bluuebunny/crossref_metadata_2025 huggingface dataset:
Probability of extinction of critical generation-dependent Galton–Watson processes
Authors: Fearn Dean H.
Published: July 2016

The limiting behavior of the probability of extinction of critical generation-dependent branching processes is determined. Lower bounds are determined for this probability. A means of classifying these branching processes is suggested.
Buffon's problem with a long needle
Authors: Diaconis Persi
Published: July 2016

A needle of length l dropped at random on a grid of parallel lines of distance d apart can have multiple intersections if l > d. The distribution of the number of intersections and approximate moments for large l are derived. The distribution is shown to converge weakly to an arc sine law as l/d→∞.
The severity of a viral host-vector epidemic
Authors: Radcliffe J.
Published: July 2016

An analogue of the deterministic pandemic threshold theorem for a measles epidemic is established for a viral host-vector epidemic. The severity of the epidemic is shown to be the unique positive solution of a pair of equations.
A non-central multivariate hypergeometric distribution arising from biased sampling with application to selective predation
Authors: Chesson Jean
Published: July 2016

Selective predation is an example of biased sampling which gives rise to a non-central multivariate hypergeometric distribution. The probability distribution function of this distribution is derived.
A note on the subcritical generalized age-dependent branching process
Authors: Doney R. A.
Published: July 2016

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat
as t → ∞, a necessary and sufficient condition for e–at
(1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.
Martingale convergence in the branching random walk
Authors: Biggins J. D.
Published: July 2016

A result like the Kesten-Stigum theorem is obtained for certain martingales associated with the branching random walk. A special case, when a ‘Malthusian parameter’ exists, is considered in greater detail and a link with some known results about the Crump-Mode model for a population is established.
Towards consensus: some convergence theorems on repeated averaging
Authors: Chatterjee S. , Seneta E.
Published: July 2016

The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.
Dynamic investment strategies for a risky R and D project
Authors: Deshmukh S. D. , Chikte S. D.
Published: July 2016

During the course of an R and D project, it is often meaningful and possible to evaluate its status, so that this information may be used for making better financing decisions over time. The project status changes stochastically due to the internal (technological) and the external (market) uncertainties, the former being partially controlled by expenditure of resources. In addition to the resource expenditure strategy, the manager must also decide when to terminate the project. Once the project is terminated, a terminal return is collected, whose value depends on the final project status. It is shown that the project should be terminated if the current status is either too low or too high to make further expenditure worthwhile. Otherwise, for an intermediate (promising) status of the project, an aggressive investment strategy is shown to be optimal. Thus, the model unifies the problems of optimally undertaking, financing and terminating an R and D project in face of various uncertainties.
Infinite dams with discrete additive inputs
Authors: Ali Khan M. S.
Published: July 2016

This paper considers an infinite dam fed by a discrete input Xt
during the time interval [t, t + 1), t = 0, 1, 2, ···. At time t – 0 there is an output Yt
= min(Zt

–1, + Xt

–1, r) from the dam leaving behind the amount Zt
= max(0, Zt

–1, + Xt

–1, r). The probability Pr(Zt
= i), i = 0, 1, 2, ··· is discussed under the strict assumption that r > 1 and the given initial condition that Z
0 = u, u = 1, 2, ···. The generating function technique has been used throughout the paper.
Asymptotic behaviour of the Ek/G/1 queue with finite waiting room
Authors: Hokstad Per
Published: July 2016

The asymptotic behaviour of the Ek/G/1 queue with finite waiting room is studied. Using a combination of the supplementary variable and phase techniques, queue length and waiting time distributions are obtained. Also the idle period distribution and mean length of the idle and busy periods are found.
A note on the non-homogeneous poisson cluster process
Authors: Westcott Mark
Published: July 2016

It is proved that, in a non-homogeneous Poisson cluster process whose cluster centre rate is λ(t), the long-term behaviour of several associated quantities is equivalent to their behaviour in a homogeneous process with centre rate Λ and the same cluster structure if λ(t)→ λ as t→∞.
Optimal strategies in measurable learning systems on metric spaces
Authors: Doberkat Ernst–Erich
Published: July 2016

A dynamic programming approach for the investigation of learning systems is taken. Making use of one-stage decision models and dynamic programs, respectively, two learning models are formulated and the existence of optimal strategies for learning in the respective models is proven.
Characterizations via conditional distributions
Authors: Berk Robert H.
Published: July 2016

