bluuebunny/crossref_metadata_2025 huggingface dataset:
Ergodicity and identifiability for random translations of stationary point processes
Authors: Mori Toshio
Published: July 2016

A bivariate point process consisting of an original stationary point process and its random translation is considered. Westcott's method is applied to show that if the original point process is ergodic then the bivariate point process is also ergodic. This result is applied to an identification problem of the displacement distribution. It is shown that if the spectrum of the original process is the real line then the displacement distribution is identifiable from almost every sample realisation of the bivariate process.
The single-server queue with random service output
Authors: Boxma O. J.
Published: July 2016

In this paper a problem arising in queueing and dam theory is studied. We shall consider a G/G*/1 queueing model, i.e., a G/G/1 queueing model of which the service process is a separable centered process with stationary independent increments. This is a generalisation of the well-known G/G/1 model with constant service rate.

Several results concerning the amount of work done by the server, the busy cycles etc., are derived, mainly using the well-known method of Pollaczek. Emphasis is laid on the similarities and dissimilarities between the results of the ‘classical’ G/G/1 model and the G/G*/1 model.
The G/M/m queue with finite waiting room
Authors: Hokstad Per
Published: July 2016

The G/M/m queue with only s waiting places is studied. We start by studying the joint distribution of the number of customers present at time t and the time elapsing until the next arrival after t. This gives the asymptotic distribution of the number of customers at the moment of an arrival and at an arbitrary moment. Then waiting time and virtual waiting time distributions are easily obtained. For the G/M/1 queue also the transient solution is given. Finally the case s = ∞ is considered.
Limiting results for carrier-borne epidemics
Authors: Saunders Roy
Published: July 2016

In this article we examine the behavior of Mm
(τ), the number of susceptibles remaining uninfected at time τ in a carrier-borne epidemic as the initial number of susceptibles m tends to infinity for fixed initial number of carriers n. We show that as m → ∞ the proportion of susceptibles process {Mm
(τ)/m : ≧ 0} converges weakly to a limiting process. The limiting process has sample functions almost surely of the form exp (– h(τ)), where the form of h(τ) is determined by the removal times of the carriers.
The Galton-Watson process conditioned on the total progeny
Authors: Kennedy Douglas P.
Published: July 2016

Let Zk
denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process {Zk, 0 ≦ k ≦ n} conditioned on the event {N =n} is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the nth generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.
Results on the intersection of randomly located sets
Authors: Streit Franz
Published: July 2016

Randomly generated subsets of a point-set A
0 in the k-dimensional Euclidean space Rk
are investigated. Under suitable restrictions the probability is determined that a randomly located set which hits A
0. is a subset of A
0
. Some results on the expected value of the measure and the surface area of the common intersection-set formed by n randomly located objects and A
0 are generalized and derived for arbitrary dimension k.
Some lower bounds for the distribution of the supremum of the Yeh-Wiener process over a rectangular region
Authors: Chan Arthur H. C.
Published: July 2016

Let W (s, t), s, t ≧ 0, be the two-parameter Yeh–Wiener process defined on the first quadrant of the plane, that is, a Gaussian process with independent increments in both directions. In this paper, a lower bound for the distribution of the supremum of W (s, t) over a rectangular region [0, S]×[0, T], for S, T > 0, is given. An upper bound for the same was known earlier, while its exact distribution is still unknown.
A bivariate distribution in regeneration
Authors: Chung Kai Lai
Published: July 2016

The joint distribution of the time since last exit, and the time until next entrance, into a unique boundary point is given in Formula (1) below. The boundary point may be replaced by a regenerative phenomenon.
A note on a singular diffusion equation in population genetics
Authors: Aase Knut
Published: July 2016

This note points out an error made in an earlier paper treating a diffusion equation for the probability distribution of gene frequencies under selection. The main interest lies in the orthogonality properties of certain eigenfunctions and the determination of the corresponding Fourier coefficients. Two particular cases of selection parameter values are considered. Approximate formulas are given for the two smallest eigenvalues for arbitrary values of the selection parameters.
Pólya sequences, binomial convolution and the union of random sets
Authors: Walkup David W.
Published: July 2016

A basic result in the theory of total positivity is that the convolution of any two Pólya frequency sequences is again a Pólya frequency sequence. The like result for binomial convolution, associated with exponential generating functions, is proved. This and similar results are used to obtain an upper bound on the probability that the union of independent random subsets of a finite set N is all of N. Parallels from the theory of reliability involving sums of random variables with increasing failure rates are noted.
Optimal replacement with semi-Markov shock models
Authors: Feldman Richard M.
Published: July 2016

Consider a system that is subject to a sequence of randomly occurring shocks; each shock causes some damage of random magnitude to the system. Any of the shocks might cause the system to fail, and the probability of such a failure is a function of the sum of the magnitudes of damage caused from all previous shocks.

