Showing papers in "Journal of Applied Probability in 1973"

The asymptotic theory of linear time series models

Abstract: A linear time-series model is considered to be one for which a stationary time series, which is purely non-deterministic, has the best linear predictor equal to the best predictor. A general inferential theory is constructed for such models and various estimation procedures are shown to be equivalent. The treatment is considerably more general than previous treatments. The case where the series has mean which is a linear function of very general kinds of regressor variables is also discussed and a rather general form of central limit theorem for regression is proved. The central limit results depend upon forms of the central limit theorem for martingales.

Weak convergence of probability measures and random functions in the function space D[0,∞)

Abstract: This paper extends the theory of weak convergence of probability measures and random functions in the function space D[0,1] to the case D [0,∞), elaborating ideas of C. Stone and W. Whitt. 7)[0,∞) is a suitable space for the analysis of many processes appearing in applied probability.

The estimation of frequency

Abstract: Very general forms of the strong law of large numbers and the central limit theorem are proved for estimates of the unknown parameters in a sinusoidal oscillation observed subject to error. In particular when the unknown frequency 0o, is in fact 0 or nt it is shown that the estimate, 0N, satisfies 0N = 0o for N ? No (w) where No (w) is an integer, determined by the realisation, w, of the process, that is almost surely finite.

Assembly-like queues

Abstract: 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 W,, 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 W, 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.

Some results on regular variation for distributions in queueing and fluctuation theory

Abstract: For the distribution functions of the stationary actual waiting time and of the stationary virtual waiting time of the GI/G/l queueing system it is shown that the tails vary regularly at infinity if and only if the tail of the service time distribution varies regularly at infinity. For sn the sum of n i.i.d. variables xi, i = 1, …, n it is shown that if E {x 1} < 0 then the distribution of sup, s 1 s 2, …] has a regularly varying tail at + ∞ if the tail of the distribution of x 1 varies regularly at infinity and conversely, moreover varies regularly at + ∞. In the appendix a lemma and its proof are given providing necessary and sufficient conditions for regular variation of the tail of a compound Poisson distribution.

On a model for storage and search

Abstract: 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.

A Gaussian Markovian process on a square lattice

Abstract: A definition of the Markovian property is given for a lattice process and a Gaussian Markovian lattice process is constructed on a torus lattice. From this a Gaussian Markovian process is constructed for a lattice in the plane and its properties are studied. MARKOVIAN PROCESSES; LATTICE PROCESSES 1. A Markov process on a torus In this paper we construct a stationary process on the square lattice formed by all the pairs of positive and negative integers, in which the random variables, Xmn,, say, are identically normally distributed and satisfy a Markovian property. We do this by first constructing a similar such process on a lattice torus and letting the size of the latter tend to infinity.

An optimal isolation policy for an epidemic

Abstract: Policies of isolating infectives in the general stochastic epidemic are considered. With costs assigned to the infection and isolation of individuals, an optimal policy is found, which at any stage minimises the expected future cost. An optimal policy is also found for the general deterministic epidemic and the two policies are compared. Finally, some numerical examples are provided. EPIDEMIC; COST; EXPECTED FUTURE SIZE; OPTIMAL ISOLATION POLICY

The heavy traffic approximation for single server queues in series

Abstract: A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

Record values and inter-record times

Abstract: First, asymptotic results for inter-record times when the CDF of the underlying LID process is not necessarily continuous are obtained, by a stochastic order argument, from known results for the continuous case. Then the asymptotic behaviour of the bivariate process of upper-record values and inter-record times is studied. Finally, assuming continuity of the underlying CDF, we derive the law of the process of total times spent in sets of states, viewing upper record values as states and inter-record times as times spent in a state, the process so viewed being a discrete time continuous state Markov jump process. The possible relevance of this result to single lane road traffic flow is indicated. RECORD VALUES; INTER-RECORD TIMES; LOG SERIES LAW; COMPOUND POISSON;

Almost sure stability of maxima

Abstract: For random variables {Xn, n ? 1} unbounded above set Mn = max {X1, X2, --, X,}. When do normalizing constants bn exist such that Mn/bn,-+ 1 a.s.; i.e., when is (Mn) a.s. stable? If {Xn) is i.i.d. then {Mn) is a.s. stable iff for all e > 0, J' j (1-F((1--ex))-1 F(dx) 1 a.s. are given and this is shown to be insufficient in general for lim infn-, Mn/bn = 1 a.s. except when I = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

On the numerical solution of Brownian motion processes

Abstract: 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.

