bluuebunny/crossref_metadata_2025 huggingface dataset:
Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector
Authors: Neuts Marcel F.
Published: April 2006

It is shown that a class of infinite, block-partitioned, stochastic matrices has a matrix-geometric invariant probability vector of the form (x0, x1,…), where xk = x0Rk, for k ≧ 0. The rate matrix R is an irreducible, non-negative matrix of spectral radius less than one. The matrix R is the minimal solution, in the set of non-negative matrices of spectral radius at most one, of a non-linear matrix equation.

Applications to queueing theory are discussed. Detailed explicit and computationally tractable solutions for the GI/PH/1 and the SM/M/1 queue are obtained.
Probability distributions for the strength of fibrous materials under local load sharing I: Two-level failure and edge effects
Authors: Harlow D. Gary , Phoenix S. Leigh
Published: April 2006

The focus of this paper is on obtaining a conservative but tight bound on the probability distribution for the strength of a fibrous material. The model is the chain-of-bundles probability model, and local load sharing is assumed for the fiber elements in each bundle. The bound is based upon the occurrence of two or more adjacent broken fiber elements in a bundle. This event is necessary but not sufficient for failure of the material. The bound is far superior to a simple weakest link bound based upon the failure of the weakest fiber element. For large materials, the upper bound is a Weibull distribution, which is consistent with experimental observations. The upper bound is always conservative, but its tightness depends upon the variability in fiber element strength and the volume of the material. In cases where the volume of material and the variability in fiber strength are both small, the bound is believed to be virtually the same as the true distribution function for material strength. Regarding edge effects on composite strength, only when the number of fibers is very small is a correction necessary to reflect the load-sharing irregularities at the edges of the bundle.
On the Markov property of the GI/G/∞ Gaussian limit
Authors: Glynn Peter W.
Published: April 2006

It is shown that the heavy-traffic Gaussian limit for GI/G/∞ queues is Markovian if and only if the service-time distribution H(t) is of the form 1-H(t) = pe–αt for α > 0 and 0 < p ≦ 1.
Control of arrivals to a stochastic input–output system
Authors: Johansen Søren Glud , Stidham Shaler
Published: April 2006

The problem of controlling input to a stochastic input-output system by accepting or rejecting arriving customers is analyzed as a semi-Markov decision process. Included as special cases are a GI/G/1 model and models with compound input and/or output processes, as well as several previously studied queueing-control models. We establish monotonicity of socially and individually optimal acceptance policies and the more restrictive nature of the former, with random rewards for acceptance and both customer-oriented and system-oriented non-linear waiting costs. Distinctive features of our analysis are (i) that it allows dependent interarrival times and (ii) that the monotonicity proofs do not rely on the standard concavity-preservation arguments.
Limit theorems for the simple branching process allowing immigration, I. The case of finite offspring mean
Authors: Pakes Anthony G.
Published: April 2006

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process.

The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.
An analysis of sex in random environments, I
Authors: Hines W. G. S. , Moore W. S.
Published: April 2006

A model of the effects of random variations in environment upon reproductive fitness of sexual and asexual members of a population is analyzed, using individual selection arguments. Conditions are found under which sexuality does result in a net selective advantage, with the conditions per locus becoming markedly less severe as the number of loci considered increases.
Maximum likelihood estimation for branching processes with immigration
Authors: Bhat B. R. , Adke S. R.
Published: April 2006

This paper establishes the strong consistency of the maximum likelihood estimators of the parameters of discrete- and continuous-time Markov branching processes with immigration. The asymptotic distributions of the maximum likelihood estimators of the parameters of a Galton–Watson branching process with immigration are also obtained.
Characterizations of Poisson traffic streams in Jackson queueing networks
Authors: Melamed Benjamin
Published: April 2006

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues.