The purpose of this paper is to derive the optimal replacement rule for such a system whose cumulative damage process is a semi-Markov process. This allows for both the time between shocks and the damage due to the next shock to be dependent on the present cumulative damage level.

Only policies within the class of control-limit policies will be considered; namely, policies with which no action is taken if the damage is below a fixed level, and a replacement is made if the damage is above that.

An example will be given illustrating the use of the optimal replacement rule.
On stochastic population models in genetics
Authors: Kelly F. P.
Published: July 2016

Trajstman (1974) has shown that two different population models used to study the number of mutant forms maintained in a population have a certain marginal stationary distribution in common. In this note a general stochastic population model is proposed which subsumes these two models and shows that their transition rates are also related.
Stability of one-dimensional systems of colliding particles
Authors: Karr Alan F.
Published: July 2016

Envision a one-dimensional system of infinitely many identical particles, in which initial particle positions constitute a Poisson random measure and the initial velocity of a particle depends only on its initial position. Given its initial conditions the system evolves deterministically, by means of perfectly elastic collisions. In this note we derive conditions for continuity of the probability laws of the system and of the particle paths, as functions of the parameters of the initial conditions. These results have the physical interpretation of stability theorems.
A modified random walk in the presence of partially reflecting barriers
Authors: Dua Saroj , Khadilkar Shobha , Sen Kanwar
Published: July 2016

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b > 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 <m <a) under different conditions.
Remarks on renewal theory for percolation processes
Authors: Smythe R. T.
Published: July 2016

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L
1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.
Limit distributions of the number of renewals and waiting times
Authors: Mohan N. R.
Published: July 2016

Let {X
n} be an infinite sequence of independent non-negative random variables. Let the distribution function of Xi
, i = 1, 2, …, be either F
1 or F
2 where F
1 and F
2 are distinct. Set Sn
= X
1 + X
2 + … + Xn
and for t > 0 define and Zt
= SN

(t)+1 – t. The limit distributions of N(t), Yt
and Zt
as t → ∞ are obtained when F
1 and F
2 are in the domains of attraction of stable laws with exponents α
1 and α
2
, respectively and Sn
properly normalised has the composition of these two stable laws as its limit distribution.
Stochastic pursuit-evasion games with information lag. II. Observation with error
Authors: Schreuer Mynda
Published: July 2016

A discrete-time, scalar, pursuit-evasion game is considered in which an evader, moving according to a stationary stochastic process, is continually being tracked by a pursuer. The pursuer's observations of the evader's positions are subject to observation error and time lag. It is shown that when the pursuer's strategy is linear in his information, the evader is able to infer the values of the observation errors (subject to the same time lag) and it is assumed that he utilises this information linearly in his evasion strategy. The payoff is taken to be the mean-square distance between pursuer and evader, the latter being subject to a quadratic constraint on his movement. The upper value and the corresponding strategies are found for all values of the variance θ of the observation error. The lower value problem is solved completely only for variances θ below a certain critical value and partially for variances above this critical value.
The extinction time of Markov branching processes
Authors: Wang Robert C.
Published: July 2016

This paper considers the expression of expectations and density function of the extinction time for a continuous-time Markov branching process.
The size of the region occupied by one type in an invasion process
Authors: Sudbury Aidan
Published: July 2016

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.
A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables
Authors: Solari M. E. , Dunnage J. E. A.
Published: July 2016

We give an expression for the expectation of max (0, S
1, …, Sn
) where Sk
is the kth partial sum of a finite sequence of exchangeable random variables X
1, …, Xn
. When the Xk
are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.
Gordin's theorem and the periodogram
Authors: Rootzén Holger
Published: July 2016

