bluuebunny/crossref_metadata_2025 huggingface dataset:
Probability in quantum mechanics
Authors: Corbett J. V.
Published: April 2006

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
Strict supercritical generation-dependent Crump–Mode–Jagers branching processes
Authors: Edler L.
Published: April 2006

The general age-dependent branching model of Crump, Mode and Jagers will be generalized towards generation-dependent varying lifespan and reproduction distributions. A system of integral and renewal equations is established for the generating functions and the first two moments of Zi(t) (the number of individuals alive at time t), if the population was initiated at time 0 by one ancestor of age 0 from generation i. Convergence in quadratic mean of Zi(t)/EZi(t) as t tends to infinity is obtained if the generation-dependent reproduction functions converge to a supercritical one. In particular, if this convergence is slow enough tγ exp (αt) is the asymptotic behavior of EZi(t) for t tending to infinity, where γ is a positive real number and α the Malthusian parameter of growth of the limiting reproduction function.
Hitting-time and occupation-time bounds implied by drift analysis with applications
Authors: Hajek Bruce
Published: April 2006

Bounds of exponential type are derived for the first-hitting time and occupation times of a real-valued random sequence which has a uniform negative drift whenever the sequence is above a fixed level. The only other assumption on the random sequence is that the increments satisfy a uniform exponential decay condition. The bounds provide a flexible technique for proving stability of processes frequently encountered in the control of queues.

Two applications are given. First, exponential-type bounds are derived for aGI/G/1 queue when the service distribution is exponential type. Secondly, geometric ergodicity is established for a certain Markov chain inwhich arises in the decentralized control of a multi-access, packet-switched broadcast channel.
Sur la transience de certaines chaines de Markov sur les permutations
Authors: Dies Jacques-Edouard
Published: April 2006

Books being placed on a shelf, a book bi is selected with probability pi, it is removed and replaced at the left-hand end of the shelf prior to next removal. The different arrangements of books on the shelf are considered as states in a Markov chain, known as a Tsetlin library. In this paper, we establish the necessary and sufficient condition of transience for some extensions of this chain.
Inner contact probabilities for convex bodies
Authors: Weil Wolfgang
Published: April 2006

LetKand L be convex bodies inwhereLcan roll freely inK.Suppose that Borel sets are painted in the boundaries ofKandL.The probability that after a random rolling ofLinKthe contact is paint-to-paint is determined and expressed by curvature measures ofKandL.
A stochastic model for sand sorting in a wind-tunnel
Authors: Barndorff-Nielsen O. , Darroch J. N.
Published: April 2006

The wind exerts a sorting effect on particles of sands and, under certain stable conditions that occur frequently but whose nature is little understood, the sorting results in log-size distributions of the hyperbolic form, first noted by R. A. Bagnold. Here, for wind-tunnel experiments a stochastic model is constructed which exhibits a sorting effect deriving from the dependence of distance travelled on the size of the single particle. Under rather specific, experimentally testable assumptions the model reproduces log-size distributions which are of the hyperbolic type and show a variation with distance along the wind tunnel that accords with experimental findings of R. A. Bagnold.
Critical sponge dimensions in percolation theory
Authors: Grimmett G. R.
Published: April 2006

In the bond percolation process on the square lattice, with let S(k) be the probability that some open path joins the longer sides of a sponge with dimensions k by a log k. There exists a positive constant α = αp such that

Consequently, the subset of the square lattice {(x, y):0 ≦ y ≦ f(x)} which lies between the curve y = f(x) and the x-axis has the same critical probability as the square lattice itself if and only if f(x)/log x → ∞ as x → ∞.
On determining absorption probabilities for Markov chains in random environments
Authors: Bourgin Richard D. , Cogburn Robert
Published: April 2006

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.
Continuity of random sequences and approximation of Markov chains
Authors: Kalashnikov V. V. , Anichkin S. A.
Published: April 2006

We derive conditions for the time-uniform continuity of random sequences with respect to variations of governing parameters, and also obtain some estimates of the modulus of continuity. We then apply the results to find conditions of continuity for Markov chains with arbitrary state space, and to construct finite approximations to them.
Towards models of screening for early detection of disease
Authors: Shahani A. K. , Crease D. M.
Published: April 2006

Screening for an early detection of disease, particularly when the screening tests have to be carried out on a number of occasions during the lifetime of a person, is a complex issue and the development of suitable models that can help in setting up good screening programmes for a community is a challenging task. In most cases the current emphasis is on screening at regular intervals of time and the frequency of screening is determined, usually in an informal way, by factors like the nature of the test and the nature of the disease. Analysis of a simple two-state model leads to a basic building-block which can be used in the construction of models for a variety of screening programmes. The use of this building block is illustrated and it is shown that the determination of non-regular screening times in a formal manner can lead to a greater efficiency.
On the asymptotic behaviour of branching processes with infinite mean
Authors: Schuh H.-J. , Barbour A. D.
Published: April 2006