A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.
Statistical inference for Markov processes when the model is incorrect
Authors: Foutz Robert V. , Srivastava R. C.
Published: April 2006

Statistical inference for Markov processes is commonly based on the maximum likelihood method of estimation and the likelihood ratio criterion for testing hypotheses. Construction of estimators and test statistics by these methods require that a model be chosen in the form of a family of transition density functions. In this paper, asymptotic properties of the maximum likelihood estimator and of the likelihood ratio statistic λn are examined when the model chosen for their construction is incorrect—that is, when no density in the model is a density for the transition probability distribution of the Markov process. It is shown that if and λn are constructed from a ‘regular’ incorrect model, then is consistent and asymptotically normally distributed and the asymptotic null distribution of −2 log λn is that of a linear combination of independent chi-squared random variables. These results are applied to propose measures of the performance of the test based on λn when the statistic is constructed from an incorrect model.
Comparing stochastic systems using regenerative simulation with common random numbers
Authors: Heidelberger Philip , Iglehart Donald L.
Published: April 2006

Suppose two alternative designs for a stochastic system are to be compared. These two systems can be simulated independently or dependently. This paper presents a method for comparing two regenerative stochastic processes in a dependent fashion using common random numbers. A set of sufficient conditions is given that guarantees that the dependent simulations will produce a variance reduction over independent simulations. Numerical examples for a variety of simple stochastic models are included which illustrate the variance reduction achieved.
Random compact convex sets which are infinitely divisible with respect to Minkowski addition
Authors: Mase Shigeru
Published: April 2006

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.
On N-person stochastic games by denumerable state space
Authors: Federgruen A.
Published: April 2006

This paper considers non-cooperative N-person stochastic games with a countable state space and compact metric action spaces. We concentrate upon the average return per unit time criterion for which the existence of an equilibrium policy is established under a number of recurrency conditions with respect to the transition probability matrices associated with the stationary policies. These results are obtained by establishing the existence of total discounted return equilibrium policies, for each discount factor α ∈ [0, 1) and by showing that under each one of the aforementioned recurrency conditions, average return equilibrium policies appear as limit policies of sequences of discounted return equilibrium policies, with discount factor tending to one.

Finally, we review and extend the results that are known for the case where both the state space and the action spaces are finite.
An application of time reversal of Markov processes to a problem of population genetics
Authors: Nagasawa Masao , Maruyama Takeo
Published: April 2006

A formula is proved for the expected value of sum of a function of gene frequency along a sample path in the past, given the present frequency. The proof and the explanation of the formula is based on the general theory of time reversal of Markov processes. Moreover, a relation between time reversal and conditional processes is discussed, and it is shown that the fictitious drift term appears when one looks back at the history of mutants given the present frequency.
Application of the Galton-Watson process to the kin number problem
Authors: Waugh W. A. O'n.
Published: April 2006

The kin number problem concerns the relationship between the distribution of the number of offspring of a randomly chosen individual in a population, and that of the number of relatives of various degrees of affinity of a randomly chosen individual (referred to as ‘Ego’). This problem is considered in terms of a population consisting of a large number of simultaneously developing Galton-Watson family trees. By time-reversal starting with the epoch at which Ego is sampled, it is shown that the number of offspring of Ego's parent (Ego plus siblings) and the number of offspring of Ego's grandparent (Ego's parent plus Ego's parent's siblings) have the same distribution, and the probability generating function (p.g.f.) is obtained in terms of the reproduction p.g.f. Further development of the method yields the joint p.g.f. of any number of generations prior, and subsequent, to that of Ego. Means and variances are obtained, and numerical examples are given based on data for the reproduction p.g.f. in two contrasting human populations. Various applications including demography and cancer research are discussed.
Limit theorems for point processes generated in a general branching process
Authors: Härnqvist Martin
Published: April 2006

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit.

Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.
Competing risks and independent minima: a marked point process approach
Authors: Arjas Elja , Greenwood Priscilla
Published: April 2006

Given an indexed family of rate functions, we construct a family of independent random variables such that their minimum and the corresponding index form a marked point having the same rates. If the rates have common discontinuities, the random variables are conditionally independent, given a random allocation of these jumps. Between any two adjacent points of a marked point process there is a similar structure. Coherent functions in system lifetime theory provide examples.

