bluuebunny/crossref_metadata_2025 huggingface dataset:
Light traffic approximations for regenerative queueing processes
Authors: Wang Chia-Li
Published: April 2006

We consider a regenerative queueing process that is (partially) generated by an embedded phase-type renewal process. We show that, under some specified conditions, a performance measure is an analytic function of the rate of the renewal process. We then develop several methods for deriving its Taylor polynomial in the renewal rate. These polynomials are asymptotically exact as the rate decreases, and, thus, are called light traffic approximations of the performance measure. We show via examples that these new methods are not only more efficient compared to existing ones, but also more versatile due to their general settings, such as to conduct perturbation analysis and study transient behavior.
Renewal-type behavior of absorption times in Markov chains
Authors: Van Cutsem Bernard , Ycart Bernard
Published: April 2006

This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). They are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.
On evaluations and asymptotic approximations of first-passage-time probabilities
Authors: Sacerdote L. , Tomassetti F.
Published: April 2006

The series expansion for the solution of the integral equation for the first-passage-time probability density function, obtained by resorting to the fixed point theorem, is used to achieve approximate evaluations for which error bounds are indicated. A different use of the fixed point theorem is then made to determine lower and upper bounds for asymptotic approximations, and to examine their range of validity.
Shapes of prisms after repeated random division
Authors: Cowan Richard
Published: April 2006

The shape of a rectangular prism in (d + 1) dimensions can be described by d quantities. In this talk, using a definition of shape which involves the ratios of edge lengths, we investigate shape distributions which are invariant when the prism is cut into two, also rectangular, prisms, with one prism retained for measurement and the other discarded. Interesting new distributions on [0, l]d arise.
Dimensions of measures under orthogonal projections
Authors: Zähle Martina
Published: April 2006

Let dimH, E be the Hausdorff dimension and dimP, E the packing dimension of the subset E of ℝn given by the unique exponent where the corresponding Hausdorff or packing measure of E jumps from infinity to zero.
A unified method to analyze overtake free queueing systems
Authors: Bertsimas Dimitris , Mourtzinou Georgia
Published: April 2006

In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = λG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional laws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional laws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek–Khinchine formula to the G//G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/∞ queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals.
Karyomorphology ofHeptacodium(Caprifoliaceae s.str.) and its phylogenetic implications
Authors: Zhang Zhao-Yang , Zhou Zhe-Kun , Gu Zhi-Jian
Published: April 2006

A karyomorphological study of the speciesHeptacodium miconioidesand five species in Caprifoliaceae of presumably related genera [AbeliaandDepelta(Linnaeeae);LycesteriaandLonicera(Caprifolieae);Weigela(Diervilleae)] reveals similarities in some karyological features such as structure of interphase nuclei, chromosome size, and condensation behaviour of chromosomes during mitotic prophase. However, differences between them in chromosome base number and morphology of metaphase chromosomes have also been revealed. The chromosome number ofHeptacodium2n= 28 is unique in Caprifoliaceae, presumably having resulted from polyploidization following hybridization between two ancestral species (possibly with chromosome base numberx= 8 or 9) bearing characters of tribes Caprifolieae and Linnaeeae, respectively. The phylogenetic position ofHeptacodiumis discussed based on combination of karyomorphology and previous morphological and molecular data.
Lectotypification of Campanula barbata L. and C. cenisia L. (Campanulaceae)
Authors: Pistarino Annalaura , Forneris Giuliana , Jarvis Charlie
Published: April 2006

The names Campanula barbata and C. cenisia were published by Carl Linnaeus in 1759 and 1763, respectively. For both names, Linnaeus referred to material he had received from the Italian botanist Carlo Allioni, and we designate lectotypes from Allioni material preserved in LINN. For C. cenisia, it has been possible to identify the locus classicus.
Typification of the name Opephyllum martensii Schmitz (Delesseriaceae, Rhodophyta)
Authors: Wynne Michael J.
Published: April 2006

Six slides labelled "Nitophyllum martensioides", located in the slide collection of Schmitz's red algal types in the Natural History Museum, London (BM), can be identified as Opephyllum martensii Schmitz (Schmitz & Hauptfleisch, 1897). Label data fully correspond to the protologue of O. martensii (collected by Martens from Mindanao, Philippines). This collection of six slides from a single gathering is, therefore, interpreted to be the holotype for the name Opephyllum martensii, superseding the neotype recently proposed by Lin & al. (2001).
Phenetic analyses of morphological traits in the Malesherbia humilis complex (Malesherbiaceae)
Authors: Gengler-Nowak Karla
Published: April 2006

