bluuebunny/crossref_metadata_2025 huggingface dataset:
Queues with delayed feedback
Authors: Foley Robert D. , Disney Ralph L.
Published: April 2006

Queues with delayed feedback have been little studied in queueing theory. Presented here is a rather complete discussion of such problems including queue length processes, busy period processes and several customer flow processes (e.g., departure processes). The case in which the delay mechanism is an M-server queue is studied in detail but it is shown later that many of the results carry over to a more general delay mechanism.
A note on multivariate Poisson flows on stochastic processes
Authors: Beutler Frederick J.
Published: April 2006

In [1], a deterministic counting rate condition is shown to be necessary and sufficient for a counting process induced on a Markov step process Z to be multivariate Poisson. We show here that the result continues to hold without Z being a Markov step process.
Some reward–penalty rules for the multi-armed bandit problem which are asymptotically optimal
Authors: Glazebrook K. D.
Published: April 2006

In the mathematical learning literature, reward–penalty rules have been studied in various decision-theoretic and game-theoretic contexts, the multi-armed bandit problem included. Here we propose an elaboration of Bather's randomised allocation indices which yields rules for the multi-armed bandit which are both reward-penalty and asymptotically optimal.
Seneta constants for the supercritical Bellman–Harris process
Authors: Schuh H.-J.
Published: April 2006

Let be a supercritical Bellman-Harris process with finite offspring mean. Cohn [17] has shown that there always exist constants Ct such that limt→∞Zt/Ct = W almost surely for some non-degenerate random variable W. In this paper we give an alternative proof, based on the study of (Zt) as a point process. Our methods are to some extent analytical and parallel Seneta's [18] and Heyde's [11] approaches in the case of the Galton–Watson process. We further identify Ct as 1/(–log Ft(–1)(γ)), where Ft(γ) = E(γzt), i.e. the norming constants found by Seneta [18] for the Galton–Watson process, apply also to the Bellman-Harris process. Finally we derive a weak law of large numbers for W, prove that W is continuous on (0,∞) and show that W has [0,∞) as its support.
Stochastic continuous-time model reference adaptive systems with decreasing gain
Authors: Aase Knut Kristian
Published: April 2006

Recursive estimation is considered for parameters of certain continuous stochastic models. Several optimality properties are shown to hold for the resulting recursive estimator, where a stochastic approximation viewpoint is taken when deriving statistical properties, like strong consistency and convergence in distribution. Applications are considered throughout, where for example explosion theory for diffusion processes is used as a modeling guide, in a particular application.
Minimizing expected makespan in stochastic open shops
Authors: Pinedo Michael L. , Ross Sheldon M.
Published: April 2006

Suppose that two machines are available to process n tasks. Each task has to be processed on both machines; the order in which this happens is immaterial. Task j has to be processed on machine 1 (2) for random time Xj (Yj) with distribution Fj (Gj). This kind of model is usually called an open shop. The time that it takes to process all tasks is normally called the makespan. Every time a machine finishes processing a task the decision-maker has to decide which task to process next. Assuming that Xj and Yj have the same exponential distribution we show that the optimal policy instructs the decision-maker, whenever a machine is freed, to start processing the task with longest expected processing time among the tasks still to be processed on both machines. If all tasks have been processed at least once, it does not matter what the decision-maker does, as long as he keeps the machines busy. We then consider the case of n identical tasks and two machines with different speeds. The time it takes machine 1 (2) to process a task has distribution F (G). Both distributions F and G are assumed to be new better than used (NBU) and we show that the decision-maker stochastically minimizes the makespan when he always gives priority to those tasks which have not yet received processing on either machine.
Aggregated Markov processes incorporating time interval omission
Authors: Ball Frank , Sansom Mark
Published: April 2006

We consider a finite-state-space, continuous-time Markov chain which is time reversible. The state space is partitioned into two sets, termed ‘open' and ‘closed', and it is only possible to observe which set the process is in. Further, short sojourns in either the open or closed sets of states will fail to be detected. We show that the dynamic stochastic properties of the observed process are completely described by an embedded Markov renewal process. The parameters of this Markov renewal process are obtained, allowing us to derive expressions for the moments and autocorrelation functions of successive sojourns in both the open and closed states. We illustrate the theory with a numerical study.
Inventory systems for perishable commodities with renewal input and Poisson output
Authors: Kaspi H. , Perry D.
Published: April 2006

