An urn contains m types of balls of unequal numbers. Let ni be the number of balls of type i, i = 1, 2, …, m. Balls are drawn with replacement until first duplication. In the case of finite memory of order k, the distribution of Ym,k, the number of drawings required, is discussed. Special cases are obtained.

In this paper, we study a similar replacement model in which the successive survival times of the system form a process with non-increasing means, whereas the consecutive repair times after failure constitute a process with non-decreasing means. The system is replaced at the time of the Nth failure since the installation or last replacement. Based on the long-run average cost per unit time, we determine the optimal replacement policy N∗ and the maximum of the long-run average reward explicitly. Under additional conditions, the policy N∗ is even optimal among all replacement policies.

Two kinds of multidimensional IFR distribution are defined by using a partial order in R n +, which is derived from a non-negative, strictly increasing function in R n +. Some closure properties under operations and an application to a shock model are discussed.

The flexibility of a notion considered by Lindley and Singpurwalla is pointed out. It is shown that their set-up can be generalized by looking at systems whose component life lengths are a priori dependent.

We consider the G/G/1 queue as an operator that maps inter-arrival times to inter-departure times of points, given the service times. For arbitrarily fixed statistics of service times, we are interested in the existence of distributions of inter-arrival times that induce identical distributions on the inter-departure times. In this note we prove, by construction, the existence of one of such distribution.

Stochastic concavity of the output process with respect to buffer sizes is established in a series of ·/M/1/B queues with loss at the first node.

We consider a tandem queueing system with m stages and finite intermediate buffer storage spaces. Each stage has a single server and the service times are independent and exponentially distributed. There is an unlimited supply of customers in front of the first stage. For this system we show that the number of customers departing from each of the m stages during the time interval [0, t] for any t ≧ 0 is strongly stochastically increasing and concave in the buffer storage capacities. Consequently the throughput of this tandem queueing system is an increasing and concave function of the buffer storage capacities. We establish this result using a sample path recursion for the departure processes from the m stages of the tandem queueing system, that may be of independent interest. The concavity of the throughput is used along with the reversibility property of tandem queues to obtain the optimal buffer space allocation that maximizes the throughput for a three-stage tandem queue.

We study the relative errors in reliability measures for the series system, under the wrong assumption of independence incorporating environmental effects, when the Marshall and Olkin model with environmental effects should be used.

It has recently been shown that for the G/G/1 queue, virtual delay and attained waiting time have the same stationary distribution. We present a sample-path derivation of this result.

The problem addressed here is related to the minimization of the total loss probability in series of finite queues at which customers are rejected if the waiting capacity is exceeded. More precisely, one is concerned with the question of determining whether or not there may exist conditions under which an increase of the loss rate at one queue, e.g. at the most upstream one, could result in a decrease of the total loss rate throughout the whole network. The answer obtained in the context of a simple model is negative.

We show that a theory for stopped two-dimensional random walks is well suited to describe cumulative shock models. Limit theorems for the lifetime/failure time of a system are provided.

In this paper we consider a network of ·/GI/1/N queues with finite-length buffers. A suitable Markov process for the time-evolution of this system is defined. This process is subsequently shown to be ergodic under the conditions of Borovkov (1987).

For an initially empty M/M/1 queue, it is shown that the transform of the expectation of the number of departures in the interval (0, t] is invariant under an interchange of arrival and service rates. However, in the GI/G/1 queue with an initial single customer, the corresponding transform does not have this symmetric property.

Singh and Jain (1989) have proved some preservation results for partial orderings of life distributions assuming that shocks occur according to a homogeneous Poisson process. It is shown that their results hold under less restrictive conditions.

Two types of conditions are discussed ensuring the equality between long-run time fractions and long-run event fractions of stochastic processes with embedded point processes. Modifications of this equality statement are considered.

We consider two variants of M/G/1 queues with exhaustive service and multiple vacations; (1) customers cannot wait for their services longer than an interval of length T, and (2) customers cannot stay in the system longer than an interval of length T . We show that the probability distribution functions of the waiting times for the two systems are given in terms of those for the corresponding M/G/1 vacation systems without any residence-time limits.