Non-identifiability of multivariate lifetime distributions from mortality data is interpreted in this context.
The convex hull of a spherically symmetric sample
Authors: Eddy William F. , Gale James D.
Published: April 2006

Using the isomorphism between convex subsets of Euclidean space and continuous functions on the unit sphere we describe the probability measure of the convex hull of a random sample. When the sample is spherically symmetric the asymptotic behavior of this measure is determined. There are three distinct limit measures, each corresponding to one of the classical extreme-value distributions. Several properties of each limit are determined.
Gradient approach for recursive estimation and control in finite Markov chains
Authors: El-Fattah Yousri M.
Published: April 2006

The problem studied is that of controlling a finite Markov chain so as to maximize the long-run expected reward per unit time. The chain's transition probabilities depend upon an unknown parameter taking values in a subset [a, b] of Rn. A control policy is defined as the probability of selecting a control action for each state of the chain. Derived is a Taylor-like expansion formula for the expected reward in terms of policy variations. Based on that result, a recursive stochastic gradient algorithm is presented for the adaptation of the control policy at consecutive times. The gradient depends on the estimated transition parameter which is also recursively updated using the gradient of the likelihood function. Convergence with probability 1 is proved for the control and estimation algorithms.
Estimation of the parameters of a semi-Markov process from censored records
Authors: Thompson M. E.
Published: April 2006

The estimation of the parameters of a discrete-time semi-Markov process is considered, when the data consist of records of a large number of individuals observed in a specified short period of time. Such a problem may arise in the modelling of intra-urban mobility. Methods of estimation of the parameters are suggested, and a numerical example discussed for the case when the observation period consists of three consecutive time points.
The point-process approach to age- and time-dependent branching processes
Authors: Kimmel Marek
Published: April 2006

The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.
An analysis of the Pólya point process
Authors: Waymire Ed , Gupta Vijay K.
Published: April 2006

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.
The rate of convergence of extremes of stationary normal sequences
Authors: Rootzén Holger
Published: April 2006

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order

where ρ is the maximal correlation, ρ =sup {0, r1, r2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.
Birth, immigration and catastrophe processes
Authors: Brockwell P. J. , Gani J. , Resnick S. I.
Published: April 2006

We consider Markov models for growth of populations subject to catastrophes. Emphasis is placed on discrete-state models where immigration is possible and the catastrophe rate is population-dependent. Explicit formulas for descriptive quantities of interest are derived when catastrophes reduce population size by a random amount which is either geometrically, binomially or uniformly distributed. Comparison is made with continuous-state Markov models in the literature in which population size evolves continuously and deterministically upwards between random jumps downward.
Stochastic theory of a fluid model of producers and consumers coupled by a buffer
Authors: Mitra Debasis
Published: April 2006

This paper analyzes, derives efficient computational procedures and numerically investigates the following fluid model which is of interest in manufacturing and communications: m producing machines supply a buffer, n consuming machines feed off it. Each machine independently alternates between exponentially distributed random periods in the ‘in service' and ‘failed' states. Producers/consumers have their own failure/repair rates and working capacities. When the buffer is either full or empty some of the machines in service are not utilized to capacity; otherwise they are fully utilized. Our main result is for the state distribution of the Markovian system in equilibrium which is the solution of a system of differential equations. The spectral expansion for its solution is obtained. Two important decompositions are obtained: the eigenvectors have the Kronecker-product form in lower-dimensional vectors; the characteristic polynomial is factored with each factor an explicitly given polynomial of degree at most 4. All eigenvalues are real. For each of various cases of the model, a system of linear equations is derived from the boundary conditions; their solution complete the spectral expansion. The count in operations of the entire procedure is O(m3n3): independence from buffer size exemplifies an important attraction of fluid models. Computations have revealed several interesting features, such as the benefit of small machines and the inelasticity of production rate to inventory. We also give results on the eigenvalues of a more general fluid model, reversible Markov drift processes.
Heavy-usage asymptotic expansions for the waiting time in closed processor-sharing systems with multiple classes
Authors: Morrison J. A. , Mitra D.
Published: April 2006

