Consider a stochastic process X(t) (t ≧ 0) taking values in a countable state space, say, {1, 2,3, …}. To be picturesque we think of X(t) as the state which a particle is in at epoch t. Suppose the particle moves from state to state in such a way that the successive states visited form a Markov chain, and that the particle stays in a given state a random amount of time depending on the state it is in as well as on the state to be visited next. Below is a possible realization of such a process.

Recently Kshirsagar and Gupta [5] obtained expressions for the asymptotic values of the first two moments of a Markov renewal process. The method they employed involved formal inversion of matrices of Laplace-Stieltjes transforms. Their method also required the imposition of a non-singularity condition. In this paper we derive the asymptotic values using known renewal theoretic results. This method of approach utilises the fundamental matrix of the imbedded ergodic Markov chain and the theory of generalised matrix inverses. Although our results differ in form from those obtained by Kshirsagar and Gupta [5] we show that they reduce to their results under the added non-singularity condition. As a by-product of the derivation we find explicit expressions for the moments of the first passage time distributions in the associated semi-Markov process, generalising the results of Kemeny and Snell [4] obtained for Markov chains.

This is the first part of a three part work, the common setting being Ed, Euclidean space of d dimensions. Many random physical phenomena, often in the form of structures, admit models which are assemblages of random s-flats in Ed. Indeed, taking s = 0, any n-sample from a d-dimensional distribution may of course be so regarded! For static phenomena generally 0 ≦ s < d ≦ 3, while 4 is to be substituted for 3 if time variation is allowed. By postulating a high degree of stochastic independence and uniformity, a variety of “simple” models is defined. However, their investigation poses problems in geometrical probability of a wide range of difficulty; the solutions of many of which are still far from complete.

The problem of allocating scientific effort between a number of competing projects is considered. A stochastic mathematical model is set up, and for the case of just two competing projects a number of results are proved which enable the best policy to be determined. An indication is given of the ways in which these results might be extended so as to deal with more general situations.

We consider sequential decision processes in which the posterior distribution of some unobservable must be included among the sufficient statistics. The technique of an earlier paper (Whittle (1964)) is applied to show that the residual loss function is necessarily homogeneous of degree one in this distribution, if certain conventions are adopted. This point leads to considerable simplification. Examples are given of the great variety of problems which can advantageously be formulated in this manner.