bluuebunny/crossref_metadata_2025 huggingface dataset:
Birth-and-death processes on the integers with phases and general boundaries
Authors: Hajek Bruce
Published: July 2016

The invariant probability distribution is found for a class of birth-and-death processes on the integers with phases and one or two boundaries. The invariant vector has a matrix geometric form and is found by solving a non-linear matrix equation and then finding an invariant probability distribution on the boundary states. Levy's concept of watching a Markov process in a subset is used to naturally decouple the computation of distributions on the boundary and interior states.
On the reduction in remaining system lifetime due to the failure of a specific component
Authors: Natvig Bent
Published: July 2016

In a previous paper, Natvig [4], we suggested a new measure of the importance of a component in a coherent system and derived some of its properties. The measure is for the case of components not undergoing repair proportional to the expected reduction in remaining system lifetime due to the failure of the component. In the present paper, we arrive at the whole distribution of this reduction in remaining system lifetime. Furthermore, for the case where components have proportional hazards, and are not repaired, a speculation of another measure is given. This measure is proportional to the derivative of the expected total lifetime of a new system with respect to the inverse of the component's proportional hazard rate.
Exponential trends of Ornstein–Uhlenbeck first-passage-time densities
Authors: Nobile A. G. , Ricciardi L. M. , Sacerdote L.
Published: July 2016

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.
Transient solution to the many-server Poisson queue: a simple approach
Authors: Parthasarathy P. R. , Sharafali M.
Published: July 2016

An elegant time-dependent solution for the number in the M/M/c queueing system is derived in a direct way.
On a Daley-Kendall model of random rumours
Authors: Pittel B.
Published: July 2016

Suppose that a certain population consists of N individuals. One member initially learns a rumour from an outside source, and starts telling it to other members, who continue spreading the information. A knower becomes inactive once he encounters somebody already informed. Daley and Kendall, who initiated the study of this model, conjectured that the number of eventual knowers is asymptotically normal with mean and variance linear in N. Our purpose is to confirm this conjecture.
Empirical Laplace transform and approximation of compound distributions
Authors: Csörgő Sándor , Teugels Jef L.
Published: July 2016

Letbe i.i.d. non-negative random variables with d.f.Fand Laplace transformL.LetNbe integer valued and independent ofIn many applications one knows that fory→ ∞ and a functionφwhere in turnτis the solution of an equationOn the basis of a sample of sizenwe derive an estimatorτnfor τ by solvingψ(τn, Ln(τn),L′n(τn), · ··) = 0 whereLnis the empirical version ofL.This estimator is then used to derive the asymptotic behaviour ofφ(y, τn, Ln(τn),L′n(τn), · ··). We include five examples, some of which are taken from insurance mathematics.
On remaining full busy periods of GI/G/c queues and their relation to stationary point processes
Authors: Ghahramani Saeed
Published: July 2016

For a GI/G/c queue, a full busy period is a period commencing when an arrival finds c − 1 customers in the system and ending when, for the first time after that, a departure leaves behind c − 1 customers in the system. We show that given a full busy period is found to be in progress at a random epoch, the remaining full busy period has the equilibrium distribution. Moreover, we demonstrate that this property is typical for a broad class of stationary random processes.
Mixing rates for Brownian motion in a convex polyhedron
Authors: Matthews Peter
Published: July 2016

For Brownian motion on a convex polyhedral subset of a sphere or torus, the rate of convergence in distribution to uniformity is studied. The main result is a method to take a Markov coupling on the full sphere or torus and create a faster coupling on the convex polyhedral subset. Upper bounds on variation distance are computed, and applications are discussed.
Equilibrium distribution of block-structured Markov chains with repeating rows
Authors: Grassmann Winfried K. , Heyman Daniel P.
Published: July 2016

In this paper we consider two-dimensional Markov chains with the property that, except for some boundary conditions, when the transition matrix is written in block form, the rows are identical except for a shift to the right. We provide a general theory for finding the equilibrium distribution for this class of chains. We illustrate theory by showing how our results unify the analysis of the M/G/1 and GI/M/1 paradigms introduced by M. F. Neuts.
On an index policy for restless bandits
Authors: Weber Richard R. , Weiss Gideon
Published: July 2016

