bluuebunny/crossref_metadata_2025 huggingface dataset:
On higher-order properties of compound geometric distributions
Authors: Willmot Gordon E.
Published: July 2016

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to thenth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of thekth moment of the time of ruin in the classical risk model.
Markov-modulated linear fluid networks with Markov additive input
Authors: Kella Offer , Stadje Wolfgang
Published: July 2016

We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.
Entropy-based measure of uncertainty in past lifetime distributions
Authors: Di Crescenzo Antonio , Longobardi Maria
Published: July 2016

As proposed by Ebrahimi, uncertainty in the residual lifetime distribution can be measured by means of the Shannon entropy. In this paper, we analyse a dual characterization of life distributions that is based on entropy applied to the past lifetime. Various aspects of this measure of uncertainty are considered, including its connection with the residual entropy, the relation between its increasing nature and the DRFR property, and the effect of monotonic transformations on it.
The number of intersections of random chords to a circle — third and fourth moments
Authors: Gates John
Published: July 2016

In this paper the distributions of the number of intersections of three, four and five random chords to a circle are obtained by a reduction technique employing Ptolemy's theorem. These results are then used to obtain the skewness and kurtosis of the number of intersections of n random chords to a circle.
On the limiting behaviour of a non-homogeneous Markovian manpower model with independent Poisson input
Authors: Vassiliou P.-C. G.
Published: July 2016

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.
Isolated nearest neighbors
Authors: Pickard David K.
Published: July 2016

For point processes, a pair of points is isolated if each is the nearest neighbor of the other, and its degree of isolation is the relative distance to the next-nearest point. The frequency with which such pairs occur and the distribution of their degree of isolation are obtained for Poisson processes in ℝ
v
. Similar results are obtained for some related models, incidentally solving a problem proposed by D. P. Shine.
Weak convergence of an autoregressive process used in modeling population growth
Authors: Cumberland W. G. , Sykes Z. M.
Published: July 2016

Under simple limiting conditions a first-order autoregressive process is shown to converge weakly to an Ornstein-Uhlenbeck process. The result is discussed in the context of modeling vital rates for biological populations in random environments.
The spatial general epidemic and locally dependent random graphs
Authors: Kuulasmaa Kari
Published: July 2016

The threshold behaviour of the stochastic spatial general epidemic model on a discrete location space is investigated by making use of the general percolation theory of McDiarmid.
Asymptotic behavior of Markov systems
Authors: Vassiliou P.-C. G.
Published: July 2016

In this paper we study the asymptotic behavior of Markov systems and especially non-homogeneous Markov systems. It is found that the limiting structure and the relative limiting structure exist under certain conditions. The problem of weak ergodicity in the above non-homogeneous systems is studied. Necessary and sufficient conditions are provided for weak ergodicity. Finally, we discuss the application of the present results in manpower systems.
Remarks on a model of competitive bidding for employment
Authors: Pakes Anthony G.
Published: July 2016

Arnold and Laguna introduced a model for income distributions in which the income of the present generation of individuals has the same distribution as the minimum of a random number Nn
of independent copies of some random variable and {Nn
} is independent. The present paper gives a fairly complete analysis of this model and a number of extensions of it.
Evolutionarily stable strategies in diploid populations with general inheritance patterns
Authors: Hines W. G. S. , Bishop D. T.
Published: July 2016

A simple argument demonstrates that the mean strategy of a diploid sexual population at evolutionary equilibrium can be expected to be an evolutionarily stable strategy (ESS) in the formal sense. This result follows under a wide set of models of genetic inheritance of strategy (including sexual selection) provided that the ESS is both attainable and maintainable.
Conditioned limit theorems for waiting-time processes of the M/G/1 queue
Authors: Hooghiemstra G.
Published: July 2016

This paper is on conditioned weak limit theorems for imbedded waiting-time processes of an M/G/1 queue. More specifically we study functional limit theorems for the actual waiting-time process conditioned by the event that the number of customers in a busy period exceeds n or equals n. Attention is also paid to the actual waiting-time process with random time index.

Combined with the existing literature on the subject this paper gives a complete account of the conditioned limit theorems for the actual waiting-time process of an M/G/1 queue for arbitrary traffic intensity and for a rather general class of service-time distributions.

