bluuebunny/crossref_metadata_2025 huggingface dataset:
The common ancestor at a nonneutral locus
Authors: Fearnhead Paul
Published: July 2016

We consider a nonneutral population genetics model with parent-independent mutations and two selective classes. We calculate the stationary distribution of the type of the common ancestor of a sample of genes from this model. The expected fitness of any ancestor (including the most recent common ancestor of any sample) is shown to be greater than the expected fitness of a randomly chosen gene from the population. The process of mutations to the common ancestor is also analysed. Our results are related to, but more general than, results obtained from diffusion theory.
On explicit and Fréchet-optimal lower bounds
Authors: Seneta Eugene , Chen John Tuhao
Published: July 2016

Ease of computation of Fréchet-optimal lower bounds, given numerical values of the binomial moments S

ij
, i, j = 1, 2, is demonstrated. A sufficient condition is given for an explicit bivariate bound of Dawson-Sankoff structure to be Fréchet optimal. An example demonstrates that in the bivariate case even the multiplicative structure of the S

ij
does not guarantee a Dawson-Sankoff structure for Fréchet-optimal bounds. A final section is used to illuminate the nature of Fréchet optimality by using generalized explicit bounds. This note is a sequel to both Chen and Seneta (1995) and Chen (1998).
Bayesian inference for Markov chains
Authors: Eichelsbacher Peter , Ganesh Ayalvadi
Published: July 2016

We consider the estimation of Markov transition matrices by Bayes’ methods. We obtain large and moderate deviation principles for the sequence of Bayesian posterior distributions.
A class of interacting marked point processes: rate of convergence to equilibrium
Authors: Torrisi G. L.
Published: July 2016

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.
Computing limiting stationary distributions of small noisy networks
Authors: Richman Fred , Winkowska-Nowak Katarzyna
Published: July 2016

The dynamics of opinion transformation is modeled by a neural network with a nonnegative matrix of connections. Noise is introduced at each site, and the limit of the stationary distributions of the resulting Markov chains as the noise goes to zero is taken as an indication of what configurations will be seen. An algorithm for computing this limit is given, and a number of examples are worked out. Some of the mathematical ideas developed, such as visible states, time scales, and a calculus of indexed probabilities, are of independent interest.
Markovian couplings staying in arbitrary subsets of the state space
Authors: López F. Javier , Sanz Gerardo
Published: July 2016

Let (X

t
) and (Y

t
) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X

t
) and (Y

t
) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.
On bounds for some optimal policies in reliability
Authors: Mi Jie
Published: July 2016

Often in the study of reliability and its applications, the goal is to maximize or minimize certain reliability characteristics or some cost functions. For example, burn-in is a procedure used to improve the quality of products before they are used in the field. A natural question which arises is how long the burn-in procedure should last in order to maximize the mean residual life or the conditional survival probability. In the literature, an upper bound for the optimal burn-in time is obtained by assuming that the underlying distribution of the products has a bathtub-shaped failure rate function; however, no lower bound is available. A similar question arises in studying replacement policy, warranty policy, and inspection models. This article gives a lower bound for the optimal burn-in time, and lower and upper bounds for the optimal replacement and warranty policies, under the same bathtub-shape assumption.
Some properties of two families of classes of life distributions
Authors: Mohan N. R. , Ravi S.
Published: July 2016

We study the closure properties of the family ℒ(α) of classes of life distributions introduced by Lin (1998) under general compounding. We define a discrete analogue of this family and study some properties.
The sojourn time distribution in an infinite server resequencing queue with dependent interarrival and service times
Authors: Huisman Tijs , Boucherie Richard J.
Published: July 2016

We consider an infinite server resequencing queue, where arrivals are generated by jumps of a semi-Markov process and service times depend on the jumps of this process. The stationary distribution of the sojourn time, conditioned on the state of the semi-Markov process, is obtained both for the case of hyperexponential service times and for the case of a Markovian arrival process. For the general model, an accurate approximation is derived based on a discretisation of interarrival and service times.
A matrix exponential form for hitting probabilities and its application to a Markov-modulated fluid queue with downward jumps
Authors: Miyazawa Masakiyo , Takada Hiroyuki
Published: July 2016

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.
Poisson traffic flow in a general feedback queue
Authors: Peköz Erol A. , Joglekar Nitindra
Published: July 2016