We investigate the optimal allocation of effort to a collection of n projects. The projects are ‘restless' in that the state of a project evolves in time, whether or not it is allocated effort. The evolution of the state of each project follows a Markov rule, but transitions and rewards depend on whether or not the project receives effort. The objective is to maximize the expected time-average reward under a constraint that exactly m of the n projects receive effort at any one time. We show that as m and n tend to ∞ with m/n fixed, the per-project reward of the optimal policy is asymptotically the same as that achieved by a policy which operates under the relaxed constraint that an average of m projects be active. The relaxed constraint was considered by Whittle (1988) who described how to use a Lagrangian multiplier approach to assign indices to the projects. He conjectured that the policy of allocating effort to the m projects of greatest index is asymptotically optimal as m and n tend to∞. We show that the conjecture is true if the differential equation describing the fluid approximation to the index policy has a globally stable equilibrium point. This need not be the case, and we present an example for which the index policy is not asymptotically optimal. However, numerical work suggests that such counterexamples are extremely rare and that the size of the suboptimality which one might expect is minuscule.
Complementary generating functions for the MX/GI/1/k and GI/My/1/K queues and their application to the comparison of loss probabilities
Authors: Miyazawa Masakiyo
Published: July 2016

A direct proof is presented for the fact that the stationary system queue length distribution just after the service completion epochs in the Mx/GI/1/k queue is given by the truncation of a measure on Z+ = {0, 1, ·· ·}. The related truncation formulas are well known for the case of the traffic intensity ρ < 1 and for the virtual waiting time process in M/GI/1 with a limited waiting time (Cohen (1982) and Takács (1974)). By the duality of GI/MY/1/k to Mx/GI/1/k + 1, we get a similar result for the system queue length distribution just before the arrival of a customer in GI/MY/1/k. We apply those results to prove that the loss probabilities of Mx/GI/1/k and GI/MY/1/k are increasing for the convex order of the service time and interarrival time distributions, respectively, if their means are fixed.
Closed-form solution for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period
Authors: Sericola Bruno
Published: July 2016

Markov process are widely used to model computer systems. De Souza e Silva and Gail [3] calculated numerically the distribution of the cumulative operational time of repairable computer systems modelled by Markovian processes, that is, the distribution of the total time during which the system was in operation over a finite observation period. An extension of their approach is presented here. A closed-form solution is obtained for the distribution of the total time spent in a subset of states of a homogeneous Markov process during a finite observation period, which is theoretically and numerically interesting. We also give an application of this result to a fault-tolerant system.
A paradox of congestion in a queuing network
Authors: Cohen Joel E. , Kelly Frank P.
Published: July 2016

In an uncongested transportation network, adding routes and capacity to an existing network must decrease, or at worst not change, the average time individuals require to travel through the network from a source to a destination. Braess (1968) discovered that the same is not true in congested networks. Here we give an example of a queuing network in which added capacity leads to an increase in the mean transit time for everyone. Self-seeking individuals are unable to refrain from using the additional capacity, even though using it leads to deterioration in the mean transit time. This example appears to be the first queuing network to demonstrate the general principle that in non-co-operative games with smooth payoff functions, user-determined equilibria generically deviate from system-optimal equilibria.
Limit theorems for maxima and crossings of a sequence of Gaussian processes and approximation of random processes
Authors: Seleznjev O. V.
Published: July 2016

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.
The superposition of independent discrete Markovian packet streams
Authors: Neuts Marcel F. , Pearce Charles E. M.
Published: July 2016

In a finite, aperiodic, irreducible Markov chain, a visit to some states generates a one; to some others, a zero, while the remaining states are considered silent, in that neither symbol is generated. States during which either a one or a zero is generated are called active states, and sojourns in the set of active states correspond to messages. The output process is called a Markovian packet stream. Informally, a stream is called thin, if the steady-state fraction of time spent in the active states is small and messages are separated by silent periods of long durations. A limit theorem for the superposition of a large number of independent, stochastically identical and appropriately thin Markovian packet streams is obtained. The class of limit processes consists of a family of stationary, integer-valued, discrete-parameter processes of dependent Poisson random variables. Some properties of the limit processes are established.
Optimal control of an M/G/1 queue with imperfectly observed queue length when the input source is finite
Authors: Wakuta Kazuyoshi
Published: July 2016