We consider an inventory system to which arrival of items stored is a renewal process, and the demand is a Poisson process. Items stored have finite and fixed lifetimes. The blood-bank model inspired this study. Three models are studied. In the first one, we assume that each demand is for one unit and unsatisfied demands leave the system immediately. Using results on this model one is able to study a model in which arrival of items is Poisson but demands are for several units, and a model in which demands are willing to wait. We compute ergodic limits for the lost demands and the lost items processes and the limiting distribution of the number of items stored. The main tool in this analysis is an analogy to M/G/1 queueing systems with impatient customers.
Dispersive ordering and monotone failure rate distributions
Authors: Bartoszewicz J.
Published: April 2006

Recently many authors (e.g. Shaked (1982), Deshpande and Kochar (1983), Sathe (1984)) have established relations between the dispersive ordering and some other partial orderings of distributions. This note presents connections which the dispersive ordering has with monotone failure rate distributions.
Extended continued fractions, recurrence relations and two-dimensional Markov processes
Authors: Pearce C. E. M.
Published: April 2006

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.
Mean transition times for the Ehrenfest urn model
Authors: Blom Gunnar
Published: April 2006

For an Ehrenfest urn model, the mean transition time from one state to another is represented as a definite integral. Simple explicit expressions are given in some special cases.
Pseudo-modular decompositions and ‘refined bounds’ for the interval reliability and the availability for binary coherent systems
Authors: Mazars N.
Published: April 2006

‘Divide and conquer’ is a traditional approach in various fields of applied mathematics. In reliability, only modules have been proposed to decompose complex coherent systems. However, a system may include no modules, except ‘trivial’ ones. This paper is the second step of a study concerned with module generalizations : in the first step, pseudo-modules have been retained as the most general coherent subsystems which can yield a complete extension of the fundamental results concerning modules; in addition, it has been proved that they concern any binary coherent system; in this paper, it is shown that pseudo-modular decompositions are the most general coherent decompositions which can yield a complete extension of all the ‘refined bounds’ for the interval reliability and the availability currently proposed in terms of modular decompositions.

This study also yields some fundamental results to extend all the ‘refined bounds’ currently proposed for multistate coherent systems, using an easier approach.
Recent investigations involving stochastic population models
Authors: Bartlett M. S.
Published: April 2006

After some introductory general remarks on recent investigations involving population models, two broad classes of stochastic model are discussed, further, viz., spatial nearest-neighbour lattice models, and doubly stochastic models.
Approximations for delays using asymptotic estimates
Authors: Burman David Y. , Smith Donald R.
Published: April 2006

Consider a general single-server queue where the customers arrive according to a Poisson process whose rate is modulated according to an independent Markov process. The authors have previously reported on limits for the average delay in light and heavy traffic. In this paper we review these results, extend them to multiserver queues, and describe some approximations obtained from them for general delays.
On a generalized finite-capacity storage model
Authors: Woolford Samuel W.
Published: April 2006

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, …}, having state space J ⊆ {1, 2, …}. If Xn, = j, then there is a random. ‘input’ Vn(j) (a negative input implying a demand) of ‘type’ j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn(k)}, for k ≠ = j. Here, the random variables Vn(j) represent instantaneous ‘inputs’ of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (zn, Xn) and (Zn, OnLn), where Zn, is the level of storage at time n, Qn is the cumulative overflow at time n, and Ln is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.
Best-choice problems involving recall and uncertainty of selection when the number of observations is random
Authors: Petruccelli Joseph D.
Published: April 2006

From one point of view this paper adds to a previous formulation of the best-choice problem (Petruccelli (1981)) the possibility that the number of available observations, rather than being known, is a bounded random variable N with known distribution. From another perspective, it expands the formulations of Presman and Sonin (1972) and Rasmussen and Robbins (1975) to include recall and uncertainty of selection of observations. The behaviour of optimal stopping rules is examined under various assumptions on the general model. For optimal stopping rules and their probabilities of best choice relations are obtained between the bounded and unbounded N cases. Two particular classes of stopping rules which generalize the s(r) rules of Rasmussen and Robbins (1975) are considered in detail.
An ordering inequality for exchangeable random variables
Authors: Watson G. S.
Published: April 2006