For independent random variablesXandY,if the conditional distribution ofXgivenX+Ysatisfies certain conditions, then the joint distribution ofXandYis a member of a certain one-parameter exponential family. Extensions fornindependent random variables are given. A characterization for independent random variables involving order statistics is also given.
Asymptotic results for non-linear processes of the McKean tagged-molecule type
Authors: Giroux Gaston
Published: July 2016

McKean's tagged-molecule process is a non-linear homogeneous two-state Markov chain in continuous time, constructed with the aid of a binary branching process. For each of a large class of branching processes we construct a similar process. The construction is carefully done and the weak homogeneity is deduced. A simple probability argument permits us to show convergence to the equidistribution (½, ½) and to note that this limit is a strong equilibrium. A non-homogeneous Markov chain result is also used to establish the geometric rate of convergence. A proof of a Boltzmann H-theorem is also established.
Variability orderings related to coverage problems on the circle
Authors: Huffer Fred
Published: July 2016

Suppose that n arcs with random lengths having distributions F
1, F
2, · ··, Fn
are placed uniformly and independently on a circle. This paper presents inequalities which tell how certain distributions and probabilities change as the variability of the distributions F
l, F
2, ··, Fn
is increased. A distribution F is considered to be more variable than G if f h(x)dF(x) ≧ h(x)dG(x) for all convex functions h.
Reordering heuristics for routing in communications networks
Authors: Topkis Donald M.
Published: July 2016

Move-to-front and move-to-back heuristics are considered for adaptively reordering routing tables in communications networks. Limiting properties are established for Markov chain models of these heuristics. The limiting distributions of the number of path attempts per call for the two heuristics and for a fixed random permutation are stochastically ordered.
The waiting-time process of a queueing system with gamma-type input and blocking
Authors: Langaris C.
Published: July 2016

In this work the first-come-first-served waiting-time process of a customer in a two-stage queueing network without intermediate waiting space is analysed. We assume that the arrivals follow the gamma distribution, the service times in the first stage are arbitrarily distributed, and the service times in the second stage are again of gamma type.

Connecting the waiting time of the (n + 1)th customer with that of the nth and locating the zeros of a certain function we derive expressions for the Laplace transform of the waiting-time distribution both in the transient and the steady state.
Convergence of quasi-stationary to stationary distributions for stochastically monotone Markov processes
Authors: Pollak Moshe , Siegmund David
Published: July 2016

It is shown that if a stochastically monotone Markov process on [0,∞) with stationary distribution H has its state space truncated by making all states in [B,∞) absorbing, then the quasi-stationary distribution of the new process converges to H as B →∞.
Stationary Poisson departure processes from non-stationary queues
Authors: Foley Robert D.
Published: July 2016

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt
/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.
Zero-sum two-person semi-Markov games
Authors: Lal Arbind K. , Sinha Sagnik
Published: July 2016

Semi-Markov games are investigated under discounted and limiting average payoff criteria. The issue of the existence of the value and a pair of stationary optimal strategies are settled; the optimality equation is studied and under a natural ergodic condition the existence of a solution to the optimality equation is proved for the limiting average case. Semi-Markov games provide useful flexibility in constructing recursive game models. All the work on Markov/semi-Markov decision processes and Markov (stochastic) games can be viewed as special cases of the developments in this paper.
New better than used in expectation processes
Authors: Lam C. Y. Teresa
Published: July 2016

In this paper, we study the new better than used in expectation (NBUE) and new worse than used in expectation (NWUE) properties of Markov renewal processes. We show that a Markov renewal process belongs to a more general class of stochastic processes encountered in reliability or maintenance applications. We present sufficient conditions such that the first-passage times of these processes are new better than used in expectation. The results are applied to the study of shock and repair models, random repair time processes, inventory, and queueing models.
Equivalence of two descriptions of the ages of alleles
Authors: Ethier S. N.
Published: July 2016

It is shown that the two descriptions of the ages of alleles corresponding to the two formulations of the stationary infinitely-many-neutral-alleles diffusion model discussed by Ethier (1990a) are equivalent.
An operation which inverts Bernoulli multiplication and associated stationary reversible Markov processes
Authors: Littlejohn R. P.
Published: July 2016

A simple operation is described which inverts Bernoulli multiplication. It is used to define two classes of stationary reversible Markov processes with general marginal distribution. These are compared to the DAR(1) process of Jacobs and Lewis (1978). LJAR(1) is used to model ovulation rate time series.
A simple deterministic epidemic in several related growing communities
Authors: Gani J.
Published: July 2016