We consider the optimal control of an M/G/1 queue with finite input source. The queue length, however, can be only imperfectly observed through the observations at the initial time and the times of successive departures. At these times, the service rate can be chosen, based on the observable histories. A service cost and a holding cost are incurred. We show that such a control problem can be formulated as a semi-Markov decision process with imperfect state information, and present sufficient conditions for the existence of an optimal stationary I-policy.
Stochastic ordering of order statistics
Authors: Barbour A. D. , Lindvall T. , Rogers L. C. G.
Published: July 2016

If Xi
, i = 1, ···, n are independent exponential random variables with parameters λ
1, · ··, λ n
, and if Yi, i = 1, ···, n are independent exponential random variables with common parameter equal to (λ1 + · ·· + λ n
)/n, then there is a monotone coupling of the order statistics X
(1), · ··, X
(n) and Y
(1), · ··, Y
(n); that is, it is possible to construct on a common probability space random variables X′
i
, Y′
i
, i = 1, ···, n, such that for each i, Y
′
(i)≦X
′
(i) a.s., where the law of the X
′

i
(respectively, the Y
′

i
) is the same as the law of the Xi
(respectively, the Yi.) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ···, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.
On the limit distribution of the number of cycles in a random graph
Authors: Takács Lajos
Published: July 2016

Let Γ
n
(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where . Denote by γ n
(p) the total number of cycles in the graph Γ
n
(p). The main object of this paper is to prove that the limit distribution of γ n
(λ /n) is a Poisson distribution with expectation if n → ∞and 0 < λ< 1.
On solutions of a stochastic integral equation of the volterra type with applications for chemotherapy
Authors: Szynal D. , Wedrychowicz S.
Published: July 2016

This paper deals with the existence of solutions of a stochastic integral equation of the Volterra type and their asymptotic behaviour. Investigations of this paper use the concept of a measure of non-compactness in Banach space and fixed-point theorem of Darbo type. An application to a stochastic model for chemotherapy is also presented.
The backlog and depletion-time process for M/G/1 vacation models with exhaustive service discipline
Authors: Keilson Julian , Ramaswamy Ravi
Published: July 2016

The vacation model studied is an M/G/1 queueing system in which the server attends iteratively to ‘secondary' or ‘vacation' tasks at ‘primary' service completion epochs, when the primary queue is exhausted. The time-dependent and steady-state distributions of the backlog process [6] are obtained via their Laplace transforms. With this as a stepping stone, the ergodic distribution of the depletion time [5] is obtained. Two decomposition theorems that are somewhat different in character from those available in the literature [2] are demonstrated. State space methods and simple renewal-theoretic tools are employed.
Response times in M/M/1 time-sharing schemes with limited number of service positions
Authors: Avi-Itzhak Benjamin , Halfin Shlomo
Published: July 2016

Two service schemes for an M/M/1 time-sharing system with a limited number of service positions are studied. Both schemes possess the equilibrium properties of symmetric queues; however, in the first one, a preempted job is placed at the end of the waiting line; while in the second one, it is placed at the head of the line. Methods for calculating the Laplace transforms and moments of the response times are presented. The variances of the response times are then compared numerically to indicate that the first scheme is superior to the second scheme. It is also indicated that in both cases the response time variance decreases when the number of service positions increases.
On the maximum of gaussian fourier series emerging in the analysis of random vibrations
Authors: Orsingher Enzo
Published: July 2016

In this paper we study the maximum of Gaussian Fourier series emerging in the analysis of random vibrations of a finite string. Evaluating the distribution of the maximal displacement corresponds to the analysis of the maximum for the related Gaussian Fourier series with independent coefficients.

When vibrations are triggered by an initial white noise disturbance (where the instantaneous form of the vibrating string is composed of three processes pieced together) we give upper bounds for the maximal displacement.