The limit processes that occur are Brownian excursion and meander, while in the case of random time index also the following limit occurs: Brownian excursion divided by an independent and uniform (0, 1) distributed random variable.
A note on the recurrence of a correlated random walk
Authors: Henderson Robin , Renshaw Eric , Ford David
Published: July 2016

A non-analytical proof of recurrence is obtained via an embedding procedure for a two-dimensional correlated lattice random walk.
The M/G/k loss system with servers subject to breakdown
Authors: Fakinos D.
Published: July 2016

This paper studies the equilibrium behaviour of the M/G/k loss system with servers subject to breakdown. Such a system has k servers, whose customers arrive in a Poisson process. They are served if there is an idle server, otherwise they leave and do not return. Each server is subject to breakdown with probability of occurrence depending on the length of the time the server has been busy since his last repair. On a breakdown the customer waits and his service is continued just after the repair of the server. Among other things, a generalization of the Erlang B-formula is given and it is shown that the equilibrium departure process is Poisson. In fact these results are obtained for the more general case where customers may balk and service and repair rates are state dependent.
The robust estimation of autoregressive processes by functional least squares
Authors: Heathcote C. R. , Welsh A. H.
Published: July 2016

The stationary autoregressive model but with a long-tailed error distribution is analysed using the method of functional least squares. A family of estimators indexed by a real parameter is obtained and uniform consistency and weak convergence established. The optimum member of the family is chosen to have minimum variance with respect to the parameter, and the parameter value chosen detects and adjusts for long-tailed error distributions. Results of a simulation are given.
Polygons formed by random lines in a plane: some further results
Authors: Tanner J. C.
Published: July 2016

Expressions are derived for the third and fourth moments of the distribution of the numbers of sides of the polygons formed by random lines in a plane. These are evaluated numerically; it seems unlikely that they can be expressed simply in terms of elementary functions or constants. Using these results, the third and fourth moments of the distribution of the perimeters of the polygons are derived.
The convexity of the mean queue size of the M/M/c queue with respect to the traffic intensity
Authors: Grassmann W.
Published: July 2016

In this paper, we show that the expected number in an M/M/c queue is convex with respect to the traffic intensity. The proof is conducted by expressing the second derivative of the expected queue size as the sum of non-negative terms.
Consistency for the least squares estimator in a transfer function model
Authors: Solo Victor
Published: July 2016

The consistency is developed under mild conditions for the least squares estimator of the parameters of a transfer function time series model.
The sojourn time in the GI/M/1 queue by processor sharing
Authors: Ramaswami V.
Published: July 2016

A queueing model of considerable interest in computer engineering is the processor-sharing model in which the server shares its fixed capacity equally among all units present in the system. Here, we derive the mean and the variance of the equilibrium sojourn time, and deduce that the variance of the sojourn time is larger for the processor-sharing model than for the corresponding FCFS model.
A renewal density theorem in the multi-dimensional case
Authors: Mode Charles J.
Published: July 2016

In this note a renewal density theorem in the multi-dimensional case is formulated and proved. Let f(
x
) be the density function of a p-dimensional random variable with positive mean vector μ and positive-definite covariance matrix Σ, let hn
(
x
) be the n-fold convolution of f(
x
) with itself, and set

Then for arbitrary choice of integers k
1, …, kp–

1 distinct or not in the set (1, 2, …, p), it is shown that under certain conditions

as all elements in the vector
x
= (x
1, …, xp
) become large. In the above expression μ‵ is interpreted as a row vector and μ a column vector. An application to the theory of a class of age-dependent branching processes is also presented.
An uncertainty function arising in sequential search theory
Authors: Posner Edward C. , Rumsey Howard
Published: July 2016

This note presents a new “uncertainty function” of finite probability distributions that arises in sequential search theory. The function arises as the finite term in an asymptotic expression for an expected search time as the allowable “error probability” approaches zero. This function is not a function of entropy; it may be of more relevance than entropy to workers in certain fields for describing the complexity of a finite distribution.
The distribution of the content of finite dams
Authors: Takács Lajos
Published: July 2016

We shall consider the following model of finite dams: In the time interval (0, ∞) water is flowing into a dam (reservoir) in accordance with a random process. Denote by χ(u) the total quantity of water flowing into the dam in the time interval (0, u). The capacity of the dam is a finite positive number m. If the dam becomes full, the excess water overflows. If the dam is not empty, there is a continuous release at a constant unit rate. Denote by η(t) the content of the dam at time t.
Stochastic approach to chemical kinetics
Authors: McQuarrie Donald A.
Published: July 2016

