We show that the solution of a quasi-renewal equation with an exponential distribution of the renewals converges at infinity and we compute explicitly the limit, hence generalizing the classical renewal theorem. We apply this result to a stochastic model of DNA replication introduced by Cowan and Chiu (1994).

Technology developed in a predecessor paper (Chen and Seneta (1996)) is applied to provide, in a unified manner, a sharpening of bivariate Bonferroni-type bounds on P(v1≥r, v2≥u) obtained by Galambos and Lee (1992; upper bound) and Chen and Seneta (1986; lower bound).

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

In this paper we consider an irreducible random walk in Z+ defined by X(m+1) = max(0, X(m) + A(m+1)) with E{A} < 0 and for an s ≥ 0 where a+ = max(0,a). Let π be the stationary distribution of X. We show that one can find probability distributions πn supported by {0,n} such that ||πn - π||1 ≤ Cn-s, where the constant C is computable in terms of the moments of A, and also that ||πn - π||1 = o(n-s). Moreover, this upper bound reveals exact for s ≥ 1, in the sense that, for any positive ε, we can find a random walk fulfilling the above assumptions and for which the relation ||πn - π||1 = o(n-s-ε) does not hold. This result is used to derive the exact convergence rate of the time stationary distribution of an M/GI/1/n queueing system to the time stationary distribution of the corresponding M/GI/1 queueing system when n tends to infinity.

The use of bias optimality to distinguish among gain optimal policies was recently studied by Haviv and Puterman [1] and extended in Lewis et al. [2]. In [1], upon arrival to an M/M/1 queue, customers offer the gatekeeper a reward R. If accepted, the gatekeeper immediately receives the reward, but is charged a holding cost, c(s), depending on the number of customers in the system. The gatekeeper, whose objective is to ‘maximize’ rewards, must decide whether to admit the customer. If the customer is accepted, the customer joins the queue and awaits service. Haviv and Puterman [1] showed there can be only two Markovian, stationary, deterministic gain optimal policies and that only the policy which uses the larger control limit is bias optimal. This showed the usefulness of bias optimality to distinguish between gain optimal policies. In the same paper, they conjectured that if the gatekeeper receives the reward upon completion of a job instead of upon entry, the bias optimal policy will be the lower control limit. This note confirms that conjecture.

In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.