In the last section we consider forced vibrations at special instants where the instantaneous form of the vibrating string has the structure of a Brownian bridge. This enables us to give the exact distribution of the maximum for the related sine series.
Product-form queueing networks with negative and positive customers
Authors: Gelenbe Erol
Published: July 2016

We introduce a new class of queueing networks in which customers are either ‘negative' or ‘positive'. A negative customer arriving to a queue reduces the total customer count in that queue by 1 if the queue length is positive; it has no effect at all if the queue length is empty. Negative customers do not receive service. Customers leaving a queue for another one can either become negative or remain positive. Positive customers behave as ordinary queueing network customers and receive service. We show that this model with exponential service times, Poisson external arrivals, with the usual independence assumptions for service times, and Markovian customer movements between queues, has product form. It is quasi-reversible in the usual sense, but not in a broader sense which includes all destructions of customers in the set of departures. The existence and uniqueness of the solutions to the (nonlinear) customer flow equations, and hence of the product form solution, is discussed.
A gaussian process with parabolic covariances
Authors: Cabaña Enrique M.
Published: July 2016

The centred, periodic, stationary Gaussian process X(z), ≧ z ≧ 1 with covariances , appears when one studies the solutions of the vibrating string equation forced by noise, corresponding to the case of a finite string with the extremes tied together. The close relationship between this process and a Brownian bridge permits us to compute the distribution of the maximum excursion of the string at particular times.
Analysis of the Luria–Delbrück distribution using discrete convolution powers
Authors: Ma W. T. , Sandri G. vH. , Sarkar S.
Published: July 2016

The Luria–Delbrück distribution arises in birth-and-mutation processes in population genetics that have been systematically studied for the last fifty years. The central result reported in this paper is a new recursion relation for computing this distribution which supersedes all past results in simplicity and computational efficiency:p0=e–m;wheremis the expected number of mutations. A new relation for the asymptotic behavior ofpn(≈c/n2) is also derived. This corresponds to the probability of finding a very large number of mutants. A formula for thez-transform of the distribution is also reported.
Dependence structure of Poisson-paced records
Authors: Bunge J. A. , Nagaraja H. N.
Published: July 2016

Let Y
0, Y
1, Y
2, ··· be an i.i.d. sequence of random variables with absolutely continuous distribution function F, and let {N(t), t ≧ 0} be a Poisson process with rate λ (t) and mean Λ(t), independent of the Yj
's. We associate Y
0 with the point t = 0, and Yj
with the jth point of N(·), j ≧ 1. The first Yj
(j ≧ 1) to exceed all previous ones is the first record value, and the time of its occurrence is the first record time; subsequent record values and times are defined analogously. For general Λ, we give the joint distribution of the values and times of the first n records to occur after a fixed time T, 0 ≦ T < ∞. Assuming that F satisfies Von Mises regularity conditions, and that λ (t)/Λ (t) → c ∈ (0, ∞) as t → ∞, we find the limiting joint p.d.f. of the values and times of the first n records after T, as T → ∞. In the course of this we correct a result of Gaver and Jacobs (1978). We also consider limiting marginal and conditional distributions. In addition, we extend a known result for the limit as the number of records
K → ∞, and we compare the results for the limit as T → ∞ with those for the limit as K → ∞.
On the optimality of LEPT and cµ rules for machines in parallel
Authors: Chang Cheng-Shang , Chao Xiuli , Pinedo Michael , Weber Richard
Published: July 2016

We consider scheduling problems with m machines in parallel and n jobs. The machines are subject to breakdown and repair. Jobs have exponentially distributed processing times and possibly random release dates. For cost functions that only depend on the set of uncompleted jobs at time t we provide necessary and sufficient conditions for the LEPT rule to minimize the expected cost at all t within the class of preemptive policies. This encompasses results that are known for makespan, and provides new results for the work remaining at time t. An application is that if the cµ rule has the same priority assignment as the LEPT rule then it minimizes the expected weighted number of jobs in the system for all t. Given appropriate conditions, we also show that the cµ rule minimizes the expected value of other objective functions, such as weighted sum of job completion times, weighted number of late jobs, or weighted sum of job tardinesses, when jobs have a common random due date.
On fast simulation of the time to saturation of slotted ALOHA
Authors: Anantharam V.
Published: July 2016

Cottrell et al. (1983) have indicated how ideas from the large deviations theory lead to fast simulation schemes that estimate the mean time taken by the slotted ALOHA protocol to saturate starting empty. Such fast simulation schemes are particularly useful when the attempt probability is small. The remaining time to saturation when the protocol has been operating for a time is more accurately described by the quasi-stationary exit time from the stable regime. The purpose of this article is to prove that the ratio of the quasi-stationary exit time to the exit time starting empty approaches 1 as the attempt probability becomes small.
Steady-state Markov chain models for the Heine and Euler distributions
Authors: Kemp A. W.
Published: July 2016