Consider a ·/G/kfinite-buffer queue with a stationary ergodic arrival process and delayed customer feedback, where customers after service may repeatedly return to the back of the queue after an independent general feedback delay whose distribution has a continuous density function. We use coupling methods to show that, under some mild conditions, the feedback flow of customers returning to the back of the queue converges to a Poisson process as the feedback delay distribution is scaled up. This allows for easy waiting-time approximations in the setting of Poisson arrivals, and also gives a new coupling proof of a classic highway traffic result of Breiman (1963). We also consider the case of nonindependent feedback delays.
Dynamic models of long-memory processes driven by Lévy noise
Authors: Anh V. V. , Heyde C. C. , Leonenko N. N.
Published: July 2016

A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Lévy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.
Periodic Ornstein-Uhlenbeck processes driven by Lévy processes
Authors: Pedersen Jan
Published: July 2016

In this paper, the class of periodic Ornstein-Uhlenbeck processes is defined. It is shown that periodic Ornstein-Uhlenbeck processes are stationary Markov random fields and the class of stationary distributions is characterized. In particular, any self-decomposable distribution is the stationary distribution of some periodic Ornstein-Uhlenbeck process. As examples, gamma periodic Ornstein-Uhlenbeck processes and Gaussian periodic Ornstein-Uhlenbeck processes are considered.
On the class of controlled branching processes with random control functions
Authors: González M. , Molina M. , Del Puerto I.
Published: July 2016

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.
Strong ergodicity for Markov processes by coupling methods
Authors: Mao Yong-Hua
Published: July 2016

In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.
On modes of long-range dependence
Authors: Heyde C. C.
Published: July 2016

This paper aims at enhancing the understanding of long-range dependence (LRD) by focusing on mechanisms for generating this dependence, namely persistence of signs and/or persistence of magnitudes beyond what can be expected under weak dependence. These concepts are illustrated through a discussion of fractional Brownian noise of index H ∈ (0,1) and it is shown that LRD in signs holds if and only if ½ < H < 1 and LRD in magnitudes if and only if ¾ ≤ H < 1. An application to discrimination between two risky asset finance models, the FATGBM model of Heyde and the multifractal model of Mandelbrot, is given to illustrate the use of the ideas.
A note on level-crossing analysis for the excess, age, and spread distributions
Authors: Katayama Tsuyoshi
Published: July 2016

In this paper, we show that the time-average distributions of excess, age, and spread are given by the solution of first-order differential equations. These differential equations can be directly derived in a simple, unified way using a general level-crossing formula based on the balance of up and down crossings on sample paths, which may be helpful for the intuitive interpretation.
Analysis of the fluid weighted fair queueing system
Authors: Guillemin Fabrice , Mazumdar Ravi , Dupuis Alain , Boyer Jacqueline
Published: July 2016

We analyse in this paper the fluid weighted fair queueing system with two classes of customers, who arrive according to Poisson processes and require arbitrarily distributed service times. In a first step, we express the Laplace transform of the joint distribution of the workloads in the two virtual queues of the system by means of unknown Laplace transforms. Such an unknown Laplace transform is related to the distribution of the workload in one queue provided that the other queue is empty. We explicitly compute the unknown Laplace transforms by means of a Wiener—Hopf technique. The determination of the unknown Laplace transforms can be used to compute some performance measures characterizing the system (e.g. the mean waiting time for each class) which we compute in the exponential service case.
On the integral of the workload process of the single server queue
Authors: Borovkov A. A. , Boxma O. J. , Palmowski Z.
Published: July 2016

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .
Denumerable-state continuous-time Markov decision processes with unbounded transition and reward rates under the discounted criterion
Authors: Guo Xianping , Zhu Weiping
Published: July 2016

In this paper, we consider denumerable-state continuous-time Markov decision processes with (possibly unbounded) transition and reward rates and general action space under the discounted criterion. We provide a set of conditions weaker than those previously known and then prove the existence of optimal stationary policies within the class of all possibly randomized Markov policies. Moreover, the results in this paper are illustrated by considering the birth-and-death processes with controlled immigration in which the conditions in this paper are satisfied, whereas the earlier conditions fail to hold.
Local limit theorems for non-critical Galton–Watson processes by or without immigration
Authors: Höpfner R.
Published: July 2016

From normal limiting distributions of suitably normed sequences of Galton–Watson processes or Galton-Watson processes with immigration, with initial states tending to ∞, we can derive local limit theorems for the transition probabilities Qn (i, j) and Pn
(i, j) in the non-critical case, when initial state i and final state j tend to ∞ with n.
Extreme values, range and weak convergence of integrals of Markov chains
Authors: Brockwell P. J. , Resnick S. I. , Pacheco-Santiago N.
Published: July 2016

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.
Dispersive ordering of distributions
Authors: Shaked Moshe
Published: July 2016