Let X1, · ··, Xn be exchangeable random variables with finite variance and two sequences of constants satisfying a1≦···≦an, b1≦···≦bn. Suppose that a′1, ···, a′n is a rearrangement of a1, ···, an and that g(x) is a non-decreasing function. Then
Perturbation of the stationary distribution measured by ergodicity coefficients
Authors: Seneta E.
Published: April 2006

It is shown that an easily calculated ergodicity coefficient of a stochastic matrix P with a unique stationary distribution πT, may be used to assess sensitivity of πT to perturbation of P.
Heredity of stationary and reversible stochastic processes
Authors: Serfozo Richard F.
Published: April 2006

When a stochastic process (a random measure, set, field, etc. on a group) is stationary, ergodic, or reversible, then certain functions of this process inherit these properties. We present sufficient conditions for this inheritance.
The behaviour of the likelihood function for ARMA models
Authors: Deistler M. , Pötscher B. M.
Published: April 2006

The paper deals with some properties of the (Gaussian) likelihood function for multivariable ARMA models. Its behaviour at the boundary of the parameter space is described; its continuity properties as well as the question of the existence of a maximum are discussed. We have not been able to show in general the existence of the maximum over the usual parameter spaces. However, the maximum always exists over a suitably enlarged parameter space (given that the data are non-degenerate), which includes parameters corresponding to processes with discrete spectral components.
Burke's theorem on passage times in Gordon–Newell networks
Authors: Daduna Hans
Published: April 2006

In a closed cycle of exponential queues where the first and the last nodes are multiserver queues while the other nodes are single-server queues, the cycle-time distribution has a simple product form. The same result holds for passage-time distributions on overtake-free paths in Gordon–Newell networks. In brief, we prove Burke's theorem on passage times in closed networks.
Perturbation analysis of a phase-type queue with weakly correlated arrivals
Authors: Latouche Guy
Published: April 2006

We consider a queue which is obtained by slightly modifying the M/PH/1 queue, so that weak correlations are introduced among the interarrival times. We show how the stationary probability distribution may be studied by a perturbation analysis.
Minimizing expected makespans on uniform processor systems
Authors: Coffman E. G. , Flatto L. , Garey M. R. , Weber R. R.
Published: April 2006

We study the problem of scheduling n given jobs on m uniform processors to minimize expected makespan (maximum finishing time). Job execution times are not known in advance, but are known to be exponentially distributed, with identical rate parameters depending solely on the executing processor. For m = 2 and 3, we show that there exist optimal scheduling rules of a certain threshold type, and we show how the required thresholds can be easily determined. We conjecture that similar threshold rules suffice for m > 3 but are unable to prove this. However, for m > 3 we do obtain a general bound on problem size that permits Bellman equations to be used to construct an optimal scheduling rule for any given set of m rate parameters, with the memory required to represent that scheduling rule being independent of the number of remaining jobs.
Optimal control of service rates in networks of queues
Authors: Weber Richard R. , Stidham Shaler
Published: April 2006

We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of ·/M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi(xi) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval , where cost ci(μ) is charged for each unit of time that the service rate μis in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are extended to an average-cost measure, and an example is included to show that in general the assumption of convex holding costs may not be relaxed. A further example shows that the optimal policy may not be monotone unless the choice of possible service rates at each queue includes 0.
State-dependent signalling in queueing networks
Authors: Henderson W. , Northcote B. S. , Taylor P. G.
Published: April 2006

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue, force a single customer to be routed through the network or leave the network respectively. They are ‘signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network.

This paper considers networks with state-dependent intensities in which signals can be either a trigger or a batch of negative customers (the batch size being determined by an arbitrary probability distribution). It is shown that such networks still have a product-form equilibrium distribution. Natural methods for state space truncation and for the inclusion of multiple customer types in the network can be viewed as special cases of this state dependence. A further generalisation allows for the possibility of signals building up at nodes.
On some exponential functionals of Brownian motion
Authors: Yor Marc
Published: April 2006

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.
On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains
Authors: Nair M. G. , Pollett P. K.
Published: April 2006

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992).