The paper puts forward steady-state Markov chain models for the Heine and Euler distributions. The models for oil exploration strategies that were discussed by Benkherouf and Bather (1988) are reinterpreted as current-age models for discrete renewal processes. Steady-state success-runs processes with non-zero probabilities that a trial is abandoned, Foster processes, and equilibrium random walks corresponding to elective M/M/1 queues are also examined.
Asymptotic analysis of extremes from autoregressive negative binomial processes
Authors: McCormick William P. , Park You Sung
Published: July 2016

It is well known that most commonly used discrete distributions fail to belong to the domain of maximal attraction for any extreme value distribution. Despite this negative finding, C. W. Anderson showed that for a class of discrete distributions including the negative binomial class, it is possible to asymptotically bound the distribution of the maximum. In this paper we extend Anderson's result to discrete-valued processes satisfying the usual mixing conditions for extreme value results for dependent stationary processes. We apply our result to obtain bounds for the distribution of the maximum based on negative binomial autoregressive processes introduced by E. McKenzie and Al-Osh and Alzaid. A simulation study illustrates the bounds for small sample sizes.
Alternating projections and interpolation of stationary processes
Authors: Pourahmadi Mohsen
Published: July 2016

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.
Extremal properties of the FIFO discipline in queueing networks
Authors: Righter Rhonda , Shanthikumar J. George
Published: July 2016

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:

(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.

(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.

We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.

Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.
L'intégrale du mouvement brownien
Authors: Lachal Aimé
Published: July 2016

Let be the Brownian motion process starting at the origin, its primitive and Ut
= (Xt+x + ty, Bt
+ y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa
), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab
), where σ α
and σ ab
denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process .

Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb
).
On Levy's theorem concerning positiveness of transition probabilities of Markov processes: the circuit processes case
Authors: Kalpazidou S.
Published: July 2016

We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.
Boundary-crossing probabilities of some random fields related to likelihood ratio tests for epidemic alternatives
Authors: Yao Qiwei
Published: July 2016

We consider the likelihood ratio tests to detect an epidemic alternative in the following two cases of normal observations: (1) the alternative specifies a square wave drift in the mean value of an i.i.d. sequence; (2) the alternative permits a square wave drift in the intercept of a simple linear regression model. To develop the approximations for the significance levels leads us to consider boundary-crossing problems of some two-dimensional discrete-time Gaussian fields. By the method which was proposed originally by Woodroofe (1976) and adapted to study maxima of some random fields by Siegmund (1988), some large deviations for the conditional non-linear boundary-crossing probabilities are developed. Some results of Monte Carlo experiments confirm the accuracy of these approximations.
Asymptotic distribution of the maximum of n independent stochastic processes
Authors: Balkema A. A. , De Haan L. , Karandikar R. L.
Published: July 2016

Limits in distribution of maxima of independent stochastic processes are characterized in terms of spectral functions acting on a Poisson point process.
On a new formula for the transient state probabilities for M/M/1 queues and computational implications
Authors: Conolly B. W. , Langaris Christos
Published: July 2016

Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.
A constraint on the random packing of disks
Authors: Cowan Richard
Published: July 2016

This paper addresses random packing of equal-sized disks in a manner such that no disk has a gap on its circumference large enough to accommodate an extra touching neighbour. This structure generalises the deterministic packing models discussed in classical geometry (Coxeter (1961), Hilbert and Cohn-Vossen (1952)). Relationships with the dual mosaic formed by joining the centres of touching disks are established. Constraints on the neighbourhood of disks and on the packing density are established.
A logarithmic reduction algorithm for quasi-birth-death processes
Authors: Latouche Guy , Ramaswami V.
Published: July 2016

Quasi-birth-death processes are commonly used Markov chain models in queueing theory, computer performance, teletraffic modeling and other areas. We provide a new, simple algorithm for the matrix-geometric rate matrix. We demonstrate that it has quadratic convergence. We show theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.
Burn-in and mixed populations
Authors: Block Henry W. , Mi Jie , Savits Thomas H.
Published: July 2016