In this article we shall present a summary of the various stochastic approaches and applications to chemical reaction kinetics, but before discussing these we first briefly introduce the basic ideas and definitions of classical or deterministic chemical kinetics. One of the basic questions to which chemists address themselves is the rate of chemical reactions, or in other words, how long it takes for a chemical reaction to attain completion, or equilibrium. Apparently the first significant quantitative investigation was made in 1850 by L. Wilhelmy [93]. He studied the inversion of sucrose (cane sugar) in aqueous solutions of acids, whose reaction is

He found empirically that the rate of decrease of concentration of sucrose was simply proportional to the concentration remaining unconverted, i.e., if S(t) is the concentration of sucrose, then

The constant of proportionality is called the rate constant of the reaction. If S
o is the initial concentration of sucrose, then

Since then an enormous number of reactions has been studied and the field of chemical kinetics is now one of the largest areas of chemical research. The importance of the field lies in the fact that it yields concise expressions for the time dependence of reactions, predicts yields, optimum economic conditions, and gives one much insight into the actual molecular processes involved. The detailed molecular picture of a reaction process is called the mechanism of the reaction.
The asymptotic behaviour of an invasion process
Authors: Kelly F. P.
Published: July 2016

Black and white cells are positioned at the vertices of a rectangular lattice. When a cell division occurs, the daughter cells are of the same colour as the parent cell; one of them replaces an adjacent cell and the other remains in the position of the parent cell. In one variant of the model it is assumed that whenever a white cell appears at the origin it is transformed into a black cell; apart from this the black and white cells are equally competitive and in particular they divide at the same rate. Initially, only the cell at the origin is black. The asymptotic behaviour of the black clone is investigated.
An application of chain-dependent processes to meteorology
Authors: Katz Richard W.
Published: July 2016

An explicit formula is derived for the variance normalizing constant in the central limit theorem for chain-dependent processes. As an application to meteorology, a specific chain-dependent process is proposed as a probabilistic model for the sequence of daily amounts of precipitation. This model is a generalization of the commonly used Markov chain model for the occurrence of precipitation.
Distributed lag regression by an almost periodic design matrix
Authors: Katzoff M. J. , Shumway R. H.
Published: July 2016

Frequency domain estimation and tests of hypotheses are considered for a general multivariate distributed lag regression model. The usual vector of regression functions is replaced by a matrix of almost periodic functions, a case for which the terms appearing in the frequency domain estimators have well-defined limits. Asymptotic distributions and consistency are established for the regression coefficients and error spectra. A test statistic is proposed for the no regression hypothesis which is asymptotically central under the null hypothesis and converges to infinity under the alternative. In the multivariate case, a product of beta-distributed variables is obtained.
Equilibrium distributions for systems of chemical reactions with applications to the theory of molecular adsorption
Authors: Orriss J.
Published: July 2016

In this paper a stochastic model is set up for a certain type of reversible chemical reaction and a solution given for the equilibrium distribution; this solution is then extended to deal with any system of chemical reactions.

Three different types of reaction are considered:

(1) Several substances Ai
react together and give a set of substances Bj. The reaction is reversible, with the substances Ai
appearing only on one side of the reaction equation and the substances Bj
only on the other.

(2) Several different reactions involving the substances Ai
and Bj
take place simultaneously, but in each reaction equation the substances Ai
can appear only on one side and the Bj
only on the other.

(3) The restriction of the sets Ai
and Bj
to different sides of the reaction equations is removed: any reaction involving any of the substances Ai
and Bj
on either side of the equation is permissible.

The paper concludes with some applications of the results to problems of molecular adsorption.
Asymptotic correlation in a queue
Authors: Craven B. D.
Published: July 2016

Let Xt
denote the waiting time of customer t in a stationary GI/G/1 queue, with traffic intensity τ; let ρn
denote the correlation between Xt
and Xt+n. For a rational GI/G/1 queue, in which the distribution of the difference between arrival and service intervals has a rational characteristic function, it is shown that, for large n, ρn
is asymptotically proportional to n–

3/2
e
–βn
, where β and the factor of proportionality are calculable. The asymptotic law n
–3/2
e–βn
applies also to the approach of the waiting-time distribution to the stationary state in an initially empty rational GI/G/1 queue, and to the correlations in the queueing systems recently analysed by Cohen [1]. Its more general applicability is discussed.
On the service time distribution and the waiting time process of a potentially infinite capacity queueing system
Authors: Hadidi Nasser
Published: July 2016