Malesherbia (Malesherbiaceae) is a small, xerophytic group endemic to the Peruvian and Chilean deserts and adjacent Argentina. Malesherbia humilis var. humilis, M. humilis var. propinqua, M. humilis var. parviflora, M. gabrielae, M. taltalina, and M. multiflora form a morphologically cohesive group native to Chile and Argentina characterized by a profusion of small flowers and a short, bushy habit. The difficulty in distinguishing among these taxa suggests that the taxonomic delimitations may not be meaningful. A phenetic study was conducted to determine if this complex is best treated as a large, polymorphic species or as a series of very similar yet distinguishable taxa. Sixteen morphological characters were examined on 113 operational taxonomic units (OTUs); the data were analyzed using UPGMA, PCA, and discriminant analyses. Five clusters were identified by the analyses. These clusters are distinguishable by minor characters and are geographically cohesive. Because of the minor nature of the differences found and the unreliability of most differences noted in the descriptions of the taxa, these five clusters are recognized as varieties of M. humilis. A key to the varieties and short descriptions of each are presented.
Novel visualisation techniques for working with multiple, overlapping classification hierarchies
Authors: Graham Martin , Watson Mark F. , Kennedy Jessie B.
Published: April 2006

A Java‐based program is presented that provides a visualisation tool for display of and comparison between classification hierarchies. Taxa, or groups of taxa, can be tracked across multiple, overlapping classifications. Recognition of congruence and dissimilarity between classifications can be visually assessed through pattern matching. Software development, testing, and operation of the tool are discussed, together with potential applications. Further information (including software) is available via the Prometheus website [www.prometheusdb.org].
Stone, plant, or animal
Authors: Nicolson Dan H.
Published: April 2006

Linnaeus organised almost everything to which he put his mind and was the first to systematically apply a binomial system to all of nature. The philosophy underlying his system of nature is that of a ladder (pyramid) leading from stones, the lowest (furthest from Man), to animals, with Man, created in the image of God, being the highest. The classification functioned like a two­dimensional map, characters of taxa being like latitude and longitude with a hierarchy of precision. It was of value in an age of discovery, although his classifications of stones and plants had short lives. The binomial system of naming, although not considered important by Linnaeus, was important in separating diagnoses/descriptions of taxa from the names of taxa. It is hoped that new philosophies will maintain the distinction between things named and their names.
Why do we name organisms? Some reminders from the past
Authors: Stevens Peter F.
Published: April 2006

The naming systems of Linnaeus and Bentham in particular are examined to clarify the relationships between naming and ideas of relationships. Linnaean binomials were adopted largely for practical reasons. Furthermore, Linnaeus proposed his names in the context of system, putting organisms in groups of 10. This allowed botanists of moderate capabilities to know at least the genera. Although binomials are names of taxa of the two lowest levels of a rank hierarchy, much of Linnaeus' work does not fit easily in the currently widely accepted view of Linnaeus as a hard­bitten essentialist. Neither Lamarck nor the later Bentham believed in a rank hierarchy, although to name organisms both used what is here called a flagged hierarchy: name terminations indicating only a set of inclusion relationships, not ranks of nature implied by a rank hierarchy. Bentham was clear that the adoption of a flagged hierarchy with groups of a particular size in theGenera plantarumwas to facilitate botanists' understanding of the system as a whole. Systematists like Bentham and Linnaeus managed information and presented classification systems simultaneously. I conclude that the lower level of Linnaeus' hierarchy is a special case of the noun + adjective combination that pervade folk classifications in particular and human language in general. Linking essentialism and "Linnaean" nomenclature is at best a red herring, thus few nineteenth­century botanists believed in a fully­developed rank hierarchy. Naming hierarchies are mostly such that at each level members belong to only one group, and this is at a higher level; most such hierarchies are fairly shallow. Historically, uninomials have seemed more attractive when generic limits were in flux, but suboptimal when relationships were more stable. Naming systems in general incorporate a substantial element of convention, emphasizing particular numbers of groups and groups of particular size; this facilitates comprehension and communication. Similar conventions will be needed whatever naming system is used.
How to chop up a tree
Authors: Brummitt Richard K.
Published: April 2006