Two distributions, F and G, are said be ordered in dispersion if F
-1(β)-F
-1(α)≦G
-1(β)-G
-1(α) whenever 0<α <β <1. This relation has been studied by Saunders and Moran (1978). The purpose of this paper is to study this partial ordering in detail. Few characterizations of this concept are given. These characterizations are then used to show that some particular pairs of distributions are ordered by dispersion. In addition to it some proofs of results of Saunders and Moran (1978) are simplified. Furthermore, the characterizations of this paper can be used to throw a new light on the meaning of the underlying partial ordering and also to derive various inequalities. Several examples illustrate the methods of this paper.
Limit theorems for out guesses with mean-guided second guessing
Authors: Hwang Jiunn Tzon , Zidek James V.
Published: July 2016

A second guesser, G2, proposes to guess ‘smaller' or ‘larger' on each of n contests, i, according as , or where Xi, i = 1, ···, n are the revealed guesses of the first guesser, G1. G2 wins contest i if his assertion about the size of the target quantity θ

i
, is more accurate than that of G1, Xi, i = 1, ···, n. Otherwise G 1 is the winner. Laws of large numbers are derived for G2's win totals for arbitrary configurations of θ

i
. Limiting distributions are obtained for the first guesser's win totals in two special cases, the first where the θ

i
are highly concentrated and the second where they are widely dispersed. G2 enjoys an enormous advantage over G1 in the former case, as intuition would suggest.
The limit distribution for maxima of ‘weighted' rth-nearest-neighbour distances
Authors: Henze Norbert
Published: July 2016

Let X
1, X
2, · ··, Xn
be independent identically distributed random points with common density f(x), taking values in a bounded region (p ≧ 1). We obtain the limit distribution, as n → ∞, for the maximum value of the suitably ‘weighted' (according to f(x)) rth-nearest-neighbour distances of Χ
1, · ··, Χ n
(r ≧ 1 fixed) provided that f(x) is bounded from below by a positive constant and a weak continuity condition holds. This is achieved by refining an argument used by the author (Henze (1981)) to derive the limit distribution in the special case r = 1. Edge-effects are eliminated by defining, for each Xi
, the distance to the boundary of G to be the ‘rth-nearest-neighbour distance' if it is smaller than the distance to the rth nearest neighbour among the remaining points. Applications to a multivariate test of goodness of fit are given.
Optimal estimation of the criticality parameter of a supercritical branching process having random environments
Authors: Pakes Anthony G. , Heyde C. C.
Published: July 2016

Let {Zn
} be a Galton-Watson process with i.i.d. random environments. This paper is concerned with estimation of π= E log Mn
, where Mn
is the. conditional expected number of offspring per nth generation individual, given the environments, when this quantity is positive. We show that, given non-extinction, {n
-1 log Z

n
} is asymptotically the most efficient estimator of πamongst a broad class of those that are linear combinations of {log Zi
: 1 ≦ j ≦ n}·
Product autoregression: a time-series characterization of the gamma distribution
Authors: Mckenzie Ed
Published: July 2016

A non-linear stationary stochastic process {Xt
} is derived and shown to have the property that both the processes {Xt
} and {log Xt
} have the same correlation structure, viz. the Markov or first-order autoregressive correlation structure. The generation of such processes is discussed briefly and a characterization of the gamma distribution is obtained.
The variance of discounted Markov decision processes
Authors: Sobel Matthew J.
Published: July 2016

Formulae are presented for the variance and higher moments of the present value of single-stage rewards in a finite Markov decision process. Similar formulae are exhibited for a semi-Markov decision process. There is a short discussion of the obstacles to using the variance formula in algorithms to maximize the mean minus a multiple of the standard deviation.
An optimal parking problem
Authors: Tamari Mitsushi
Published: July 2016

The decision-maker drives a car along a straight highway towards his destination and looks for a parking place. When he finds a parking place, he can either park there and walk the distance to his destination or continue driving. Parking places are assumed to occur in accordance with a Poisson process along the highway. The decision-maker does not know the distance Y to his destination exactly in advance. Only an a priori distribution is assumed for Y and cases of typically important distribution are examined. When we take as loss the distance the decision-maker must walk and wish to minimize the expected loss, the optimal stopping rule and the minimum expected loss are obtained. In Section 3 a generalization to the cases of a non-homogeneous Poisson process and a renewal process is considered.
The control of a finite dam
Authors: Attia F. A. , Brockwell P. J.
Published: July 2016