In 1968 M. I. Gordin proved a very strong central limit theorem for stationary, ergodic sequences by means of approximation with martingales. In the present paper Gordin's theorem is generalized to cover also the periodogram of a stationary sequence, and the restriction of ergodicity is removed. It is noted that known central limit theorems for stationary processes can often be generalized to the periodogram by means of this result.
A functional equation and its application to the characterization of the Weibull and stable distributions
Authors: Wang Y. H.
Published: July 2016

The Cauchy functional equation Φ(x+y) = Φ(x) + Φ(y) is generalized to the form, assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by, for allx,y≧ 0;(2) a characterization of the symmetricα-stable distribution by the equidistribution of linear statistics.
Sufficient optimality conditions for control-limit policy in a semi-Markov process
Authors: Gertsbach I.
Published: July 2016

A finite-state semi-Markov process (SMP) with penalties is considered. A property which is similar to an increasing-hazard-rate property for a Markov chain is defined for an SMP. The SMP is controlled by shifts from the stateEitoimmediately after a transition has occurred. Conditions are given which guarantee that the optimal stationary Markovian policy belongs to a subclass of control-limit policies.
Maximum likelihood estimation for stochastic rth -order reactions
Authors: Mullooly John P.
Published: July 2016

In this paper we derive the probability distributions of the number of molecules and of the lifetime of a molecule in a stochastic rth-order system by direct evaluation of probabilities, avoiding the use of differential-difference equations. Maximum likelihood estimation of the rate constant is based on an observation of the level of the system at time t > 0. We find the asymptotic solution of the likelihood equation for a large initial number of molecules. By comparison with the numerical solution of the likelihood equation, the asymptotic estimator is shown to be a satisfactory approximation for second order reactions which are far from completion.
Equilibrium distributions of polymers on sites which are linearly arrayed
Authors: Littlewood B. , Verrall J. L.
Published: July 2016

Certain long molecules can be regarded as linear arrays of chemically active sites. During chemical reaction, adjacent groups of sites may link to form dimers, trimers, etc. This paper considers the case when all such groups are unstable, and gives equilibrium distributions for the composition of the molecules as functions of the rates of formation and decay of the linked groups.
A bivariate birth-death process which approximates to the spread of a disease involving a vector
Authors: Griffiths D. A.
Published: July 2016

A simple model for a bivariate birth-death process is proposed. This model approximates to the host-vector epidemic situation. An investigation of the transient process is made and the mean behaviour over time is explicitly found. The probability of extinction and the behaviour of the process conditional upon extinction are examined and the probability distribution of the cumulative population size to extinction is found. Appropriate circumstances are suggested under which the model might possibly be applied to malaria. The host-vector model is classified within a general class of models which represent large population approximations to epidemics involving two types of infectives.
On the quasi-stationary distributions of the GI/M/1 queue
Authors: Kyprianou E. K.
Published: July 2016

This paper studies the existence, in a stable GI/M/1 queue, of the limit as t → ∞ of the distribution of the virtual waiting time process at time t conditioned on the event that at no time in the interval [0, t] the queue has become empty. The conditional limit distribution obtained when the traffic intensity is strictly less than one is the weighted sum of an exponential and a gamma distribution. Similar conditional limit distributions are obtained for the queue size process and the waiting time process as defined by Prabhu (1964).
A storage model in which the net growth-rate is a Markov chain
Authors: Brockwell P. J.
Published: July 2016

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).
Study of a network of serial and non-serial servers with phase type service and finite queueing space
Authors: Arya Krishan Lall
Published: July 2016

The paper develops the steady-state solution of a finite space queueing system wherein each of the two non-serial servers is separately in series with two non-serial servers. It is assumed that the arriving units of the same type may demand a different number of service phases. Poisson arrivals and exponential service times are assumed at all the four channels of the system. Service of units is completed on a first-come, first-served basis at each channel. The steady-state solution for infinite queueing space is obtained as a special case of finite queueing space.
Characterization of certain populations by independence of order statistics
Authors: Galambos János
Published: July 2016

A method, making use of characteristic functions, is presented to prove theorems on characterization of distributions by independence of order statistics. Since exact independence in practice is rarely achieved, much attention is being given in the literature to theorems on determining the distributions for which certain statistics are ‘almost’ independent. The technique of this new line of investigation is strongly dependent on the fact that for the case of exact independence there is a solution in terms of characteristic functions. The present work was guided by this fact. In addition to the new method, Theorem 2 appears to be new.
Best asymptotic value of the error exponent with random coding on discrete memoryless channels
Authors: Kambo N. S. , Singh Samar
Published: July 2016