Over the past 50 years it has been pointed out with increasing frequency that our traditional Linnaean system of classification and nomenclature is incompatible with a phylogenetic system which recognises only monophyletic groups. Dividing up an evolutionary tree into mutually exclusive families, genera, and species which are all monophyletic is a logical impossibility. Darwin had emphasised that evolution is descent with modification. The rise of cladistic thinking in the last 40 years has promoted an obsession with monophyletic taxa, with classification based solely on descent at the expense of modification. Despite strong psychological pressures on a generation of biologists who have been brought up on the dogma of monophyly, the Hennigian view of classification is now increasingly seen as illogical and out­of­date. Some are therefore supporting thePhyloCode, which is based on a logical position but is impractical for general purpose classification and communication since it recognises no ranks and abandons binomials. Others still cling to the nonsensical concept of recognising families, genera, species, etc., and all being monophyletic. Linnaean classification is the optimal tool for cataloguing biodiversity and will inevitably be maintained, but this requires recognition of paraphyletic taxa and some rethinking of the practice and purposes of biological classification. Those who want a classification to recognise only monophyletic taxa should adopt an appropriate nomenclatural system such as is offered by thePhyloCode. To do otherwise will tend to lead to bad taxonomy.
Relationships among genera in Saniculoideae and selected Apioideae (Umbelliferae) inferred from nrITS sequences
Authors: Valiejo-Roman Carmen M. , Terentieva Elena I. , Samigullin Tagir H. , Pimenov Michael G.
Published: April 2006

The internal transcribed spacers (ITS1 and ITS2) of 18S­26S nuclear ribosomal DNA were newly sequenced for eight species of Umbelliferae (six species from subfamily Saniculoideae: Actinolema macrolema, Astrantia minor, Eryngium giganteum, E. coeruleum, Hacquetia epipactis, and Lagoecia cuminoides, two species from subfamily Hydrocotyloideae: Dickinsia hydrocotyloides and Azorella trifurcata), as well as Hohenackeria exscapa, a species of uncertain position in the family. Phylogenetic analyses of new data, plus previously reported sequences of 52 other species using neighbor­joining, maximum parsimony, and maximum likelihood methods yielded similar results: (1) Actinolema is sister to Astrantia corresponding to Drude's treatments; (2) in Astrantia, molecular divergence is revealed between sects. Astrantiella (A. minor) and Astrantia (A. major, A. maxima); (3) Eryngium appears to be paraphyletic; (4) Hacquetia might be treated as a part of Sanicula; and (5) Lagoecia is very distant from all other Saniculoideae and close to some genera of Apioideae. Our results correspond to matK data previously published: (1) Hohenackeria forms a clade with Bupleurum, in a position near the base of the Apioideae tree; (2) Azorella is sister to a large cluster uniting all Saniculoideae and Apioideae, being slightly closer to them than to the Hydrocotyle­Araliaceae clade; (3) Dickinsia is very distant from phenetically similar Hydrocotyle, falling within a large cluster of Apioideae, but also including Lagoecia and Naufraga.
The Global Taxonomy Initiative in Africa
Authors: Klopper Ronell R. , Smith Gideon F. , Chikuni Augustin C.
Published: April 2006

The Conference of Parties (COP) to the Convention on Biological Diversity (CBD) has recognised the currently existing taxonomic impediment to ensure the conservation and management of the world's biodiversity. In response, COP has formulated the concept of a Global Taxonomy Initiative (GTI) to promote a concerted effort among international funding agencies, national and sub­national governments, and non­governmental bodies. The GTI Africa Regional Workshop was held at the Kirstenbosch National Botanical Garden, Claremont, Cape Town, South Africa, from 27 February to 1 March 2001. The Workshop was attended by a total of 43 delegates, representing 32 countries (23 African) and 36 institutions or organisations. The Kirstenbosch Declaration was compiled during the Workshop and summarises the major decisions and proposals made by the delegates at the Workshop. A Final Report summarising the discussions and decisions from the GTI Africa Regional Workshop is being prepared and will be distributed to all interested parties. A taxonomic needs assessment for Africa has been conducted as part of this project. The results from this first ever comprehensive taxonomic needs assessment covering plants, animals and micro­organisms for the African continent as a unit, are reported on here and forms an important part of the Final Report. The needs assessment highlights the prominent taxonomic impediment currently existing in the continent. Taxonomic capacity building is urgently needed in Africa; therefore, African taxonomic institutions can benefit immensely from the Global Taxonomy Initiative.
Lectotypification of the Linnaean names in Viburnum L. (Viburnaceae)
Authors: Malécot Valéry , McNeill John
Published: April 2006