The long-run average cost per unit time of operating a finite dam controlled by a Pl
M
policy (Faddy (1974), Zuckerman (1977)) is determined when the cumulative input process is (a) a Wiener process with drift and (b) the integral of a Markov chain. It is shown how the cost for (a) can be obtained as the limit of the costs associated with a sequence of input processes of the type (b).
A functional form for a particular coefficient of ergodicity
Authors: Tan Choon-Peng
Published: July 2016

Seneta, in a recent paper, presented a general treatment of the concept of ‘coefficient of ergodicity', τ (P), for a finite stochastic matrix P. In this paper, a functional form for τ
∞(P) in terms of the attributes of P is determined. It is shown that, by increasing the dimension of P, τ
∞(Ρ) can assume any large value. In view of this, τ
∞ will be practically useful only in the case τ
∞(P) ≦ τ
1(P), where τ
1(P) is the well-known Dobrushin or delta coefficient.
Comparison of percolation probabilities
Authors: Rüschendorf Ludger
Published: July 2016

Some recent results of Oxley and Welsh (1979) and McDiarmid (1981) concerning Bernoulli percolation on clutters are generalized. Our results allow to consider quantitative aspects of percolation on graphs and clutters.
An alternate proof for the representation of discrete distributions by equiprobable mixtures
Authors: Dieter U.
Published: July 2016

The paper contains a simple proof that every finite discrete distribution is the equiprobable mixture of r distributions, each of which has at most a mass points.
Recognition of triangles and quadrilaterals by chord length distribution
Authors: Gates J.
Published: July 2016

It is shown how a triangle can be recognised from the odd moments of the length of random chords and how a quadrilateral can be recognised from the derivative of its chord length density.
A note on extended block replacement policy with used items
Authors: Murthy D. N. P. , Nguyen D. G.
Published: July 2016

The extended block replacement policy with used items as advocated by Tango (1978) uses used items of age T only although the policy creates used items of lower age. In this note we modify Tango's policy so as to make use of all used items created by the policy.
The threshold behaviour of epidemic models
Authors: Ball Frank
Published: July 2016

We provide a method of constructing a sequence of general stochastic epidemics, indexed by the initial number of susceptibles N, from a time-homogeneous birth-and-death process. The construction is used to show strong convergence of the general stochastic epidemic to a birth-and-death process, over any finite time interval [0, t], and almost sure convergence of the total size of the general stochastic epidemic to that of a birth-and-death process. The latter result furnishes us with a new proof of the threshold theorem of Williams (1971). These methods are quite general and in the remainder of the paper we develop similar results for a wide variety of epidemics, including chain-binomial, host-vector and geographical spread models.
Sequential random displacements of points in an interval
Authors: Mannion David
Published: July 2016

An array of points {Z
1, Z
2, …, Zn–

1} in the interval V = [0, L], is such that 0 ≦ Z
1, ≦ Z
2 ≦ … ≦ Z

n–1, ≦ L. One of the points is chosen at random (Zk
, say, with probability p
k) and displaced to a new position within the interval [Zk

–1, Zk

+ 1], the position again chosen at random according to a probability distribution Gk. We derive some results concerning the limiting distribution of the array after a succession of such displacements. If Gk
is a uniform distribution, it appears that the number of displacements necessary to open up a gap between at least one pair of adjacent points of size at least γ is O(ρ n
), n →∞, where ρ = L/(L – γ).
Coefficients of ergodicity with respect to vector norms
Authors: Tan Choon-Peng
Published: July 2016

The stationary distribution may be used to estimate the rate of geometric convergence to ergodicity for a finite homogeneous ergodic Markov chain. This is done by invoking the spectrum localization property of a new class of ergodicity coefficients defined with respect to column vector norms for the transition matrix P. Explicit functional forms in terms of the entries of P are obtained for these coefficients with respect to the l∞
and l
1, norms, and comparison in performance with various known coefficients is made with the aid of numerical examples.
Stochastic barriers for the Wiener process
Authors: Park C. , Beekman J. A.
Published: July 2016

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ T

W(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ T

W(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.
Parameter values of ARMA models minimising the one-step-ahead prediction error when the true system is not in the model set
Authors: Kabaila Paul
Published: July 2016

In This paper we answer the following question. Is there any a priori reason for supposing that there is no more than one set of ARMA model parameters minimising the one-step-ahead prediction error when the true system is not in the model set?
A note on conditioned random walk
Authors: Doney R. A.
Published: July 2016

This note is concerned with N, the time at which a random walk first exits from [0,∞), and M, the maximum of the random walk up to time N. In the case that the random walk has zero mean and finite variance, simple proofs are given of asymptotic estimates for P{M > x}, P{N ≦ ux
2|M > x} and P{M ≦ v √n|N > n}.
The first repetition of a pattern in a symmetric Bernoulli sequence
Authors: Karnin Ehud D.
Published: July 2016