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of closed queueing networks in heavy usage, which in practical terms means that the processor is utilized more than about 80 per cent. This paper extends recent work by Mitra and Morrison [10] on the same system in normal usage. The closed system has a CPU operating under the processor-sharing (‘time-slicing’) discipline and a bank of terminals. The presence of multiple job-classes allows distinctions in the user’s behavior in the terminal and in the service requirements. This work is primarily applicable to the case of large numbers of terminals. We give an effective method for calculating, for the equilibrium waiting time, the first and second moments and the leading term in the asymptotic approximation to the distribution. Our results are in the form of asymptotic expansions in inverse powers of , where N is a large parameter. The expansion coefficients depend on the classical parabolic cylinder functions.
A variant of the Ehrenfest model
Authors: Van Beek K. W. H. , Stam A. J.
Published: April 2006

The Ehrenfest model is modified by drawing r balls at a time. The stationary distribution is the same as for r = 1.
On the limit behavior of certain quantities in a subcritical storage model
Authors: Puri Prem S. , Tollar Eric S.
Published: April 2006

The limit behavior of the content of a subcriticai storage model defined on a semi-Markov process is examined. This is achieved by creating a renewal equation using a regeneration point (i0,0) of the process. By showing that the expected return time to (i0, 0) is finite, the conditions needed for the basic renewal theorem are established. The joint asymptotic distribution of the content of the storage at time t and the accumulated amount of the unmet (lost) demands during (0,t) is then established by showing the asymptotic independence of these two.
Aggregation of independent Paretian random variables
Authors: Roehner Bertrand , Winiwarter Peter
Published: April 2006

Empirical Paretian distributions play an important role in urban demography, size distributions of firms and income distributions; hence the addition of Paretian random variables is of interest. First, we give the asymptotic behavior (for large values of the variable) of the density function of a sum of n independently distributed Paretian variables. We then obtain the limiting distribution of an infinite sum of (i.i.d) Paretian variables and link our results with the theory of stable distributions.
A note on explosiveness of Markov branching processes
Authors: Grey D. R.
Published: April 2006

It is shown that no condition of the form E[g(N)] =∞, where N denotes a typical family size, is sufficient for explosiveness of a Markov branching process.
Robust population models with application in genetic and epidemic theory
Authors: Heyde C. C.
Published: April 2006

From the point of view of practical usefulness, the results of modelling exercises should not be too sensitive to the detailed assumptions built into the model. Consequently, it is a great advantage if structural assumptions are kept to a minimum.
Open networks of queues: their algebraic structure and estimating their transient behavior
Authors: Massey William A.
Published: April 2006

We develop the mathematical machinery in this paper to construct a very general class of Markovian network queueing models. Each node has a heterogeneous class of customers arriving at their own Poisson rate, ultimately to receive their own exponential service requirements. We add to this a very general type of service discipline as well as class (node) switching. These modifications allow us to model in the limit, service with a general distribution. As special cases for this model, we have the product-form networks formulated by Kelly, as well as networks with priority scheduling. For the former, we give an algebraic proof of Kelly's results for product-form networks. This is an approach that motivates the form of the solution, and justifies the various needs of local and partial balance conditions.

For any network that belongs to this general model, we use the operator representation to prove stochastic dominance results. In this way, we can take the transient behavior for very complicated networks and bound its joint queue-length distribution by that for M/M/1queues.
Etude d'une file GI/G/1 à service autonome (avec vacances du serveur)
Authors: Fricker C.
Published: April 2006

The purpose of this letter is to study a modified GI/G/1 queueing system in which the server becomes unavailable for (independent) random periods each time he is free. This problem was first studied by Gelenbe and Iasnogorodski [6] who obtained the stationary law of the waiting time of a customer. We construct a simple probabilistic model coupling a G//G/1 queue with an autonomous server (in Borovkov's terminology [1]) with a GI/G/1 queue of classical type having the same characteristics, to compare them stochastically. We prove that the waiting time is a Markov chain, using a renewal process property which has not previously been noted.
On random rotations
Authors: Roberts P. H. , Winch D. E.
Published: April 2006

