bluuebunny/crossref_metadata_2025 huggingface dataset:
The limit behaviour of the maximum of random variables with applications to outlier-resistance
Authors: Mathar Rudolf
Published: July 2016

We consider degenerate limit laws for the sequence {Xn, n
}
n

(N
of successive maxima of identically distributed random variables. It turns out that the concentration of Xn, n
for large n can be determined in terms of a tail ratio of the underlying distribution function F. Applications to the outlier-behaviour of probability distributions are given.
The dependence of sojourn times on service times in tandem queues
Authors: Cao Xi-Ren
Published: July 2016

In this paper we study a series of servers with exponentially distributed service times. We find that the sojourn time of a customer at any server depends on the customer's past history only through the customer's interarrival time to that server. A method of calculating the conditional probabilities of sojourn times is developed.
The statistical analysis of direct repeats in nucleic acid sequences
Authors: Shukla Rakesh , Srivastava R. C.
Published: July 2016

Sequence symmetries in DNA and RNA are being discovered at an increasing rate. Conjectures and hypotheses are being proposed for their possible structural and functional role in the nucleic acid. In this paper a probability model is studied which evaluates the probabilities of various repeats occurring by chance alone. Expressions are derived for the mean and variance of the statistics employed. The central limit theorem for dependent trials is used to obtain the asymptotic distributions. An indication is given of how to use the model to search for various gene amplification events in the evolutionary history of the sequences.
The kin number problem in a multitype Galton–Watson population
Authors: Joffe A. , Waugh W. A. O'n.
Published: July 2016

The kin number problem in its simplest form is that of the relationship between sibship sizes and offspring numbers. The fact that the distributions are different, and the relationship between the two, is well known to demographers. It is important in such applications as estimating fertility from sibship rather than offspring counts. Further studies have been made, concerning relatives of other degrees of affinity than siblings, but these did not usually yield joint distributions. Recently this aspect of the problem has been studied in the framework of a Galton–Watson process (Waugh (1981), Joffe and Waugh (1982)).

In these studies the population is treated as monotype. Applications such as pedigree studies of diseases require a multitype approach (in the example, two types: victims and others). In this paper such a study is undertaken. The existence of more than one type complicates the time-reversal used in the previous studies, and raises questions about the way in which the focal individual (called ‘Ego') is sampled. These are dealt with, and joint distributions obtained, under a number of sampling schemes which might arise naturally.
Some sufficient conditions for non-ergodicity of markov chains
Authors: Szpankowski Wojciech
Published: July 2016

Some sufficient conditions for non-ergodicity are given for a Markov chain with denumerable state space. These conditions generalize Foster's results, in that unbounded Lyapunov functions are considered. Our criteria directly extend the conditions obtained in Kaplan (1979), in the sense that a class of Lyapunov functions is studied. Applications are presented through some examples; in particular, sufficient conditions for non-ergodicity of a multidimensional Markov chain are given.
Selection of order of observation in optimal stopping problems
Authors: Hill Theodore P. , Hordijk Arie
Published: July 2016

In optimal stopping problems in which the player is free to choose the order of observation of the random variables as well as the stopping rule, it is shown that in general there is no function of all the moments of individual integrable random variables, nor any function of the first n moments of uniformly bounded random variables, which can determine the optimal ordering. On the other hand, several fairly general rules for identification of the optimal ordering based on individual distributions are given, and applications are made to several special classes of distributions.
The queue GI/Hm/s in continuous time
Authors: De smit Jos H. A.
Published: July 2016

This is an extension of our previous paper [4] on the queue GI/Hm/s. In that paper we have derived results for the actual waiting time, the number of customers in the system at arrival epochs and the number of customers during a busy cycle. Here we obtain results for the virtual waiting time and the number of customers in the system at arbitrary times.
Estimating volumes from systematic hyperplane sections
Authors: Cruz-Orive Luis-M.
Published: July 2016

A theory for volume estimation from independent, unbounded plane probes is available. In practice, however, probes cannot be unbounded and independent at the same time, hence the interest of systematic sectioning. Previous empirical studies on real specimens showed that the mean square error of a volume ratio estimator obtained from m systematic sections behaved roughly as m–3 for small m. This drastic increase in efficiency with respect to independent sectioning prompted us to develop some theoretical models in this paper. We study specimens consisting of two ellipsoids in Rn. In virtue of an invariance result, an ellipsoid–ellipsoid model can be studied via a simple model consisting of two concentric n-balls. Explicit results are obtained for n = 1 and n = 3. In the latter case the bias and the mean square error of the relevant volume ratio estimator are both shown to be of O (m
–4).
Accurate and asymptotic results for distributions of scan statistics
Authors: Gates D. J. , Westcott M.
Published: July 2016