This paper first considers a simple deterministic epidemic in a growing population. It then examines a simple deterministic epidemic in several related growing communities, where infection is transmitted between individuals both within and across the communities. An explicit parametric solution is derived for the number of susceptibles in these communities as a function of time.
A quasi-likelihood approach to estimating parameters in diffusion-type processes
Authors: Heyde C. C.
Published: July 2016

Estimation of parameters in diffusion models is usually handled by maximum likelihood and involves the calculation of a Radon–Nikodym derivative. This methodology is often not available when minor changes are made to the model. However, these complications can usually be avoided and results obtained under more general conditions using quasi-likelihood methods. The basic ideas are explained in this paper and are illustrated through discussion of the Cox–Ingersoll–Ross model and a modification of the Langevin model.
On the MX/G/∞ queue by heterogeneous customers in a batch
Authors: Cong Tang Dac
Published: July 2016

We consider the Mx/G/∞ queue in which customers in a batch belong to k different types, and a customer of type i requires a non-negative service time with general distribution function Bi
(s) (1 ≦ i ≦ k). The number of customers in a batch is stochastic. The joint probability generating function of the number of customers of type i being served at a fixed time t > 0 is derived by the method of collective marks.
On some percolation results of J. M. Hammersley
Authors: Oxley J. G. , Welsh D. J. A.
Published: July 2016

We examine how much classical percolation theory on lattices can be extended to arbitrary graphs or even clutters of subsets of a finite set. In the process we get new short proofs of some theorems of J. M. Hammersley. The FKG inequality is used to get an upper bound for the percolation probability and we also derive a lower bound. In each case we characterise when these bounds are attained.
The interchangeability of ·/M/1 queues in series
Authors: Weber Richard R.
Published: July 2016

A series of queues consists of a number of · /M/1 queues arranged in a series order. Each queue has an infinite waiting room and a single exponential server. The rates of the servers may differ. Initially the system is empty. Customers enter the first queue according to an arbitrary stochastic input process and then pass through the queues in order: a customer leaving the first queue immediately enters the second queue, and so on. We are concerned with the stochastic output process of customer departures from the final queue. We show that the queues are interchangeable, in the sense that the output process has the same distribution for all series arrangements of the queues. The ‘output theorem' for the M/M/1 queue is a corollary of this result.
Feynman integral and phase space probability
Authors: Gilson J. G.
Published: July 2016

The relation between “forward” and “backward” definitions of quantum velocity is discussed. A connection between the Feynman Integral and Wigner's phase space distributions which was noted in an earlier paper is explored further and some new and useful formulae are derived.
Passage-time generating functions for continuous-time finite Markov chains
Authors: Darroch J. N. , Morris K. W.
Published: July 2016

Let S denote a subset of the states of a finite continuous-time Markov chain and let Y(a) denote the time that elapses until a weighted sum of a time units have been spent in S. Formulae are derived for the generating functions of Y(a) and of Y(a + b) – Y(b).
A note on the asymptotic distribution of the traffic-time-average in a GI/G/∞ with bulk arrivals
Authors: Narasimham T.
Published: July 2016

The asymptotic distribution of the traffic average in the case of M/G/∞ with bulk arrivals has been proved in [7] to be normal with mean μCμPμF
and variance where and are the variances and μP
and μc
are the means of the service-time and the batch-size respectively, μF
being the mean inter-arrival time. We prove in this note that the asymptotic normality holds even in the more general case of a G//G/∞ with bulk arrivals.
The finite dam II
Authors: Roes P. B. M.
Published: July 2016

A weir of capacity K is considered in which the water inflow is a process with stationary independent increments. Unless the weir is empty, there is a continuous release of water at unit rate; if K is finite the weir may become full in which case the excess water overflows instantaneously. A weir for which K is infinite will be referred to as infinite dam. For the latter the transient behaviour is well known if the input possesses a second moment (cf. e.g., Prabhu [7]) and serves as the starting point for the present paper. This result is first extended to yield the Laplace transform (L.T.) of the trivariate Laplace-Stieltjes transform (L.S.T.) of the content v(t) at time t, the input X(t) in (0, t) and the total time d(t) in the interval (0, t) during which the dam is dry. (Incidentally, the last two quantities, for relevant time intervals, will be carried throughout.) Then we use a relation between the latter and the L.S.T. of the expected number of downward level y crossings of the v(t) process established in Roes [9]. Since the dam processes considered are Markov processes, we have therewith the L.S.T. of the renewal function of the renewal process imbedded at level y. From this, one finds the L.S.T.'s of first entrance and taboo first entrance times (for their definition see introduction). Next we calculate the first skip times for the infinite dam from the first entrance times and the L.T. of the L.S.T. of v(t). It is then a routine matter to determine the taboo first skip times. From the (taboo) first entrance and skip times we derive the first entrance times for the finite dam, which in turn lead to the renewal functions of the renewal processes imbedded in the finite dam content process v*(t) and hence to the transient behaviour of the finite dam.