Burn-in is a procedure used for eliminating weak components in a mixed population. In this paper we focus on general mixed populations. Three types of results are established. First, it is shown that any mixed population displays a type of monotonicity property which is appropriate for burn-in. Second, it is shown that if, asymptotically, components have constant failure rates, then the mixed population will also asymptotically have a constant failure rate and this will correspond to the rate of the strongest subpopulation of the mixture. Finally, it is shown for a reasonable cost function, that if one mixture distribution dominates another in a strong sense, the resulting mixture of the dominant distribution will have larger optimal burn-in time.
Sur une procédure de branchement déterministe et ses dérivées aléatoires
Authors: Huillet Thierry , Kłopotowski Andrzej
Published: July 2016

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.
On transient behaviour of a nearest-neighbour birth-death process on a lattice
Authors: Granovsky Boris L. , Rozov Liat
Published: July 2016

We provide the explicit expression for the mean coverage function of a generalized voter model on a regular lattice and establish a characterization of the class of the above processes. As a result, we derive the exact rate of convergence of the considered processes to the steady state. We also prove the existence of different processes with the same mean coverage function on a given lattice.
On Cowan and Mecke's Markov chain
Authors: Liu Huaigao
Published: July 2016

In the paper, we discuss the properties of Cowan and Mecke's Markov chain P in general form. We give some criteria for determining whether the chain P is transient, recurrent null or recurrent non-null, and for the chain P to have invariant measures.
On the asymptotic distribution of the maximum number of infectives in epidemic models by immigration
Authors: Abramov V. M.
Published: July 2016

This paper considers the asymptotic distribution of the maximum number of infectives in an epidemic model by showing that, as the initial number of susceptibles converges to infinity, the process of infectives converges almost surely to a birth and death process. The model studied here is more general than usual (see e.g. Bailey (1975), Bharucha-Reid (1960), Keilson (1979)) in that it incorporates immigration and the limiting birth and death process is non-linear. The main novelty of the present paper is the martingale approach used to prove the above-mentioned convergence.
Transient solutions for denumerable-state Markov processes
Authors: Hsu Guang-Hui , Yuan Xue-Ming
Published: July 2016

The algorithm for the transient solution for the denumerable state Markov process with an arbitrary initial distribution is given in this paper. The transient queue length distribution for a general Markovian queueing system can be obtained by this algorithm. As examples, some numerical results are presented.
Renewal processes with random numbers of delays: application to a conception and birth model
Authors: Lange Kenneth , Johnson Norman J.
Published: July 2016

Asymptotic formulas and Laplace–Stieltjes transforms are derived for the first two moments of a renewal process with a random number of delays. These are simplified when all the delays follow the same distribution. An asymptotic occupancy result is also derived for two-stage renewal processes with random numbers of delays. As an example, a demographic model of conception and birth is discussed. This model represents the sequence of live births to a woman as a renewal process. If the woman practises birth control after achieving her desired family composition, the renewal process has a random number of delays.
Stochastically perturbed limit cycles
Authors: Holland Charles J.
Published: July 2016

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.
Chaînes colorées: trois extensions d'une formule de P. Nelson
Authors: Letac Gérard
Published: July 2016

Nelson [9], [10] has computed the generating function of return probabilities to the initial state for a particular Markov chain on permutations of three objects. The formula obtained is

The present paper studies three distinct Markov chains generalizing the Nelson chain: the so-called three-coloured chain, with some birth-and-death processes on ℤ as a particular case, a chain on a graph close to the graph of the edges of a cube, and the daisy library. Two other themes piece together these chains: the notion of coloured chain and the technique of computation by additive processes.
Asymptotic inference for an Ising lattice III. Non-zero field and ferromagnetic states
Authors: Pickard David K.
Published: July 2016

In previous papers (1976), (1977) limit theorems were obtained for the classical Ising model in the absence of an external magnetic field, thereby providing a basis for asymptotic inference. The present paper extends these results to arbitrary external magnetic fields. Statistical inference for this model is important because its nearest-neighbour interactions provide a natural first approximation to spatial interaction among binary variables located on square lattices.