We derive asymptotic forms for the distributions of k-point scan statistics as the interval L under study becomes infinite, while k and the window length are held fixed. In the Poisson case the intensity is also held fixed. In the uniform case the number of points N becomes infinite and N/L tends to a limit, representing a limiting intensity. These results are made explicit for k = 3, and in the Poisson case provide approximations which are typically accurate to six or seven figures, even for small L.
Adaptive approach to some stopping problems
Authors: Tamaki Mitsushi
Published: July 2016

This paper mainly considers the adaptive version of two typical stopping problems, i.e., the parking problem and the secretary problem with refusal. In the first problem, while driving towards a destination, we observe the successive parking places and note whether or not they are occupied. Unoccupied spaces are assumed to occur independently, with probability p. The second problem is to select the best applicant from a population, where each applicant refuses an offer with probability 1 – p. We assume beta prior for p in advance. As time progresses, we update our belief for p in a Bayesian manner based on the observed states of the process. We derive several monotonicity properties of the value function and characterize the optimal strategy in either problem. We also attempt to relax the same probability condition in the classical parking problem.
Asymptotic analysis for a production–inventory control model with renewal arrivals and exponential demands
Authors: De kok A. G.
Published: July 2016

We consider a production–inventory problem in which the production rate can be dynamically adjusted to cope with random fluctuations in demand. Customers arrive according to a renewal process, and the customer's demand is assumed to be exponentially distributed. Excess demand is backlogged. The production is controlled by a two-critical-number rule that prescribes which one of the two possible production rates must be used. Tractable expressions are given for several services measures including the fraction of demand backlogged. The analysis is based on the results for hitting probabilities in random walks, where the jump distribution has an exponential right or left tail.
A nearly independent, but non-strong mixing, triangular array
Authors: Andrews Donald W. K.
Published: July 2016

The condition of strong mixing for triangular arrays of random variables is a common condition of weak dependence. In this note, it is shown that this condition is not as general as one might believe. In particular, it is shown that there exist triangular arrays of first-order autoregressive random variables which converge almost surely to an independent identically distributed sequence of random variables and for which the central limit theorem holds, but which are not strong mixing triangular arrays. Hence, the strong mixing condition is more restrictive than desired.
Some exact distributions of a last one-sided exit time in the simple random walk
Authors: Chao Chern-Ching , Slivka John
Published: July 2016

For each positive integer n, let Sn
be the nth partial sum of a sequence of i.i.d. random variables which assume the values +1 and −1 with respective probabilities p and 1 – p, having mean μ= 2p − 1. The exact distribution of the random variable , where sup Ø= 0, is given for the case that λ > 0 and μ+ λ= k/(k + 2) for any non-negative integer k. Tables to the 99.99 percentile of some of these distributions, as well as a limiting distribution, are given for the special case of a symmetric simple random walk (p = 1/2).
The effect of increasing service rates in a closed queueing network
Authors: Shanthikumar J. George , Yao David D.
Published: July 2016

In this paper we study the equilibrium behavior of the queue lengths in a product-form closed queueing network when the service rates at a subset of stations (nodes) are increased. Univariate and multivariate likelihood ratio orderings as well as multivariate stochastic ordering of the queue lengths are established to indicate the effect of increasing the service rates. Relations among these orderings are also developed.
Markov chains with binomial time change
Authors: Siegrist Kyle
Published: July 2016

The effect of a binomial time change on a given Markov chain is studied. Results are obtained for the hitting times, potential operator, and transience and recurrence properties of the time-changed chain. The limiting behavior is considered as the binomial parameter approaches 0 and the time variable approaches∞.
A family of bounds for the transient behavior of a Jackson network
Authors: Massey William A.
Published: July 2016

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.
Birth and death processes in random environments with feedback
Authors: Cornez Richard
Published: July 2016

A generalization of a birth and death chain in a random environment (Y
n
, Z
n

) is developed allowing for feedback to the environmental process (Y
n

). The resulting process is then known as a birth and death chain in a random environment with feedback. Sufficient conditions are found under which the (Z
n

) process goes extinct almost surely or has strictly positive probability of non-extinction.
On extreme-value theory in the presence of a trend
Authors: De Haan L. , Verkade E.
Published: July 2016

Extreme-value theory is considered in the context of independent but not identically distributed random variables. The departure from the case of identical distributions comes from a trend added to i.i.d. observations. Unlike previous authors we consider trends that do not completely destroy the asymptotic behaviour from the i.i.d. case.
Transient renewal processes in the subexponential case
Authors: Murphree Emily S.
Published: July 2016