The advantage of the present approach over the one given in Roes [8] is that it is entirely probabilistic and avoids involved analytic arguments. As a result, the question of uniqueness of the solution does not arise, while more insight is obtained in the structure. The L.S.T. of several first entrance times and first skip times have been derived by Cohen [2] for compound Poisson input.
On some limit theorems for the GI/G/1 queue
Authors: Hooke John A.
Published: July 2016

For a GI/G/1 queue a limit theorem is obtained for the total input of work during (0, t]. This result is then used to obtain similar theorems for the waiting and idle times and for busy periods initiated by large service loads. Some of the results contained in the paper have recently been proved by other authors in more general settings. The intent of this work is to show how they may be obtained in lesser generality using simpler techniques.
Weak convergence theorems for priority queues: preemptive-resume discipline
Authors: Whitt Ward
Published: July 2016

We shall consider a single-server queue with r priority classes of customers and a preemptive-resume discipline. In this system customers are served in order of their priority while customers of the same priority are served in order of their arrival. Higher priority customers, immediately upon arrival, replace lower priority customers at the server, while customers displaced in this way return to the server before any other customers of the same priority receive service. When a displaced customer returns to the server, his remaining service time is the uncompleted portion of his original service time (cf. Jaiswal (1968)).
High density shot noise and gaussianity
Authors: Papoulis A.
Published: July 2016

The distance from Gaussianity of the shot noise processis considered, wheretiare the random times of a Poisson process with average densityλ(t).WithF(x) the distribution function ofx(t) andG(x) that of a normal process with the same mean and variance asx(t) it is shown thatwhereIf the processx(t) is stationary with λ(t) =λandh(t, τ) =h(t – τ) and the functionh(t) is bandlimited by ωc, then the above yields
Asymptotic final size distribution of the multitype Reed–Frost process
Authors: Scalia-Tomba Gianpaolo
Published: July 2016

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.
Probabilistic analysis of optimum partitioning
Authors: Karmarkar Narendra , Karp Richard M. , Lueker George S. , Odlyzko Andrew M.
Published: July 2016

Given a set ofnitems with real-valued sizes, the optimum partition problem asks how it can be partitioned into two subsets so that the absolute value of the difference of the sums of the sizes over the two subsets is minimized. We present bounds on the probability distribution of this minimum under the assumption that the sizes are independent random variables drawn from a common distribution. For a large class of distributions, we determine the asymptotic behavior of the median of this minimum, and show that it is exponentially small.
Optimal search for a randomly moving object
Authors: Weber R. R.
Published: July 2016

It is desired to minimize the expected cost of finding an object which moves back and forth between two locations according to an unobservable Markov process. When the object is in location i (i = 1, 2) it resides there for a time which is exponentially distributed with parameter λ1 and then moves to the other location. The location of the object is not known and at each instant until it is found exactly one of the two locations must be searched. Searching location i for time δ costs ciδ and conditional on the object being in location i there is a probability α
i
δ + o(δ) that this search will find it. The probability that the object starts in location 1 is known to bé p
1(0). The location to be searched at time t is to be chosen on the basis of the value of p
1(t), the probability that the object is in location 1, given that it has not yet been discovered. We prove that there exists a threshold Π such that the optimal policy may be described as: search location 1 if and only if the probability that the object is in location 1 is greater than Π. Expressions for the threshold Π are given in terms of the parameters of the model.
An asymptotic formula for the Bayes risk in discriminating between two Markov chains
Authors: Nagaev A. V.
Published: July 2016

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.
Priority queueing model with changeover times and switching threshold
Authors: Deng Yonglu , Tan Jiqing
Published: July 2016

The paper studies a single-server two-queue priority system with changeover times and switching threshold. The server serves queue 1 exhaustively and does not remain at an empty queue if the other one is non-empty. It immediately switches from queue 2 to queue 1 when the length of the latter reaches some level M. Whenever service is changed from one queue to the other a changeover time is required. Arrivals are Poisson, service times and changeover times are independent and exponentially distributed. Using an analytic method we obtain the steady-state joint probability generating function of the lengths of the two queues. By means of this probability generating function some performance measures of the system such as mean length of queue and mean delay can be calculated.
Recent results in information theory
Authors: Kotz Samuel
Published: July 2016

Information theory, in the strict sense, is a rapidly developing branch of probability theory originating from a paper by Claude E. Shannon in theBell System Technical Journalin 1948,in which anew mathematical model ofcommunications systems was proposed and investigated.