In [1] the authors dealt with a particular queueing system in which arrivals occurred in a Poisson stream and the probability differential of a service completion was μσn
when the queue contained n customers. Much of the theory could not be carried out further analytically for a general σn
, which is a purely n-dependent quantity. To carry the analysis further to the extent of finding the “effective” service time and the waiting time distribution when σn
is a linear function of n, (which is considered to be rather general and sufficient for practical purposes), constitutes the subject matter of this paper.
Optimal dispatching of a poisson process
Authors: Ross Sheldon M.
Published: July 2016

Items arrive at a processing plant at a Poisson rate λ. At time T, all items are dispatched from the system. An intermediate dispatch time is to be chosen to minimize the total wait of all items. It is shown that if the dispatch time must be chosen at time 0 then T/2 not only minimizes the expected total wait but it also maximizes the probability that the total wait is less than a for every a > 0. If the intermediate dispatch time is allowed to be a (random) stopping time, then it is shown that the policy which dispatches at time t iff N(t) > λ(T – t) is optimal, where N(t) denotes the number of items present at time t. The distribution of the optimal dispatch time and the optimal expected total wait are determined. A generalization to the case of a non-homogeneous Poisson process, a time lag, and batch arrivals is given. Finally, the case where the process goes on indefinitely and any number of dispatches are allowed (at a cost K per dispatch) is considered.
Sur L'enveloppe convexe des nuages de points aleatoires dans Rn
. I
Authors: Raynaud H.
Published: July 2016

L'enveloppe convexe des nuages de points dans Rn
a donné lieu à un certain nombre de travaux. Dans deux articles ([5] et [6]), Rényi et Sulanke ont proposé et résolu un certain nombre de problèmes. Une loi de probabilité étant donnée dans R
2, on procède aux tirages indépendants de N points suivant cette loi.
The recovery of the root of a tree
Authors: Haigh John
Published: July 2016

A population which develops as a linear birth process from one individual gives rise to a family tree of size N which shows the relationships of the N members to each other. If this tree is given, but with no a priori information of which point is the root (the original member) we seek to utilize our knowledge of the stochastic manner in which the tree grew to estimate this root. A maximum likelihood estimate is obtained, and the probabilities that this estimate is the true kth point, or a member of the true rth generation are obtained as functions of N. As N → ∞ these probabilities converge to non-zero values for any fixed k, r, which are surprisingly large—for example, when k is 1, the limiting value is 1 — loge2 ≈ .30685.
Maximal branching processes and ‘long-range percolation’
Authors: Lamperti John
Published: July 2016

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij
of the chains we will study are to be of the form

where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.
On Wiener–Ito representation and the best linear predictors for bilinear time series
Authors: Terdik Gy. , Rao T. Subba
Published: July 2016

This paper gives the necessary and sufficient condition for the second-order stationarity of lower triangular bilinear models; this condition is in terms of norms of higher-order transfer functions. The spectral density function of the process is evaluated in terms of the transfer functions, and it is shown that the second-order structure is similar to a linear ARMA model with uncorrelated errors. The best linear predictor is obtained and it is shown that the variance of the prediction error is always greater than the optimal prediction error variance obtained from the bilinear model. An expression for the bispectral density function is also obtained.
Resolvent operators of Markov processes and their applications in the control of a finite dam
Authors: Attia F. A.
Published: July 2016

The resolvent operators of the following two processes are obtained: (a) the bivariate Markov process W = (X, Y), where Y(t) is an irreducible Markov chain and X(t) is its integral, and (b) the geometric Wiener process G(t) = exp{B(t} where B(t) is a Wiener process with non-negative drift μ and variance parameter σ2. These results are then used via a limiting procedure to determine the long-run average cost per unit time of operating a finite dam where the input process is either X(t) or G(t). The system is controlled by a policy (Attia [1], Lam [6]).
On queues with periodic inputs
Authors: Bambos Nicholas , Walrand Jean
Published: July 2016

We consider a single-server queue with a periodic and ergodic input. It is shown that if the traffic intensity is less than 1, then the waiting time process is asymptotically periodic. Limit theorems associated with the asymptotic behavior of the queue are also proven. The results are then extended to acyclic networks of queues with periodic inputs. Particular cases of these results had been previously obtained for a single queue with periodic Poisson arrival input process and with independent and identically distributed service times.
Shepp statistic for Markov chains application to a long-run average cost criterion
Authors: Ksir Brahim
Published: July 2016

This paper is a generalization to Markov chains of the work of Shepp [6] in the i.i.d case. Shepp studies the limiting values of the averages Tn
= (Sn

