This paper deals with the blood-bank model; namely, an inventory system in which both arrival of items and demand are stochastic and items stored have finite lifetimes. We assume that the arrival and demand processes are independent Poisson processes. We use an analogy with queueing models with impatient customers to obtain some of the important characteristics of the system.

We discuss a method of obtaining invariance relations in complex systems by using the theory of point processes. New formulae are given for obtaining them generally, and in particular in many-stage models such as tandem and network queues. The formulae are shown to be useful by applications to a many-server queue and a tandem queue. Stochastic inequalities in a tandem queue are also discussed using the invariance relations obtained.

This paper is concerned with the assessment of the statistical efficiency of proposed regenerative simulation methods. We compare the efficiency of the ‘marked job' and ‘labelled jobs' methods for estimation of passage times in multiclass networks of queues with general service times. Using central limit theorem arguments, we show that the confidence intervals constructed for the expected value of a general function of the limiting passage time using the labelled jobs method are shorter than those obtained from the marked job method. This is consistent with intuition since the labelled jobs method extracts more passage-time information from a fixed-length simulation run.

In this paper we generalize the minimal repair replacement model introduced by Barlow and Hunter (1960). We assume that there is available information about the underlying condition of the system, for instance through measurements of wear characteristics and damage inflicted on the system. We assume furthermore that the system failure rate and the expected cost of a repair/replacement at any point of time are adapted to this information. At time t = 0 a new system is installed. At a stopping time T, based on the information about the condition of the system, the system is replaced by a new and identical one, and the process is repeated. Failures that occur before replacement are rectified through minimal repair. We assume that a minimal repair changes neither the age of the system nor the information about the condition of the system. The problem is to find a T which minimizes the total expected discounted cost. Under appropriate conditions an optimal T is found. Some generalizations and special cases are given.

In this note we establish an inequality between the maximal coefficient of correlation and the φ -mixing coefficient which is symmetric in its arguments. Motivated by this inequality, we introduce a mixing coefficient which is the product of two φ -mixing coefficients.

We also study an invariance principle under conditions imposed on this new mixing coefficient. As a consequence of this result it follows that the invariance principle holds when either the direct-time process or its time-reversed process is φ -mixing; when both processes are φ-mixing the invariance principle holds for sequences of L2-integrable random variables under a mixing rate weaker than that used by Ibragimov.

We prove a conjecture of Arnold (1968) which simplifies the determination of an optimal bound on absorption probability originally due to Moran (1960).

Consider first-passage percolation on the square lattice. Hammersley and Welsh, who introduced the subject in 1965, conjectured that the expected minimum travel time from (0, 0) to (n, 0) along paths contained in the cylinder is always non-decreasing in n. However, when the bonds have time-coordinate 1 with probability p and 0 with probability 1 – p (0 < P < 1), then, for p sufficiently small, we get a counterexample.

Consider a processor-sharing queue placed in a quasi-reversible network in equilibrium. This note explains why the expected sojourn time of a customer in such a queue is proportional to his service time.

A distribution F is less dispersed than a distribution G if for all .

We generalize a characterization of dispersive ordering of Shaked (1982) concerning sign changes of Fc – G, where Fc is a translate of F. We then use this generalization plus total positivity to develop a simple proof of a characterization of dispersive distributions due to Lewis and Thompson (1981); a distribution H is dispersive if

It is possible to deduce some information about the time constant μ without using Wierman and Reh's (1978) renewal theorem.

An M/G/∞ queue that is cleared whenever the level Q is exceeded plays the role of the outstanding orders process in an inventory system. The stationary version of the process is examined and an interesting property of the variance is illustrated.

In a recent paper (Yang et al. (1982)) a model for proofreading was proposed in which a reader has a fixed probability p of detecting a misprint in a document containing a Poisson number of errors. This note points out that the conclusions derived from such a model can be extremely misleading if the probability of detecting a misprint varies from one misprint to another.

Let X1, X2, · ·· be a stationary sequence of random variables and E1, E2, · ··, EN mutually exclusive events defined on k consecutive X's such that the probabilities of the events have the sum unity. In the sequence Ej1, Ej2, · ·· generated by the X's, the mean waiting time from an event, say Ej1, to a repetition of that event is equal to N (under a mild condition of ergodicity). Applications are given.