One of the central innovations of this model was in regarding the prime components of a communications system (the source of messages and the communication channel) as probabilistic entities. Shannon also proposed a quantitative measure of the amount of information based on his notion of entropy and proved the basic theorem of this theory concerning the possi bility of reliable transmission of information over a particular class of noisy channels.
Simple random walks on semi-groups of nilpotency class two
Authors: Dowson D. C.
Published: July 2016

One of the earliest known distributions is that of the Binomial distribution which arises from a Bernoulli sequence defined on two symbols (or generators) a and b. The corresponding limit theorem is that of Demoivre and Laplace and states (in an obvious notation) that (r – np)/√npq converges to the standard Normal distribution N(0,1). If the generators do not commute the situation is a good deal more complicated and in order to say very much about the sequences generated we must be able to put them in some simple canonical form. One case in which this can certainly be done is when the two symbols generate a semi-group of nilpotency class two. This means that although ba ≠ ab, we do have ba = ab (b,a) where (b, a) is a symbol which commutes with both a and b. Each sequence can then be expressed in the form aαbβ
(b,a)
γ
. In this paper we examine first the conditional distribution of γ given α and β for Bernoulli sequences in the symbols a and b and obtain central limit theorems when γ is appropriately normed. We then consider the more general problem of the m-generator semi-group of nilpotency class two and obtain the corresponding multi-dimensional central limit theorem in the case where the probability measure is discrete and is distributed over the generators.
Inequalities for branching processes
Authors: Heathcote C. R. , Seneta E.
Published: July 2016

If F(s) is the probability generating function of a non-negative random variable, the nth functional iterate Fn
(s) = Fn–

1 (F(s)) generates the distribution of the size of the nth generation of a simple branching process. In general it is not possible to obtain explicit formulae for many quantities involving Fn
(s), and this paper considers certain bounds and approximations. Bounds are found for the Koenigs-type iterates lim
n→∞
m
−n
{1−Fn
(s)}, 0 ≦ s ≦ 1 where m = F′
(1) < 1 and F′′
(1) < ∞; for the expected time to extinction and for the limiting conditional-distribution generating function limn→∞{Fn
(s) − Fn
(0)} [1 – Fn
(0)]–1. Particular attention is paid to the case F(s) = exp {m(s − 1)}.
Dynamic programming, pattern recognition and location of faults in complex systems
Authors: Bellman Richard
Published: July 2016

In a previous paper devoted to an application of dynamic programming to pattern recognition [1], we pointed out that some identification problems could be regarded as generalized trajectory processes. The functional equation technique [2] could then be employed to obtain an analytic formulation of the determination of optimal search techniques. In many cases, however, (for example, in chess or checkers), a straightforward use of the functional equation is impossible because of dimensionality difficulties. In circumventing these obstacles to effective computational solution, we employed a decomposition technique which we called “stratification” [1, 3]. In this paper, we present a different way of avoiding the dimensionality problem, based upon the concept of “extended state variable”. To indicate the utility of the concept, we shall apply it to the problem of finding a fault in a complex system.
On infinite server queues with batch arrivals
Authors: Shanbhag D. N.
Published: July 2016

The queueing system studied in this paper is the one in which

(i)

there are an infinite number of servers,

(ii)

initially (at t = 0) all the servers are idle,

(iii)

one server serves only one customer at a time and the service times are independent and identically distributed with distribution function B(t) (t > 0) and mean β(< ∞),

(iv)

the arrivals are in batches such that a batch arrives during (t, t + δt) with probability λ(t)δt + o(δt) (λ(t) > 0) and no arrival takes place during (t, t + δt) with the probability 1 –λ(t)δt + o(δt),

(v)

the batch sizes are independent and identically distributed with mean α(< ∞), and the probability that a batch size equals r is given by a
r(r ≧ 1),

(vi)

the batch sizes, the service times and the arrivals are independent.
On the algebra of queues
Authors: Kingman J. F. C.
Published: July 2016