A transient renewal process based on a sequence of possibly infinite waiting times is defined. The process is studied when the (rescaled) distribution of the waiting times belongs to the subexponential class of distributions. In this case, even conditional on all waiting times observed by time t being finite, the distributions of the forward and backward delays at t are asymptotically degenerate. Also, the conditional moments of the number of events by time t converge to the same finite limits as the unconditional moments.
Recursions and limit theorems for the strength and lifetime distributions of a fibrous composite
Authors: Kuo Chia-Chyuan , Phoenix S. Leigh
Published: July 2016

A composite material is a parallel arrangement of stiff brittle fibers in a flexible matrix. Under load fibers fail, and the loads of failed fibers are locally redistributed onto nearby survivors through the matrix. In this paper we develop a new technique for computing the probability of failure under a previously studied model of the failure process. A recursion and limit theorem are obtained which apply separately to static strength and fatigue lifetime depending on the composite loading and the probability model for the failure of individual fibers under their own loads. The limit theorem yields an approximation for the distribution function for composite lifetime which is of the form 1 – [1 – W(t)]

mn

where W(t) is a characteristic distribution function and mn is the composite volume, reflecting a size effect. A similar result holds also for static strength. In both cases such a result was conjectured several years ago. This limit theorem is obtained from the recursion upon applying a key theorem in the theory of the renewal equation. In the proofs three technical conditions arise which must be verified in specific applications. In the case of static strength these conditions are quite easy to verify, but in the case of fatigue lifetime the verification is generally difficult, and entails considerable numerical computation.
Lifetime regression models based on a functional equation of physical nature
Authors: Castillo Enrique , Galambos Janos
Published: July 2016

There are a number of ad hoc regression models for the statistical analysis of lifetime data, but only a few examples exist in which physical considerations are used to characterize the model. In the present paper a complete characterization of a regression model is given by solving a functional equation recurring in the literature for the case of a fatigue problem. The result is that, if the lifetime for given values of the regressor variable and the regressor variable for a given lifetime are both Weibull variables (assumptions which are well founded, at least as approximations, from extreme-value theory in some concrete applications), there are only three families of (conditional) distribution for the lifetime (or for the regressor variable). This model is then applied to a practical problem for illustration.
Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues
Authors: Stadje Wolfgang
Published: July 2016

For a sequence ξ

1
, ξ

2
, · ·· of i.i.d. random variables let X

0
= 0 and X
k

= max(X
k

–1
+ ξ
k
, 0) for k = 1, 2, ···. Let . These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties of N
x

are derived.
A stochastic death process with elimination by pairing
Authors: Faddy M. J. , Esty Warren W.
Published: July 2016

A stochastic death process in which particles of two types combine and are removed from the system at a rate proportional to the product of their numbers is derived and analyzed. Exact expressions for the probability distribution and moments are given and an approximating diffusion is obtained. Numerical calculations show close agreement between the exact solution and this diffusion even for small numbers of particles, and reveal the relatively low level of stochastic variation.
A note on continuous search algorithms
Authors: Garey L. E. , Gupta R. D.
Published: July 2016

Continuous random search methods with an average complexity given by O(log(1/ε)) for ε → 0 where ε is a given accuracy were presented in a recent paper. In this article an example of an O(log log(1/ε)) method is presented and illustrated.
On the solution of compartmental processes and their use in modelling stages of infection of a disease with reference to onchocerciasis
Authors: Saporu F. W. O.
Published: July 2016

The solution of a hierarchicaln-compartmental homogeneous process with multiple-valued migration rates is obtained using Laplace transform techniques. Such models are suggested for modelling the infection process of endemic diseases because of their property of gamma-distributed compartmental residence times in each ofncompartments. Its relevance to the onchocerciasis infection process (with and without the imposition of controls) is discussed and application to age-prevalence data (having only two infective states into which an individual can be classified) is undertaken.
Expected population size in the generation-dependent branching process
Authors: Biggins J. D. , Götz Thomas
Published: July 2016

A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered.
On the optimality of static priority policies in stochastic scheduling on parallel machines
Authors: Kämpke Thomas
Published: July 2016

n jobs are to be preemptively scheduled for processing on n machines. The machines may have differing speeds and the jobs have processing requirements which are distributed as independent exponential random variables with different means. Holding cost g(U) is incurred per unit time that the set of uncompleted jobs is U and it is desired to minimize the total expected holding cost which is incurred until all jobs are complete. We show that if g satisfies certain simple conditions then the optimal policy is one which takes the jobs in the order 1, 2, ···, n and assigns each uncompleted job in turn to the fastest available machine. In the special case in which the objective is to minimize the expected weighted flowtime, where there is a holding cost of wi
while job i is incomplete, the sufficient condition is simply w1 ≧ … ≧ wn
and λ1 w1 ≧ … ≧ λn wn
.
A note on waiting times in systems with queues in parallel
Authors: Blanc J. P. C.
Published: July 2016

