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.

Let {xt} be a discrete-time multivariate stationary process possessing an infinite autoregressive representation and let ΓB(k), ΓF(k) and Γ be the block Toeplitz covariance matrices of xB(k) = [x′–1, x′–2, · ··, x′–k]′, xF(k) = [x′1, x′2 · ·· x′k] and x = [·· ·x′–2, x′–1, x′0, x′1, x′2 · ··]′ respectively, where k ≧ 1, is finite or infinite. Also let φ m,n(j) and δm,n(u) be the coefficients of xt+ j and xt– u respectively in the linear least-squares interpolator of xt from xt+ 1, · ··, xt+ m; xt− 1, · ··, xt– n, where m, n ≧ 0, 0 ≦ j ≦ m, 0 ≦ u ≦ n are integers, zt(m, n) denote the interpolation error and τ2(m, n) = E[zt(m, n)zt(m, n)′]. A physical interpretation for the components of ΓB(k)–1, ΓF(k)–1 and Γ–1 is given by relating these components to the φm,n(j) δm,n(u) and τ2(m, n). A similar result is shown to hold also for the estimators of ΓB(k)–l and the interpolation parameters when these have been obtained from a realization of length T of {xt}. Some of the applications of the results are considered.

We prove in the setting of the system of unbounded spins developed by Lebowitz and Presutti, a strong variational principle analogous to Lanford and Ruelle's for discrete spins. We establish this result for finite-range regular and superstable potentials satisfying a condition which enables us to define the thermodynamic energy of the system.

Let γ t and δ t denote the residual life at t and current life at t, respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as holds for a single positive integer n, then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t, we find that for some fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t.

The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained exponential model? We present some general results for this constrained model and then apply them to obtain concrete answers in the edge-triangle model with fixed density of edges.

A comparison theorem is stated for Markov processes in Polish state spaces. We consider a general class of stochastic orderings induced by a cone of real functions. The main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators imply ordering of the processes. Several applications to convex type and to dependence orderings are given. In particular, Liggett's theorem on the association of Markov processes is a consequence of this comparison result.

Hunt and Kurtz [9] consider a loss network as the number of circuits and the offered traffics become large. They prove a functional law of large numbers for such a network and illustrate their results with some simple examples. In this paper we apply their results to slightly more complicated examples to illustrate other, and sometimes surprising, behaviour of the loss networks in heavy traffic. The networks we consider operate under somewhat unusual routing rules but this is to enable us to produce the behaviour in networks with only a few links. In larger, real-world networks it is likely that much more ‘natural' and intuitively appealing routing rules could produce similar undesirable behaviour.

Suppose that there is a sequence of programs or jobs that are scheduled to be executed one after another on a computer. A program may terminate its execution because of the failure of the computer, which will obliterate all work the computer has accomplished, and the program has to be run all over again. Hence, it is common to save the work just completed after the computer has been working for a certain amount of time, say y units. It is assumed that it takes a certain time to perform a save. During the saving process, the computer is still subject to random failure. No matter when the computer failure occurs, it is assumed that the computer will be repaired completely and the repair time will be negligible. If saving is successful, then the computer will continue working from the end of the last saved work; if the computer fails during the saving process, then only unsaved work needs to be repeated. This paper discusses the optimal work size y under which the long-run average amount of work saved is maximized. In particular, the case of an exponential failure time distribution is studied in detail. The properties of the optimal age-replacement policy are also derived when the work size y is fixed.

In a correlated random walk (CRW) the probabilities of movement in the positive and negative direction are given by the transition probabilities of a Markov chain. The walk can be represented as a Markov chain if we use a bivariate state space, with the location of the particle and the direction of movement as the two variables. In this paper we derive explicit results for the following characteristics of the walk directly from its transition probability matrix: (i) n -step transition probabilities for the unrestricted CRW, (ii) equilibrium distribution for the CRW restricted on one side, and (iii) equilibrium distribution and first-passage characteristics for the CRW restricted on both sides (i.e., with finite state space).