The more one studies the vast and ever-growing literature of the theory of queues, the more one is bewildered by the wide variety of mathematical techniques which different authors have used to analyse the same or similar problems. And yet it seems that, beneath the superficial diversity, there is a deeper unity obscured by the special devices and notations characteristic of the different approaches. It is the thesis of this paper that there is indeed such a unity, and that it is best appreciated by observing that the central results of queueing theory, despite the analytical and combinatorial accretions which, by historical accident, they have acquired, are essentially algebraic in character.
A Mendelian Markov process with multinomial transition probabilities
Authors: Khazanie R. G. , McKean H. E.
Published: July 2016

The utility of the moments of the binomial distribution in finding the n-stage transition probabilities was indicated by Khazanie and McKean (1966). In this paper, maintaining the same breeding structure as in the two-allele case, it is proposed to extend the technique to cover the multiallelic case in general, with specific emphasis on the triallelic case. The existing literature on the multiple allelic case is at best scanty (see for example Kimura (1955)). Section 6 will briefly touch on a further extension to multiple loci which segregate independently.
On age dependent branching processes
Authors: Weiner Howard
Published: July 2016

This paper deals with asymptotic properties of various models of age dependent branching processes, relying heavily on Harris [3].
On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states
Authors: Seneta E. , Vere-Jones D.
Published: July 2016

Distributions appropriate to the description of long-term behaviour within an irreducible class of discrete-time denumerably infinite Markov chains are considered. The first four sections are concerned with general reslts, extending recent work on this subject. In Section 5 these are applied to the branching process, and give refinements of several well-known results. The last section deals with the semi-infinite random walk with an absorbing barrier at the origin.
A note on recent research in geometrical probability
Authors: Moran P. A. P.
Published: July 2016

The subject of geometrical probability has recently shown signs of renewed vigour which is partly due to the attractive nature of the many unsolved problems which it contains, and partly the ever widening field of applications. These applications are so diverse and so scattered in the literatures of different branches of science that when Professor M.G. Kendall and I published a monograph on the subject in 1963 some interesting papers were ignored. Others have since appeared and the purpose of the present paper is to survey, in a very brief manner, work not described in that book. In fact the present bibliography, which contains 94 items, is disjoint from the bibliography in our joint monograph.
On the conditional expected value of contributions from a renewal process
Authors: James M. F.
Published: July 2016

The problem discussed in this paper arose from the study of the effects of unresolved resonances on neutron cross-sections, but it is considered here in more general terms.

Events in a modified renewal process occur at successive intervals x
1, x
2, …, and the ith event has associated with it a parameter Γ
i
. The random variables xi
and Γ
i
are all independent, and their probability distribution functions are known. Each event contributes to two quantities F(u) and G(v) measured at u and v respectively. The value of the total contribution of all events to G(v) is assumed to be known from observation: this paper gives formulae for the mean value of F(u) conditional on this known value of G(v).
Stationary joint distributions arising in the analysis of the M/G/1 queue by the method of the imbedded markov chain
Authors: Jenkins J. H.
Published: July 2016

Probability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.
Equilibrium distributions for an open migration process
Authors: Whittle P.
Published: July 2016

We consider a system of colonies in which migration, immigration and emigration take place according to a fairly general rule. Necessary and sufficient conditions for equilibrium are given in the theorem of Section (3). The equilibrium distribution of colony numbers is determined by Equations (7) and (9); the colony sizes are thus statistically independent in equilibrium.

Attention is also paid to the case where one colony becomes a “bottleneck” to circulation, when equilibrium changes character
On the dynamic programming approach to Pontriagin's maximum principle
Authors: Morton Richard
Published: July 2016

Suppose that the state variables x = (x
1,…,x

n
)′

where the dot refers to derivatives with respect to time t, and u ∊ U is a vector of controls. The object is to transfer x to x
1 by choosing the controls so that the functional

takes on its minimum value J(x) called the Bellman function (although we shall define it in a different way). The Dynamic Programming Principle leads to the maximisation with respect to u of

and equality is obtained upon maximisation.
On a property of the variance of the waiting time of a queue
Authors: Tambouratzis D. G.
Published: July 2016

In this note, we consider a queueing system under any discipline which does not affect the distribution of the number of customers in the queue at any time. We shall show that the variance of the waiting time is a maximum when the queue discipline is “last come, first served”. This result complements that of Kingman [1] who showed that, under the same assumptions, the mean waiting time is independent of the queue discipline and the variance of the waiting time is a minimum when the customers are served in the order of their arrival.
Products of 2 × 2 stochastic matrices with random entries
Authors: Van Assche Walter
Published: July 2016

The limit of a product of independent 2 × 2 stochastic matrices is given when the entries of the first column are independent and have the same symmetric beta distribution. The rate of convergence is considered by introducing a stopping time for which asymptotics are given.
On a property of finite-state birth and death processes
Authors: Branford A. J.
Published: July 2016