Numerical data are presented concerning the mean and the standard deviation of the waiting-time distribution for multiserver systems with queues in parallel, in which customers choose one of the shortest queues upon arrival. Moreover, a new numerical method is outlined for calculating state probabilities and moments of queue-length distributions. This method is based on power series expansions and recursion. It is applicable to many systems with more than one waiting line.
Augmented truncations of infinite stochastic matrices
Authors: Gibson Diana , Seneta E.
Published: July 2016

We consider the problem of approximating the stationary distribution of a positive-recurrent Markov chain with infinite transition matrix P, by stationary distributions computed from (n × n) stochastic matrices formed by augmenting the entries of the (n × n) northwest corner truncations of P, as n →∞.
Spectral structure of the first-passage-time densities for classes of Markov chains
Authors: Kijima Masaaki
Published: July 2016

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.
Simulated annealing methods with general acceptance probabilities
Authors: Anily S. , Federgruen A.
Published: July 2016

Heuristic solution methods for combinatorial optimization problems are often based on local neighborhood searches. These tend to get trapped in a local optimum and the final result is often heavily dependent on the starting solution. Simulated annealing methods attempt to avoid these problems by randomizing the procedure so as to allow for occasional changes that worsen the solution. In this paper we provide probabilistic analyses of different designs of these methods.
On the best order of observation in optimal stopping problems
Authors: Gilat David
Published: July 2016

For optimal stopping problems in which the player is allowed to choose the order of the random variables as well as the stopping rule, a notion of order equivalence is introduced. It is shown that different (non-degenerate) distributions cannot be order-equivalent.

This result unifies and generalizes two theorems of a similar nature recently obtained by Hill and Hordijk (1985).
On a first passage problem in stochastic storage systems with total release
Authors: Ten Hoopen M. , Reuver H. A.
Published: July 2016

The deleting or delay effect which a recurrent point process exerts upon another recurrent point process (or process with stationary independent increments) is treated. The concept arises in the statistical analysis of neuronal firing patterns, but may also be useful in applications to storage systems in operations research.
The joint distribution of the virtual waitingtime and the residual busy period for the M/G/1 queue
Authors: Neuts Marcel F.
Published: July 2016

In the M/G/1 queue, the joint distribution of the virtual waitingtime and the residual duration of the busy period at any time t, is given by a formula analogous to the one for the virtual waitingtime by itself. In the limit, as t → ∞, the limiting joint distribution is given by a formula of the Pollaczek-Khintchine type.
Some models for interaction of renewal processes related to neuron firing
Authors: Coleman R. , Gastwirth J. L.
Published: July 2016

One area of science in which probabilistic models are beginning to play a role is in neuron firing. The firing process may be thought of in terms of electrical potentials: there is a potential difference across the membrane surrounding a neuron; this potential difference is determined by the concentrations of sodium, potassium, and chlorine ions on the two sides of the membrane. When electrical impulses arrive along the many input fibers, the potential difference changes; when it crosses a certain threshold, the neuron “fires”, that is, an impulse is discharged along the axon. Arriving impulses are of two types, those which raise and those which lower the potential inside the neuron; these are called stimuli and inhibitors, respectively. The mechanism by which stimuli and inhibitors interact in changing the neuron potential is not well understood; however, it appears that some of the stimuli cause the neuron to respond, or fire, and the inhibitors play a role in preventing the other stimuli from causing a response. The ability of stimuli to cause responses is further weakened by the decay, in the absence of arriving impulses, of the transmembrane potential toward a “resting level”, the level to which the potential is reset after a firing. While several models for the firing process have been proposed ([3], [4], [5], [6], [7], [11]), the present paper is devoted to introducing some modifications into a model proposed by Ten Hoopen and Reuver ([9], [10]). In particular, we place a time limit on the ability of inhibitors to prevent a stimulus from producing a response. Hopefully, these modifications bring their model closer to physiological reality.
The Trivariate Distribution of the Maximum Queue Length, the Number of Customers Served and the Duration of the Busy Period for the M/G/1 Queueing System
Authors: Enns E.G.
Published: July 2016

In the study of the busy period for a single server queueing system, three variables that have been investigated individually or at most in pairs are:

1.

The duration of the busy period.

2.