Lectotypifications for names proposed by Linnaeus in Viburnum (Viburnaceae) have been made for only three of the ten entities described in 1753 and 1762. Lectotypifications of the remaining seven names are proposed with comments on the three previously designated herbarium sheets.
Lectotypification of Lobaria amplissima (Scop.) Forssell (Lobariaceae, Ascomycotina)
Authors: Burgaz Ana Rosa , Tretiach Mauro
Published: April 2006

The top figure of tab. 46 in Micheli’s Nova Genera Plantarum is selected as lectotype, and a specimen from his herbarium (FI‐M) as epitype of the name of the lichen, Lobaria amplissima (Scop.) Forssell.
On the dynamic programming approach to Pontriagin's maximum principle
Authors: Morton Richard
Published: April 2006

Suppose that the state variables x = (x1,…,xn)′

where the dot refers to derivatives with respect to time t, and u ∊ U is a vector of controls. The object is to transfer x to x1 by choosing the controls so that the functional

takes on its minimum value J(x) called the Bellman function (although we shall define it in a different way). The Dynamic Programming Principle leads to the maximisation with respect to u of

and equality is obtained upon maximisation.
Diffusion approximations in collective risk theory
Authors: Iglehart L. Donald
Published: April 2006

Collective risk theory is concerned with the random fluctations of the total assets, the risk reserve, of an insurance company. Consider a company which only writes ordinary insurance policies such as accident, disability, fire, health, and whole life. The policyholders pay premiums regularly and at certain random times make claims to the company. A policyholder's premium, the gross risk premium, is a positive amount composed of two components. The net risk premium is the component calculated to cover the payments of claims on the average, while the security risk premium, or safety loading, is the component which protects the company from large deviations of claims from the average and also allows an accumulation of capital. When a claim occurs the company pays the policyholder a positive amount called the positive risk sum.
An identity for Markovian replacement processes
Authors: Mandl Petr
Published: April 2006

A Markovian process with rewards is considered. The interference into its development consists in instantaneous changes of state called replacements. It is shown that, for any replacement policy not depending on the future, the expected reward is a linear function of the expected replacement numbers. The application of this result is illustrated by examples.
On the maximum of sums of random variables and the supremum functional for stable processes
Authors: Heyde C.C.
Published: April 2006

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S0= 0, Sn = Σ i=1nXi, n ≧ 1, and Mn = max0 ≦ k ≦ nSk. In the case where the Xi are such that Σ1∞n−1Pr(Sn > 0) < ∞, we have limn→∞Mn = M which is finite with probability one, while in the case where Σ1∞n−1Pr(Sn < 0) < ∞, a limit theorem for Mn has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1∞n−1Pr(Sn < 0) < ∞, Σ1∞n−1Pr(Sn > 0) < ∞ (the case of oscillation of the random walk generated by the Sn) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn in the case of oscillation.
Random paths through convex bodies
Authors: Coleman Rodney
Published: April 2006

Several mechanisms under which the randomness of straight line paths through convex bodies can arise are described. Some general results are given relating four of these mechanisms, and the corresponding distributions of the lengths of the straight line paths are found for the circle, the rectangle, and the cube.
Linear processes are nearly Gaussian
Authors: Mallows C. L.
Published: April 2006

Let U denote the set of all integers, and suppose that Y = {Yu; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au; u ∈ U} be a sequence of real numbers with Σuau2 = 1. Then Xu = ΣwawYu–w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu) = 0, E(Xu2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au| is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy ≦ g maxu |au | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w1, … wn), (Xw1, … Xwn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filter a is studied.
A diffusion model for the control of a dam
Authors: Bather J. A.
Published: April 2006

A previous paper [2] was concerned with the determination of optimal policies for restocking an inventory which is continuously depleted by a random process of demands. The purpose of the present paper is to develop a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process. This reversal of the roles of input and output does not, by itself, change the character of the problem. But the consideration of set-up costs for ordering replacements, which leads to inventory policies of the (s, S) type, has no counterpart here. It is natural to regard the dam as a device for smoothing out random fluctuations in a flow of water and, under utility assumptions which reflect this attitude, it follows that the optimal output rate is a continuous function of the level of water in the reservoir. Our main object is to determine this function.
Extended criterion for comparison of empirical distributions
Authors: Percus Ora Engleberg , Percus Jerome K.
Published: April 2006

