Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity λ. Let Pk denote the probability that the first device will survive the first k shocks and let denote such a probability for the second device. Let and denote the survival functions of the first and the second device respectively. In this note we show that some partial orderings, namely likelihood ratio ordering, failure rate ordering, stochastic ordering, variable ordering and mean residual-life ordering between the shock survival probabilities and are preserved by the corresponding survival functions and .

In a recent paper by Shanthikumar and Sumita (1987), it is conjectured that the ergodic sojourn time of a customer in G/IFR/1 queues is minimized by FIFO (first in, first out) discipline in the sense of increasing and convex ordering. This paper shows that their conjecture is true. In fact, FIFO discipline minimizes for any non-decreasing and convex function f, where N is the (constant) number of arrivals, θ (k) is the customer identity of the kth departing customer, and an and τ n denote the arriving and departing times of the nth customer, respectively.

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

A system is discussed which is subject to a conditional Poisson stream of failures, whose intensity function representing the cumulative damage level is a Markov chain.

The IFR property of the stochastic process governing a one-component system supported by an inactive standby and a repair facility when the lifetime of one component and the repair time of the other component are dependent, is established. We solve the problem of selecting repair rates to maximize the steady-state availability for given component failure rates when a lower bound for the MTBF and upper bounds for the steady-state expected number of repairs of the components per unit time and expected number of failures of the system per unit time are given.

In a given set, the elements are compared pairwise. The number W of similar pairs is studied, that is, the number of pairs with a certain property in common. Under certain conditions, W has, approximately, a Poisson distribution. Examples are considered connected with the birthday problem and with a circle problem involving DNA breakages.

Let dx(t) = y(t) dt, where y(t) is a one-dimensional Wiener process. In this note, we obtain a formula for the moment-generating function of y(T), where T is the 1/2-winding time about the origin of the integrated Wiener process x(t).

We show for the three-node Jackson network studied in [3] that a customer’s sojourn times in nodes 1 and 3 are positively correlated. We actually prove a stronger result, that the two sojourn times are associated random variables. Our proof uses a stochastic ordering argument similar to that in [4].

Reich (1957) proved that the sojourn times in two tandem queues are independent when the first queue is M/M /1 and the second has exponential service times. When service times in the first queue are not exponential, it has been generally expected that the sojourn times are not independent. A proof for the case of deterministic service times in the first queue is offered here.

In this work we investigate under what circumstances the equilibrium distribution of the numbers of groups of various sizes in a certain M/G/k group-arrival group-departure loss system can be obtained in a closed product form.

A system is repaired on failure. With probability p, it is returned to the ‘good as new' state (perfect repair) and with probability 1 – p, it is returned to the functioning state, but is only as good as a system of age equal to its age at failure (imperfect repair). In this article, we develop replacement policies for a deteriorating system with imperfect maintenance. The successive survival times and consecutive repair times form a geometric process which is stochastically non-increasing or non-decreasing respectively. Explicit expressions are obtained for the long-run expected cost under two kinds of replacement policies based on the working age of the system and the number of imperfect repairs before a replacement.

We consider the same circuit switching problem as in Mitra [1]. The calculation of the blocking probabilities is reduced to finding the partition function for a closed exponential pseudo-network with L−1 customers. This pseudo-network differs from that in [1] in one respect only: service rates at nodes 1, 2, …, p depend on the queue length. The asymptotic expansion developed in [1] follows from our exact expression for the partition function.