A simple proof is given of the result that the ‘overflow' from a finite-state birth and death process is a renewal stream characterized by hyperexponential inter-event times. Our structure is utilized to give a converse result that any hyperexponential renewal stream can be so produced as the overflow from a finite-state birth and death process.
A simulation study of random caps on a sphere
Authors: Solomon H. , Sutton C.
Published: July 2016

This paper describes the computer simulation of a coverage problem in geometric probability, that of placing random caps on the surface of a sphere. The simulation results were compared with exact values, where known, and the differences were negligible. This suggested the use of simulation results to assess several approximation formulas in the literature.
Subcritical branching processes in a two-state random environment, and a percolation problem on trees
Authors: Dekking F. M.
Published: July 2016

We determine the decay rate of the survival probability of subcritical branching processes in a two-state random environment, where one state is subcritical, the other supercritical. This result is applied to obtain the asymptotic behavior (asn→∞) of the number of different words of lengthnoccurring on the binary, and generally theb-ary, tree with Bernoulli percolation.
Transition probabilities for some ‘special' diffusions
Authors: Beneš Václav E. , Karatzas Ioannis
Published: July 2016

We study the transition probabilities of the diffusions dXt
= (1 – exp(Xt
))dt + dWt
and dXt
= – tanh Xtdt + dWt
, in terms of special functions of mathematical physics (confluent hypergeometric and Legendre functions, respectively).
On an optimal selection problem of Cowan and Zabczyk
Authors: Bruss F. Thomas
Published: July 2016

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.
Random cyclic transformations of points
Authors: Veitch J. G. , Watson G. S.
Published: July 2016

We consider the action of independent and identically distributed n × n circulants S
1, S
2, … on V = [v
1, …, vn
] whose columns are the positions of n points in ℝ
d
. The positions of the n points after m transformations are the columns of W
(m) = VS
1 … Sm
. We describe, in several ways, the shape of the configuration of the points W
(m) as m →∞. When n = 3, 4 and d = 2, a special discussion in terms of Moebius transformations is given.
The risk-sensitive certainty equivalence principle
Authors: Whittle Peter
Published: July 2016

A risk-sensitive certainty equivalence principle is deduced, expressed in Theorem 1, for a model with linear dynamics and observation rules, Gaussian noise and an exponential-quadratic criterion of the form (2). The senses in which one is now to understand certainty equivalence and the separation principle are discussed.
The geometricity of the limiting distributions in queues and dams
Authors: Phatarfod R. M.
Published: July 2016

There are a number of cases in the theories of queues and dams where the limiting distribution of the pertinent processes is geometric with a modified initial term — herein called zero-modified geometric (ZMG). The paper gives a unified treatment of the various cases considered hitherto and some others by using a duality relation between random walks with impenetrable and with absorbing barriers, and deriving the probabilities of absorption by using Waldian identities. Thus the method enables us to distinguish between those cases where the limiting distribution would be ZMG and those where it would not.
The first-order autoregressive Mittag–Leffler process
Authors: Jayakumar K. , Pillai R. N.
Published: July 2016

The first-order autoregressive semi-Mittag-Leffler (SMLAR(1)) process is introduced and its properties are studied. As an illustration, we discuss the special case of the first-order autoregressive Mittag-Leffler (MLAR(1)) process.
Note on the strong limiting behaviour of busy periods in GI/G/1 queues under heavy traffic
Authors: Steinebach Josef , Zhang Hanqin
Published: July 2016

In this note, the strong limiting behaviour of busy periods in GI/G/1 queues is studied under the condition that the traffic intensity equals unity.
The asymptotic theory of concomitants of order statistics
Authors: David H. A. , Galambos J.
Published: July 2016

In a random sample of n pairs (X

r
, Y

r
), r = 1, 2, …, n, drawn from a bivariate normal distribution, let Xr

:n
be the rth order statistic among the Xr
and let Y
[r:n] be the Y-variate paired with Xr

:n
. The Y[r:n]
, which we call concomitants of the order statistics, arise most naturally in selection procedures based on the Xr

:n
. It is shown that asymptotically the k quantities
k fixed, are independent, identically distributed variates. In addition, putting Rt,n
for the number of integers j for which , the asymptotic distribution and all moments of n–

1
Rt, n
are determined for t such that t/n → λ with 0 < λ < 1.
The spectral density of a Markov process
Authors: Craven B. D.
Published: July 2016