+ f(n) – Sn
)/f(n) where Sn
= X0
+ X
1+ · ·· + Xn
, X
0 = 0, n = 1, 2, ···, is a sum of mutually independent and identically distributed random variables. The function f takes positive integer values and non-decreasingly tends to infinity. Here we take a class of functions f in central position f(n) = [c log n], c > 0, n = 1, 2, ···. There are many refinements of the function f in the i.i.d case [1], [2]. Here we consider the more general case where X
1
, · ··, Xn
is an irreducible and recurrent Markov chain. The state space of the chain is either compact or countable.
Looking forwards and backwards in a bisexual moran model
Authors: Kämmerle K.
Published: July 2016

In this paper a bisexual Moran model is introduced. The population consists of N pairs of individuals. At times t = 1, 2, ·· ·two individuals are born, who ‘choose their parents randomly' and independently of each other. Then one of the pairs is removed and replaced by the two individuals born at that instant.

The extinction probability of the descendants of a single pair and the number of ancestors of a whole generation are studied. A limit result for large population sizes has been derived by diffusion approximation methods.
An ergodic L
2-theorem for simulated annealing in bayesian image reconstruction
Authors: Winkler Gerhard
Published: July 2016

An ergodic L
2-theorem for inhomogeneous Markov chains covering simulated annealing with or without constraints and stochastic relaxation with or without constraints arising in Bayesian image reconstruction is proved. The derivation is self-contained.
Further results on ASTA for general stationary processes and related problems
Authors: Miyazawa Masakiyo , Wolff Ronald W.
Published: July 2016

We consider the equivalence of state probabilities of a general stationary process at an arbitrary time and at embedded epochs of a given point process, which is called ASTA (Arrivals See Time Averages). By using an event-conditonal intensity, we give necessary and sufficient conditions for ASTA for a large class of state sets, which determines a state distribution. We do not need any additional assumptions except that the general process has left-hand limits at all points of time. Especially, for a stationary pure-jump process with a point process, ASTA is obtained for all state sets. As an application of those results, Anti-PASTA is obtained for a pure-jump Markov process and a certain class of GSMP (Generalized Semi-Markov Processes), where Anti-PASTA means that ASTA implies that the arrival process is Poisson.
Availability of a series system with replacement and repair
Authors: Fawzi B. B. , Hawkes A. G.
Published: July 2016

We consider two models for the reliability of a series system of components with spares, serviced by a single service facility. Replacement of failed components by a spare has pre-emptive priority over repair of the failed components; the models differ in whether interrupted repairs resume from where they left off, or restart from the beginning. We find the availability of such systems: numerical results indicate that this is insensitive to the shape of the repair-time and replacement-time distributions. We also obtain the distribution of system downtime for these models.
A note on the total size distribution of carrier-borne epidemic models
Authors: Ball Frank
Published: July 2016

The coupling method of Ball (1986) is extended to carrier-borne epidemics, thus providing a new proof of a result of Daniels (1972) concerning the total size distribution of Downton's stochastic carrier-borne epidemic. The generalization to multipopulation carrier-borne epidemics is immediate.
The M/G/1 processor sharing queue as the almost sure limit of feedback queues
Authors: Resing J. A. C. , Hooghiemstra G. , Keane M. S.
Published: July 2016

In the paper a probabilistic coupling between the M/G/1 processor sharing queue and the M/M/1 feedback queue, with general feedback probabilities, is established. This coupling is then used to prove the almost sure convergence of sojourn times in the feedback model to sojourn times in the M/G/1 processor sharing queue. Using the theory of regenerative processes it follows that for stable queues the stationary distribution of the sojourn time in the feedback model converges in law to the corresponding distribution in the processor sharing model. The results do not depend on Poisson arrival times, but are also valid for general arrival processes.
Sums of i.i.d. random variables and an application to the explosion criterion for markov branching processes
Authors: Schuh H.-J.
Published: July 2016

We give necessary and sufficient conditions for in terms of , where Sn
is the sum of n i.i.d. random variables with values in]0, ∞[, and A ≧ 0. We use these results to give a probabilistic proof of the ‘explosion criterion' for continuous-time Markov branching processes, which is usually shown analytically.
Asymptotic properties of stereological estimators of volume fraction for stationary random sets
Authors: Mase Shigeru
Published: July 2016

We shall discuss asymptotic properties of stereological estimators of volume (area) fraction for stationary random sets (in the sense of Matheron) under natural and general assumptions. Results obtained are strong consistency, asymptotic normality, and asymptotic unbiasedness and consistency of asymptotic variance estimators. The method is analogous to the non-parametric estimation of spectral density functions of stationary time series using window functions. Proofs are given for areal estimators, but they are also valid for lineal and point estimators with slight modifications. Finally we show that stationary Boolean models satisfy the relevant assumptions reasonably well.
Simple derivations of the invariance relations and their applications
Authors: Miyazawa Masakiyo
Published: July 2016