The paper deals with the asymptotic behaviour of infinite mean Galton–Watson processes (denoted by {Zn}). We show that these processes can be classified as regular or irregular. The regular ones are characterized by the property that for any sequence of positive constants {Cn}, for which a.s. exists, The irregular ones, which will be shown by examples to exist, have the property that there exists a sequence of constants {Cn} such that In Part 1 we study the properties of {Zn/Cn} and give some characterizations for both regular and irregular processes. Part 2 starts with an a.s. convergence result for {yn(Zn)}, where {yn} is a suitable chosen sequence of functions related to {Zn}. Using this, we then derive necessary and sufficient conditions for the a.s. convergence of {U(Zn)/Cn}, where U is a slowly varying function. The distribution function of the limit is shown to satisfy a Poincaré functional equation. Finally we show that for every process {Zn} it is possible to construct explicitly functions U, such that U(Zn)/en converges a.s. to a non-degenerate proper random variable. If the process is regular, all these functions U are slowly varying. The distribution of the limit depends on U, and we show that by appropriate choice of U we may get a limit distribution which has a positive and continuous density or is continuous but not absolutely continuous or even has no probability mass on certain intervals. This situation contrasts strongly with the finite mean case.
Convergence theorems for empirical Lorenz curves and their inverses
Authors: Goldie Charles M.
Published: April 2006

The Lorenz curve of the distribution of ‘wealth’ is a graph of cumulative proportion of total ‘wealth’ owned, against cumulative proportion of the population owning it. This paper uses Gastwirth's definition of the Lorenz curve which applies to a general probability distribution on (0, ∞) having finite mean; thus it applies both to a ‘population’ distributionand to empirical distributions obtained on sampling. The Lorenz curves of the latter are proved to converge, with probability 1, uniformly to the former, and similarly for their inverses. Modified Lorenz curves are also defined, which treat atoms ofdifferently, and these and their inverses are proved strongly consistent. Functional central limit theorems are then proved for empirical Lorenz curves and their inverses, under condition thatbe continuous and have finite variance. A mild variation condition is also needed in some circumstances. If the support ofis connected, the weak convergence is relative toC[0, 1] with uniform topology, otherwise toD[0, 1] withM1topology. Selected applications are discussed, one being to the Gini coefficient.
Statistical spatial series modelling
Authors: Tjøstheim Dag
Published: April 2006

A random spatial series is a collection of random variablesF(xi, · · ·,xn) depending on several spatial coordinates (x1, · · ·,xn). An attempt is made to construct a statistical second-order theory of such series when (x1, · · ·,xn) varies over a regular cartesian lattice. Using the properties of the linear (Hilbert) space associated with the series, the concepts of innovation and purely non-deterministic (p.n.d.) series are introduced. For a p.n.d. seriesF(x1, · · ·,xn) a unilateral representation is obtained in terms of a white innovations seriesZ(y1, · · ·,yn) whereThe representatation is specialized to the homogeneous case and we discuss spectral conditions for p.n.d. The familiar time-series condition ∫ logf(λ)dλ > –∞ on the spectral densityfis necessary but not sufficient. A sufficient condition is stated. Motivated by the p.n.d. unilateral representation results we define unilateral arma (autoregressive-moving average) spatial series models. Stability and invertibility conditions are formulated in terms of the location of zero sets of polynomials relative to the unit polydisc inCn, and a rigorous shift operator formalism is established. For autoregressive spatial series a Yule–Walker-type matrix equation is formulated and it is shown how this can be used to obtain estimates of the autoregressive parameters. It is demonstrated that under mild conditions the estimates are consistent and asymptotically normal.
Applying the method of phases in the optimization of queuing systems
Authors: Langen Hans-Joachim
Published: April 2006

A new device in the optimization of queuing systems is introduced by using the method of phases. Non-exponential queues under control are considered with respect to the expected discounted reward criterion. For models with hyper-Erlang distributions equivalent phase-type systems are established. Approximation results for Markov decision models allow the extension to the case of general distribution functions. The approach is demonstrated by finding the form of an optimal policy for the GI/M/c queue with customer admission and batch arrival as well as for the GI/M/1 queue with interarrival time control.
Limit theorems for the simple branching process allowing immigration, II. The case of infinite offspring mean
Authors: Barbour A. D. , Pakes Anthony G.
Published: April 2006

This paper presents some limit theorems for the simple branching process allowing immigration, {Xn}, when the offspring mean is infinite. It is shown that there exists a function U such that {e–nU/(Xn)} converges almost surely, and if s = ∑ bj, log+U(j) < ∞, where {bj} is the immigration distribution, the limit is non-defective and non-degenerate but is infinite if s = ∞.