Generalizing a technique given by Wolfowitz, we calculate a formula for the asymptotic value of the error exponent with random coding on a discrete memoryless channel. We then evaluate this formula analytically for low rates and show that the exact value given by this formula agrees with the random coding exponent of Gallager and Fano. This proves that their exponent, which was only known to be a bound for low rates, gives the exact value and this brings out the inherent limitation of the random coding method.
Contributions to the theory of optimal resource allocation
Authors: Laska E. , Meisner M. , Siegel C.
Published: July 2016

A problem in optimal resource allocation is considered for n jobs with identically distributed service times admitting a monotone hazard function. If the hazard function is increasing, it is shown that the procedure of allocating the full resource individually to each job until its completion minimizes the expected completion time of the jth job. The procedure which at any instant of time equally allocated the resource among all of the remaining jobs is shown to minimize the expected total cumulative waiting time if the hazard is decreasing.
The continuity of the single server queue
Authors: Kennedy Douglas P.
Published: July 2016

In many applications of queueing theory assumptions of either Poisson arrivals or exponential service times are made. The implicit assumption is that if the actual arrival process approximates a Poisson process and the service times are close to exponential, then the quantities of interest in the real queueing system (viz. the virtual waiting time, queue length, idle times, etc.), will approximate those of the idealized model. The continuity of the single server queue acting as functionals of the arrival and service processes is established. The proof involves an application of the theory of weak convergence of probability measures on metric spaces.
The area under the infectives trajectory of the general stochastic epidemic
Authors: Downton F.
Published: July 2016

If the cost of an epidemic is a linear function of the number of persons infected at any time, the total cost depends upon the area under the infectives trajectory. This note uses a combinatorial argument to show that for the general stochastic epidemic the distribution of this area may be expressed in terms of the probabilities of the ultimate size of the epidemic.
On dams with additive inputs and a general release rule
Authors: Çinlar E. , Pinsky M.
Published: July 2016

An infinite capacity dam subject to an additive input process and a general release rule is considered. The input process can have infinitely many jumps in any finite interval, and the rate of release is r(x) when the dam content is x. The content is constructed as the increasing limit of a sequence of Markov processes and the convergence is shown to be almost surely uniform over finite intervals. The limit process is shown to be a standard Markov process and its characteristic equation is computed.
On queues involving batches
Authors: Mohanty S. G.
Published: July 2016

It has been demonstrated by Takács in a series of papers and in his book that combinatorial methods can be successfully applied to derive certain probability distributions in queueing processes. In this paper, we further illustrate the usefulness of combinatorial techniques and determine the stochastic law of the busy period in two queueing systems particularly involving batches. It may be of interest to note that queues involving batches have been dealt with in [3] and [7].
Limits for the superposition of m-dimensional point processes
Authors: Whitt Ward
Published: July 2016

To obtain a limit with independent components in the superposition of m-dimensional point processes, a condition corresponding to asymptotic independence must be included. When this condition is relaxed, convergence to limits with dependent components is possible. In either case, convergence of finite distributions alone implies tightness and thus weak convergence in the function space D[0, ∞) × … × D[0, ∞).
An example on geometric ergodicity of a finite Markov chain
Authors: Teugels Jozef L.
Published: July 2016

A discrete time Markov chain with (n + 1) states is constructed in which there are n states having different decay parameters for geometric ergodicity.
Assembly-like queues
Authors: Harrison J. Michael
Published: July 2016

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn
cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn
under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.
Diffusion limits for congestion models
Authors: Mcneil Donald R.
Published: July 2016

Heuristic methods yield simple limiting results for the behaviour of input-output systems under conditions of heavy traffic. It is assumed that the input is time-dependent, while the output rate depends entirely on the content of the system. It is shown that the content, when suitably normalized and scaled, behaves like a (non-stationary) Ornstein-Uhlenbeck process.
On the numerical solution of Brownian motion processes
Authors: Anderssen R. S. , Hoog F. R. De , Weiss R.
Published: July 2016

A new class of finite difference methods based on the concept of product integration is proposed for the numerical solution of the systems of weakly singular first kind Volterra equations which arise in the study of Brownian motion processes.
Estimation of the mean of a stationary time series by sampling
Authors: Brillinger David R.
Published: July 2016

