The branching random walk model is generalized towards generation-dependent displacement and reproduction distributions. Asymptotic theory of branching random walk in varying environments from the L2 point of view is given. If Zn(x) is the number of nth-generation particles to the left of x, then under appropriate conditions for suitably chosen xn, Zn (xn)/Zn (+∞) converges in L2 completely to a limiting distribution. Sufficient conditions for almost sure convergence are given. As a corollary an analogue of the central limit theorem for the proportion of particles of the nth generation in time interval In in the age-dependent Crump–Mode–Jagers process is obtained.

This paper considers a class of Markov chains on a bivariate state space , whose transition probabilities have a particular ‘block-partitioned' structure. Examples of such chains include those studied by Neuts [8] who took E to be finite; they also include chains studied in queueing theory, such as (Nn, Sn) where Nn is the number of customers in a GI/G/1 queue immediately before, and Sn the remaining service time immediately after, the nth arrival.

We show that the stationary distribution Πfor these chains has an ‘operator-geometric' nature, with , where the operator S is the minimal solution of a non-linear operator equation. Necessary and sufficient conditions for Πto exist are also found. In the case of the GI/G/1 queueing chain above these are exactly the usual stability conditions.

G1/G/1 QUEUE; PHASE-TYPE; INVARIANT MEASURE; FOSTER'S CONDITIONS

A construction is given of a process in which X(t) represents the content at time t of a dam whose cumulative input process is a Lévy process with measure v and whose release rate at time t is r(X(t)). It is assumed only that r(0) = 0 and that r is strictly positive and left-continuous with strictly positive finite right limits on (0,∞). The sample-paths of X are shown to satisfy the storage equation

The process X is analyzed using renewal theory and stochastic comparison techniques, and necessary and sufficient conditions are found in terms of v and r for X to have a stationary distributionπ. These generalize previous results which were obtained under the assumption that v is finite. Conditions for Πto have an atom at 0 are considered in some detail, and related results on the positivity of the expected occupation time of level 0 are given.

Necessary and sufficient conditions for the existence of Πare expressed in terms of the existence of non-negative integrable solutions of certain integral equations and conditions are given under which such solutions are necessarily stationary densities for X. A simple sufficient condition for X to have a stationary distribution is found in terms of and in the case when r is non-decreasing the condition is shown to be also necessary. Finally some examples are considered; these show that the results described above unify various known conditions in special cases, and confirm several conjectures in the related literature.

One inherent weakness of traditional reliability theory is that the system and the components are always described just as functioning or failed. However, recent papers by Barlow and Wu (1978) and El-Neweihi et al. (1978) have made significant contributions to start building up a theory for a multistate system of multistate components. Here the states represent successive levels of performance ranging from a perfect functioning level down to a complete failure level. In the present paper we will give two suggestions of how to define a multistate coherent system. The first one is more general than the one introduced in the latter paper, the results of which are, however, extendable. (This is also true for a somewhat more general model than ours, treated in independent work by Griffith (1980).) Furthermore, some new definitions and results are given (which trivially extend to the latter model). Our second model is similarly more general than the one introduced in Barlow and Wu (1978), the results of which are again extendable. In fact we believe that most of the theory for the traditional binary coherent system can be extended to our second suggestion of a multistate coherent system.