The number of customers served during the busy period.

3.

The maximum number of customers in the queue during the busy period.
Bounds for Certain Branching Processes
Authors: Pollak Edward
Published: July 2016

We consider a branching process for which the offspring distribution has the generating function f(t) and mean f
'(1) = m < 1. The probability that a line descended from an individual still survives in generation n is asymptotically of the form cmn. A method is derived whereby good bounds for c may be obtained. This method makes use of the first three moments of the distribution of offspring.
The Markovian self-exciting process
Authors: Oakes David
Published: July 2016

The self-exciting point process with exponential exciting function is investigated using the immigration birth representation of the process. The counting distribution is derived explicitly and some simpler interval properties are given. In particular, it is shown that the serial covariances of the interval sequence decrease monotonically to zero.
On a class of parabolic differential equations driven with stochastic point processes
Authors: Gopalsamy K. , Bharucha-Reid A. T.
Published: July 2016

This paper is concerned with the solution of an initial and boundary value problem for a parabolic differential equation driven by a stochastic point process.
Asymptotic distributions for the Ornstein-Uhlenbeck process
Authors: Beekman John A.
Published: July 2016

This paper gives the asymptotic distributions, as the time period grows infinite, of the first exit times above a fixed constant and from upper and lower constant boundaries for the Ornstein-Uhlenbeck stochastic process. The results of a large amount of numerical analysis illustrate the asymptotic forms.
Branching processes with varying and random geometric offspring distributions
Authors: Keiding Niels , Nielsen John E.
Published: July 2016

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.
A supplement to the strong law of large numbers
Authors: Heyde C. C.
Published: July 2016

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that |n
−1 Σ
i=1

n

X

i
− μ| > ∊, N
(∊), is finite a.s. It is known, furthermore, that EN
(∊) < ∞ if and only if EX
1
2 < ∞. Here it is shown that if EX
1
2 < ∞ then ∊
2
EN
(∊)
→ var X
1 as ∊ → 0.
Weak convergence in an appointment system
Authors: Anderson C. W.
Published: July 2016

It is assumed that customers at a service facility have appointments at times 0,1,2, … for which they may be unpunctual by random amounts or may never arrive at all. A weak convergence theorem is proved for the process which counts the number of arrivals. This makes it possible to carry over the results of Iglehart and Whitt (1970a) to obtain heavy traffic functional limit theorems for queues with arrivals by appointment.
Busy period of a finite queue with phase type service
Authors: Rosenlund Stig I.
Published: July 2016

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in two phases, during the first of which a place in the waiting room is occupied. Starting from a determinant expression for the Laplace-Stieltjes transform of the busy period given in [2] we obtain contour integral expressions for the transform and the expectation, thereby generalising a result by Cohen [1]. This is effected by developing the determinants along the first row.
Recognizing both the maximum and the second maximum of a sequence
Authors: Tamaki M.
Published: July 2016

We consider the situation in which the decision-maker is allowed to have two choices and he must choose both the best and the second best from a group of N applicants. The optimal stopping rule and the maximum probability of choosing both of them are derived.
Exponential ergodicity in Markovian queueing and dam models
Authors: Tuominen Pekka , Tweedie Richard L.
Published: July 2016

We investigate conditions under which the transition probabilities of various Markovian storage processes approach a stationary limiting distribution π at an exponential rate. The models considered include the waiting time of the M/G/1 queue, and models for dams with additive input and state-dependent release rule satisfying a ‘negative mean drift' condition. A typical result is that this exponential ergodicity holds provided the input process is ‘exponentially bounded'; for example, in the case of a compound Poisson input, a sufficient condition is an exponential bound on the tail of the input size distribution. The results are proved by comparing the discrete-time skeletons of the Markov process with the behaviour of a random walk, and then showing that the continuous process inherits the exponential ergodicity of any of its skeletons.
Convergence to equilibrium in a traffic model with restricted passing
Authors: Unkelbach Hans Dieter
Published: July 2016

A road traffic model with restricted passing, formulated by Newell (1966), is described by conditional cluster point processes and analytically handled by generating functionals of point processes.

