Let {Xt}t≧0 (t may be discrete or continuous) be a random process whose finite-dimensional distributions are of exponential type. The first-passage time inf{t:Xt≧f(t)}, where f(t) is a positive, continuous function, such that f(t)= o(t) as t↑∞, is considered. The problem of finiteness of its moments is solved for both the case that {Xt}t≧0 has stationary independent increments as well as the case in which no assumptions are made about stationarity and independence for the increments of the process. Applications to sequential estimation are also given.

Using the characterization of point processes having the order statistic property we prove that the only renewal process that has the order statistic property is the Poisson process.

We consider the first-passage times of continuous-time Markov chains. Based on the approach of generalized inverse, moments of all orders are derived and expressed in simple, explicit forms in terms of the ‘fundamental matrix'. The formulas are new and are also efficient for computation.

Discrete renewal theory is generalized to study the occurrence of a collection of patterns in random sequences, where a renewal is defined to be the occurrence of one of the patterns in the collection which does not overlap an earlier renewal. The action of restriction enzymes on DNA sequences provided motivation for this work. Related results of Guibas and Odlyzko are discussed.

The asymptotic distribution of the ‘coefficient of cross-association' between two sequences of nominal data is investigated, under the assumption that the sequences have been generated by means of a specific (n − 1)-parameter Markov-chain model, frequently used by geologists. A practical approximate formula for calculating the standard deviation of this distribution is presented, and the region of its applicability examined.

Suppose Y is the strength of a system subjected to a sequence of stresses X1, X2, X3, · ··. Let N(t) be the number of stresses occurring by time t and let T be the lifetime of the system. Then the survival distribution of is given by where and . In this note we aim at determining from the above model various non-parametric classes of life distributions which are of interest in reliability theory.

A distribution R is ‘more dispersed' than a distribution P if any two quantiles of R are more widely separated than the corresponding quantiles of P. Lewis and Thompson (1981) prove that a non-degenerate distribution P is strongly unimodal if and only if it is ‘dispersive', i.e. if P∗ Q1 is more dispersed than P∗ Q2 whenever Q1 is more dispersed than Q2. For P admitting a positive Lebesgue density, we prove that P is strongly unimodal if and only if P ∗ Q is more dispersed than P for every distribution Q. Hence the latter property also characterizes the dispersivity of P.

In this paper, we give some upper (lower) bounds on where the Xi's are real-valued random variables. Some applications are given.

We describe a simple relation between the asymptotic behavior of the variance and of the expected number of distinct sites visited during a correlated random walk. The relation is valid for multistate random walks with finite variance in dimensions 1 and 2. A similar relation, valid in all dimensions, exists between the asymptotic behavior of the variance and of the probability of return to the origin.

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji on machine Mj is an independent random variable Ti. It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti, (i are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.

The condition of strong mixing for triangular arrays of random variables is a common condition of weak dependence. In this note, it is shown that this condition is not as general as one might believe. In particular, it is shown that there exist triangular arrays of first-order autoregressive random variables which converge almost surely to an independent identically distributed sequence of random variables and for which the central limit theorem holds, but which are not strong mixing triangular arrays. Hence, the strong mixing condition is more restrictive than desired.

A series of bounds for the availability and unavailability in a fixed time interval, I, for a system of maintained, interdependent components are given in Natvig (1980) in the traditional binary case, and in Funnemark and Natvig (1985) in the multistate case. For the special case of independent components the only assumption needed to arrive at these bounds is that the marginal performance process of each component is associated in I. When these processes are Markovian and binary, a sufficient condition for this to hold is given by Esary and Proschan (1970). In the present paper we generalize this condition to the multistate case, and give an equivalent and much more convenient condition in terms of the transition intensities.

It is shown that any phase-type distribution can be represented as a proper mixture of two distinct phase-type distributions. Using different terms, it is shown that the class of phase-type distributions does not include any extreme ones. A similar result holds for the subclass of upper-triangular phase-type distributions.

We consider the problem of minimizing the long-run average cost per unit time of operating a finite dam in the class of the policies of the following type. Assume that the dam is initially empty, the release rate is kept at 0 until the dam storage increases to λ, and as soon as this occurs, water is released at rate M, then the output rate is kept at M as long as the dam storage is more than τ and it must be decreased to 0 if the dam storage becomes τ. We assume that the input of water into the finite dam is a Wiener process with non-negative drift μ and variance parameter σ2. There is a cost in increasing the output rate from 0 to M as well as in decreasing the rate from M to 0 and whenever the dam storage is below level a, there is a penalty cost per unit time depending on the level. A reward is given for each unit of water released. In this paper, the long-run average cost per unit time is determined, and therefore the optimal policy can be found numerically.

Limit processes for sequences of partial sums of non-linear regression residuals are obtained under assumptions on regressor functions imposed by Jennrich (1969). The limit process and covariance kernel are calculated explicitly for functions of exponential type.

This paper is concerned with the determination of the optimal release rate for a continuous storage system. The input to the store is assumed to be a compound Poisson process (with non-negative jumps), and its content is released either at a constant rate or at a linear rate. Rewards are collected at an output-dependent rate and are continuously discounted at a constant rate. The problem consists in finding the optimal release rate which maximizes the infinite-horizon expected discounted return. This problem is solved by deriving an explicit solution for the expected discounted return, from which the optimal release rate can be found.

First-exit-time problems for Brownian motion have been studied extensively because of their theoretical importance as well as their practical applications. Except for a very few special cases such as straight-line barriers, the distribution (or density) of the first-exit time cannot be expressed in a closed form. In general the distribution and the density appear as solutions of Volterra integral equations. To solve such an equation analytically, some regularity conditions are needed for the barrier function including differentiability. This paper gives various integral equations involving the distribution and the density of the first-exit time for any sectionally continuous barrier, and then shows how to solve those integral equations numerically.

Consider n jobs and m machines. The m machines are identical and set up in parallel. All n jobs are available at t = 0 and each job has to be processed on one of the machines; any one can do. The processing time of job j is Xj, a random variable with distribution Fj. The sequence in which the jobs start with their processing is predetermined and preemptions are not allowed. We investigate the effect of the variability of the processing times on the expected makespan and the expected time to first idleness. Bounds are presented for these quantities in case the distributions of the processing times of the jobs are new better (worse) than used.