In a symmetric sequence of Bernoulli trials we define each successive l outcomes as a pattern, and look at the first time that any pattern repeats. Asymptotic expressions, as l →∞, of the distribution and the expectation of this waiting time are derived.
Stabilité de la récurrence nulle pour certaines chaines de Markov perturbées
Authors: Cocozza-Thivent C. , Kipnis C. , Roussignol M.
Published: July 2016

We investigate how the property of null-recurrence is preserved for Markov chains under a perturbation of the transition probability. After recalling some useful criteria in terms of the one-step transition nucleus we present two methods to determine barrier functions, one in terms of taboo potentials for the unperturbed Markov chain, and the other based on Taylor's formula.
On interpoint distances for planar Poisson cluster processes
Authors: Kryscio Richard J. , Saunders Roy
Published: July 2016

For stationary Poisson or Poisson cluster processes ξ on R2
we study the distribution of the interpoint distances using the interpoint distance function and the nearest-neighbor indicator function . Here Sr
(x) is the interior of a circle of radius r having center x, I(t) is that subset of D which has x ∊ D and St
(x) ⊂ D and χ is the usual indicator function. We show that if the region D ⊂ R2
is large, then these functions are approximately distributed as Poisson processes indexed by and , where µ(D) is the Lebesgue measure of D.
A useful ageing property based on the Laplace transform
Authors: Klefsjö Bengt
Published: July 2016

The class of life distributions for which , where , and , is studied. We prove that this class is larger than the HNBUE (HNWUE) class (consisting of those life distributions for which for x ≧ 0) and present results concerning closure properties under some usual reliability operations. We also study some shock models and a certain cumulative damage model. The class of discrete life distributions for which for 0 ≦ p ≦ 1, where , is also studied.
On learning and the evolutionarily stable strategy
Authors: Hines W. G. S. , Bishop D. T.
Published: July 2016

In evolutionarily stable strategy models, transmission of one's strategy to one's offspring has routinely been assumed to be exact, whether this transmission is genetic or by instruction. If inexact transmission of strategy is possible, however, the nature of the mode of transmission becomes important. This paper demonstrates analytically that the possibility of learning one's strategy can increase rather than decrease eventual strategy diversity and that the resulting population mean strategy can differ appreciably from an evolutionarily stable strategy. These results are found to be consistent with simulation results reported in the literature.
Comparison results for diffusions conditioned on positivity
Authors: Day Martin V.
Published: July 2016

We consider a diffusion process on the reals subject to the conditional probability that the process is positive from t = 0 to the present. We establish comparison results between the conditioned diffusion and a second unconditioned Markov diffusion. One result allows the initial process to be non-Markov before conditioning. A stronger comparison theorem is shown to hold in the Markov case.
Some limit theorems for clustered occupancy models
Authors: Ammann Larry P.
Published: July 2016

Most generalizations of the classical occupancy model involve non-homogeneous shot assignment probabilities, but retain the independence of the individual shot assignments. Hence, these models are associated with non-homogeneous Poisson processes. The present article discusses a generalization in which the shot assignments are not independent, but which result in clustering of the shots. Conditions are given under which this clustered occupancy model converges to a Poisson cluster process. Limiting distributions for the number of empty cells are obtained for various allocation intensities when the total number of shots is deterministic as well as random. In particular, it is shown that when the allocation is sparse, then the limiting distribution of the number of empty cells is compound Poisson.
On the interpretation of random fluctuations in competing chemical systems
Authors: Hall Peter
Published: July 2016

Several stochastic models have been proposed to describe the kinetic theory of reversible chemical reactions. However, in macroscopic systems the effects of stochastic variability are often outweighed by mean effects. In the present paper we show that some observed phenomena can be explained quite adequately by a stochastic model in which the stochastic variability is not negligible in comparison with mean effects. Our argument involves approximations to a stochastic model for competing chemical reactions.
A secretary problem with backward solicitation and uncertain employment
Authors: Choe K. I. , Bai D. S.
Published: July 2016

A secretary problem which allows the applicant to refuse an offer of employment with a fixed probability and admits backward solicitations of previous interviewees with known probability of successful solicitation is considered. The optimal strategy that maximizes the probability of employing the best applicant is derived. Two types of probability of successful solicitation, constant and geometric, are discussed in detail.
A note on the convexity of performance measures of M/M/c queueing systems
Authors: Lee Hau Leung , Cohen Morris A.
Published: July 2016

Convexity of performance measures of queueing systems is important in solving control problems of multi-facility systems. This note proves that performance measures such as the expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.
On conditional passage time structure of birth-death processes
Authors: Sumita Ushio
Published: July 2016