LetX(t), – ∞ <t< ∞, be a stationary time series with meancx. Let 0 <τ1<τ2 < … <τN≦Tdenote A given sampling times in the interval (0,T]. We determine the asymptotic distribution of the estimate [X(τ1) + … +X(τN)]/Nofcxwhen the sampling times are fixed, satisfying a form of generalised harmonic analysis requirement, and when the sampling times are the times of events of a stationary point process independent of the seriesX(t). The results obtained may be viewed as non-standard central limit theorems.
An application of the generating function to the solution of a genetic problem
Authors: Diamantis Basil
Published: July 2016

An application of the generating function to genetics has provided the probability distribution of homozygotes obtained by self-fertilization, in the case of two linked genetic loci. The individual probabilities are functions of the frequency of genetic recombination between the two loci.
Maximum likelihood estimation for stochastic rth-order reactions. II
Authors: Mullooly John P.
Published: July 2016

In this paper we consider maximum likelihood estimation of the rate constant for stochastic rth-order reactions based on observation of the level of the system at time t > 0. Conditions are found for which the waiting time until the nth. reaction event is asymptotically normal, as the initial number of molecules and the number of reaction events become large. This distributional result is used to derive an approximate estimator which is shown for the second-order case to be close to the exact maximum likelihood estimate over a wide range of percentage completion.
Asymptotic properties of the stationary measure of a Markov branching process
Authors: Yang Y. S.
Published: July 2016

The asymptotic properties of the unique stationary measure of a Markov branching process will be given. In the critical case with finite variance, the result can be deduced from a result for discrete time processes of Kesten, Ney and Spitzer (1966) where the proof makes use of a stronger assumption than the finiteness of variance. For the continuous time case where the stationary measure has an explicit form, we can use the discrete renewal theorem which takes care of the infinite variance case as well.
Observations on a result in sequential analysis by P. Whittle
Authors: Gittins J. C.
Published: July 2016

In a series of papers Whittle has proposed locating the optimal boundary between the continuation region and the stopping region, in the no-overshoot approximation, for a class of sequential analysis problems by using the criterion that the gradient of the objective function must be continuous at the boundary. It is shown here that the result originally proposed by Whittle is not true in the generality which he supposed, and an indication is given of some of the circumstances in which it does hold.
A note on Kendall's autoregressive series
Authors: Brown T. J. , Hardin J. C.
Published: July 2016

Realizations of the autoregressive series considered by Kendall are generated using both Gaussian and uniformly distributed random variables. Estimates of the autocorrelation and power spectrum of the process obtained from these realizations are compared with the theoretical expressions derived by Bartlett. Agreement is well within the theoretical variance.
A more general form of a theorem of Crofton
Authors: Ruben H. , Reed W. J.
Published: July 2016

Let Dj
be a domain in nj
,-dimensional Euclidean space, for j = 1, …, k. Suppose that for each j = 1,…, k, Nj
points are chosen independently at random in Dj. A theorem, which is an extension of a theorem of Crofton, is proved about the expected value of functions of the points.
The initial geographical spread of host-vector and carrier-borne epidemics
Authors: Radcliffe J.
Published: July 2016

The deterministic and stochastic models developed by Bartlett (1956) to describe the initial spatial spread of an epidemic such as measles are extended to host-vector and carrier-borne epidemics such as malaria and typhoid in which more than one type of individual is involved. If the epidemic does not die out, then the behaviour predicted by the deterministic model and an examination of the mean distributions in the stochastic model is a wave of infection spreading out from the initial source of infection.
On a solution of the migration process and the application to a problem in epidemiology
Authors: Raman S. , Chiang C. L.
Published: July 2016

In this paper we derive a successive method of solution to a general class of models pertaining to the investigation of the migration process. A specific application of the results to a problem arising in a study of the epidemiology of leprosy has been considered. The application to explorative studies in biological modelling using an interactive computing mode has been discussed.
Channel capacities for list codes
Authors: Ahlswede Rudolf
Published: July 2016

In the present paper we demonstrate that the concept of a list code is from a mathematical point of view a more canonical notion than the classical code concept (list size one) in that it allows a unified treatment of various coding problems. In particular we determine for small list sizes the capacities of arbitrarily varying channels.
On the optimal search for a target whose motion is a Markov process
Authors: Saretsalo Lauri
Published: July 2016