Some results for dams with Markovian inputs

Abstract: The paper considers the dam problem with Markovian inputs, with special reference to the serial correlation coefficient of the input process. An input model is proposed which by giving particular values to the parameters makes the stationary distribution of the inputs one of the standard discrete distributions. The probabilities of first emptiness before overflow are first obtained by using the Markovian analogue of Wald's Identity. From these, the stationary distributions of the dam content are obtained by a duality argument. Both, finite and infinite dams are considered. MARKOVIAN INPUT OF LINEAR REGRESSIVE KIND; MARKOVIAN ANALOGUE OF WALD'S IDENTITY; PROBABILITIES OF EMPTINESS; STATIONARY DISTRIBUTIONS OF DAM CONTENT; DEPENDENCE ON SERIAL CORRELATION COEFFICIENT

Optimal Issuing Policies under Stochastic Field Lives

Abstract: : Consider a stockpile of n items where the ith item has a rating (r sup i), i = 1,...,n. An item with rating (r sup i) if released to the field at time 0 will have a random field life distributed as x sub (r sub i); X sub (r sub i), i = 1,...,n is a collection of random variables which are increasing in r in the sense of monotone likelihood ratio. An item with rating (r sup i) if kept in stockpile until time t and then released to the field will have a field life distributed as X sub (r sub i)d(t) where d(t) is a non-random function. Items are to be issued from the stockpile to the field until the stockpile is depleted. The ith issued item is placed in the field immediately upon the death in the field of the (i-1)st issued item. The problem studied is to find the order of item issue which maximizes in some sense the total field life obtained from the stockpile. (Author)

Erlang's formula and some results on the departure process for a loss system

Abstract: The present paper investigates the limiting distribution of the number of busy channels (queue size) and the remaining lengths of holding times at an epoch of departure for a loss system with general holding times and exponentially distributed interarrival times. Further, it is established that for this loss system in the limit an interdeparture interval length is independent of the queue size at the end of the interval and is distributed according to an exponential distribution with mean X-1. It is also seen that in the limit interdeparture times are mutually independent. LOSS SYSTEM WITH POISSON ARRIVALS, ERLANG'S FORMULA, LIMITING DEPARTURE PROCESS; REMAINING HOLDING TIMES; BUSY CHANNELS

The simple branching process with infinite mean. I

Abstract: The simple branching process {Z,) with mean number of offspring per individual infinite, is considered. Conditions under which there exists a sequence {p,} of positive constants such that p,, log (1 +Z,) converges in law to a proper limit distribution are given, as is a supplementary condition necessary and sufficient for p, constant cn as n-- oo, where 0 < c < 1 is a number characteristic of the process. Some properties of the limiting distribution function are discussed; while others (with additional results) are deferred to a sequel. BRANCHING PROCESS; GALTON-WATSON PROCESS; INFINITE MEAN; CONVEXITY;

An extension of a theorem concerning an interesting Markov chain

Abstract: In a single-shelf library of N books we suppose that books are selected one at a time and returned to the kth position on the shelf before another selection is made. Books are moved to the right or left as necessary to vacate position k. The probability of selecting each book is assumed to be known, and the N! arrangements of the books are considered as states of an ergodic Markov chain for which we find the stationary distribution.

On the expected range and expected adjusted range of partial sums of exchangeable random variables

Abstract: Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range or adjusted range of partial sums of random variables. In this paper, formulas for the expected values of the range and adjusted range of partial sums of exchangeable random variables are presented. Such formulas are based on an elegant result given in Spitzer (1956). Some consequences of the aforementioned formulas are discussed. EXPECTED ADJUSTED RANGE OF PARTIAL SUMS OF EXCHANGEABLE RANDOM VARIABLES; ADJUSTED RANGE IN STORAGE THEORY; HURST PHENOMENON; STABLE DISTRIBUTIONS

Clone-selection and optimal rates of mutation

Abstract: The paper employs methods of multitype branching processes to evaluate the probability of survival of mutable clones under environmental conditions which are unfavorable to the original parent of the clone. When other factors are taken to be constant, the long-term survival probability of a clone is implicitly demonstrated as a function of the intrinsic rate of mutation carried by this clone. The existence of a mutation rate which maximizes clone survival probability is shown and the effects of environmental deterioration on this optimal rate are studied. Finally, rigorous quantitative results are obtained for the classical situation of a Poisson distribution of offspring numbers. These results are then applied to the biological problem of indirect selection (Eshel (1972)).

On the optimal search for a target whose motion is a Markov process

Abstract: 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 optimnality 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). OPTIMAL SEARCH; MARKOV PROCESS MODEL FOR MOVING TARGETS; CLASSICAL DETECTION LAW; TERMINATION OF MARKOV PROCESSES, NECESSARY CONDITION FOR OPTIMAL SEARCH DENSITY Summary Consider a target moving in a measurable space (E,Y) according to a Markov process X = (x,, C, A~,, PS,,) in a search field with the density 2(t, y). We assume the classical form of the instantaneous probability of detection. The aim is to select such a search density that the probability of non-detection becomes as small as possible. A necessary condition for the optimality, which generalizes the result by Hellman (1972), is derived.

A limit theorem for priority queues in heavy traffic

Abstract: A single server, two priority queueing system is studied under the heavy traffic condition where the system traffic intensity is either at or near its critical value. An approximation is developed for the transient distribution of the low priority customers' virtual waiting time process. This result is stated formally as a limit theorem involving a sequence of systems whose traffic intensities approach the critical value.

Interconnected population processes

Abstract: This paper investigates the effect of migration between two colonies each of which undergoes a simple birth and death process. Expressions are obtained for the first two moments and approximate solutions are developed for the probability generating function of the colony sizes.