates of these events are simply the first n points of a Markov sequence suitably scaled so that the average density of the process is independent of n. The second-order properties of Pn are discussed and sufficient conditions are found for Pn to converge in distribution to a Poisson process as n →∞. A simple example involving the cardioid distribution is described.
Estimation of the variance for the multitype Galton–Watson process
Authors: Nanthi K.
Published: April 2006

This paper is concerned with the estimation of the variance for the multitype Galton-Watson process X = {Xn = (Xn(1),…, Xn(p)); n ≧ 0}. Two estimators for the variance matrix are obtained and asymptotic results for the estimators are given. The first is a maximum likelihood estimator based upon knowledge of individual offspring sizes, the second estimator is based on parent-offspring type combination counts only. Estimators for the asymptotic variances of the Asmussen and Keiding estimator and Becker estimator are also proposed.
On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions
Authors: König Dieter , Miyazawa Masakiyo , Schmidt Volker
Published: April 2006

For several queueing systems, sufficient conditions are given ensuring that from the coincidence of some time-stationary and customer-stationary characteristics of the number of customers in the system such as idle or loss probabilities it follows that the arrival process is Poisson.
A note on the convexity of performance measures of M/M/c queueing systems
Authors: Lee Hau Leung , Cohen Morris A.
Published: April 2006

Convexity of performance measures of queueing systems is important in solving control problems of multi-facility systems. This note proves that performance measures such as the expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.
The stable pedigrees of critical branching populations
Authors: Nerman Olle
Published: April 2006

The asymptotic sizes and compositions of critical general branching processes conditioned on non-extinction are treated. First, results are given for processes counted with random characteristics allowed to depend on the whole daughter processes, then these are used to derive conditional limit theorems for the pedigrees of randomly sampled individuals among populations counted with 0–1-valued characteristics. The limiting stable pedigrees have nice independence structures and are easily derived from the original branching laws.
On an integral equation for first-passage-time probability densities
Authors: Ricciardi L. M. , Sacerdote L. , Sato S.
Published: April 2006

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.
A property of longtailed distributions
Authors: Embrechts Paul , Omey Edward
Published: April 2006

We investigate sufficient conditions so that is subexponential. Here F is a distribution function on [0, ∞[, with finite mean. Some applications to risk theory and rates of convergence in renewal theory are given.
On a stopping problem involving refusal and forced stopping
Authors: Yasuda Masami
Published: April 2006

Although the usual optimal stopping problem is described as a Markov decision process with two decisions, stop and continue, we shall consider a model which distinguishes the observer's strategy from the system's two decisions. The observer can select a strategy defined on an action space, and the decision of the system to stop or continue is determined by a prescribed conditional probability. For this model, it may happen that the strategy (a) to stop is refused, or (b) to continue is forcibly stopped. This is a slight modification of the one-dimensional stopping problem by involving refusal and forced stopping. The model is motivated by the uncertain secretary choice problem of Smith (1975) and the multivariate stopping problem of Kurano, Yasuda and Nakagami (1980), (1982).
Stochastic barriers for the Wiener process
Authors: Park C. , Beekman J. A.
Published: April 2006

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ TW(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ TW(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.
A stability property of the Ewens sampling formula
Authors: Sawyer Stanley
Published: April 2006

An error bound for convergence to the Ewens sampling formula is given where the population size or mutation rate may vary from generation to generation, or the population is not yet at equilibrium. An application is given to a model of Hartl and Campbell about selectively-equivalent subtypes within a class of deleterious alleles, and a theorem is proven showing that the size of the deleterious class stays within bounds sufficient to apply the first result. Generalizations are discussed.
Characterizations of exponential distributions by independent non-stationary record increments
Authors: Pfeifer Dietmar
Published: April 2006

A non-homogeneous version of the classical record process is presented which allows two different characterizations of exponential distributions by independent non-stationary record increments. A connection with the interarrival times of the corresponding record counting process (which is pure birth) is also pointed out.
The risk-sensitive homing problem
Authors: Kuhn Jonathan
Published: April 2006

The ‘homing' optimal control problem, described in Whittle and Gait (1970), is given a risk-sensitive formulation. It is shown that the reduction of an optimally controlled homing problem to the treatment of an uncontrolled process, demonstrated by Whittle and Gait, can be achieved in the risk-sensitive case. Two scalar problems are analyzed in detail.
The shortest queue problem
Authors: Halfin Shlomo
Published: April 2006

A Poisson stream of customers arrives at a service center which consists of two single-server queues in parallel. The service times of the customers are exponentially distributed, and both servers serve at the same rate. Arriving customers join the shortest of the two queues, with ties broken in any plausible manner. No jockeying between the queues is allowed. Employing linear programming techniques, we calculate bounds for the probability distribution of the number of customers in the system, and its expected value in equilibrium. The bounds are asymptotically tight in heavy traffic.
The maximum in critical Galton–Watson and birth and death processes
Authors: Kämmerle K. , Schuh H.-J.
Published: April 2006

In this paper recent results by Weiner [10] on Mn:= max{Z0, · ··, Zn} are strengthened and generalized, where (Zn)n is a critical Galton–Watson branching process with finite and positive offspring variance and Z0 ≡ 1. It is shown that

Explicit asymptotic bounds are given for with . If (Zn)n has a linear fractional offspring distribution, it can be embedded in a critical birth and death process (Ẑ t)t. Using martingale methods one obtains thereof.

These results generalize to the case Z0 ≡ k.
Scheduling jobs with stochastically ordered processing times on parallel machines to minimize expected flowtime
Authors: Weber R. R. , Varaiya P. , Walrand J.
Published: April 2006

A number of jobs are to be processed using a number of identical machines which operate in parallel. The processing times of the jobs are stochastic, but have known distributions which are stochastically ordered. A reward r(t) is acquired when a job is completed at time t. The function r(t) is assumed to be convex and decreasing in t. It is shown that within the class of non-preemptive scheduling strategies the strategy SEPT maximizes the expected total reward. This strategy is one which whenever a machine becomes available starts processing the remaining job with the shortest expected processing time. In particular, for r(t) = – t, this strategy minimizes the expected flowtime.
Convex ordering of sojourn times in single-server queues: extremal properties of FIFO and LIFO service disciplines
Authors: Shanthikumar J. George , Sumita Ushio
Published: April 2006

In this paper, the extremal properties of the ergodic sojourn times in G/G/1queues under various service disciplines are studied in terms of the convex ordering. It is shown that among work-conserving non-preemptive service disciplines that are service time independent, the FIFO and the LIFO service disciplines provide the minima and the maxima, respectively, of the ergodic sojourn times for any G/G/1 queue. For G/M/1 queues, this class of work-conserving service disciplines is extended to include preemptive/resume disciplines. In this case the FIFO and LIFO-P (preemptive/resume LIFO) service disciplines attain the minima and maxima, respectively. These extend results of Durr (1971), Kingman (1962) and a recent result of Ramaswami (1984). Further results are obtained for G/Em/1 and G/D/1 queues.
Optimum replacement of a system subject to shocks
Authors: Abdel-Hameed Mohamed
Published: April 2006

A system is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement time for the system. Boland and Proschan [4] consider periodic replacement of the system and give sufficient conditions for the existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. [3] establish similar results assuming that cost structure is time dependent, still requiring that the shock process is a non-homogeneous Poisson process. We show via a sample path argument that the results of [3] and [4] hold for any counting process whose jump size is of one unit magnitude.