We will consider the optimal search for a target whose motion is a Markov process. The classical detection law leads to the use of multiplicative functionals and the search is equivalent to the termination of the Markov process with a termination density. A general condition for the optimality is derived and for Markov processes in n-dimensional Euclidean space with continuous transition functions we derive a simple necessary condition which generalizes the result of Hellman (1972).
Extremal processes and record value times
Authors: Resnick Sidney I.
Published: July 2016

Let {Xn
, n ≧ 1} be i.i.d. and Yn
= max {X
1,…, Xn
}. Xj
is a record value of {Xn
} if Yj > Yj–

1 The record value times are Ln, n ≧ 1 and inter-record times are Δ
n
, n ≧ 1. The known limiting behavior of {Ln
} and {Δn
} is close to that of a non-homogeneous Poisson process and an explanation of this is obtained by embedding {Yn
} in a suitable extremal process which jumps according to a non-homogeneous Poisson process.
An expansion for the distribution function of a random sum
Authors: Marsh L. M.
Published: July 2016

The Edgeworth expansion gives an indication of the rate of convergence of the distribution function of the sum of a fixed number of random variables to the normal distribution. A similar expansion is given here for the distribution function of the sum of a random number N of random variables, when the probability generating function of N takes a special form.
On the distribution of inanimate marks over a linear birth-and-death process
Authors: Morgan Byron J. T.
Published: July 2016

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived.
Convergence of general branching processes and functionals thereof
Authors: Jagers Peter
Published: July 2016

With each individual in a branching population associate a random function of the age. Count the population by the values of these functions. Different choices yield different processes. In the supercritical case a unified treatment of the asymptotics is possible for a wide class, including for example the number of individuals having some random age dependent property or integrals of branching processes. As an application, the demographic concept of average age at childbearing is given a rigorous interpretation.
On the distribution of attributes in organizations
Authors: Störmer Horand
Published: July 2016

The paper deals with a stochastic organization model described by a generalized Poisson input process and given probabilities of entering individuals obtaining certain attributes in this organization. The distribution of attributes at any given instant is derived. Examples illustrate the application of obtained results.
On infinite dams with inputs forming a stationary process
Authors: Tin Pyke , Phatarfod R. M.
Published: July 2016

This paper considers a dam of infinite capacity with a discrete-valued stationary input process and a unit release whenever possible. It is shown how, by suitable manipulations of the equation governing the dam content process, the stationary distribution of the dam being empty can be obtained, as also can (with a few additional assumptions) the expected value of the dam content in the stationary case. Results obtained are applied to particular cases of input — independent and identical, Markov, Bivariate Markov and moving-average.
Runs of a discrete time regenerative phenomenon
Authors: Imhof J. P.
Published: July 2016

The limiting behavior of the number of runs up to time n of a regenerative phenomenon on the positive integers is related to the one of the number of regenerations. Applications are considered in fluctuation theory, and for the GI/G/1 queue.
Multi-server assembly queues
Authors: Crane Michael A.
Published: July 2016

A model is studied in which each of several servers assembles finished products consisting of N different input items. Items of each type arrive independently at the assembly station and are grouped into N-tuples consisting of one item of each type. N-tuples are assembled into finished products by the servers on a first come-first-served basis. The model is analyzed by means of the theory of weak convergence, and functional limit theorems are obtained for appropriately normalized random functions induced by the queue size processes. The limits are expressed as functionals of multi-dimensional Wiener processes, with ordinary central limit theorems obtained as corollaries.
A note on models using the branching process with immigration stopped at zero
Authors: Seneta E. , Varé S. Ta
Published: July 2016

The Galton-Watson process with immigration which is time-homogeneous but not permitted when the process is in state 0 (so that this state is absorbing) is briefly studied in the subcritical and supercritical cases. Results analogous to those for the ordinary Galton-Watson process are found to hold. Partly-new techniques are required, although known end-results on the standard process with and without immigration are used also. In the subcritical case a new parameter is found to be relevant, replacing to some extent the criticality parameter.
Partial barrier-absorption probabilities for the Wiener process
Authors: Park C. , Schuurmann F. J.
Published: July 2016

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The techniques of computing probabilities of the type are well known. The main purpose of this paper is to present ways of finding barrier-absorption probabilities when the barrier function is defined only on sub-intervals of [0, T].
A threshold theorem for the Reed-Frost chain-binomial epidemic
Authors: Ball Frank
Published: July 2016