In this paper we consider stochastic recursive equations of sum type, , and of max type, , where Ai, bi, and b are random, (Xi) are independent, identically distributed copies of X, and denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal Ls-metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

In this article, we consider the limit behavior of the hazard rate function of mixture distributions, assuming knowledge of the behavior of each individual distribution. We show that the asymptotic baseline function of the hazard rate function is preserved under mixture.

In this paper we propose a general bivariate random effect model with special emphasis on frailty models and environmental effect models, and present some stochastic comparisons. The relationship between the conditional and the unconditional hazard gradients are derived and some examples are provided. We investigate how the well-known stochastic orderings between the distributions of two frailties translate into the orderings between the corresponding survival functions. These results are used to obtain the properties of the bivariate multiplicative model and the shared frailty model.

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. We prove that if the random variable X2:n – X1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980).

This note gives a new strong stationary time (SST) for reversible finite Markov chains. A modification of the initial distribution is represented as a mixture of distributions which have eigenvector interpretations, and for which good simple SSTs exist. This provides some insight into the relationship between SSTs and eigenvalues. Connections to duality and the threshold phenomenon are discussed.

For homogeneous, two-dimensional random field ξ(t), t ∈ R2 we develop the ‘half' spectral theory sufficient to rigorously define its envelope η (t). We then specialise to the case of ξ Gaussian, which implies η is Rayleigh, and consider the mean value of a certain characteristic of the sets {t:η(t) ≧ u} (u ≧ 0). From this we deduce some qualitative information about the sample path behaviour of the Rayleigh field η .

Let 0 ≦ T1 ≦ T2 ≦ ·· · represent the epochs in time of occurrences of events of a point process N(t) with N(t) = sup{k : Tk ≦ t}, t ≧ 0. Besides certain mild conditions on the process N(t) (see Conditions (A1)– (A3) in the text) we assume that for every k ≧ 1, as t →∞, the vector (t – TN(t), t – TN(t)–1, · ··, t – TN(t)–k+1) converges in law to a k-dimensional distribution which coincides with that of a random vector ξ k = (ξ1, · ··, ξ k) necessarily satisfying P(0 ≦ ξ1 ≦ ξ2 ≦ ·· ·≦ ξk) = 1. Let R(t) be an arbitrary function defined for t ≧ 0, satisfying 0 ≦ R(t) ≦ 1, ∀0 ≦ t <∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that

The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role of R(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of the GI/G/∞ queue apparently not studied before.

We use Gurland's (1957) definition and notation for generalized distributions, i.e., given random variables Xi with probability generating functions gi(s), i = 1, 2, 3, if g3(s) = g1[g2(s)], we say that X3 is X1 generalized by X2 and write X3~ X1VX2.

An explicit formula for an invariant measure of a time-reversible Markov chain is presented. It is based on a characterization of time reversibility in terms of the transition probabilities alone.

This paper discusses a renewal process whose time development between renewals is described by a Markov process. The process may be controlled by choosing the times at which renewal occurs, the objective of the control being to maximise the long-term average rate of reward. Let γ ∗ denote the maximum achievable rate. We consider a specific policy in which a sequence of estimates of γ ∗ is made. This sequence is defined inductively as follows. Initially an (a priori)estimate γo is chosen. On making the nth renewal one estimates γ ∗ in terms of γo, the total rewards obtained in the first n renewal cycles and the total length of these cycles. γ n then determines the length of the (n + 1)th cycle. It is shown that γ n tends to γ ∗ as n tends to∞, and that this policy is optimal.

The time at which the (n + 1)th renewal is made is determined by solving a stopping problem for the Markov process with continuation cost γ n per unit time and stopping reward equal to the renewal reward. Thus, in general, implementation of this policy requires a knowledge of the transition probabilities of the Markov process. An example is presented in which one needs to know essentially nothing about the details of this process or the fine details of the reward structure in order to implement the policy. The example is based on a problem in biology.

A characterization theorem of the exponential distribution by Ahsanullah and Rahman is examined in contrast to existing results by several other authors.