In the literature, various methods have been studied for obtaining invariance relations, for example, L = λW (Little's formula), in queueing models. Recently, it has become known that the theory of point processes provides a unified approach to them (cf. Franken (1976), König et al. (1978), Miyazawa (1979)). This paper is also based on that theory, and we derive a general formula from the inversion formula of point processes. It is shown that this leads to a simple proof for invariance relations in G/G/c queues. Using these results, we discuss a condition for the distribution of the waiting time vector of a G/G/c queue to be identical with that of an M/G/c queue.
Mutations, perturbations and evolutionarily stable strategies
Authors: Hines W. G. S.
Published: July 2016

The changes in diversity of competitive strategies in a Maynard Smith population model with mixed strategies are related to the changes in population mean strategy. The effects of slight mutations in strategy frequencies, and of slight perturbations of the contest payoff rules are then investigated, and found to increase and decrease diversity respectively (to a third-order approximation). A relation among mutational effects, payoff perturbation effects and stable population diversity is suggested.
Limit theorems for uniform distributions on spheres in high-dimensional euclidean spaces
Authors: Stam A. J.
Published: July 2016

If X = (X
1, · ··, Xn
) has uniform distribution on the sphere or ball in ℝ with radius a, then the joint distribution of , ···, k, converges in total variation to the standard normal distribution on ℝ. Similar results hold for the inner products of independent n-vectors. Applications to geometric probability are given.
Asymptotic results for the best-choice problem with a random number of objects
Authors: Yasuda Masami
Published: July 2016

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.
On weak convergence within the hnbue family of life distributions
Authors: Basu Sujit K. , Bhattacharjee Manish C.
Published: July 2016

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.
Reversibility of Markov chains with applications to storage models
Authors: Ōsawa Hideo
Published: July 2016

This paper studies the reversibility conditions of stationary Markov chains (discrete-time Markov processes) with general state space. In particular, we investigate the Markov chains having atomic points in the state space. Such processes are often seen in storage models, for example waiting time in a queue, insurance risk reserve, dam content and so on. The necessary and sufficient conditions for reversibility of these processes are obtained. Further, we apply these conditions to some storage models and present some interesting results for single-server queues and a finite insurance risk model.
The critical branching process with immigration stopped at zero
Authors: Ivanoff B. Gail , Seneta E.
Published: July 2016

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).
On a spectral density estimate obtained by averaging periodograms
Authors: Dahlhaus Rainer
Published: July 2016

A spectral density statistic obtained by averaging periodograms over overlapping time intervals is considered where the periodograms are calculated using a data window. The asymptotic mean square error of this estimate for scale parameter windows is determined and, as an example, it is shown that the use of the Tukey–Hanning window leads partially to a smaller mean square error than a window suggested by Kolmogorov and Zhurbenko. Furthermore the Tukey–Hanning window is independent of the unknown spectral density, which is not the case for the Kolmogorov–Zhurbenko window. The mean square error of this estimate is also less than the mean square error of commonly used window estimates. Finally, a central limit theorem for the estimate is established.
A queueing system with impatient customers
Authors: De Kok A. G. , Tijms H. C.
Published: July 2016

A queueing situation often encountered in practice is that in which customers wait for service for a limited time only and leave the system if not served during that time. This paper considers a single-server queueing system with Poisson input and general service times, where a customer becomes a lost customer when his service has not begun within a fixed time after his arrival. For performance measures like the fraction of customers who are lost and the average delay in queue of a customer we obtain exact and approximate results that are useful for practical applications.
Exact convergence rate of an Erdös-Rényi strong law for moving quantiles
Authors: Deheuvels Paul , Steinebach Josef
Published: July 2016

Consider a sequence U
1, U
2
, · ·· of i.i.d. uniform (0, 1)-random variables. For fixed α ∈ (0, 1), let U(n, K) denote the [Kα]th order statistic of the subsample Un

+1, · ··, Un

+K
, and set . Book and Truax (1976) proved the following analogue of the Erdös-Rényi (1970) strong law of large numbers: for α < u < 1 and C = C(α, u) such that −1/C = αlog(u/α)+ (1 – α)log((l – u)/(1 –α)), it holds almost surely that