When s = ∞, limit theorems are found for {U(Xn)} which involve a slowly varying non-linear norming.
On multitype branching processes in a random environment
Authors: Tanny David
Published: April 2006

This paper is concerned with the growth of multitype branching processes in a random environment (mbpre). It is shown that, under suitable regularity conditions, the process either explodes of becomes extinct. A classification theorem is given delineating the cases of explosion or extinction. Furthermore, it is shown that the process grows at an exponential rate on its set of non-extinction provided the process is stable. Criteria is given for non-certain extinction of the mbpre to occur, and an example shows that the stability condition cannot be removed. The method of proof used, in general, is direct probabilistic computation rather than the classical functional iteration techniques. Growth theorems are first proved for increasing mbpre and subsequently transferred to general mbpre using the associated mbpre and the reduced mbpre.
Choosing the best of the current crop
Authors: Campbell Gregory , Samuels Stephen M.
Published: April 2006

A best choice problem is presented which is intermediate between the constraints of the ‘no-information’ problem (observe only the sequence of relative ranks) and the demands of the ‘full-information’ problem (observations from a known continuous distribution). In the intermediate problem prior information is available in the form of a ‘training sample’ of sizemand observations are the successive ranks of thencurrent items relative to their predecessors in both the current and training samples.

Optimal stopping rules for this problem depend on m andnessentially only throughm + n; and, asm/(m + n) →t, their success probabilities,P*(m, n), converge rapidly to explicitly derived limitsp*(t) which are the optimal success probabilities in an infinite version of the problem. For fixedn, P*(m, n) increases withmfrom the ‘no-information’ optimal success probability to the ‘full-information’ value for sample sizen.And astincreases from 0 to 1,p*(t) increases from the ‘no-information’ limite–1≍ 0·37 to the ‘full-information’ limit ≍0·58. In particularp*(0·5) ≍ 0·50.
Exact sampling distributions from the infinite neutral alleles model
Authors: Griffiths R. C.
Published: April 2006

The exact transient sampling distribution is found for the infinite neutral alleles model with mutation, thus extending the stationary distribution found by Ewens (1972). The first eigenfunction in the sampling distribution depends only on the homozygosity. In a large sample the expected number of allele types is close to the expected number in a stationary distribution and depends little on the initial frequencies or the time of sampling. The expected number of types in a sample for a non-stationary population is tabulated.

The sampling distribution from a population with no mutation is found. The probability that the population is monomorphic given that the sample contains only one type is tabulated, where initially the population has a large number of alleles with equal frequencies.

A study is made of the joint distribution of the allele frequencies in two samples taken timetapart from a stationary population. Of particular interest is the number of allele types in common in two such samples. The distribution of the number of types in common in a population viewed at timetapart is also found and tabulated for different mutation rates and time. The expected frequency of the allele types in common is tabulated.

The distribution of allele frequencies in two divergent populations of common ancestry is shown to be the same as the distribution of frequencies in a single population at two time points 2tapart. A sufficient statistic for the time of divergence is shown to be the paired frequencies of the allele types in common. The expected waiting time until the populations have no allele types in common is tabulated.
A natural class of multilocus recombination processes and related measures of crossover interference
Authors: Karlin Samuel , Liberman Uri
Published: April 2006

Various classifications and representations of multilocus recombination structures are delimited. A class of recombination distributions called the count–location chiasma process is parametrized by a distribution of the number of crossover events and for such crossover events by a conditional distribution of crossover locations. A number of properties and examples of this recombination structure are developed connecting orderings among the recombination mapping functions and the nature of interference.
The geometry of random drift V. Axiomatic derivation of the WFK diffusion from a variational principle
Authors: Antonelli Peter L.
Published: April 2006

An axiomatic derivation of the Wright–Fisher–Kimura (wfk) diffusion model for genetic drift is given using a variational principle. This is analogous to the characterization of the standard normal distribution in terms of an isoperimetric problem in the calculus of variations where the integrand is Shannon's information measure. We, on the other hand, give a Fisher-type information-theoretic interpretation of the variational principle largely motivated by the geometric approach to statistical likelihood theory due to S. V. Huzurbazzar, B. R. Rao and A. W. F. Edwards. In our process theory, the Ricci curvature tensor plays the role of information matrix. Ultimately, it is proved to be a normalized matrix of second partial derivates of the pseudodensity associated with the Christoffel velocity field. The proofs use classical projective differential geometry and depend on previous work in this series on the geometry of random drift.
Comments on the age distribution of Markov processes
Authors: Pakes Anthony G. , Tavaré Simon
Published: April 2006