For a Markov process in discrete time, having finite covariances, both the transient behaviour and the correlation structure may be summed up in a single generating function, here called the key function. If the process is stationary, and the key function possesses a suitable analytic extension, then the process possesses a continuous spectral density, which can be calculated from the key function. A converse result, and some results for the non-stationary process, are also obtained. The calculation is discussed in more detail for the waiting-time process in the GI/G/1 queue.
An asymptotically efficient test for the bundle strength of filaments
Authors: Sen Pranab Kumar
Published: July 2016

Based on a Wiener process approximation, a sequential test for the bundle strength of filaments is proposed and studied here. Asymptotic expressions for the OC and ASN functions are derived, and it is shown that asymptotically the test is more efficient than the usual fixed sample size procedure based on the asymptotic normality of the standarized form of the bundle strength of filaments, studied earlier by Daniels (1945), and Sen, Bhattacharyya and Suh (1973).
Continuous stochastic games
Authors: Sobel Matthew J.
Published: July 2016

Nonzero-sum N-person stochastic games are a generalization of Shapley's two-person zero-sum terminating stochastic game. Rogers and Sobel showed that an equilibrium point exists when the sets of states, actions, and players are finite. The present paper treats discounted stochastic games when the sets of states and actions are given by metric spaces and the set of players is arbitrary. The existence of an equilibrium point is proven under assumptions of continuity and compactness.
Non-linear matrix integral equations of Volterra type in queueing theory
Authors: Purdue Peter
Published: July 2016

The use of a branching process argument in complex queueing situations often leads to a discussion of a non-linear matrix integral equation of Volterra type. By the use of a fixed point theorem we show these equations have a unique solution.
Two mean values which characterize the Poisson process
Authors: Çinlar Erhan , Jagers Peter
Published: July 2016

The Poisson process enjoys two special properties: the mean forward recurrence time at time t does not depend on t, and the mean backward recurrence time at time t is the “mean” of the interval distribution truncated at t. Poisson process is the only renewal process with these properties.
A class of almost nowhere differentiable stationary Gaussian processes which are somewhere differentiable
Authors: Davies P. L.
Published: July 2016

A stationary Gaussian process is exhibited with the following property: the covariance function of the process is not differentiable at the origin and yet almost all the sample paths of the process are differentiable in a set of points of the power of the continuum. The process provides a counter example to a statement of Slepian.
A model for queues in series
Authors: Sharma O. P.
Published: July 2016

This paper studies the stationary behaviour of a finite space queueing model consisting of r queues in series with multi-server service facilities at each queue. Poisson input and exponential service times have been assumed. The model is suitable for phase-type service as well as service with waiting allowed before the different phases. In the case of single-server queues explicit expressions for certain probability distributions, parameters and a steady-state solution for infinite queueing space have been obtained.
On a model for storage and search
Authors: Burville P. J. , Kingman J. F. C.
Published: July 2016

Books are arranged on a shelf. After use, a book is returned to the left hand end of the shelf. Expressions are derived for the probability that, in equilibrium, a given book is in a given position on the shelf, in terms of the frequencies with which the different books are demanded.
LOW PREVALENCE OF INTESTINAL SCHISTOSOMIASIS AMONG FISHERFOLK LIVING ALONG THE RIVER NILE IN NORTH-WESTERN UGANDA: A BIOSOCIAL INVESTIGATION
Authors: Pearson Georgina
Published: July 2016

Mass drug administration has been less successful as a technique for controlling intestinal schistosomiasis (S. mansoni) than anticipated. In Uganda, the mass distribution of praziquantel has been provided to populations at risk of infection since the early 2000s, but prevalence mostly remains high. This is the case, for example, at locations in north-western and south-eastern Uganda. However, there is a remarkable exception. Among Madi fishing populations and their immediate neighbours, living close to the border with South Sudan, the rate of infection has dropped dramatically. A parasitological survey carried out at twelve fishing sites in 2013 identified only three cases of S. mansoni among 383 adults tested. This article asks: why is the prevalence of S. mansoni so low among fisherfolk in northern Uganda? Taking a biosocial approach, it suggests that the mass distribution of drugs, free of charge, has had an impact. However, the low prevalence of infection cannot be attributed to this alone. Other important factors may also have contributed to the decline in infection. These include changing fishing livelihoods, local attitudes to public health interventions, access to water and sanitation facilities, hygiene practices and the use of anti-malarial treatments. Above all, the article highlights the importance of investigating both social and biological dimensions of infection simultaneously, and of recognizing the local complexities of sustainably treating this debilitating parasitic disease.