A body receives a sequence of rotations through a fixed angle about an axis whose direction is arbitrary. The probability distribution governing the resulting orientation of the body is determined. The problem is generalized to the case where the axis of each individual rotation makes the same angle, Θ, with an axis fixed in the body but is otherwise random. The resulting distribution is shown, in the case , to reduce to the Roberts-Ursell distribution for random walk on a sphere. Some diffusion limits are examined.
Mutual and shared neighbor probabilities: finite- and infinite-dimensional results
Authors: Schilling M. F.
Published: April 2006

Let X1, ···, Xn be i.i.d. random variables defined in ℝd having common continuous density f(x), and let Rij be the rank of Xj in the ordered list of distances from X¡. Both the mutual neighbor probabilities p1(r, s) = P(R12 = r, R21 = s) and the neighbor-sharing probabilities p2(r, s) = P(R13 = r, R23 = s) are studied from an asymptotic viewpoint. Infinite-dimensional limits are found for both situations and take particularly simple forms. Both cases exhibit considerable stability across dimensions and thus are well approximated by their infinite-dimensional values. Tables are provided to support the results given.
The hazards of optimal proofreading
Authors: Tierney Luke
Published: April 2006

In a recent paper (Yang et al. (1982)) a model for proofreading was proposed in which a reader has a fixed probability p of detecting a misprint in a document containing a Poisson number of errors. This note points out that the conclusions derived from such a model can be extremely misleading if the probability of detecting a misprint varies from one misprint to another.
Finite birth-and-death models in randomly changing environments
Authors: Gaver D. P. , Jacobs P. A. , Latouche G.
Published: April 2006

An efficient computational approach to the analysis of finite birth-and-death models in a Markovian environment is given. The emphasis is upon obtaining numerical methods for evaluating stationary distributions and moments of first-passage times.
On conditional Ornstein–Uhlenbeck processes
Authors: Salminen P.
Published: April 2006

It is well known that the law of a Brownian motion started from x > 0 and conditioned never to hit 0 is identical with the law of a three-dimensional Bessel process started from x. Here we show that a similar description is valid for all linear Ornstein–Uhlenbeck Brownian motions. Further, using the same techniques, it is seen that we may construct a non-stationary Ornstein–Uhlenbeck process from a stationary one.
Asymptotic analysis for interactive oscillators of the van der Pol type
Authors: Narita Kiyomasa
Published: April 2006

We consider the N-oscillator system of the van der Pol type, which contains a small positive parameter ε multiplying the non-linear damping and the random disturbance. For a formulation of the output we take the solution X(t) = (Xi(t))i=1···,N of the system of 2N-dimensional stochastic differential equations. Rotating each component Xi(t) about the origin of the plane by an angle t, we find that on time scales of order 1/ε together with sufficiently large N each Xi(t) behaves as the equi-ultimately bounded solution of an equation of the McKean type admitting a stationary probability distribution.
Note sur un modele de file GI/G/1 a service autonome (avec vacances du serveur)
Authors: Fricker Christine
Published: April 2006

Keilson and Servi introduced in [5] a variation of a GI/G/1 queue with vacation, in which at the end of a service time, either the server is not idle, and he starts serving the first customer in the queue with probability p, or goes on vacation with probability 1 – p, or he is idle, and he takes a vacation. At the end of a vacation, either customers are present, and the server starts serving the first customer, or he is idle, and he takes a vacation. The case p = 1, called the GI/G/1/V queue, was studied analytically by Gelenbe and Iasnogorodski [3] (see also [4]) and then by Doshi [1] and Fricker [2] who obtained stochastic decomposition results on the waiting-time of the nth customer extending the law decomposition result of [3]. Keilson and Servi [5] give a more complete analytic method of treating both the GI/G/1/V model and the Bernoulli vacation model: instead of the waiting time, they use a bivariate process at the service and vacation initiation epochs and the waiting-time distribution is computed as a conditional distribution of the above. In this note the law decomposition result is obtained from a reduction to the GI/G/1/V model with a modified service-time distribution just using the waiting time, with simple path arguments so that by [1] and [2] stochastic decomposition results are valid, which extend the result of [5].
A limit theorem for the reliability of a consecutive-k-out-of-n system
Authors: Papastavridis S.
Published: April 2006