Previous work on the concept of a limiting conditional age distribution of a discrete-state continuous-time Markov process with one absorbing state is generalised. The generalisation allows this process to have a finite number of absorbing states and the associated return process to have an arbitrary initial distribution on the transient states of the absorbing process. If the return process is ρ-recurrent, possesses the strong ratio limit property and satisfies some further requirements then the limiting age distribution exists. The proof of this result requires a new representation of the ρ-invariant measure of the return process.

The following examples are treated, (a) finite state space birth-death processes, (b) Markov branching processes and the linear death process, and (c) the linear birth and death process with killing.
Insensitive average residence times in generalized semi-Markov processes
Authors: Barbour A. D. , Schassberger R.
Published: April 2006

For a broad class of stochastic processes, the generalized semi-Markov processes, conditions are known which imply that the steady state distribution of the process, when it exists, depends only on the means, and not the exact shapes, of certain lifetime distributions entering the definition of the process. It is shown in the present paper that this insensitivity extends to certain average and conditional average residence times. Particularly interesting applications can be found in the field of networks of queues.
Brownian motion and a sharply curved boundary
Authors: Barbour A. D.
Published: April 2006

Daniels (1974) reduced the problem of approximating the distribution of the maximum size of a closed epidemic to that of finding the distribution of max0≦t≦2{W(t) –N1/2c(t)}, wherecis a smooth function with a unique minimum of 0 att= 1, and he derived an approximation to this distribution which he showed to be accurate to orderN–1/4. In this paper, his approximation is shown to be accurate to orderN–1/3, and a refined approximation is given which is accurate to orderN–1/2logN.The new approximation is still normal, and its accuracy is similar to that of the original approximation of a discrete process by the Wiener process.
Insensitivity in queueing systems
Authors: Burman David Y.
Published: April 2006

It is well known that the stationary distribution of the number of busy servers in the Erlang blocking system (M/G/c/c) depends on the service-time distribution only through its mean. This insensitivity property is shared by several other queueing systems. In this paper, we give simple sufficient conditions for determining if this insensitivity property holds for general queueing systems and related stochastic models. The conditions involve determining whether the solution of the stationary Markovian flow equations also solves certain restricted flow equations. The proof that these conditions are sufficient is direct and elementary.
Markov properties of diffusion local time: a martingale approach
Authors: McGill Paul
Published: April 2006

This paper uses martingale calculus in order to study the Markov properties of diffusion local time first discovered by Ray and Knight. The approach enables us to calculate the laws of the processes involved and is easily modified to deal with conditioning with respect to the excursion σ-fields.
An infinite-alleles version of the simple branching process
Authors: Griffiths R. C. , Pakes Anthony G.
Published: April 2006

Individuals in a population which grows according to the rules defining the simple branching process can mutate to novel allelic forms. We obtain limit theorems for the number of alleles present in any generation, the total number of alleles ever seen and the number of the generation containing the last mutation event.

In addition we define a notion of frequency spectrum for each generation as the expected number of alleles having a given number of representatives. As the generation number increases we prove the existence of a limiting notion of the frequency spectrum and discuss its upper tail behaviour. Our results here are incomplete and we make some conjectures which are supported by informal argument and specific examples.
Thinning of point processes—covariance analyses
Authors: Chandramohan Jagadeesh , Foley Robert D. , Disney Ralph L.
Published: April 2006

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.
Distribution properties of the system failure time in a general shock model
Authors: Shanthikumar J. G. , Sumita Ushio
Published: April 2006

In this paper we study some distribution properties of the system failure time in general shock models associated with correlated renewal sequences (Xn, Yn) . Two models, depending on whether the magnitude of the nth shock Xn is correlated to the length Yn of the interval since the last shock, or to the length of the subsequent interval to the next shock, are considered. Sufficient conditions under which the system failure time is completely monotone, new better than used, new better than used in expectation, and harmonic new better than used in expectation are given for these two models.
On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls
Authors: Falin G. I.
Published: April 2006

We propose a simple and efficient method of obtaining sufficient conditions for the existence of a stationary regime for multichannel fully available queueing systems with repeated calls.
Normal convergence of multidimensional shot noise and rates of this convergence
Authors: Heinrich Lothar , Schmidt Volker
Published: April 2006