We prove a threshold theorem for the Reed–Frost chain-binomial model which is analogous to the threshold theorem of Williams (1971) for the general stochastic epidemic. We show that when the population size is large a ‘true epidemic’ occurs with a non-zero probability if and only if an initial infective individual infects on average more than one susceptible individual.
A note on the evaluation of first-passage-time probability densities
Authors: Ricciardi L. M. , Sato S.
Published: July 2016

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.
On the continuous-review (s, S) inventory model under compound renewal demand and random lead times
Authors: Sahin Izzet
Published: July 2016

Binomial moments of the time-dependent and limiting distributions of inventory deficit are derived for a continuous-review (s, S) inventory system under the assumption that interarrival times, demand sizes, and lead times are independent sequences of independent, identically distributed random variables. Explicit expressions for the limiting distribution are also given in some special cases of practical interest. The approach used follows that of Finch [2] who investigated the system under the additional assumption of unit demands.
Convergence of age and type distributions in multitype critical Bellman-Harris processes
Authors: Murthy K. Rama
Published: July 2016

The age and type distributions (conditioned on non-extinction) in a critical Bellman-Harris process are shown to converge in probability to a deterministic distribution under a finite first-moment hypothesis on the offspring distribution and a tightness condition on the lifetime distributions.
The moment ratios of particle size distributions in some simple growth models
Authors: MacGillivray H. L.
Published: July 2016

Important parameters of particle size distributions in dispersed systems in engineering and related fields are ratios of moments and inverse powers of these ratios, known as mean sizes. The variation in these parameters is examined for the simplest growth model in which the size distribution is translated, and the results for this process considered in relation to the problems of models of other growth processes. For initial size distributions with monotone hazard rate, the results are particularly significant, and the properties of the normalised moments of other distributions are also considered.
Lower bounds for the critical probability in percolation models with oriented bonds
Authors: Gray Lawrence , Wierman John C. , Smythe R. T.
Published: July 2016

In completely or partially oriented percolation models, a conceptually simple method, using barriers to enclose all open paths from the origin, improves the best previous lower bounds for the critical percolation probabilities.
Dependent thinning of point processes
Authors: Isham Valerie
Published: July 2016

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.
The effect of variability in the GI/G/s queue
Authors: Whitt Ward
Published: July 2016

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.
Time-dependent solution of the logistic model for population growth in random environment
Authors: Prajneshu
Published: July 2016

The exact time-dependent solution as well as the stationary solution of the logistic model for population growth with varying carrying capacity is worked out in both the Stratonovich and Ito calculi by solving the forward Kolmogorov equation.
A counterpart of the Borel-Cantelli lemma
Authors: Bruss F. Thomas
Published: July 2016

The general part of the Borel-Cantelli lemma says that for any sequence of events (An) defined on a probability space (Ω, Σ,P), the divergence of ΣnP(An) is necessary forP(Ani.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the Anare independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced byfor some. We will see that this property of (An) may frequently be assumed without loss of generality. We also disclose a useful duality which allows straightforward conclusions without selecting independent sequences. A simple random walk example and a new result in the theory ofϕ-branching processes will show the tractability of the method.
The application of the Hankel transformation to spherically symmetric coverage problems
Authors: Schroeter Gerhard
Published: July 2016

We derive an integral expression for the expected coverage of an n-dimensional target when the value density function of the target, the damage function, and the distribution of the damage function center around the target center are spherically symmetric. This integral expression is transformed by means of Parseval's theorem for Hankel transforms to an integral of the product of the Hankel transforms of order ½n − 1 of all underlying functions.
Extinction probability, regularity and asymptotic growth of Markovian populations
Authors: Lenz Norbert
Published: July 2016

The distribution of the maximum and the extinction probability for a Markovian population is derived. Asymptotic growth is described, using the sequence of sojourn times. A regularity criterion for the processes under consideration exists under certain assumptions. For a class of processes with specific population-dependent transition rates the asymptotic behaviour is given explicitly.
On the normal convergence of a class of simple batch epidemics
Authors: Langberg Naftali A.
Published: July 2016

A group of n susceptible individuals exposed to a contagious disease is considered. It is assumed that at each instant in time one or more susceptible individuals can contract the disease.