A consecutive-k-out-of-n system consists of n identical and linearly ordered components. The system will fail if and only if at least k consecutive components fail. Let Tn be the system&s lifetime. Then, under very general conditions we prove that there is a positive constant a, so that the distribution of the random variable n(1/ka)Tn converges to a Weibull distribution, as n→∞
Analyticity of Poisson-driven stochastic systems
Authors: Zazanis Michael A.
Published: April 2006

Let ψ be a functional of the sample path of a stochastic system driven by a Poisson process with rate λ . It is shown in a very general setting that the expectation of ψ,Eλ [ψ], is an analytic function of λ under certain moment conditions. Instead of following the straightforward approach of proving that derivatives of arbitrary order exist and that the Taylor series converges to the correct value, a novel approach consisting in a change of measure argument in conjunction with absolute monotonicity is used. Functionals of non-homogeneous Poisson processes and Wiener processes are also considered and applications to light traffic derivatives are briefly discussed.
Local Maxima and the Expected Euler Characteristic of Excursion Sets of χ 2, F and t Fields
Authors: Worsley K. J.
Published: April 2006

The maximum of a Gaussian random field was used by Worsley et al. (1992) to test for activation at an unknown point in positron emission tomography images of blood flow in the human brain. The Euler characteristic of excursion sets was used as an estimator of the number of regions of activation. The expected Euler characteristic of excursion sets of stationary Gaussian random fields has been derived by Adler and Hasofer (1976) and Adler (1981). In this paper we extend the results of Adler (1981) to χ2, F and t fields. The theory is applied to some three-dimensional images of cerebral blood flow from a study on pain perception.
A Central Limit Theorem for Cumulative Processes
Authors: Roginsky Allen L.
Published: April 2006

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.
Coverage of a square lattice by an inclined rectangle
Authors: Rosen Dennis
Published: April 2006

By use of a method of dissection, a formula is derived for the variance in the number of lattice points covered by a rectangle of arbitrary size, lying on a square lattice of unit spacing and inclined to the lattice at an angle of which the tangent is a rational fraction.
The Hawk–Dove game as an average-cost problem
Authors: McNamara J. M. , Merad S. , Collins E. J.
Published: April 2006

This paper considers a version of the Hawk–Dove game of Maynard Smith and Price (1973) in which animals compete for a sequence of food items. Actions may depend on an animal's energy reserves. Costs and transition probabilities under a given policy depend on the mean level of aggressiveness, p, of the rest of the population. We find the optimal policy for a single animal under an average cost criterion when ρ is constant over time. We then consider the whole interacting population when individual members follow the same stationary policy. It is shown that the mean aggressiveness, p, asymptotically approaches a limiting value in this population. We then consider the existence of evolutionarily stable strategies for the population. It is shown that such strategies always exist but may not be unique.
A queueing process with the possibility of customers becoming servers
Authors: Huang Wen-Jang , Puri Prem S.
Published: April 2006

A new queueing system called G/G/{p} is introduced and studied. In this queue, unlike standard queues, the customers after being served are allowed to become servers themselves. More precisely, at the completion of his service each customer is assumed to become a server with probability p or leave the system with probability 1 – p, independent of everything else. We make some comparisons about the waiting times and queue sizes among different queueing systems. We also study the joint distribution of the queue size, the number of servers and the number of departures at time t for exact and asymptotic behavior for large t.
Generalised eigenproblems arising in aggregated markov processes allowing for time interval omission
Authors: Jalali Assad , Hawkes Alan G.
Published: April 2006