Using a representation formula expressing the mixed cumulants of realvalued random variables by corresponding moments, sufficient conditions are given for the normal convergence of suitably standardized shot noise assuming that the generating stationary point process is independently marked and Brillinger mixing and that its intensity tends to ∞. Furthermore, estimates for the rate of this normal convergence are obtained by exploiting a general lemma on probabilities of large deviations and on the rate of normal convergence.
Plane polygons and a conjecture of Blaschke’s
Authors: Waksman Peter
Published: April 2006

The Radon transform of a plane domain is a random variable assigning to each line in the plane the chord length of its intersection with the domain. The probability distribution of this random variable does not characterize the domain, but it is shown to characterize a sufficiently asymmetric convex polygon. Under weaker assumptions, a convex polygon is characterized by this distribution, up to a finite number of rearrangements.
Approximations for compound Poisson and Pólya processes
Authors: Embrechts Paul , Jensen Jens L. , Maejima Makoto , Teugels J. L.
Published: April 2006

Suppose Xi≧0 are i.i.d., i = 1, 2, ···. We derive a saddlepoint approximation for P{∑N(t)k=1Xk> y} as y→∞ and t is fixed, where N(t), t≧0, is either a Poisson or a Pólya process. These results are then compared and contrasted with the well-known Esscher approximation.
Semi-Markov shock models with additive damage
Authors: Posner M. J. M. , Zuckerman D.
Published: April 2006

We examine a replacement model for a semi-Markov shock model with additive damage. Sufficient conditions are given for the optimality of control limit policies. The paper generalizes and unifies previous research in the area.

In addition, we investigate in detail the practical modelling and computational aspects of the replacement problem using a semi-Markov modelling structure.
Stationary representation of queues. II
Authors: Szczotka Władysław
Published: April 2006

The paper is a continuation of [7]. One of the main results is as follows: if the sequence (w, v, u) is asymptotically stationary in some sense then (l, w, v, u) is asymptotically stationary in the same sense. The other main result deals with an asymptotic behaviour of the vector of the queue size and the waiting time in the heavy-traffic situation. This result resembles a formula of the Little type.
A note on Cowan’s process of space division
Authors: Mecke J.
Published: April 2006

Relations of a Markov chain describing the repeated splitting of polygons to processes known from queueing theory are pointed out.
Time series analysis and adaptive filtering in hydrology
Authors: Delleur J. W.
Published: April 2006

Most time series models in hydrology are used for river flow forecasting, for generation of synthetic data sequences or for the study of physical characteristics underlying the hydrological processes. The models are formulated as linear stochastic difference equations. Three phases are considered for the selection of a model based on a satisfactory representation of a given empirical time series: identification, estimation and validation. Several criteria have been proposed for the selection of the order of ARMA models. The Akaike information criterion (Ale) is popular among hydrologists, but the posterior probability criterion has the advantage of optimality and asymptotic consistency. There are numerous applications of AR or ARMA models to annual streamflow series which are stationary. Seasonal, monthly, weekly or daily streamflow series are cyclically stationary and generally exhibit periodicities in the mean and variance and possibly in the autocorrelation structure. Removal of the periodicity has been accomplished by fitting harmonic series or by subtracting the seasonal mean and dividing by the seasonal standard deviation, and a time series model is then fitted to the residual series. Alternatively, ARMA models with time-varying coefficients are also used. The multiplicative ARlMA model of Box and Jenkins is less frequent in hydrology because of the difficulty in the identification of the parameter structure. Multivariate models are used when river flows at different sites are considered. Parameter estimation in multivariate time series models can become cumbersome because of the dimensionality of the problem. Often the covariance matrix of the noise term is not known in advance and limited information estimates are used. Multivariate models have been used for annual and monthly series. Disaggregation models have been used to subdivide a yearly series into monthly or weekly series or to disaggregate a main river flow into tributary flows while maintaining certain space and time cross-correlations. The aggregation of monthly into yearly time series has been shown to improve the parameter estimation of the yearly series. Hydrologic time series occasionally exhibit changes in level due to natural or man-made causes such as forest fires, volcanic eruption, climatological change, urbanization etc. These situations can be treated making use of intervention analysis.
A new approach to the M/G/1 processor-sharing queue
Authors: Schassberger R.
Published: April 2006

The M/G/1 processor-sharing queue is studied by way of an approximating sequence of models featuring a round-robin discipline and operating in discrete time. In particular, residence-time distributions of jobs are derived.
A note on cycle times in tree-like queueing networks
Authors: Harrison P. G.
Published: April 2006