In view of the Deheuvels–Devroye (1983) improvements of the original Erdös-Rényi law, we determine the lim inf and lim sup of

where K = [C log N]. This improves (∗), showing that it holds with a best-possible convergence rate of order O(log log N/log N). Using the quantile transformation the result can be extended to a general i.i.d. sequence X
1, X
2, · ·· with d.f. F satisfying a strict monotonicity condition.
Some remarks on the Rayleigh process
Authors: Giorno V. , Nobile A. G. , Ricciardi L. M. , Sacerdote L.
Published: July 2016

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.
Optimal replacement times — a general set-up
Authors: Aven Terje , Bergman Bo
Published: July 2016

For a large class of replacement models for stochastically deteriorating systems the optimality criteria of total expected discounted cost and long-run (expected) average cost per unit time have a common structure. In the present paper a formal description of this structure is given and the optimal rule is determined. A so-called ‘λ -minimization technique' is applied. This method is discussed in general terms.
An adaptive multistage queueing system
Authors: Viscolani Bruno
Published: July 2016

The system provides each customer with a service made up of a random number of stages in sequence. Arrivals are Poisson, and the stage times are independent exponential random variables. The number of stages of a particular service depends on the customer's random demand and on the arrival process, in a way which is aimed at preventing the queue from growing fast while matching the customer's demand as well as possible.
A note on the asymptotic eigenvalues and eigenvectors of the dispersion matrix of a second-order Stationary Process on a d-dimensional Lattice
Authors: Martin R. J.
Published: July 2016

A sufficiently large finite second-order stationary time series process on a line has approximately the same eigenvalues and eigenvectors of its dispersion matrix as its counterpart on a circle. It is shown here that this result can be extended to second-order stationary processes on a d-dimensional lattice.
Optimal control of a finite dam: Wiener process input
Authors: Yeh Lam , Hua Lou Jiann
Published: July 2016

The problem of optimal control of a finite dam in the class of policies has been considered by Lam Yeh [6], [7]. In this paper, by using the first Dynkin formula, the same problems of specifying an optimal policy in the class of the policies to minimize the expected total discounted cost as well as the long-run average cost are considered. Both the expected total discounted cost and long-run average cost are determined explicitly, and then the optimal policy can be found numerically, Also, we obtain the transition density function and the resolvent operator of a reflecting Wiener process.
On limiting behavior of ordinary and conditional first-passage times for a class of birth-death processes
Authors: Sumita U.
Published: July 2016

LetN(t) be a birth-death process on 𝒩= {0, 1, 2, ·· ·} governed by the transition ratesλn> 0 (n≧ 0) andμn> 0 (n≧ 1) whereλn→λ> 0 andμn→μ> 0 asn→ ∞ andρ=λ/μ. LetTmnbe the first-passage time ofN(t) frommtonand defineIt is shown that, whenconverges in distribution toTBP(μ,λ)asn → ∞whereTΒΡ (μ,λ)is the server busy period of anM/M/1 queueing system with arrival rateμand service rateλ. CorrespondinglyT0n/E[T0n] converges to 1 with probability 1 asn→∞. Of related interest is the conditional first-passage timemTrnofN(t) from r tongiven no visit tomwherem < r < n.As we shall see, the conditional first-passage time ofN(t) can be viewed as an ordinary first-passage time of a modified birth-death processM(t) governed bywhereare generated fromλnandμn. Furthermore it is shown that forandwhile forandThis enables one to establish the relation between the limiting behavior of the ordinary first-passage times and that of the conditional first-passage times.
The control of a finite dam with penalty cost function: Markov input rate
Authors: Attia F. A.
Published: July 2016

The long-run average cost per unit time of operating a finite dam controlled by a policy (Lam Yeh (1985)) is determined when the cumulative input process is the integral of a Markov chain. A penalty cost which accrues continuously at a rate g(X(t)), where g is a bounded measurable function of the content, is also introduced. An example where the input rate is a two-state Markov chain is considered in detail to illustrate the computations.
Busy-period analysis of a correlated queue with exponential demand and service
Authors: Langaris Christos
Published: July 2016

In this paper we investigate the server's busy period in a single-server queueing situation in which the interarrival interval T preceding the arrival of a customer and his service time S are assumed correlated. A closed-form expression is obtained for the Laplace transform bn
(z) of the joint probability and probability density function of the busy period duration and the number of customers served in it. Some numerical values are given showing the effect of correlation between T and S.
Epidemics with carriers: The large population approximation
Authors: Pettigrew Hugh M. , Weiss George H.
Published: July 2016

This paper applies the constant population approximation to the study of epidemics which involve more than a single type of infective. An example of this would be a situation in which both clinically infected individuals and subclinically infected individuals or carriers are present.