We consider a continuous-time Markov chain in which one cannot observe individual states but only which of two sets of states is occupied at any time. Furthermore, we suppose that the resolution of the recording apparatus is such that small sojourns, of duration less than a constant deadtime, cannot be observed. We obtain some results concerning the poles of the Laplace transform of the probability density function of apparent occupancy times, which correspond to a problem about generalised eigenvalues and eigenvectors. These results provide useful asymptotic approximations to the probability density of occupancy times. A numerical example modelling a calcium-activated potassium channel is given. Some generalisations to the case of random deadtimes complete the paper.
A note on the multitype measure branching process
Authors: Li Zeng-Hu
Published: April 2006

The existence of a class of multitype measure branching processes is deduced from a single-type model introduced by Li [8], which extends the work of Gorostiza and Lopez-Mimbela [5] and shows that the study of a multitype process can sometimes be reduced to that of a single-type one.
Existence of non-trivial quasi-stationary distributions in the birth-death chain
Authors: Ferrari Pablo A. , Martínez Servet , Picco Pierre
Published: April 2006

We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.
The µc-rule is not optimal in the second node of the tandem queue: a counterexample
Authors: Hordijk Arie , Koole Ger
Published: April 2006

In this note we give a counterexample which shows that the µc-rule is not optimal in the second node of the tandem queue. This counterexample contradicts the interchange argument in Nain [1] and Nain et al. [2].
Modeling of repairable systems
Authors: Ebrahimi Nader
Published: April 2006

The reliability of many stochastic repairable systems depends on several characteristics that are time dependent. In this paper, we develop general repair models for a repairable system by using auxiliary stochastic processes which describe physical characteristics of the system and derive various properties of resulting models. We also obtain an inference procedure to assess the number of failures and expected number of failures for our proposed models by observing auxiliary processes.

Clearly, our inference procedure is different from the traditional approach, where successive times of occurrence of failures are observed.
The Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Parent Tetrahedron
Authors: Mannion David
Published: April 2006

We solve a problem proposed by V. Klee (1969). He asked for a calculation of κ, the expected value of V, the volume of a daughter tetrahedron whose vertices are chosen at random (i.e. independently and uniformly) in the interior of a parent tetrahedron of unit volume. We discover:

We also calculate the second, fourth and sixth moments of V.
The Early and Final States of an Epidemic in a Large Heterogeneous Population by a Small Initial Number of Infectives
Authors: Butler Steven M.
Published: April 2006

This paper describes the early and final properties of a general S–I–R epidemic process in which the infectives behave independently, each infective has a random number of contacts with the others in the population, and individuals vary in their susceptibility to infection. For the case of a large initial number of susceptibles and a small (finite) initial number of infectives, we derive the threshold behavior and the limiting distribution for the final state of the epidemic. Also, we show strong convergence of the epidemic process over any finite time interval to a birth and death process, extending the results of Ball (1983). These complement some results due to Butler (1994), who considers the case of a large initial number of infectives.
Continuous-time Markov chains in a random environment, with applications to ion channel modelling
Authors: Ball Frank , Milne Robin K. , Yeo Geoffrey F.
Published: April 2006

We study a bivariate stochastic process {X(t)} = Z(t))}, where {XE(t)} is a continuous-time Markov chain describing the environment and {Z(t)} is the process of interest. In the context which motivated this study, {Z(t)} models the gating behaviour of a single ion channel. It is assumed that given {XE(t)}, the channel process {Z(t)} is a continuous-time Markov chain with infinitesimal generator at time t dependent on XE(t), and that the environment process {XE{t)} is not dependent on {Z(t)}. We derive necessary and sufficient conditions for {X(t)} to be time reversible, showing that then its equilibrium distribution has a product form which reflects independence of the state of the environment and the state of the channel. In the special case when the environment controls the speed of the channel process, we derive transition probabilities and sojourn time distributions for {Z(t)} by exploiting connections with Markov reward processes. Some of these results are extended to a stationary environment. Applications to problems arising in modelling multiple ion channel systems are discussed. In particular, we present ways in which a multichannel model in a random environment does and does not exhibit behaviour identical to a corresponding model based on independent and identically distributed channels.
Stationary distributions under mutation-selection balance: structure and properties
Authors: Bürger Reinhard , Bomze Immanuel M.
Published: April 2006

