The optimal selection of a maximum of a sequence with the possibility of ties is considered. The object is to examine each observation in the sequence of known length n and, based only on the relative rank among predecessors, either to stop and select it as a maximum or to continue without recall. Rules which maximize the probability of correctly selecting a maximum from a sequence with ties are investigated. These include rules which randomly break ties, rules which discard tied observations, and minimax rules based on the atoms of a discrete distribution function. If the sequence is random from F, a random distribution function from a Dirichlet process prior with non-atomic parameter, optimal rules are developed. The limiting behavior of these rules is studied and compared with other rules. The selection of the parameter of the Dirichlet process regulates the ties.

Section 1 of this paper reviews some works related to reliability evaluation of systems without repair. The assumption that element failure rates are low enables one to obtain an expression for the main term in the asymptotic representation of system reliability function. Section 2 is devoted to repairable systems. The main index of interest in reliability is the time to the first system failure. A typical situation in reliability is that the repair time is much smaller than the element lifetime. This ‘fast repair' property leads to an interesting phenomenon, that for many renewable systems the time to system failure converges in distribution, under appropriate norming, to an exponential random variable. Some basic theorems explaining this fact are presented and a series of typical examples is considered. Special attention is paid to reviewing the works describing the exponentiality phenomenon in birth-and-death processes. Some important aspects of computing the normalizing constants are considered, among them the role played by the so-called ‘main event'. Section 2 also reviews various bounds on the deviation from exponentiality. Section 3 gives brief comments on some works and techniques related to asymptotic reliability analysis. In particular, attention is paid to the works presenting upper and lower bounds on the reliability function.

A considerable part of this review is based on sources originally published in Russian.

We develop the mathematical machinery in this paper to construct a very general class of Markovian network queueing models. Each node has a heterogeneous class of customers arriving at their own Poisson rate, ultimately to receive their own exponential service requirements. We add to this a very general type of service discipline as well as class (node) switching. These modifications allow us to model in the limit, service with a general distribution. As special cases for this model, we have the product-form networks formulated by Kelly, as well as networks with priority scheduling. For the former, we give an algebraic proof of Kelly's results for product-form networks. This is an approach that motivates the form of the solution, and justifies the various needs of local and partial balance conditions.

For any network that belongs to this general model, we use the operator representation to prove stochastic dominance results. In this way, we can take the transient behavior for very complicated networks and bound its joint queue-length distribution by that for M/M/1queues.

The M/G/1 processor-sharing queue is studied by way of an approximating sequence of models featuring a round-robin discipline and operating in discrete time. In particular, residence-time distributions of jobs are derived.

Russell (1983) considers the distribution properties of N(a), the minimum value of n for which the sum Sn of n random variables, which are i.i.d. uniformly on (0, 1), exceeds the constant a, .

Cycle-time distribution is shown to take the form of a linear combination of M Erlang-N density functions in a cyclic queueing network of M servers and N customers. For paths of m servers in tree-like networks, the components in the more complex linear combination are convolutions of Erlang-N with at most m − 1 negative exponentials.