In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.
Stochastic monotonicity and conditional Monte Carlo for likelihood ratios
Authors: Glasserman Paul
Published: April 2006

Likelihood ratios are used in computer simulation to estimate expectations with respect to one law from simulation of another. This importance sampling technique can be implemented with either the likelihood ratio at the end of the simulated time horizon or with a sequence of likelihood ratios at intermediate times. Since a likelihood ratio process is a martingale, the intermediate values are conditional expectations of the final value and their use introduces no bias.

We provide conditions under which using conditional expectations in this way brings guaranteed variance reduction. We use stochastic orderings to get positive dependence between a process and its likelihood ratio, from which variance reduction follows. Our analysis supports the following rough statement: for increasing functionals of associated processes with monotone likelihood ratio, conditioning helps. Examples are drawn from recursively defined processes, Markov chains in discrete and continuous time, and processes with Poisson input.
Cumulative shock models
Authors: Gut Allan
Published: April 2006

We show that a theory for stopped two-dimensional random walks is well suited to describe cumulative shock models. Limit theorems for the lifetime/failure time of a system are provided.
Convergence to collinearity of a sequence of random triangle shapes
Authors: Mannion David
Published: April 2006

A sequence of random triangles is constructed by choosing successively the three vertices of one triangle at random in the interior of its predecessor. A way is found to prove that the shapes of these triangles converge, almost surely, to collinear shapes, thus closing a gap in one of the central arguments of Mannion [5]. The new approach is based on a representation of the triangle process by a sequence of products of i.i.d. random matrices. We succeed in calculating the corresponding Lyapounov exponent.
The invariant distribution of a sequence of random collinear triangle shapes
Authors: Mannion David
Published: April 2006

The shape of the nth in a sequence of random triangle shapes is (xn, yn). It was shown in [8] and [9] that yn → 0, almost surely, as n → ∞. In this paper we show that x„ converges in distribution, as n→∞, to a random variable x, and we find the probability distribution of x. The convergence in law of xn enables us to complete an argument used in one part of [9].
‘Wald's Lemma' for sums of order statistics of i.i.d. random variables
Authors: Bruss F. Thomas , Robertson James B.
Published: April 2006

Let X1, X2, · ··, Xn be positive i.i.d. random variables with known distribution function having a finite mean. For a given s ≥0 we define Nn = N(n, s) to be the largest number k such that the sum of the smallest k Xs does not exceed s, and Mn = M(n, s) to be the largest number k such that the sum of the largest k X's does not exceed s. This paper studies the precise and asymptotic behaviour of E(Nn), E(Mn), Nn, Mn, and the corresponding ‘stopped' order statistics and as n →∞, both for fixed s, and where s =sn is an increasing function of n.
A note on the geometric ergodicity of a Markov chain
Authors: Chan K. S.
Published: April 2006

It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n(a + bg(x)) for some constants a, b> 0 and ρ < 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.
The transient blocking probabilities in M/M/N loss systems via large deviations
Authors: Choukri T.
Published: April 2006

By using large devaitions theory, we give asymptotic formulas for the transient blocking probabilities of M/M/N/N and M (with finite Poissonian sources) M/N/N queues.
A central limit theorem by remainder term for renewal processes
Authors: Roginsky Allen L.
Published: April 2006

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).
Moderate- and large-deviation probabilities in actuarial risk theory
Authors: Slud Eric , Hoesman Craig
Published: April 2006

A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.
Dispersion of particle systems in Brownian flows
Authors: Zirbel Craig L. , Çinlar Erhan
Published: April 2006

We study the dispersion of a collection of particles carried by an isotropic Brownian flow in Of particular interest are the center of mass and the centered spatial second moments. Their asymptotic behavior depends strongly on the spatial dimension and the largest Lyapunov exponent of the flow. We use estimates for the pair separation process to give a fairly complete picture of this behavior as t → ∞. In particular, for incompressible flows in two dimensions, we show that the variance of the center of mass grows sublinearly, while dispersion relative to the center of mass grows linearly.
Asymptotic distribution theory for Hoare's selection algorithm
Authors: Grübel Rudolf , Rösler Uwe
Published: April 2006