A generating function technique is used to determine the probability that the deviation between two empirical distributions drawn from the same population lies within a given band a specified number of times. We also treat the asymptotic problem of very large sample size, and obtain explicit expressions when the relative number of failures is very small or very large.
Linear-Intercept distributions do not characterize plane sets
Authors: Mallows C. L. , C. Clark J. M.
Published: April 2006

All sets mentioned in this paper will be assumed to be appropriately measurable without further comment, and events of probability zero will be neglected. We start with a fixed (large, convex) subset R of the plane, termed the “retina”.
A quantum-mechanical central limit theorem
Authors: Cushen C. D. , Hudson R. L.
Published: April 2006

The concepts of distribution operator, stochastic independence, convergence in distribution and normal distribution are formulated for pairs of canonically conjugate quantum-mechanical momentum and position operators. It is shown that if the sequence (pn, qn), n = 1, 2, ··· is stochastically independent and identically distributed with finite covariance and zero mean then the sequence of pairs of canonical observables

converges in distribution to a normal limit distribution.
A note on the estimation of the mean of a homogeneous random field
Authors: Skalsky Michael
Published: April 2006

An important problem, arising in connection with the estimation of mathematical expectation of a homogeneous random field X(x1, ···, xn) in Rn by means of the arithmetic mean of observed values, is to determine the number of observations for which the variance of the estimate attains its minimum. Vilenkin [2] has shown, that in the case of a stationary random process X(x) such a finite number exists, provided that the covariance function satisfies certain conditions.
A review of transient behavior in regular diffusion and birth-death processes. Part II
Authors: Keilson J.
Published: April 2006

A previous paper, hereafter referred to as “I”, reviewed the structure of a class of “regular” diffusion and birth-death processes in one dimension. Such processes were seen to have a basic simplicity manifesting itself in two ways: (1) The ergodic processes have an ergodic distribution that one may write down at once; (2) All such processes have transition functions for which a simple spectral representation is available. The simple spectral structure affords theoretical insight into the structure of related passage time densities, and numerical techniques for their evaluation. In particular, a variational principle is available for estimating the principal eigenvalues and eigenvectors.
On a characteristic property of stationary streams without aftereffect
Authors: Mejzler D.
Published: April 2006

Let us consider a stream of homogeneous random events which we call signals and let us denote by the probability that exactly k signals will appear in the time interval . We assume that the stream is not empty, i.e. the probability P0(a, t) is not identically equal to 1.
On extreme stable laws and some applications
Authors: Eaton Morris L. , Morris Carl , Rubin Herman
Published: April 2006

It is shown that the extreme stable laws have one-sided moment generating functions with interesting mathematical forms. The fact that one of these forms, zcz, is a moment generating function is used to establish two interesting statistical results; first, that rr is a moment sequence for a density with decreasing failure rate, and secondly, that the likelihood ratio test for testing a simple null hypothesis in a multinomial distribution is admissible and Bayes.
Stability Theorems for Solutions to the Optimal Inventory Equation
Authors: Boylan S. Edward
Published: April 2006

In a previous paper, [1] it was shown that a solution, f(x) will exist for the optimal inventory equation

(where f(y − z) = f(0), y < z) provided:

1.

g(x) ≧ 0, x ≧ 0;

2.

0 < a < 1;

3.

h(x) is monotonically nondecreasing, h(0) = 0;

4.

F is a distribution function on [0, ∞);

(In [1], 1–4 were denoted collectively as (A).)

and either

5a.

g(x) is continuous for all x ≧ 0;

5b.

limx→∞g(x) = ∞;

5c

h(x) is continuous for all x > 0 (Theorem 2 of [1]);

or

6.

g(x) is uniformly continuous for all x ≧ 0 (Theorem 3 of [1]).
Additional quasi-stationary distributions for semi-Markov processes
Authors: Flaspohler David C. , Holmes Paul T.
Published: April 2006

Consider a semi-Markov process X(t) defined on a subset of the non-negative integers with zero as an absorbing state and the non-zero states forming an irreducible class with exit to zero being possible. Conditions are given for the existence of the limits:

where Xj(t) is the amount of time prior to time t spent in state j.