Let N(t) be a birth-death process on N = {0,1,2,· ··} governed by the transition rates λ
n
> 0 (n ≧ 0) and μ η
> 0 (n ≧ 1). Let
mTm
be the conditional first-passage time from r to n, given no visit to m where m <r < n. The downward conditional first-passage time
nTm
is defined similarly. It will be shown that , for any λ
n
> 0 and μ η
> 0. The limiting behavior of is considerably different from that of the ordinary first-passage time where, under certain conditions, exponentiality sets in as n →∞. We will prove that, when λ
n
→ λ > 0 and μ η
→ μ > 0 as n → ∞with ρ = λ /μ < 1, one has as r → ∞where T
BP(λ,μ) is the server busy period of an M/M/1 queueing system with arrival rate λand service rate μ.
Critical Galton–Watson processes by decreasing state-dependent immigration
Authors: Mitov K. V. , Yanev N. M.
Published: July 2016

This paper deals with the Foster–Pakes model for Galton–Watson branching processes allowing immigration whenever the number of particles is 0. In the critical case we investigate the asymptotic behaviour of the probability of non-extinction, of the expectation and of the variance, and obtain different types of limit theorems depending on the temporally-decreasing sizes of the immigrants.
Geometric rate of growth in population-size-dependent branching processes
Authors: Klebaner F. C.
Published: July 2016

We consider a branching-process model {Zn
}, where the law of offspring distribution depends on the population size. We consider the case when the means mn
(mn
is the mean of offspring distribution when the population size is equal to n) tend to a limit m > 1 as n →∞. For a certain class of processes {Zn
} necessary conditions for convergence in L
1 and L
2 and sufficient conditions for almost sure convergence and convergence in L
2 of Wn = Zn/mn
are given.
Random motion on binary trees
Authors: Frank Douglas H. , Durham Stephen
Published: July 2016

This paper presents a model for random motion on binary trees. The symmetric model is defined first, and then a model with drift is given. Formulas for absorption probabilities and expected absorption times are presented in both cases.
Optimal replacement for self-repairing shock models by general failure rate
Authors: Gottlieb Gary , Levikson Benny
Published: July 2016

A device is subject to a series of shocks which cause damage and eventually failure will occur at the time of arrival of one of the shocks. In between the shocks, the device is partially repaired as the cumulative damage decreases as some Markov process. The device must be replaced upon failure at some cost but it can also be replaced before failure at a lower cost. We consider the general case where the failure rate need not be increasing and replacement can be made at any time. The form of the optimal replacement policy is found and fairly general conditions are given for which a control limit policy is optimal.
On the M/G/1 queue by additional inputs
Authors: Ott Teunis J.
Published: July 2016

A single-server queueing system is studied, the input into which consists of the sum of two independent stochastic processes. One of these is an ‘M/G' type input process, the other a much more general process which need not be Markov. There are two types of busy period, depending on which arrival process started the busy period. Stochastic monotonicity results are derived and it is found that under a stationarity-like condition the probability of being in a busy period which started with an ‘M/G' arrival is independent of time and is the same it would be with the ‘M/G' process as only input process. Also, distributional results are obtained for the virtual waiting-time process, and these results are used to reduce the study of a single-server queueing system with as input the sum of independent ‘M/G' and ‘GI/G' input streams to the study of a related GI/G/1 queueing system.

The purpose of this paper is to pave the way for a study of an M/G/1 queueing system with periodic arrivals of additional work, and for optimal scheduling of maintenance processes in certain real-time computer systems.
Recurrence processes
Authors: McCullagh Peter
Published: July 2016

A recurrence process is the process of forward recurrence times from the points of one renewal process to the next subsequent event in a second independent renewal process. Univariate and multivariate recurrence processes are examined. Exponential recurrence processes are investigated in greater detail.
Multidimensional point processes and random closed sets
Authors: Baudin Michel
Published: July 2016

Multidimensional random point processes are shown to be a special case of Matheron's random closed sets. This theory, and in particular Choquet's theorem, are then used to prove a general result on heterogeneous Poisson processes and a uniqueness property of mixed Poisson processes with gamma mixing distributions.
Polynomially additive entropies
Authors: Ebanks Bruce R.
Published: July 2016

This paper introduces a property of entropies called polynomial additivity, which includes the additivity properties of the Shannon entropy and of entropies of all degrees. The forms of all such entropies with the sum property and range of infinite cardinality are found. Of these, the only ones with the measurable sum property are affine transformations of the Shannon entropy, the entropies of all degrees, and the length of the distribution.
On the local stability of an evolutionarily stable strategy in a diploid population
Authors: Hines W. G. S. , Bishop D. T.
Published: July 2016