The traffic distributions in either space or time are in equilibrium, if the fast cars form a Poisson process with constant intensity combined with Poisson-distributed queues behind the slow cars (Brill (1971)). It is shown that this state of equilibrium is stable, which means that this state will be reached asymptotically for general initial traffic distributions. Furthermore the queues behind the slow cars dissolve asymptotically like independent Poisson processes with diminishing rate, also independent of the process of non-queuing cars. To get these results limit theorems for conditional cluster point processes are formulated.
A note on the extinction rates of some birth–death processes
Authors: Bramson Maury , Griffeath David
Published: July 2016

In this note we are concerned with the rate of extinction of certain continuous-time birth-death processes on the positive integers with absorption at 0. The class we deal with includes birth-death processes with mean holding time h(i) at i such that h (i)∼ i–α
as i →∞, 0 ≦ α< 1. In general, our result estimates to within a constant multiple the probability of non-extinction by time t. For h(i)∼ i–α
, the result states that the probability of non-extinction is of order t−1/(2-α)
We give an application to interacting particle systems.
The actual waiting time of each customer in a GI/G/1 queue
Authors: Le Minh Do
Published: July 2016

This paper studies a generalization of theGI/G/1queueing system in which the inter-arrival times are not necessarily identically distributed and there is a random set-up time for customers who arrive when the server is idle. A recursive scheme is derived to obtain the distribution of the actual waiting time of each customer in the system.
Further results of replacement problem of a parallel system in random environment
Authors: Nakagawa Toshio
Published: July 2016

We have recently discussed the replacement problem of a parallel system in a random environment. This paper extends the same replacement problem for the following two cases which are more plausible: (i) The probability that an operating component fails by the j th shock depends on the number j. (ii) The replacement cost is a linear function of failed components. The expected cost of the above model is obtained. A numerical example is finally presented when the probability of failure time of a component is a negative binomial distribution.
Applications of martingale theory to some epidemic models
Authors: Picard Philippe
Published: July 2016

The purpose of this paper is to give some very simple applications of martingales to epidemics. The results are all connected with stopping times T (for instance the classical end of epidemic) and include the expression of the joint generating function Laplace transform of and and simple relations between moments of these three variables. (Here Xt
and Yt
respectively denote the numbers of susceptibles and carriers.) We also give several relations between different types of epidemics. Although this paper only deals with Downton's model, some of the methods are still valid for more general models.
Normal approximations for binary lattice systems
Authors: Kryscio Richard J. , Saunders Roy , Funk Gerald M.
Published: July 2016

Consider an array of binary random variables distributed over an m
1(n) by m
2(n) rectangular lattice and let Y
1(n) denote the number of pairs of variables d, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditions Y(n) = (Y
1(n), · ··, Yr
(n)) is asymptotically multivariate normal for n large and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.
Imbedded and non-imbedded stationary characteristics of queueing systems with varying service rate and point processes
Authors: König D. , Schmidt V.
Published: July 2016

In this paper a unified approach is used for proving relationships between customer-stationary and time-stationary characteristics of service systems with varying service rate and point processes. This approach is based on an intensity conservation principle for general stationary continuous-time processes with imbedded stationary marked point processes. It enables us to work under weaker independence assumptions than usual in queueing theory.
Invariant imbedding and dams with Markovian input rate
Authors: Brockwell P. J. , Pacheco-Santiago N.
Published: July 2016

The method of invariant imbedding, used extensively in astrophysics and in neutron transport theory, is applied to the analysis of dams whose input rate is a finite state-space continuous-parameter Markov chain. Passage-time distributions are considered as functions of the dam capacity. When the release rate is constant the method yields first-order non-linear differential equations with initial conditions obtained from the easily derived behaviour of a dam with zero capacity. Solutions of these equations are given. Some possible correlation functions for the increments of the cumulative input process are examined when the underlying Markov chain is stationary. Finally the analysis is extended to deal with dams having content-dependent release-rate of the form
Relations between queue-size and waiting-time distributions
Authors: Mori Masao
Published: July 2016

Two types of representations for relation between queue-size and waiting-time distributions are studied. By using these, an incomplete but conceptually nice generalization of Pollaczek–Khinchine formula for mean waiting time forM/G/cis obtained.
Bounds for the expected delays in some tandem queues
Authors: Niu Shun-Chen
Published: July 2016

Tandem queues are analyzed. An upper bound for the stationary expected delay in front of the second server is found for a sequence of two queues in tandem where the first server has deterministic service times, the second server has general service distribution, and the arrival process is an arbitrary renewal process. The result is extended to the case of n queues in tandem where all the servers except the last one have constant service times.
Bounds on the velocity of spread of infection for a spatially connected epidemic process
Authors: Faddy M. J. , Slorach I. H.
Published: July 2016

The simple (non-spatial) stochastic epidemic is generalised to allow infected individuals to move forward through a system of spatially connected colonies C
1, C
2, C
3, ·· ·each containing susceptible individuals. Upper and lower bounding processes are considered, to establish bounds on the asymptotic velocity of forward spread of the infection through these spatially connected colonies. These bounds are shown to be asymptotically equivalent under certain conditions, and some simulations reveal other features of the process.
A semi-Markov model for a multigrade population with Poisson recruitment
Authors: McClean Sally I.
Published: July 2016

We consider a multigrade population with semi-Markov transitions between grades, Poisson arrivals to each grade, and departures from each grade. For this model the joint distribution of the numbers in each grade at any time is found, and the limiting distributions shown to be independently Poisson; this extends a previous result for a multigrade population with Markov transitions and Poisson recruitment.