The limits (which are independent of the initial state) are evaluated when the sufficient conditions are satisfied.
Traffic noise as a filtered Markov renewal process
Authors: Marcus Allan H.
Published: April 2006

Traffic noise depends significantly on statistical properties of highway flow. The noise heard by an off-highway observer is the sum of the contributions of all the vehicles on the highway. We assume that vehicle spacings on a single-lane infinite straight road form a Markov renewal process (MRP) with N states or vehicle types. The noise impact is then a two-sided filtered MRP. We give explicit formulae for the mean and variance in the case N = 2 and exponential headways. Jewell's (1965) study (cars vs. trucks in the southbound curb lane of U.S. 40) gives parameters of the MPR for a numerical example. Even with only a modest degree of truck clustering and variability in vehicle spacings and noise emission, the variation in noise level is much greater than usually predicted. The coefficient of variation of arithmetic noise intensity is greater than unity at distances from the roadway less than 700 feet. The logarithmic noise level parameters L50 and L10 usually computed in engineering practice may be in error by 2 to 3 decibels if these sources of variability are ignored.
Almost sure stability of maxima
Authors: Resnick Sidney I. , Tomkins R. J.
Published: April 2006

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.
Order statistics of partial sums of mutually independent random variables
Authors: Pollaczek Felix
Published: April 2006

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.
The distribution of the distance between random points
Authors: Alagar Vangalur S.
Published: April 2006

This paper considers the distribution of distance between random points and shows how the distribution can be found when the points are chosen uniformly and independently in a hypersphere or in two adjacent unit squares. The value of a powerful extension of the classical Crofton technique is illustrated here for solving such geometric probability problems. This method is quite different from those employed by Hammersley and Oser.
Stochastic formation of hierarchies
Authors: Helms L. L.
Published: April 2006

In this paper, N particles el, · ··, eN occupying positions in finite state spaces S1, · ··, SN, respectively, are considered along with an element of a finite space S N+1 of hierarchies which describe the organization of the particles into pairs, pairs of pairs, etc. A stochastic process {Xt: t 0} on a probability measure space (Ω, , P) is a hierarchic process if it is a right-continuous Markov jump process taking on values in the state space Sj and having the property that the (N + 1)th component of Xt can jump from a hierarchy to a successor or antecedent of that hierarchy. Asymptotic distributions of perturbed hierarchic processes, bilateral processes, and unilateral processes are determined in terms of an interaction function and the asymptotic distributions of the particles in the absence of any interaction.
A note on the subcritical generalized age-dependent branching process
Authors: Doney R. A.
Published: April 2006

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.
Optimum replacement policies with delay
Authors: Nakagawa Toshio , Osaki Shunji
Published: April 2006

Replacement theory for equipment has been investigated by several authors. This paper introduces the ‘delay’ time preparing for replacement, derives the expected cost per unit time, and discusses the optimum replacement policies under several conditions. Three special cases are discussed and numerical examples are presented.
A stochastic model for polymer degradation
Authors: Srinivasan S. K. , Mehata K. M.
Published: April 2006

The stochastic model for breaking of molecular segments proposed by Bithell is analysed and some results relating to the distribution of the number of fragments are obtained by using a slightly more general model which allows multiple ruptures. The product density technique is employed to derive the mean and mean square number of segments at any time t and the number of segments with length greater than y at time of production.
Stability of one-dimensional systems of colliding particles
Authors: Karr Alan F.
Published: April 2006

Envision a one-dimensional system of infinitely many identical particles, in which initial particle positions constitute a Poisson random measure and the initial velocity of a particle depends only on its initial position. Given its initial conditions the system evolves deterministically, by means of perfectly elastic collisions. In this note we derive conditions for continuity of the probability laws of the system and of the particle paths, as functions of the parameters of the initial conditions. These results have the physical interpretation of stability theorems.
The generalised state-dependent queue: the busy period Erlangian
Authors: Conolly B. W.
Published: April 2006

A continued fraction representation is presented of the Laplace transform of the generating function of the fundamental joint probability and density of busy period length measured in customers served and duration in time. The setting is the single server Erlang queueing system where the parameters of negative exponentially distributed arrival and service times have a general dependence on instantaneous system state.
The area under the infectives trajectory of the general stochastic epidemic
Authors: Downton F.
Published: April 2006

