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.

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.

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.

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.

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.

Stopping rules in the context of proofreading are studied. Previous results by Yang et al. [6] are corrected and a computational procedure to obtain an optimal rule is presented.

Consider a fixed set of n events. Let S1 denote the sum of the probabilities of occurrence of each event and S2 the sum of the probabilities of occurrence of each of the pairs of events. Using a dual linear programming method a sharp upper bound is derived for the probability of occurrence of at least m out of the n events when S1 and S2 are given. The bound improves one of the inequalities obtained by Sathe et al. (1980).

A number of extensions of the class preservation results in the Esary, Marshall and Proschan shock model have been investigated by various authors in recent years, most recently by Ghosh and Ebrahimi. In this note, generalizations are obtained using different methods for the IFR and NBUE cases when the shocking process is a regular continuous-time Markov process with stationary transition probabilites.

Jobs arrive at a buffer from which there are several parallel routes to a destination. A socially optimal policy is one which minimizes the average delay of all jobs, whereas an individually optimal policy is one which, for each job, minimizes its own delay, with route preference given to jobs at the head of the buffer. If there is a socially optimal policy for a system with no arrivals, which can be implemented by each job following a policy γ in such a way that no job ever utilizes a previously declined route, then we show that such a γ is an individually optimal policy for each job. Moreover γ continues to be individually optimal even if the system has an arbitrary arrival process, subject only to the restriction that past arrivals are independent of future route-traversal times. Thus, γ is an individually optimal policy which is insensitive to the nature of the arrival process. In the particular case where the times to traverse the routes are exponentially distributed with a possibly different mean time for each of the parallel routes, then such an insensitive individually optimal policy does in fact exist and is moreover trivially determined by certain threshold numbers. A conjecture is also made about more general situations where such individually optimal policies exist.