This model is particularly applicable to manpower planning. The inclusion of semi-Markov transitions allows us to take into account existing knowledge of the distribution of length of service until an individual leaves his firm.
Note an a death process
Authors: De Vries S. A. , Stam A. J. , Reddingius J.
Published: July 2016

The forward equations for the death process in ℕ2 with Q-matrix determined by q(i, j; i − 1, j) = αij, i ≧ 1, q(i, j; i, j − 1) = μj, j ≧ 1 are solved by giving a special interpretation to this Q-matrix.
A limit theorem for the maximum term in a particular EARMA(1, 1) sequence
Authors: Chernick Michael R.
Published: July 2016

The EARMA(1, 1) process was described by Jacobs and Lewis (1977). Chernick (1978) showed that the limit for the maximum term is the same as for a sequence of independent, identically distributed exponential random variables when the parameter ρ is less than 1. When ρ = 1, a different limit theorem is obtained. The resulting limit distribution is not an extreme-value type. It is, however, of the general form given by Galambos (1978). The sequence is exchangeable.
Modelling yeast cell growth using stochastic branching processes
Authors: Green P. J.
Published: July 2016

This paper aims to demonstrate that the general Crump–Mode–Jagers branching process may be used in a natural way to model the asymmetric growth of budding yeast cells. The models obtained are generalisations of the deterministic model proposed by Hartwell and Unger (1977): all the results that are derived in that paper may be obtained using branching-process methods, but such methods also apply when account is taken of the biologically obvious fact that the various phases of the cell growth are of random rather than fixed duration. In their full generality, branching processes involve more parameters than can be estimated by experiment, but we present below a special case in which this problem is not likely to arise.

A recent paper, Lord and Wheals (1980), discusses more of the biological background than is appropriate here. In the present paper, we show how certain statistical procedures for our model may be developed.
Place probabilities in normal order statistics models for horse races
Authors: Henery R. J.
Published: July 2016

Independent observations X
0, X
1…, X

N+1 are drawn from each of N populations whose distribution functions F(x – θi
) have means θ i
, 0 ≦ i < N, and we wish to calculate the probability P

k;N
that X
0 is the k th largest observation. For normal populations an approximation is given for P

K;N
based on a Taylor series expansion in the θ 's. If F(x) has an increasing failure rate, as is the case for the normal, an upper bound can be given for the ‘win' probability P
1;N
Some moment relations for normal order statistics are also given.
Bounds on the distance of a mixture from its parent distribution
Authors: Shaked Moshe
Published: July 2016

In a series of recent papers, Heyde (1975), Heyde and Leslie (1976), Hall (1979) and Brown (1980) obtained upper bounds on the uniform distance of a scale mixture from its parent distribution. Using a different technique we obtain further bounds which are more meaningful and superior in some applications. The new technique is then applied to obtain bounds on the uniform distance of a location mixture from its parent distribution. Comparison of the new bounds and the earlier ones is given.
The random walk associated by the game of roulette
Authors: Hill James M. , Gulati Chandra M.
Published: July 2016

The random walk arising in the game of roulette involves an absorbing barrier at the origin; at each step either a unit displacement to the left or a fixed multiple displacement to the right can occur with probabilities q and p respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series explicit expressions are deduced for the probabilities of the player's capital at the nth step.
Record values and the exponential distribution
Authors: Dallas A. C.
Published: July 2016

Let X
1, X
2, · ·· be i.i.d. and Yk
= max{X
1, X
2, · ··, X
k}. Then Xj
is a record value of the sequence if Yj > Yj

-1). Denoting record value times by Ln, n = 1, 2, ···, we prove that the independence of XLj

− XLi

and XLi

with Lj
> characterizes the exponential distribution.
Processor-sharing of two parallel lines
Authors: Konheim Alan G. , Meilijson Isaac , Melkman Abraham
Published: July 2016

An exponential server splits its service capacity between two independent Poisson streams of customers, unless one queue is empty, in which case the full service capacity is granted to the other queue. We present the stationary joint generating function of the two queue sizes explicitly in the form of an elliptic integral.
Obituary: Félix Pollaczek
Authors: Cohen J. W.
Published: July 2016