If the cost of an epidemic is a linear function of the number of persons infected at any time, the total cost depends upon the area under the infectives trajectory. This note uses a combinatorial argument to show that for the general stochastic epidemic the distribution of this area may be expressed in terms of the probabilities of the ultimate size of the epidemic.
Time and customer processes in queues with stationary inputs
Authors: Miyazawa Masakiyo
Published: April 2006

Two types of processes occurring in queues with stationary inputs are considered. They are called ‘time processes’ and ‘customer processes’. Sufficient conditions for the convergence of sample averages and the existence of limiting distributions for each type of processes are given. The results generalize those of Loynes (1962). The results are applied to three queueing processes and the Little's formula L = λW is obtained under rather general conditions.
The probability distribution of the duration of the game in the classical ruin problem
Authors: Karni Edi
Published: April 2006

This paper develops simple expressions for the exact probability distribution of the duration of the game, and the probability distribution of the duration of the game conditional upon its termination with the gambler's ruin, in the classical ruin problem.
The convergence of a branching Brownian motion used as a model describing the spread of an epidemic
Authors: Wang Frank J. S.
Published: April 2006

A spatial epidemic process where the individuals are located at positions in the Euclidean space R2 is considered. The infective individuals, with an infection period that is exponentially distributed with parameter µ, move in R2 according to a Brownian motion with a diffusion coefficient σ2. The susceptible individuals may also move. But we shall use the approximation that they remain unchanged in numbers and therefore assume that the averaged ‘density' of susceptibles per unit area is the same throughout space and time. The transition probability rate of infection of a susceptible in the infinitesimal element of area dy by an infective in dx is assumed to be a function h(x – y |) of the distance | x – y | between x and y. Then our process can be considered as a two-dimensional birth and death Brownian motion. Let be the number of infective individuals in the set D at time t and . The almost everywhere convergence of the random variables to a limit random variable W(D) is established.
Superposition and decomposition of stationary point processes
Authors: Ito Yoshifusa
Published: April 2006

A recursion formula is obtained by rearranging Lawrance's (1973) result concerning the superposition of independent stationary point processes for which there exist joint probability density functions for the intervals between successive points. When these component point processes are identically distributed, the formula can in principle be inverted to describe their probability structure given that of the superposition process.
A versatile Markovian point process
Authors: Neuts Marcel F.
Published: April 2006

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.

Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.
An upper bound for the mean of Yaglom's limit
Authors: Evans Lawrence S.
Published: April 2006

For a single-type Galton—Watson branching process with mean less than one and finite second moment, we establish an upper bound for the mean of the associated Yaglom limit. This bound is attained if and only if the generating function of the process is linear.
A bounded growth population subjected to emigrations due to population pressure
Authors: Trajstman A. C.
Published: April 2006

A model is presented for a bounded growth population subjected to random-sized emigrations that occur due to population pressure.

The deterministic growth component examined in detail is defined by a Prendiville process. Results are obtained for the times between emigration events and for the population increase between emigrations. Some information is obtained about the mean time to extinction and also for the mean population level when the emigration-size distribution is negative exponential.
Subexponential distributions and dominated-variation tails
Authors: Goldie Charles M.
Published: April 2006

For a distribution function F on (0,∞), regular variation of its tail is known to imply that F is subexponential. Let be merely of dominated variation. This note shows that F need not be subexponential, and investigates which of the known necessary conditions for subexponentiality become sufficient when insisted upon for such an F.
An application of chain-dependent processes to meteorology
Authors: Katz Richard W.
Published: April 2006

An explicit formula is derived for the variance normalizing constant in the central limit theorem for chain-dependent processes. As an application to meteorology, a specific chain-dependent process is proposed as a probabilistic model for the sequence of daily amounts of precipitation. This model is a generalization of the commonly used Markov chain model for the occurrence of precipitation.
Testing for autocorrelation and Akaike's criterion
Authors: Hannan E. J.
Published: April 2006

The problem initially considered is that of testing whether ρ = 0 in a model y(n) = x(n)+ η (n), x(n) = ρx(n – 1) + ξ (n) where only y(n) is observed and η (n), ξ (n) are white noise. This is equivalent to distinguishing between an ARMA (1, 1) model and white noise. The asymptotic distribution of the likelihood ratio criterion is derived. This is shown to be of an unusual form. This result is then used to discuss the asymptotic properties of Akaike's procedure for estimating (p, q) in an ARMA (p, q) model. If p0, q0 are the true values and p0 < P, q0< Q, when P, Q are the maximum values considered, then it is shown that, in a certain asymptotic sense, the procedure is sure to overestimate p0, q0. However, the asymptotic situation may be very far from that relevant in a practical case. The relevance of overestimation is briefly discussed.
On the waiting time of a two-stage queueing system with blocking
Authors: Langaris Christos , Conolly Brian
Published: April 2006

An analysis is given of the first-come-first-served waiting-time process in stages 1 and 2 of a two-stage service system with k and n parallel service channels in the first and second stages respectively, and m intermediate waiting places (k, n ≧ 1, m ≧ 0).