We investigate the asymptotic behaviour of the distribution of the number of comparisons needed by a quicksort-style selection algorithm that finds the lth smallest in a set of n numbers. Letting n tend to infinity and considering the values l = 1, ···,n simultaneously we obtain a limiting stochastic process. This process admits various interpretations: it arises in connection with a representation of real numbers induced by nested random partitions and also in connection with expected path lengths of a random walk in a random environment on a binary tree.
Bayes type estimators for the empty space function
Authors: Hansen Martin B⊘gsted
Published: April 2006

In the statistical analysis of random closed sets the distribution of the distance from a fixed point to the nearest point of the random set (the empty space function) is an important tool, and therefore development of efficient estimators is crucial.
Central limit theorem for a class of random measures associated with germ-grain models
Authors: Heinrich Lothar , Molchanov Ilya S.
Published: April 2006

We introduce a family of stationary random measures in the Euclidean space generated by so-called germ-grain models. The germ-grain model is defined as the union of i.i.d. compact random sets (grains) shifted by points (germs) of a point process. This model gives rise to random measures defined by the sum of contributions of non-overlapping parts of the individual grains. The corresponding moment measures are calculated and an ergodic theorem is presented. The main result is the central limit theorem for the introduced random measures, which is valid for rather general independently marked germ-grain models, including those with non-Poisson distribution of germs and non-convex grains. The technique is based on a central limit theorem for β-mixing random fields. It is shown that this construction of random measures includes those random measures obtained by the so-called positive extensions of intrinsic volumes. In the Poisson case it is possible to prove a central limit theorem under weaker assumptions by using approximations by m-dependent random fields. Applications to statistics of the Boolean model are also discussed. They include a standard way to derive limit theorems for estimators of the model parameters.
Inequalities for the cells of mixed stationary Poisson hyperplane tessellations
Authors: Mecke Joseph
Published: April 2006

Let Y be a mixed stationary Poisson hyperplane tessellation with finite intensity in the d-dimensional Euclidean space and Y′ a stationary Poisson hyperplane tessellation with the same intensity and directional distribution as Y. Denote by D0, D0′ the cells of Y, Y′ respectively containing a fixed point x0 and by V(D0), V(D0′) their volumes.
Non-Skip-Free M/G/1 and G/M/1 Type Markov Chains
Authors: Gail H. R. , Hantler S. L. , Taylor B. A.
Published: April 2006

For Markov chains of M/G/1 type that are not skip-free to the left, the corresponding G matrix is shown to have special structure and be determined by its first block row. An algorithm that takes advantage of this structure is developed for computing G. For non-skip-free M/G/1 type Markov chains, the algorithm significantly reduces the computational complexity of calculating the G matrix, when compared with reblocking to a system that is skip-free to the left and then applying usual iteration schemes to find G. A similar algorithm to calculate the R matrix for G/M/1 type Markov chains that are not skip-free to the right is also described.
Clumping in multitype-branching trees
Authors: López-Mimbela J. Alfredo , Wakolbinger Anton
Published: April 2006

We investigate the ‘clumping versus local finiteness' behavior in the infinite backward tree for a class of branching particle systems in ℝd with symmetric stable migration and critical ‘genuine multitype' branching. Under mild assumptions on the branching we establish, by analysing certain ergodic properties of the individual ancestral process, a critical dimension dc such that the (measure-valued) tree-top is almost surely locally finite if and only if d > dc. This result is used to obtain L1-norm asymptotics of a corresponding class of systems of non-linear partial differential equations.
On the undiscounted tax problem with precedence constraints
Authors: Glazebrook K. D.
Published: April 2006

A single machine is available to process a collection of jobs J, each of which evolves stochastically under processing. Jobs incur costs while awaiting the machine at a rate which is state dependent and processing must respect a set of precedence constraints Γ. Index policies are optimal in a variety of scenarios. The indices concerned are characterised as values of restart problems with the average reward criterion. This characterisation yields a range of efficient approaches to their computation. Index-based suboptimality bounds are derived for general processing policies. These bounds enable us to develop sensitivity analyses and to evaluate scheduling heuristics.
The herbarium and botanical art of Hendrik Elingsz van Rijgersma
Authors: Ehn Mia , Zanoni Thomas A.
Published: April 2006

