Double-stranded DNA molecules can be damaged by enzymic action or radiation, in a manner which creates randomly-located single-stranded breaks (nicks). The accumulation of these leads eventually to the double-stranded breakage of the molecule, because two opposite-strand nicks within a critical distance of each other establish conditions for breakage. We study the random variable N, defined as the number of nicks needed for double-stranded breakage to occur. We develop an asymptotic theory which is needed for practical computations.

In this paper a necessary and sufficient condition for the dispersive ordering in dilation sense is given by a convex function which is called the dispersive function and characterizes the distribution function. Some interesting properties of the ordering follow from this result.

Direct and simple proofs are given of the equivalence of partial balance over a set of states and the fact that the corresponding equilibrium distribution is insensitive to nominal sojourn time in that set and independent of its past and remaining duration. By a counter-example it is shown that solely insensitivity is not sufficient for partial balance.

We consider a consecutive-k-out-of-n system situated in a random environment where shocks occur and cause components to fail with certain probabilities. Under certain reasonable assumptions we compute the moment generating function of the time of failure of the system and the distribution of the number of shocks required to cause failure of the system.

Consider two exponential single-server queues in tandem and suppose that service rates of customer n are λ n and μ n respectively. In this note, a simple and direct proof is given of the fact that the departure process from the tandem queue is statistically unaffected when the service rates are interchanged if λ n – μn is independent of n. The proof is based only on the memoryless property of exponential distributions.

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.

These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

Time-reversal arguments can be used to re-derive (and slightly generalize) previous results for sojourn-time distributions in product-form queuing networks.