Félix Pollaczek was born on 1 December 1892 in Vienna, Austria; he was the descendant of a well-known mid-European Jewish family which for several centuries had counted many learned men amongst its members. He became a Frenchman by naturalization in 1947. He obtained his scientific education and training at the Latin grammar school of Vienna, and at the Technical Universities of Vienna and Brno (Czechoslovakia); he later studied mathematics at the University of Berlin. A master's degree in electrical engineering in 1920 from Brno was followed by a doctoral degree in mathematics in 1922 from Berlin with a thesis on number theory. This was concerned with the fields generated by the Ith and 12th roots of one, I being an irregular prime number. His first paper, on number theory, dates from 1917.
The effect of differential reproductive rates on the survival of a gene represented in a founder population
Authors: Heyde C. C.
Published: July 2016

Gene survival in a population which increases without density dependence is considered using a generalization of the Moran model for haploid individuals in which we have differential reproductive rates. It is shown under very mild conditions that ultimate fixation is certain. This provides some mathematical support for the phenomenon of reduced genetic variability in isolated populations established from founder groups.
On generalized functions
Authors: Rennie B. C.
Published: July 2016

There are described in the literature many spaces of what are variously described as generalized functions, distributions, or improper functions. This article introduces another. The new space is like that of M. J. Lighthill in containing the Fourier transform of every element and in having a particularly simple theory of trigonometric and Fourier series; also it is constructed in a somewhat similar way. The new space breaks away from the tradition of every element being, for some n, the nth derivative of an ordinary function, and, for example, the exponential function and its Fourier transform are in the space.
The ‘ideal’ estimation equation
Authors: Bartlett M. S.
Published: July 2016

The Cramér–Rao inequality for the variance of an unbiassed estimator is first recalled, and the value of the ‘ideal' equation of estimation existing if the minimum bound is reached illustrated by examples. When this estimation equation is not available, the more general inequality due to Kiefer is more relevant. The analogous estimation equation corresponding to the attaining of the more general minimum bound is again illustrated by examples; and the general theory, which is also available for more than one parameter, is indicated for any location and/or scaling problem.
A transformation for normal extremes
Authors: Daniels H. E.
Published: July 2016

A transformation is introduced to stabilize the variance of the largest value in a sample from a normal distribution. The transformed distribution is found to approximate closely to the limiting Gumbel form for all sample sizes. The practical implications of the result are discussed.
Critical phenomena in semi-infinite systems
Authors: Hammersley J. M.
Published: July 2016

This paper discusses the asymptotic behaviour, and in particular the critical singularities, of the generating functions for Pólya walks with a given fraction of visits to a hyperplane when the walk is or is not constrained to be on one side of that hyperplane. The three-dimensional case has applications in physics and chemistry.
Simple trigonometric models for narrow-band stationary processes
Authors: Hasofer A. M.
Published: July 2016

Over a finite interval, a Gaussian stationary process can be approximated by a finite trigonometric sum, and the error introduced by the approximation can be exactly bounded, as far as the distribution of the upper tail of the maximum is concerned. A simple case is exhibited, where a narrow band process is well approximated by means of a two-term trigonometric representation.
Testing for autocorrelation and Akaike's criterion
Authors: Hannan E. J.
Published: July 2016

The problem initially considered is that of testing whether ρ = 0 in a model y(n) = x(n)+ η (n), x(n) = ρx(n – 1) + ξ (n) where only y(n) is observed and η (n), ξ (n) are white noise. This is equivalent to distinguishing between an ARMA (1, 1) model and white noise. The asymptotic distribution of the likelihood ratio criterion is derived. This is shown to be of an unusual form. This result is then used to discuss the asymptotic properties of Akaike's procedure for estimating (p, q) in an ARMA (p, q) model. If p
0, q
0 are the true values and p
0 < P, q
0< Q, when P, Q are the maximum values considered, then it is shown that, in a certain asymptotic sense, the procedure is sure to overestimate p
0, q
0
. However, the asymptotic situation may be very far from that relevant in a practical case. The relevance of overestimation is briefly discussed.
A new family of point processes which are characterized by their second moment properties
Authors: Newman David S.
Published: July 2016

The most widely used probabilistic model of a series of “events” occurring randomly in a time or space continuum is the Poisson process. It is the proper basis for statistical analysis when general information about the rate of occurrence of events is the primary purpose of an investigation, or when the assumption of independent occurrences is reasonable and valid. A unified treatment of the statistical methods for estimating parameters in stationary and non-stationary Poisson models, and tests of the Poisson assumption against various alternatives, is contained in the lucid monograph by Cox and Lewis [6].