A herbarium collected by H. E. van Rijgersma in the Caribbean island St. Martin 1865–1868 and 1875 has been found at the Swedish Museum of Natural History (S), Stockholm. The other known collection of van Rijgersma's plants was housed in Berlin (B), but was destroyed in World War II. Specimens collected by him are also kept in Leiden (L). Many of the Stockholm sheets have original illustrations by van Rijgersma in either black and white or in full watercolour, or both. A checklist of specimens and illustrations is presented at http://www.nrm.se/fbo/hist/rijgersma/rijgersma.html.en
Ecogeographic surveys as tools for analyzing potential reproductive isolating mechanisms: an example using Solanum juglandifolium Dunal, S. ochranthum Dunal, S. lycopersicoides Dunal, and S. sitiens I. M. Johnston
Authors: Smith Stacey D. , Peralta Iris Edith
Published: April 2006

An ecogeographic survey was completed for two pairs of South American Solanum species: (1) S. juglandifolium and S. ochranthum, and (2) S. lycopersicoides and S. sitiens. The purpose of this survey was to characterize the distribution, ecology, and phenology of these species and to screen for factors that might constitute premating reproductive isolating mechanisms between each pair. Passport data from 276 herbarium specimens were entered into a database, which was subsequently analyzed to determine the ecogeographic distribution and phenology of the surveyed species. The differences between species uncovered by the survey were then considered in the context of reproductive isolation. As S. juglandifolium and S. ochranthum were found to have overlapping ecogeographic preferences and phenology, it was postulated that postmating isolating mechanisms form the principal barrier to hybridization between the two species. Unlike S. juglandifolium and S. ochranthum, S. lycopersicoides and S. sitiens differed markedly in ecogeographic distribution. The species have been successfully crossed several times artificially, so the differences in distribution found in this survey probably contribute to maintaining species barriers between the two. In summary, this ecogeographic survey provided a useful method for analyzing the potential for interspecific gene flow between closely related taxa.
Xerophyta seinei (Velloziaceae), a distinctive new species from Zimbabwe
Authors: Behnke H.-Dietmar , Kramer Klaus , Hummel Eric
Published: April 2006

Xerophyta seinei Behnke, K. Kramer & E. Hummel sp. nov. (Velloziaceae), a new species from the Rhodes Matopos National Park in southwestern Zimbabwe, is distinguished by persistent leaf blades concealing the stems, an apex of dark­green leaves with lustrous and glutinous upper surfaces, and a lamina with glands on both surfaces and with non­transcurrent vascular bundles. Its distinctness from other Xerophyta species is also substantiated by specific sieve­element plastids (with a rectangular protein crystal in addition to the angular ones common to Velloziaceae) and by an rbcL­analysis. The morphology and leaf anatomy of the new species are compared to those of other members of the family and, in particular, of Xerophyta species known from the area.
Two new species of Campanulaceae from South Africa
Authors: Hong De-Yuan
Published: April 2006

Two new species in Campanulaceae, Theilera capensis D.Y. Hong and Craterocapsa alfredica D.Y. Hong, are described from South Africa. Theilera capensis is different from T. guthriei in having oblong leaves on short branches, corolla lobes nearly as long as the tube, filaments conspicuously dilated, and an annular disc. Craterocapsa alfredica is relatively close to C. insizwae, but differs from it in having ovate or ovate‐lanceolate leaves, narrow‐lanceolate calyx lobes, and a 3‐lobed stigma.
Typification of names of lichen taxa described by G. Sampaio and some others, deposited in Porto Herbarium (PO)
Authors: Paz-Bermúdez Graciela , Aguiar-Branco Helena , Folhadela Elisa
Published: April 2006