Cycle-time distribution is shown to take the form of a linear combination of M Erlang-N density functions in a cyclic queueing network of M servers and N customers. For paths of m servers in tree-like networks, the components in the more complex linear combination are convolutions of Erlang-N with at most m − 1 negative exponentials.
Integer-valued branching processes with immigration
Authors: Steutel F. W. , Vervaat W. , Wolfe S. J.
Published: April 2006

The notion of self-decomposability for -valued random variables as introduced by Steutel and van Harn [10] and its generalization by van Harn, Steutel and Vervaat [5], are used to study the limiting behaviour of continuous-time Markov branching processes with immigration. This behaviour provides analogues to the behaviour of sequences of random variables obeying a certain difference equation as studied by Vervaat [12] and their continuous-time counterpart considered by Wolfe [13]. An application in queueing theory is indicated. Furthermore, discrete-state analogues are given for results on stability in the processes studied by Wolfe, and for results on self-decomposability in supercritical branching processes by Yamazato [14].
Asymptotic expansions of moments of the waiting time in closed and open processor-sharing systems with multiple job classes
Authors: Mitra Debasis , Morrison J. A.
Published: April 2006

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of queueing networks. We give effective methods for computing, for each of possibly several job-classes, the second moment of the equilibrium waiting time for closed systems as well as for open systems. Both open and closed systems have a CPU operating under the processor-sharing (‘time-slicing') discipline in which service-time requirements may depend on job-class. The closed system also includes a bank of terminals grouped according to job-classes, with the class structure allowing distinctions in the user's behavior in the terminal. In the contrasting open system, the job streams submitted to the CPU are Poisson with rate parameters dependent on job-classes.

Our results are exact for the open system and, for the closed system, in the form of an asymptotic series in inverse powers of a parameter N. In fact, the result for open networks is simply the first term in the asymptotic series. For larger closed systems, the parameter N is larger and thus fewer terms of the series need be computed to achieve a desired degree of accuracy. The complexity of the calculations for the asymptotic expansions is polynomial in number of classes and, importantly, independent of the class populations. Only the results on the single-class systems, closed and open, were previously known.
The cyclic queue with one general and one exponential server
Authors: Boxma O. J.
Published: April 2006

This paper considers the two-stage cyclic queueing model consisting of one general (G) and one exponential (M) server. The strong connection between the present model and the M/G/1 model (with finite waiting room) is exploited to yield the joint distribution of the successive response times of a customer at the G queue and the M queue. This result reveals a surprising phenomenon: in general there is a difference between the joint distribution of the two successive response times at (first) the G queue and (then) the M queue, and the joint distribution of the two successive response times at (first) the M queue and (then) the G queue.

Another associated result is an expression for the cycle-time distribution. Special consideration is given to the case that the number of customers in the system tends to ∞, while the mean service times tend to 0 at an inversely proportional rate.
Shape distributions for planar triangles by dual construction
Authors: Gates John
Published: April 2006

A random triangle in the plane is constructed using three independent elements from a convex set of lines. Expressions are given to calculate the shape distribution from the internal width function of the line set. Two examples are given together with their maximum angle distributions; a simple inequality implies a zero collinearity constant in general. A relationship between the shape distribution and inter-line angle distribution is given.
ESS modelling of diploid populations II: stability analysis of possible equilibria
Authors: Hines W. G. S.
Published: April 2006

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, a companion paper explored relevant features of the probability simplex of allelic frequencies for a population which is genetically homogeneous except possibly at a single locus.

The Shahshahani metric is modified in this paper to produce a measure of distance near an arbitrary frequency F in the allelic simplex which can be used when some alleles are given zero weight by F. The equation of evolution for the modified metric can then be used to show that certain sets of frequencies (corresponding to equilibrium mean strategies) act as local attractors, as long as the mean strategies corresponding to those sets are non-singular or even, in most cases, singular. We identify conditions under which the measure of distance from an initial frequency to a nearby set of equilibrium frequencies corresponding to exceptional mean strategies might increase, either temporarily or for a protracted length of time.
Stochastic models of environmental pollution
Authors: Kallianpur G. , Xiong J.
Published: April 2006

In this paper, we consider several stochastic models arising from environmental problems. First, we study pollution in a domain where undesired chemicals are deposited at random times and locations according to Poisson streams. The chemical concentration can be modeled by a linear stochastic partial differential equation (SPDE) which is solved by applying a general result. Various properties, especially the limit behavior of the pollution process, are discussed. Secondly, we consider the pollution problem when a tolerance level is imposed. The chemical concentration can still be modeled by a SPDE which is no longer linear. Its properties are investigated in this paper. When the leakage rate is positive, it is shown that the pollution process has an equilibrium state given by the deterministic model treated in [2]. Finally, the linear filtering problem is considered based on the data of several observation stations.
On the maximal entropy property for ARMA processes and ARMA approximation
Authors: Huang Dawei
Published: April 2006