The evolutionarily stable strategy for a given payoff matrix contest, although originally determined in terms of a haploid population, has been shown elsewhere to correspond to an equilibrium of the mean strategy of a diploid population. In this note, the equilibrium is shown to be locally stable for diploid populations. This local stability is demonstrated primarily by relating the behaviour of the perturbed diploid population to one, or in some cases two, associated haploid populations.
Probability models for liberation
Authors: Davy Pamela J.
Published: July 2016

An index lying between 0 and 1 is presented to describe the degree of liberation (or separation) of one component of a particulate material. It is shown how the index is related to the covariance function of the material, the distribution of shapes and sizes of the particles and to the interaction between fracture surfaces and the structure of the material. The variation of the index with the extent of crushing is investigated, together with the problem of stereological estimation. The index is evaluated for some particular probability models.
Cyclic behaviour and asymptotic stability of non-homogeneous Markov systems
Authors: Vassiliou P.-C. G.
Published: July 2016

In this paper we study the cyclic behaviour of non-homogeneous Markov systems, i.e. the behaviour of the system under the assumption of periodic sequences of transition matrices, input probabilities, output probabilities and total numbers in the system. We provide a general theorem for the limiting structure of such a system under the cyclic behaviour. We also study the asymptotic stability of non-homogeneous Markov systems and theorems are given which characterize asymptotic stability. An application of the above results is given for a British firm, BS.
The characterization of distributions by order statistics and record values — a unified approach
Authors: Deheuvels Paul
Published: July 2016

It is shown that, in some particular cases, it is equivalent to characterize a continuous distribution by properties of records and by properties of order statistics. As an application, we give a simple proof that if two successivejth record valuesandassociated to an i.i.d. sequence are such thatandare independent, then the sequence has to derive from an exponential distribution (in the continuous case). The equivalence breaks up for discrete distributions, for which we give a proof that the only distributions such thatXk, nandXk+1,n–Xk, nare independent for somek≧ 2 (whereXk, nis thekth order statistic ofX1, ···,Xn) are degenerate.
Self-organizing files with dependent accesses
Authors: Lam Kin , Leung Ming-Ying , Siu Man-Keung
Published: July 2016

We analyze certain self-organizing filing techniques when accesses are assumed to be dependent on each other. The stream of requests for accessing records in a file is modelled as a Markov chain. A general framework is introduced to obtain the asymptotic search cost of a memory-free self-organizing heuristic. The move-to-front heuristic is studied in detail. A formula for the asymptotic search cost, which generalizes that in the case of independent accesses, is obtained. Numerical examples on the performance of the transposition heuristic are provided, and compared with that of the move-to-front heuristic.
The sojourn-time distribution in the M/G/1 queue by processor sharing
Authors: Ott Teunis J.
Published: July 2016

This paper gives, in the form of Laplace–Stieltjes transforms and generating functions, the joint distribution of the sojourn time and the number of customers in the system at departure for customers in the general M/G/1 queue with processor sharing (M/G/1/PS).

Explicit formulas are given for a number of conditional and unconditional moments, including the variance of the sojourn time of an ‘arbitrary' customer.
The autostrada queueing problem
Authors: Conolly B. W.
Published: July 2016

The model considered in this note has been referred to by Haight (1958), Kingman (1961) and Flatto and McKean (1977) as two queues in parallel. Customers choose the shorter of the two queues which are otherwise independent. This system is known to be inferior to a single queue feeding the two servers, but how much? Some elementary considerations provide a fresh perspective on this awkward boundary-value problem. A procedure is proposed for the solution in the context of finite waiting-room size and some comparisons are made with the single-queue system and an independent two-queue system.
A service system with two stages of waiting and feedback of customers
Authors: Ali Osman M. E. , Neuts Marcel F.
Published: July 2016

Customers initially enter a service unit via a waiting room. The customers to be served are stored in a service room which is replenished by the transfer of all those in the waiting room at the points in time where the service room becomes empty. At those epochs of transfer, positive random numbers of ‘overhead customers' are also added to the service room. Algorithmically tractable expressions for the stationary distributions of queue lengths and waiting times at various embedded random epochs are derived. The discussion generalizes an earlier treatment by Takács in several directions.
Relationships between order statistics and record values and some characterization results
Authors: Gupta Ramesh C.
Published: July 2016