Gonçalo Sampaio authored 103 new names of lichens, comprising one new genus, 60 new species, 14 new infraspecific taxa, 25 new combinations and two nomina nova. In addition, he proposed 42 "names" that were not validly published, 18 of these being simple nomina nuda and 21 apparently representing invalid attempts to publish new combinations. The type material is deposited in the Herbarium of University of Porto (PO), except in three cases. Apart from a few specimens from Spain, the material derives from Portugal. For the names of all new taxa the type collections are listed, either a holotype or a lectotype (usually here designated). In addition, a few lichen names authored by Maurice Bouly de Lesdain or William Nylander and based on Portuguese material preserved at PO are listed with indications of type.
Some central limit analogues for supercritical Galton-Watson processes
Authors: Heyde C. C.
Published: April 2006

It is possible to interpret the classical central limit theorem for sums of independent random variables as a convergence rate result for the law of large numbers. For example, if Xi, i = 1, 2, 3, ··· are independent and identically distributed random variables with EXi = μ, var Xi = σ2 < ∞ and then the central limit theorem can be written in the form

This provides information on the rate of convergence in the strong law as . (“a.s.” denotes almost sure convergence.) It is our object in this paper to discuss analogues for the super-critical Galton-Watson process.
Some condensation processes of McKean type
Authors: Knight Frank B.
Published: April 2006

In a short article McKean [2] inaugurated the study of a class of time-inhomogeneous Markov processes which arise from the reaction of a single particle (the “tagged” particle) with an evolving environment of other particles. Other more recent references on these processes are [1], [4] and [5], in which some existence and uniqueness theorems for the solution of the resulting non-linear equation are obtained. We confine attention here to a very particular type of these processes which arises when two interacting particles simply coalesce, and the mass is taken as the process variable.
Queues with time-dependent arrival rates. III — A mild rush hour
Authors: Newell G. F.
Published: April 2006

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.
Further existence theorems for the optimal inventory equation
Authors: Boylan Edward S.
Published: April 2006

For given functions, g(x), h(x), F(x) and a constant, a, f(x) is said to be a solution of the optimal inventory equation if

where f(y - z) = f(0) when z > y. (The same convention holds in similar equations below.) A method of deriving a solution to (1) is to define inductively a sequence of approximating functions, fn(x), by:
A generalized unimodality
Authors: Olshen Richard A. , Savage Leonard J.
Published: April 2006

Khintchine (1938) showed that a real random variable Z has a unimodal distribution with mode at 0 iff Z ~ U X (that is, Z is distributed like U X), where U is uniform on [0, 1] and U and X are independent. Isii ((1958), page 173) defines a modified Stieltjes transform of a distribution function F for w complex thus:

Apparently unaware of Khintchine's work, he proved (pages 179-180) that F is unimodal with mode at 0 iff there is a distribution function Φ for which . The equivalence of Khintchine's and Isii's results is made vivid by a proof (due to L. A. Shepp) in the next section.
A queueing system with non-recurrent input and batch servicing
Authors: Pearce C.
Published: April 2006

We consider a queueing system in which arrivals occur at times , and after every kth arrival a servicing of k arrivals is begun. We assume that the number of servers is infinite. Initially, at t0= 0, the system is empty and the arrival process {tn} is about to start. The batch service times are independently and identically distributed with distribution function No assumption is made about the process {tn}.
Recurrence for Markov processes on N lines
Authors: Pinsky Mark
Published: April 2006

Let Λ = R1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ(t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.
The inter-arrival times of accepted customers in an M/G/1 queue with finite capacity
Authors: Gilchrist Robert
Published: April 2006

This note draws attention to a practical problem of interest to physicists, namely the distribution of the inter-arrival times of those customers accepted by the M/D/1/K + 1 queue. The form of the distribution is found for all K and the generalisation to M/G/1/K + 1 is discussed. The approach follows the contour integral method described by Cohen.
The severity of a viral host-vector epidemic
Authors: Radcliffe J.
Published: April 2006

An analogue of the deterministic pandemic threshold theorem for a measles epidemic is established for a viral host-vector epidemic. The severity of the epidemic is shown to be the unique positive solution of a pair of equations.
A characterization of the Poisson process
Authors: Samuels S. M.
Published: April 2006

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.

