Conservative mutations (those which do not change the amino acid coded for a codon) are discussed. In particular, formulas are given for the computation of the probability that a mutation is conservative and the expected value of this probability if the genetic code were randomly permuted, assuming wobble and non-wobble conditions.

A predominance of certain ‘sequence systemmetries’ in DNA/RNA has led to various conjectures about the possible structural/functional role these symmetries might play in nucleic acid sequences. De Wachter employed a binomial probability model to compare the observed number of ‘direct repeats’ with those expected in a random sequence. Counting of direct repeats essentially leads to a sequence of m-dependent trials. We develop a stochastic model for studying various types of symmetries. Expressions for means and variances of the statistics employed are derived. The asymptotic distributions are obtained using the central limit theorem for m-dependent random variables. It is proposed that each sequence pattern be examined separately for its chance occurrence as opposed to what de Wachter suggests, i.e., clumping of all patterns together. It is also shown how our model can be used to detect various gene-amplification events, if any, in nucleic acid sequences. Finally, for certain types of patterns, it is indicated how the theory of recurrent events can be used to get a better handle on the analysis of direct repeats.

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when

The critical time tc when diverges corresponds to a critical branching process, while post-critical times t> tc correspond to supercritical branching processes.

Motivated by the problem of solid catalytic particle attrition during chemical reactions we formulate a continuous-time Markov model to describe the shattering of a particle when it is assumed that particles can be classified into a small number of types by size. We then obtain a recursive expression for the joint probability generating function of the count of the different types of particle at time t and derive a system of differential equations for the mean number of particle counts and a system of matrix differential equations for the covariance matrix of the particle counts. Solutions to these differential equations are presented in an important special case.

Let be a double sequence of random variables such that there exists a ‘dual' sequence satisfying , and that can be expressed as a sum of independent random variables. If (suitably centered and rescaled) is approximately normally distributed as k and N → ∞ in some fashion, we can use this fact to obtain a remainder-term estimate for the asymptotic normality of as n and N → ∞ in some prescribed manner. The result in the general theorem is used in two specific situations: (i) classical occupancy where the balls can fall through the boxes, (ii) a capture-recapture problem where tagging affects catchability.

From one point of view this paper adds to a previous formulation of the best-choice problem (Petruccelli (1981)) the possibility that the number of available observations, rather than being known, is a bounded random variable N with known distribution. From another perspective, it expands the formulations of Presman and Sonin (1972) and Rasmussen and Robbins (1975) to include recall and uncertainty of selection of observations. The behaviour of optimal stopping rules is examined under various assumptions on the general model. For optimal stopping rules and their probabilities of best choice relations are obtained between the bounded and unbounded N cases. Two particular classes of stopping rules which generalize the s(r) rules of Rasmussen and Robbins (1975) are considered in detail.

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.