The existence and properties of a general ARMA (p, q) process, whose autocovariances, up to lag p, and impulse coefficients, up to lag q, coincide with some given values, are shown. A closed-form solution is obtained. Based on this, the maximal entropy property for ARMA process and the relation and difference between ARMA maximal entropy approximation and extended Padé approximation are discussed. This method may be used for the design of digital filters and for ARMA spectral estimation.
Regenerative derivatives of regenerative sequences
Authors: Glasserman Paul
Published: April 2006

Given a parametric family of regenerative processes on a common probability space, we investigate when the derivatives (with respect to the parameter) are regenerative. We primarily consider sequences satisfying explicit, Lipschitz recursions, such as the waiting times in many queueing systems, and show that derivatives regenerate together with the original sequence under reasonable monotonicity or continuity assumptions. The inputs to our recursions are i.i.d. or, more generally, governed by a Harris-ergodic Markov chain. For i.i.d. input we identify explicit regeneration points; otherwise, we use coupling arguments. We give conditions for the expected steady-state derivative to be the derivative of the steady-state mean of the original sequence. Under these conditions, the derivative of the steady-state mean has a cycle-formula representation.
On the settling time of the congested GI/G/1 queue
Authors: Stamoulis George D. , Tsitsiklis John N.
Published: April 2006

We analyze a stable GI/G/1 queue that starts operating at time t = 0 with N0 ≠ 0 customers. First, we analyze the time required for this queue to empty for the first time. Under the assumption that both the interarrival and the service time distributions are of the exponential type, we prove that , where λ and μ are the arrival and the service rates. Furthermore, assuming in addition that the interarrival time distribution is of the non-lattice type, we show that the settling time of the queue is essentially equal to N0/(μ –λ); that is, we prove that

where is the total variation distance between the distribution of the number of customers in the system at time t and its steady-state distribution. Finally, we show that there is a similarity between the queue we analyze and a simple fluid model.
The Service Time Properties of an Unreliable Server Characterize the Exponential Distribution
Authors: Khalil Z. , Dimitrov B.
Published: April 2006

Consider the total service time of a job on an unreliable server under preemptive-repeat-different and preemptive-resume service disciplines. With identical initial conditions, for both cases, we notice that the distributions of the total service time under these two disciplines coincide, when the original service time (without interruptions due to server failures) is exponential and independent of the server reliability. We show that this fact under varying server reliability is a characterization of the exponential distribution. Further we show, under the same initial conditions, that the coincidence of the mean values also leads to the same characterization.
Weak convergence of discrete scattering processes
Authors: Horváth Lajos
Published: April 2006

We show that the discrete scattering process converges weakly to a time-changed Wiener process.
Scheduling two classes of exponential jobs on parallel processors: structural results and worst-case analysis
Authors: Chang Cheng-Shang , Nelson Randolph , Pinedo Michael
Published: April 2006

In this paper, we consider scheduling problems with m machines in parallel and two classes of job. We assume that all jobs are present at time 0 and there are no further arrivals. The service times of class 1 (2) jobs are independent and exponentially distributed with mean . Each class 1 (2) job incurs a cost c1 (c2) per unit of time until it leaves the system. The objective is to minimize the expected total cost, that is the expected weighted sum of completion times. We show that the optimal policy among all preemptive policies is of threshold type. Based on these structural results, we also show that the ratio of the expected weighted sum of completion times under the cµ-rule to that under the optimal rule is less than 1·71.
The distribution of apparent occupancy times in a two-state Markov process in which brief events cannot be detected
Authors: Jalali Assad , Hawkes Alan G.
Published: April 2006

We consider a two-state Markov process in which the resolution of the recording apparatus is such that small sojourns, of duration less than some constant deadtime τ, cannot be observed: the so-called time interval omission problem. We express the probability density of apparent occupancy times in terms of an exponential and infinitely many damped oscillations. Using a finite number of these gives an extremely accurate approximation to the true density for all except small values of the time t.
Simulating level-crossing probabilities by importance sampling
Authors: Lehtonen T. , Nyrhinen H.
Published: April 2006