The most interesting behaviour occurs in zero field and at or beyond the critical point. In this case, the central limit result for nearest-neighbour interactions requires an unusual norming, the limiting variances may depend on the nature of the boundary conditions, and there cannot be any central limit result for external magnetic field. The implications of these phenomena for statistical inference are also discussed. In particular, the maximum likelihood estimator of magnetic field is not consistent. Rather it appears to have a non-trivial asymptotic distribution.
Some more results for the stable M/G/1 queue in heavy traffic
Authors: Ott Teunis J.
Published: July 2016

In a previous paper the author derived some results describing the tail-behavior of the busy time distribution function B of a stable M/G/1 queue in heavy traffic, i.e., some results were given for B (t), t large. In this paper some complementary results are given, in the same heavy-traffic situation, for B (t), t small.
On the rate of convergence of normal extremes
Authors: Hall Peter
Published: July 2016

Let Yn
denote the largest of n independent N(0, 1) variables. It is shown that if the constants an
and bn
are chosen in an optimal way then the rate of convergence of (Yn
– bn
)/an
to the extreme value distribution exp(–e–x
), as measured by the supremum metric or the Lévy metric, is proportional to 1/log n.
Basic analysis of a head-of-the-line priority queue with renewal type input
Authors: Schassberger R.
Published: July 2016

A single server is fed by a renewal stream of individual customers. These are of type k with probability πk, k = 1, …, N, and are all served individually. Upon completion of a service the server proceeds immediately with a customer of the lowest type (= highest priority) present, if any. Service times for type k are drawn from a general distribution function Bk
(t) concentrated on (0, ∞).

We lay the foundations for a broad analysis of the model.
Information density phenomena and random packing
Authors: Dolby James L. , Solomon Herbert
Published: July 2016

A density model for various phenomena with densities much smaller than those explored by the Zipf model is developed. The density of spherical random packing in n dimensions serves as a model for relative frequency of length of monosyllabic words in English and also for the relative frequency of degree holders in the U.S. at various levels of educational attainment.
Multivariate extremal processes generated by independent non-identically distributed random variables
Authors: Weissman Ishay
Published: July 2016

Letbe thekth largest amongXn1, …,Xn[nt], whereXni= (Xi– an)/bn, {Xi} is a sequence of independent random variables andbn> 0 andanare norming constants. Suppose that for eachconverges in distribution. Then all the finite-dimensional laws ofconverge. The limiting process is represented in terms of a non-homogeneous two-dimensional Poisson process.
Note on a clustering problem
Authors: Saperstein B.
Published: July 2016

Consider the clustering statistic, k* defined to be the maximum number of 1's to appear within any m consecutive positions in a random arrangement of a 1's and M–a 0's. Pr(k* – k) was found for the case k ≧ ½ a by the author [4], and then for m/M = 1/L, L an integer, by Naus [3]. Here, we generalise Naus's method to obtain the distribution of k* under no special restrictions on a, k, m or M.
A modified erlang approach to approximating GI/G/1 queues
Authors: Marchal William G. , Harris Carl M.
Published: July 2016

This paper presents a modification to the traditional technique of approximating general queueing problems by the Erlang distributions. It presents both theory and the results of some numerical testing.
Excursions above a fixed level by n-dimensional random fields
Authors: Adler Robert J.
Published: July 2016

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.
On the equivalence of certain Markov chains
Authors: Vit Pedro
Published: July 2016

It is shown that an irreducible and aperiodic Markov chain can be altered preserving irreducibility without altering the nature of the chain in the sense that, the modified chain is transient (recurrent) if and only if the original chain is transient (recurrent). Furthermore, it is shown by means of a counterexample that ergodicity (null-recurrence) is not preserved.

An interesting application of this result is a simple proof of Pakes's generalization of Foster's criterion for a chain to be recurrent.
The capacity of an intersection with non-stationary traffic
Authors: Wegmann Helmut
Published: July 2016

The average number of vehicles being able to enter an intersection per time unit from a minor road with a stop or yield sign — the capacity of the intersection — depends on the density of the traffic stream on the major road. In case the time-process of the major road traffic at the intersection is a non-homogeneous Poisson process with a periodic intensity function the capacity is calculated and compared with the capacity in the homogeneous case.
Heavy traffic theory for queues with several servers. I
Authors: Köllerström Julian
Published: July 2016

Queues with several servers are examined here, in which arrivals are assumed to form a renewal sequence and successive service times to be mutually independent and independent of the arrival times. The first-come-first-served queue discipline only is considered. An asymptotic formula for the equilibrium waiting time distribution is obtained under conditions of heavy traffic.
Asymptotic properties of certain functionals of the periodogram
Authors: Davis Herbert T.
Published: July 2016