We derive equations for the expected numbers of clinically infected individuals and carriers at any time t for the model with zero latent period and infectious periods having negative exponential distributions. From these equations we derive conditions under which a unimodal incidence curve can result, and expressions for the expected total epidemic size.
Prediction of a noise-distorted, multivariate, non-stationary signal
Authors: Sobel Eugene
Published: July 2016

This paper deals with the problem of predicting or extracting a certain type of non-stationary sequence or signal from another sequence composed of the signal and a stationary noise. The noise is assumed to be uncorrelated with the signal. Although all sequences in the body of this paper are multivariate, for purposes of discussion, this introduction will deal with only the univariate case.
Time dependence of queues with semi-Markovian services
Authors: Çinlar Erhan
Published: July 2016

A single server queueing system with Poisson input is considered. There are a finite number of types of customers and the service time of the nth customers depends on the types of the nth and the (n – l)th customers. The time dependence of the queue size process will be studied, (it will be clear how the methods of the paper can be applied to other processes of interest,) and limiting as well as transient results will be given.
Random mating in a population of cyclic size
Authors: Chia A. B.
Published: July 2016

The well-known Wright's genetic model is extended to a population whose size varies cyclically. Explicit expressions are obtained for the absorption probability and the rate of approach to homozygosity in the case where there is no selection. When selection operates, diffusion approximations are obtained for the various quantities of interest.
Infinite dams with inputs forming a Markov chain
Authors: Khan M. S. Ali , Gani J.
Published: July 2016

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].
Testing for uniformity on a compact homogeneous space
Authors: Beran R. J.
Published: July 2016

This paper applies the invariance principle to the problem of testing a distribution on a compact homogeneous space for uniformity. The notion of using a reduction by invariance in such a situation is due to Ajne[1], who considers tests invariant under rotation on a circle. In his paper, he derives the distribution of the maximal invariant and gives the general form of the most powerful invariant test for uniformity on the circle.
Markov population processes
Authors: Kingman J.F.C.
Published: July 2016

The processes of the title have frequently been used to represent situations involving numbers of individuals in different categories or colonies. In such processes the state at any time is represented by the vector n = (n
1, n
2, …, nk
), where nt
is the number of individuals in the ith colony, and the random evolution of n is supposed to be that of a continuous-time Markov chain. The jumps of the chain may be of three types, corresponding to the arrival of a new individual, the departure of an existing one, or the transfer of an individual from one colony to another.
Sur un theoreme de Lewis et la decomposition en facteurs premiers de la loi rectangulaire
Authors: Tortrat A.
Published: July 2016

Un théorème de Lewis [3] montre que les seuls facteurs indécomposables de la mesure de Lebesgue m sur S = [–π, π] sont certaines lois discrètes πn,r
(r est premier et divise n), et que les décompositions en facteurs premiers étudiées dans [5] constituent toutes les décompositions possibles (en facteurs premiers).1 De ces décompositions se déduisent toutes les propriétés de divisibilité de m.
2 Nous développons ce point de vue et posons des problèmes analogues pour le cas de la circonférence γ.
First-Order Conservative Processes with Multiple Latent Roots
Authors: Radcliffe J. , Staff P. J.
Published: July 2016

There are now many examples in various fields where the behaviour of ‘particles' as exhibited by their transition from one state to another is described by a multidimensional stochastic process. The linear migration model in which particles move independently of one another through a number of states has been dealt with by Bartlett (1949). This process has been used by Siegert (1949) in considering the approach to equilibrium of non-interacting gas molecules and by Krieger and Gans (1960) and Gans (1960) to examine the distribution of a multicomponent system disturbed from its equilibrium distribution and relaxing by first-order processes to another equilibrium. The correspondence between the deterministic model based on the principle of mass action and the stochastic model has been discussed by Darvey and Staff (1966) in the context of unimolecular multicomponent chemical reactions.
A Note on Absorption Probabilities for a Random Walk Between a Reflecting and an Absorbing Barrier
Authors: Hardin J.C. , Sweet A.L.
Published: July 2016

Generating functions for the absorption probabilities for a random walk on the integers {0,1, …, b}, where 0 is an absorbing barrier and b a semi-reflecting barrier have been obtained by Weesakul [1] and Neuts [2]. However, determination of explicit expressions for the absorption probabilities from the generating functions is generally quite difficult. In this note, two cases where this is possible are presented.