Let X1, X2, · ·· be independent and identically distributed random variables such that ΕΧ1 < 0 and P(X1 ≥ 0) ≥ 0. Fix M ≥ 0 and let T = inf {n: X1 + X2 + · ·· + Xn ≥ M} (T = +∞, if for every n = 1,2, ···). In this paper we consider the estimation of the level-crossing probabilities P(T <∞) and , by using Monte Carlo simulation and especially importance sampling techniques. When using importance sampling, precision and efficiency of the estimation depend crucially on the choice of the simulation distribution. For this choice we introduce a new criterion which is of the type of large deviations theory; consequently, the basic large deviations theory is the main mathematical tool of this paper. We allow a wide class of possible simulation distributions and, considering the case that M →∞, we prove asymptotic optimality results for the simulation of the probabilities P(T <∞) and . The paper ends with an example.
A simple coupling of renewal processes
Authors: Lindvall T.
Published: April 2006

We use a simple coupling to prove the classical result that the renewal function U of a zero-delayed renewal process satisfies U(t) – λ . t→λ2μ2/2 as t→∞ if the life-length distribution is of non-lattice type and has finite first and second moments μ and μ2 respectively; λ is the renewal intensity, and is equal to 1/μ.
Limit theorem and large deviation principle for the Voronoi tessellation generated by a Gibbs point process
Authors: Kuroda Koji , Tanemura Hideki
Published: April 2006

The Voronoi tessellation generated by a Gibbs point process is considered. Using the algebraic formalism of polymer expansion, the limit theorem and the large deviation principle for the number of Voronoi vertices are proved.
Some notes on two types of minimal repair
Authors: Finkelstein M. S.
Published: April 2006

Two types of minimal repair are discussed. After statistical minimal repair, the state of a system is statistically identical to what it was just before the failure. Physical minimal repair restores the failed unit to its exact physical condition before the failure. Several examples are given.
The Computation of Average Optimal Policies in Denumerable State Markov Decision Chains
Authors: Sennott Linn I.
Published: April 2006

This paper studies the expected average cost control problem for discrete-time Markov decision processes with denumerably infinite state spaces. A sequence of finite state space truncations is defined such that the average costs and average optimal policies in the sequence converge to the optimal average cost and an optimal policy in the original process. The theory is illustrated with several examples from the control of discrete-time queueing systems. Numerical results are discussed.
On the approximation of a ball by random polytopes
Authors: Küfer K.-H.
Published: April 2006

Let be a sequence of independent and identically distributed random vectors drawn from the d-dimensional unit ball Bd and let Xn be the random polytope generated as the convex hull of a1,· ··, an. Furthermore, let Δ(Xn): = Vol (BdXn) be the volume of the part of the ball lying outside the random polytope. For uniformly distributed ai and 2 we prove that the limiting distribution of Δ(Xn)/Ε (Δ (Xn)) for n → ∞ (satisfies a 0–1 law. In particular, we show that Var for n → ∞. We provide analogous results for spherically symmetric distributions in Bd with regularly varying tail. In addition, we indicate similar results for the surface area and the number of facets of Xn.
Potentially unlimited variance reduction in importance sampling of Markov chains
Authors: Andradóttir Sigrún , Heyman Daniel P. , Ott Teunis J.
Published: April 2006

We consider the application of importance sampling in steady-state simulations of finite Markov chains. We show that, for a large class of performance measures, there is a choice of the alternative transition matrix for which the ratio of the variance of the importance sampling estimator to the variance of the naive simulation estimator converges to zero as the sample path length goes to infinity. Obtaining this ‘optimal’ transition matrix involves computing the performance measure of interest, so the optimal matrix cannot be computed in precisely those situations where simulation is required to estimate steady-state performance. However, our results show that alternative transition matrices of the form Q = P + E/T, where P is the original transition matrix and T is the sample path length, can be expected to provide good results. Moreover, we provide an iterative algorithm for obtaining alternative transition matrices of this form that converge to the optimal matrix as the number of iterations increases, and present an example that shows that spending some computer time iterating this algorithm and then conducting the simulation with the resulting alternative transition matrix may provide considerable variance reduction when compared to naive simulation.
On procrustean mean shapes and the shape of the means
Authors: Le Hulling
Published: April 2006

Two sets of k labelled points, or configurations, in ℝm are defined to have the same shape if they differ only in translation, rotation and scaling. An important matter in practice is the estimation of the shape of the means; the shape determined by the means of data on the vertices of configurations. However, statistical models for vertices-based shapes always involve some unknown samplewise nuisance parameters associated with ambiguity of location, rotation and scaling. The use of procrustean mean shapes for a finite set of configurations, which are usually formulated directly in terms of their vertices, will enable one to eliminate these nuisance parameters.