The progress of this epidemic is modeled by a stochastic process Xn
(t), t in [0,∞) representing the number of infective individuals at time t. It is shown that Xn
(t), with the suitable standardization and under a mild condition, converges in distribution as n → ∞to a normal random variable for all t in (0, t
0), where t
0 is an identifiable number.
Availability measures for a two-state system
Authors: Baxter Laurence A.
Published: July 2016

The use of alternating renewal processes to model the behaviour of a repairable system is discussed, and properties of two measures of availability are reviewed. It is shown how the introduction of forward recurrence times to failure enables these measures to be generalised.
On occupation times for quasi-Markov processes
Authors: Bondesson Lennart
Published: July 2016

In this note the joint distribution for the times in an interval [0, t] spent in the states 1, 2, ···, N in a standard quasi-Markov process of order N is considered. An expression for the Laplace transform with respect to t of the Laplace–Stieltjes transform of this joint distribution is derived.
A simultaneous characterization of the Poisson and bernoulli distributions
Authors: Kimeldorf George , Plachky Detlef , Sampson Allan R.
Published: July 2016

Let N, X
1, X
2, · ·· be non-constant independent random variables with X
1, X
2, · ·· being identically distributed and N being non-negative and integer-valued. It is shown that the independence of and implies that the Xi
's have a Bernoulli distribution and N has a Poisson distribution. Other related characterization results are considered.
Stochastic stable population growth
Authors: Lange Kenneth , Holmes William
Published: July 2016

In classical demographic theory, the age structure of a population eventually stabilizes, and the population as a whole grows at a geometric rate. It is possible to prove stochastic analogues of these results if vital rates fluctuate according to a stationary stochastic process. The approach taken here is to study the action of random matrix products on random vectors. This permits the application of Hilbert's projective metric and leads to considerable simplification of the ergodic and central limit theory of population growth.
Predicting the benefit of screening for disease
Authors: Dubin Neil
Published: July 2016

To evaluate the benefits and risks associated with screening for disease, a model is developed to characterize the changes in incidence and survival distributions effected by a screening program. Screening is presumed to increase survival by resulting in diagnosis of disease at earlier stages. All disease states in the model are observable, thus facilitating application to empirical data. An example of such an application using data from a breast cancer detection project is given for the case of one screening for two-stage disease having Weibull-distributed diagnosis times.
On excess life in certain renewal processes
Authors: Arnold Barry C. , Groeneveld Richard A.
Published: July 2016

Excess life distributions for discrete renewal processes may be computed by using elementary discrete Markov chain concepts involving absorption probabilities. Excess life distributions in general may then be obtained by approximating the renewal process under study by a suitably chosen sequence of discrete renewal processes. The technique is illustrated in the cases of renewal processes with interarrival distributions which are a linear combination of two exponentials and uniform [0, 1]. A related algorithm is described for computer generated approximations of excess life distributions corresponding to continuous interarrival time distributions with an increasing c.d.f. Conditions for convergence of this algorithm are examined.
A strong approximation for some non-stationary complex Gaussian processes
Authors: Breuer Peter
Published: July 2016

A strong approximation theorem is proved for some non-stationary complex-valued Gaussian processes and an explicit rate of convergence is achieved. The result answers a problem raised by S. Csörgő.
Multi-species population models and evolutionarily stable strategies
Authors: Hines W. G. S.
Published: July 2016

The possibility that interspecific contest tactics might be able to stabilize two species predator-prey populations is investigated using a matrix model of the population. It is found that if the interspecific tactics have no influence on intraspecific encounters, the non-trivial equilibrium population composition is necessarily unstable. The result provides some insight into a reported instability of a predator-prey population with ‘stable' strategies.
A note on the normalised moments of distributions with non-monotonic hazard rate
Authors: MacGillivray H. L.
Published: July 2016

For distributions of non-negative random variables with a monotonic hazard rate, it is well known that the normalised moments have the same sign-regular property as the distribution. This note extends this correspondence in properties for some common distributions with non-monotonic hazard rates, linking a change in sign-regular properties.
An estimate for the tails of the distribution of the supremum for a class of stationary multiparameter Gaussian processes
Authors: Cabaña E. M. , Wschebor M.
Published: July 2016

Using Slepian processes as a standard of comparison, estimates are given for the probability that a centered multiparameter stationary Gaussian process reaches a constant barrier u on a subset of the parameter domain.