A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.
A bivariate birth-death process which approximates to the spread of a disease involving a vector
Authors: Griffiths D. A.
Published: April 2006

A simple model for a bivariate birth-death process is proposed. This model approximates to the host-vector epidemic situation. An investigation of the transient process is made and the mean behaviour over time is explicitly found. The probability of extinction and the behaviour of the process conditional upon extinction are examined and the probability distribution of the cumulative population size to extinction is found. Appropriate circumstances are suggested under which the model might possibly be applied to malaria. The host-vector model is classified within a general class of models which represent large population approximations to epidemics involving two types of infectives.
On the quasi-stationary distributions of the GI/M/1 queue
Authors: Kyprianou E. K.
Published: April 2006

This paper studies the existence, in a stable GI/M/1 queue, of the limit as t → ∞ of the distribution of the virtual waiting time process at time t conditioned on the event that at no time in the interval [0, t] the queue has become empty. The conditional limit distribution obtained when the traffic intensity is strictly less than one is the weighted sum of an exponential and a gamma distribution. Similar conditional limit distributions are obtained for the queue size process and the waiting time process as defined by Prabhu (1964).
Random collision processes and their limiting distribution using the discrimination information
Authors: Shōichi Nishimura
Published: April 2006

By analogy with statistical mechanics random collision processes are considered. In two cases we derive their limiting distributions using discrimination information. In one case the limiting distribution is unique and in the other one the family of limiting distributions is an exponential family.
Approximations to densities in geometric probability
Authors: Solomon H. , Stephens M. A.
Published: April 2006

Many random variables arising in problems of geometric probability have intractable densities, and it is very difficult to find probabilities or percentage points based on these densities. A simple approximation, a generalization of the chi-square distribution, is suggested, to approximate such densities; the approximation uses the first three moments. These may be theoretically derived, or may be obtained from Monte Carlo sampling.

The approximation is illustrated on random variables (the area, the perimeter, and the number of sides) associated with random polygons arising from two processes in the plane. Where it can be checked theoretically, the approximation gives good results. It is compared also with Pearson curve fits to the densities.
A semi-Markov model for a multigrade population with Poisson recruitment
Authors: McClean Sally I.
Published: April 2006

We consider a multigrade population with semi-Markov transitions between grades, Poisson arrivals to each grade, and departures from each grade. For this model the joint distribution of the numbers in each grade at any time is found, and the limiting distributions shown to be independently Poisson; this extends a previous result for a multigrade population with Markov transitions and Poisson recruitment.

This model is particularly applicable to manpower planning. The inclusion of semi-Markov transitions allows us to take into account existing knowledge of the distribution of length of service until an individual leaves his firm.
Plate and line segment processes
Authors: Fava N. A. , Santaló L. A.
Published: April 2006

Random processes of convex plates and line segments imbedded in R3 are considered in this paper, and the expected values of certain random variables associated with such processes are computed under a mean stationarity assumption, by resorting to some general formulas of integral geometry.
Cumulative constrained sojourn times in semi-Markov processes with an application to pensionable service
Authors: Sahin Izzet
Published: April 2006

This paper is concerned with the characterization of the cumulative pensionable service over an individual's working life that is made up of random lengths of service in different employments in a given industry, under partial coverage, transferability, and a uniform vesting rule. This characterization uses some results that are developed in the paper involving a functional and cumulative constrained sojourn times (constrained in the sense that if a sojourn time is less than a given constant it is not counted) in semi-Markov processes.
On the normal convergence of a class of simple batch epidemics
Authors: Langberg Naftali A.
Published: April 2006

A group of n susceptible individuals exposed to a contagious disease is considered. It is assumed that at each instant in time one or more susceptible individuals can contract the disease.

The progress of this epidemic is modeled by a stochastic process Xn(t), t in [0,∞) representing the number of infective individuals at time t. It is shown that Xn(t), with the suitable standardization and under a mild condition, converges in distribution as n → ∞to a normal random variable for all t in (0, t0), where t0 is an identifiable number.
Weak convergence of an iterated renewal process
Authors: Sen Pranab Kumar
Published: April 2006

Asymptotic normality and the weak invariance principle are studied for an iterated renewal process of order A martingale decomposition is used in this context to provide a simple proof.