The similarity between order statistics and record values motivated us, in this paper, to investigate some relationships between the order statistics and the record values. These relationships are employed to characterize a continuous distribution by some moment properties of the spacings of the record values, and hence obtain characterizations of the exponential distribution. Some of the well-known results follow trivially.
Optimal number of units for a parallel system
Authors: Nakagawa Toshio
Published: July 2016

This paper considers a problem of determining the optimal number of units for a parallel redundant system. The optimal number N∗ to minimize the expected cost is given by a unique solution of equations. When the system is replaced before failure at time T, the optimal N∗ and T∗ are discussed, and a computing procedure for obtaining these values is specified.
On a stochastic model of an epidemic
Authors: Ridler-Rowe C. J.
Published: July 2016

The epidemic model considered here, first given by Bartlett (see for example [2]), provides for the immigration of new susceptibles and infectives, as well as describing the spread of infection to susceptibles already present and the removal of infectives. The epidemic curve, relating the numbers of susceptibles and infectives, has been studied for certain cases by Bartlett [1], Kendall [6] and others, and provides a motivation for the results given here. With the aid of criteria given by Reuter [8], [9], the main question considered is the asymptotic behaviour of the mean duration of the epidemic. The behaviour of the limits of the transition probabilities pij
(t) as t → ∞ is also investigated.
A persistency problem connected with a point process
Authors: Elfving G.
Published: July 2016

Imagine a man owning a commodity, e.g., a house, which is for sale. Offers at varying amounts are coming in every now and then. The longer he postpones selling the more he loses because of deterioration, interest losses, or the like. At each offer he must decide whether to accept it or wait for a better one. (A more picturesque example would be that of a girl scrutinizing successive suitors.)
A limit theorem for random walks with drift
Authors: Heyde C. C.
Published: July 2016

Let Xi
, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables. Write and for x ≧ 0 define

M(x) + 1 is then the first passage time out of the interval (– ∞, x] for the random walk process Sn.
A mathematical model for delays caused by right-turning vehicles at an uncontrolled intersection
Authors: Reid D. H.
Published: July 2016

It is apparent that much of the congestion in road traffic is due to interference between vehicles at intersections, and this has led to the appearance of several mathematical discussions of the problems of delay at intersections of various types (Cohen and Stam (1963), Cohen and de Lange (1965), Hawkes (1965), (1966), Tanner (1962), Weiss and Maradudin (1962)). Comparatively little work has appeared on the problem of delay caused by vehicles turning right* across an opposing stream of vehicles, although Hawkes (1966) and Newell (1959) discuss this problem explicitly, and the models considered by all of these authors may be regarded as having some application to the right-turning problem.
Waitingtimes between record observations
Authors: Neuts Marcel F.
Published: July 2016

If Δ
r
denotes the waitingtime between the (r − 1)st and the rth upper record in a sequence of independent, identically distributed random variables with a continuous distribution, then it is shown that Δ
r
satisfies the weak law of large numbers and a central limit theorem.

This theorem supplements those of Foster and Stuart and Rényi, who investigated the index Vr
of the rth upper record.

Qualitatively the theorems establish the intuitive fact that for higher records, the waitingtime between the last two records outweighs even the total waitingtime for previous records. This explains also why the asymptotic normality of logVr
is very inadequate for approximation purposes—Barton and Mallows.
Diffusion out of a triangle
Authors: Smith W. , Watson G. S.
Published: July 2016

A problem in molecular genetics called for the following result. If a particle is dropped at random on a right-angled isosceles triangle, and thus allowed to execute symmetrical Brownian motion, the chance that it will first leave the triangle via the hypotenuse is 0.41062. An inequality and two methods of deriving this probability are illustrated. The most useful source of solutions to diffusion problems of this kind is the torsion problem of elasticity.
The Galton-Watson process with mean one
Authors: Seneta E.
Published: July 2016

Let Zn
be the number of individuals in the n-th generation of a simple (discrete, one-type) Galton-Watson process, descended from a single ancestor. Write and assume, as usual, that 0 < F(0) < 1. It is well known that the p.g.f. of Zn
is Fn
(s), the n-th functional iterate of F(s), and that if the process is construed as a Markov chain on the states 0, 1, 2,…, then its n-step transition probabilities are given by
Queues with moving average service times
Authors: Pearce C.
Published: July 2016

A model for the service time structure in the single server queue is given embodying correlations between contiguous and near-contiguous service times. A number of results are derived in the case of Poisson arrivals both for equilibrium and the transient state. In particular, Kendall's (equilibrium) result P (a departure leaves the queue empty) = 1 — (mean service time)/(mean inter-arrival time) is found still to hold good. The effect of the correlation on the mean and variance of the equilibrium queue length distribution is examined in a simple case.