The asymptotic properties of the periodogram of a weakly stationary time series for the triangular array of fundamental frequencies is studied. For linear Gaussian processes, results are obtained relating the asymptotic distribution of certain Riemann sums of the periodogram of the process to those of the periodogram of the innovation process.
Limit theorems for sums of chain-dependent processes
Authors: O'Brien G. L.
Published: July 2016

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.
On the equivalence of certain truncated point processes
Authors: Vit Pedro
Published: July 2016

A truncated point process is defined in terms of a point process and a truncation mechanism independent of the process. The equivalence between two different types of truncation is shown, where the truncations occur after a geometric number of events have occurred or after an exponential time has elapsed, some examples are given.
Note on the reversible counters system of Lampard
Authors: Phatarfod R. M.
Published: July 2016

The paper considers an extension to the reversible counters system of Lampard [1]. In Lampard's model the input processes are two independent Poisson processes; this results in a gamma Markov sequence for the time-interval between successive output pulses and a negative binomial Markov sequence for the counts at the times of out-put pulses. We consider the input process to be a bivariate Poisson process and show that the first out-put process given above is not affected, while the second out-put-process becomes of a type studied in the theory of branching processes.
An unbiased estimator for the drift of a stopped Wiener process
Authors: Ferebee Brooks
Published: July 2016

An unbiased estimator is given for the drift of a Wiener process which is observed up to a stopping time. For first-passage times across a moving boundary the estimator can be calculated by a simple modification of standard computer programs for computing the first-passage density. Examples and interpretations of the estimator are given.
On the number of uniform random variables which must be added to exceed a given level
Authors: Russell K. G.
Published: July 2016

This note obtains the distribution of N(a), the number of independent random variables each distributed uniformly on (0, 1) which must be added for the sum first to exceed some positive constant a.
On the fluctuation of stochastically monotone Markov chains and some applications
Authors: Cohn Harry
Published: July 2016

A Borel–Cantelli-type property in terms of one-step transition probabilities is given for events like {|Xn

+1| > a + ε, |Xn|≦a}, a and ε being two positive numbers. Applications to normed sums of i.i.d. random variables with infinite mean and branching processes in varying environment with or without immigration are derived.
Survival functions induced by stochastic covariate processes
Authors: Myers Lawrence E.
Published: July 2016

A vector X of patient prognostic variables is modeled as a linear diffusion process with time-dependent, non-random, continuous coefficients. The instantaneous force of mortality (hazard function) operating on the patient is assumed to be a time-dependent, continuous quadratic functional of the prognostic vector. Conditional on initial data X
0, the probability of surviving T units of time is expressed in terms of the solution of a Riccati equation, which can be evaluated in closed form if the coefficients of the process and the hazard are constant. This conditional expectation does not preserve proportional hazards.
Simpler proofs of two threshold theorems for a general stochastic epidemic
Authors: Rajarshi M. B.
Published: July 2016

We give here fairly elementary proofs for the threshold theorems due to Williams (1971) and Whittle (1955). Our proofs are based on an application of the reflection principle through the ballot problem and the exact distribution of the size of the epidemic as derived by Foster (1955). Williams's threshold theorem is extended to an epidemic with multiple introduction of cases.
Some aspects of the non-asymptotic behaviour of a two-dimensional invasion process
Authors: Downham D. Y. , Fotopoulos S. B.
Published: July 2016

Normal and abnormal cells are positioned at the vertices of a regular two-dimensional lattice. Abnormal cells divide k times as fast as normal cells. Whenever a cell divides, the daughter is the same type as the parent and replaces an adjacent cell. The Kolmogorov forwards and backwards equations are derived, and then used to obtain bounds for the distribution function of the time when all the abnormal cells are forced from the plane. These bounds are used to comment on the non-asymptotic variance of the number of abnormal cells at a given time and on a method of estimating k.
Markov chains in small time intervals
Authors: Rosenlund Stig I.
Published: July 2016

For a time-homogeneous continuous-parameter Markov chain we show that as t → 0 the transition probability pn,j
(t) is at least of order where r(n, j) is the minimum number of jumps needed for the chain to pass from n to j. If the intensities of passage are bounded over the set of states which can be reached from n via fewer than r(n, j) jumps, this is the exact order.