Although the method of analysis is straightforward the details are intricate and require careful study of the location of the zeros of a high-degree polynomial.

The analysis paves the way for an extensive study of the numerical effect on waiting time of blocking in commonly encountered systems of this nature. ‘Effective service time' in stage 1, defined so as to include blocked time, is considered separately.
A note on models using the branching process with immigration stopped at zero
Authors: Seneta E. , Varé S. Ta
Published: April 2006

The Galton-Watson process with immigration which is time-homogeneous but not permitted when the process is in state 0 (so that this state is absorbing) is briefly studied in the subcritical and supercritical cases. Results analogous to those for the ordinary Galton-Watson process are found to hold. Partly-new techniques are required, although known end-results on the standard process with and without immigration are used also. In the subcritical case a new parameter is found to be relevant, replacing to some extent the criticality parameter.
The accuracy of the diffusion approximation to the expected time to extinction for some discrete stochastic processes
Authors: Grasman J. , Ludwig D.
Published: April 2006

Asymptotic approximations and numerical computations are used to estimate the accuracy of the diffusion approximation for the expected time to extinction for some stochastic processes. The results differ for processes with a continuant transition matrix (e.g. a birth and death process), and those with a noncontinuant transition matrix (e.g. a non-linear branching process). In the latter case, the diffusion equation does not hold near the point of exit. Consequently, high-order corrections do not result in substantial improvement over the diffusion approximation.
A note on Norton's theorem for queuing networks
Authors: Walrand J.
Published: April 2006

We propose a simple proof of Norton's theorem for multiclass quasireversible networks.
General shock models associated with correlated renewal sequences
Authors: Shanthikumar J. G. , Sumita U.
Published: April 2006

In this paper we define and analyze a general shock model associated with a correlated pair (Xn, Yn) of renewal sequences, where the system fails when the magnitude of a shock exceeds (or falls below) a prespecified threshold level. Two models, depending on whether the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length Yn of the subsequent interval until the next shock, are considered. The transform results, an exponential limit theorem, and properties of the associated renewal process of the failure times are obtained. An application in a stochastic clearing system with numerical results is also given.
Sums of i.i.d. random variables and an application to the explosion criterion for markov branching processes
Authors: Schuh H.-J.
Published: April 2006

We give necessary and sufficient conditions for in terms of , where Sn is the sum of n i.i.d. random variables with values in]0, ∞[, and A ≧ 0. We use these results to give a probabilistic proof of the ‘explosion criterion' for continuous-time Markov branching processes, which is usually shown analytically.
Spectral structure of the first-passage-time densities for classes of Markov chains
Authors: Kijima Masaaki
Published: April 2006

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.
Extinction probabilities in predator–prey models
Authors: Hitchcock S. E.
Published: April 2006

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.
The departure process for the M/M/1 queue
Authors: Hubbard John R. , Pegden Claude Dennis , Rosenshine Matthew
Published: April 2006

The problem of determining the probability of j departures during the time interval (0, t) from an M/M/l queue empty at t = 0 is considered. A closed-form solution is obtained. It is shown that this solution is unique and invariant under interchanging the arrival rate and service rate. Finally, sample computational representations of the solution are developed and results of a simple computation are provided.
On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary
Authors: Ricciardi L. M. , Sacerdote L.
Published: April 2006

We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.
ANOVA models with random effects: an approach via symmetry
Authors: Speed T. P.
Published: April 2006

The standard ANOVA models with random effects for multi-indexed arrays of random variables with an arbitrary nesting structure on the indices are considered from the viewpoint of symmetry. It is found that the covariance matrix of such an array has sufficient symmetry to permit viewing the usual components of variance as a generalised spectrum and the linear models of random effects as a generalised spectral decomposition.