We give upper bounds on the critical values for oriented percolation and some interacting particle systems by computing their behavior on small finite sets.

A generalization of the block replacement policy (BRP) is proposed and analysed. Under such a policy, an operating system is preventively replaced at times kT (k = 1, 2, 3, ···), independently of its failure history. At failure an operating system is either replaced by a new or a used one or minimally repaired or remains inactive until the next planned replacement. The cost of the ith minimal repair of the new subsystem at age y depends on the random part C(y) and the deterministic part ci(y). The mathematical model is defined and general analytical results are obtained.

For a G/G/l queueing system let Xt be the number of customers present at time t and Yt(Zt) be the time elapsed since the last arrival of a customer (the last completion of a service) at time t. Let τl be the time until the number of customers in the sustem is reduced from j to j – l, given that X0 = j ≧ l, Y0 = y, Z0 = z. For the joint distribution of τl and Yτl and the Laplace transforms of the τl intergral equations are derived. Under slight conditions these integral equations have unique solutions which can be determined by standard methods. Our results offer a method for calculating the busy period distribution which is completely different from the usual fluctuatuion theoretic approach.

Motivated by work of Garsia and Lamperti we consider null-recurrent renewal sequences with a regularly varying tail and seek information about their rate of convergence to zero. The main result shows that such sequences subject to a monotonicity condition obey a limit law whatever the value of the exponent α is, 0 < α < 1. This monotonicity property is seen to hold for a large class of renewal sequences, the so-called Kaluza sequences. This class includes moment sequences, and therefore includes the sequences generated by reversible Markov chains. Several subsidiary results are proved.

The notion of a recursive causal graph is introduced, hopefully capturing the essential aspects of the path diagrams usually associated with recursive causal models. We describe the conditional independence constraints which such graphs are meant to embody and prove a theorem relating the fulfilment of these constraints by a probability distribution to a particular sort of factorisation. The relation of our results to the usual linear structural equations on the one hand, and to log-linear models, on the other, is also explained

By amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an α-theory is developed for general semi-Markov processes. It is shown that α-transient, α-recurrent and α-positive recurrent processes can be defined, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for α-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).