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.
Symmetric queues with batch departures and their networks
Authors: Miyazawa Masakiyo , Wolff Ronald W.
Published: April 2006

Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.
A diffusion model for Bookstein triangle shape
Authors: Kendall Wilfrid S.
Published: April 2006

This reports on work in progress, developing a dynamical context for Bookstein's shape theory. It shows how Bookstein's shape space for triangles arises when the landmarks are moved around by a particular Brownian motion on the general linear group of (2 × 2) invertible matrices. Indeed, suppose that the random process G(t) ∈ GL(2, ℝ) solves the Stratonovich stochastic differential equation dsG = (dsB)G for a Brownian matrix B (independent Brownian motion entries). If {x1 x2, x3} is a fixed (non-degenerate) triple of planar points then Xi(t) = G(t)xi; determines a triple {X1 X2, X3} whose shape performs a diffusion which can be shown to be Brownian motion on the hyperbolic plane of negative curvature − 2.
Statistical studies of the spatial distribution of galaxies
Authors: Martínez Vicent J. , Pons-Bordería María Jesús
Published: April 2006

The statistical study of the clustering patterns formed by the three-dimensional distribution of galaxies is one of the most important observational clues to discriminate between rival theories of structure formation in the Universe.
Mixed curvature measures and a translative integral formula
Authors: Rataj Jan
Published: April 2006

Let X, Y be two sets of positive reach in ℝd. The translative integral formula says that, for 0 ≦ k ≦ d − 1 and bounded Borel subsets A, B ε ℝd,

where is the curvature measure (of order k) of X and is the mixed curvature measure of the sets X, Y and order r, S [1]. The mixed curvature measures are introduced by means of rectifiable currents, which leads to a relatively simple proof of (1). The proof needs an additional assumption on X, Y assuring that also reach (X ∩ Yz) > 0 for almost all z. This assumption is satisfied automatically for convex bodies, in dimension 2, or for sets with a sufficiently smooth boundary. Using the additivity of mixed curvature measures, (1) can be extended to unions of sets of positive reach.
Estimation of the intensity of stationary flat processes
Authors: Schladitz Katja
Published: April 2006

The starting point for this work was the following astonishing observation regarding stationary line processes in the plane.
Rotational versions of the Crofton formula
Authors: Jensen Eva B. Vedel
Published: April 2006

Inspired by recent developments in stereology, rotational versions of the Crofton formula are derived. The first version involves rotation averages of Minkowski functionals. It is shown that for the special case where the Minkowski functional is surface area, the rotation average can be expressed in terms of hypergeometric functions. The second rotational version of the Crofton formula solves the ‘opposite’ problem of finding functions with rotation averages equal to the Minkowski functionals. For the case of surface area, hypergeometric functions appear again. The second type of rotational Crofton formula has applications in local stereology. As a by-product, a formula involving mixed volumes is found.
Solving the inverse problem for measures using iterated function systems: a new approach
Authors: Forte B. , Vrscay E. R.
Published: April 2006

We present a systematic method of approximating, to an arbitrary accuracy, a probability measure µ on x = [0,1]q, q 1, with invariant measures for iterated function systems by matching its moments. There are two novel features in our treatment. 1. An infinite set of fixed affine contraction maps on , w2, · ·· }, subject to an ‘ϵ-contractivity' condition, is employed. Thus, only an optimization over the associated probabilities pi is required. 2. We prove a collage theorem for moments which reduces the moment matching problem to that of minimizing the collage distance between moment vectors. The minimization procedure is a standard quadratic programming problem in the pi which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0, 1] are presented.
Convergence of stochastic algorithms: from the Kushner–Clark theorem to the Lyapounov functional method
Authors: Fort Jean-Claude , Pagès Gilles
Published: April 2006

In the first part of this paper a global Kushner–Clark theorem about the convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can ‘read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic optimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algorithm is relaxed using a new approach based on a path-dependent Lyapounov functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of application of the theoretical results.