This paper generalizes previous results for sojourn-time distributions along so-called overtake-free routes in product-form networks of queues.

A sampling procedure involving an urn with red and white balls in it is studied. Initially, the urn contains n balls, r of them being white. At each step, a white ball is removed, and one more ball is selected at random, painted red (if it was white before) and put back into the urn. R. F. Green proposed this scheme in 1980 as a stochastic model of cannibalistic behavior in a biological population, with red balls interpreted as cannibals. Of primary interest is the distribution of Xnr, the terminal number of red balls. A study of R. F. Green and C. A. Robertson led them to conjecture that, for r = 1 and n →∞, Xnr is asymptotically normal with mean ≈ n exp(–1) and variance ≈ n(3 exp(–2) –exp(–1)). In this paper we prove that the conjecture — its natural extension, in fact — is true. Namely, for r/n bounded away from 1, Xnr is shown to be asymptotically normal with mean ≈ n exp(ρ – 1) and variance ≈ n exp[2(ρ – 1)] (ρ2– 3ρ + 3 – exp(l – ρ)); ρ = r/n.

Gilbert and Mosteller (1966) derive the probability of obtaining the best secretary when a fixed number of potential employees of varying quality are interviewed sequentially by an employer. This paper extends these results by deriving the probability of selecting an applicant of arbitrary rank under a variety of circumstances concerning the objective of employers and the amount of information employers have about applicant quality.

The constancy of the conditional expectation of some appropriate functions of order statistics on some others, is used to characterize the geometric distribution among the discrete distributions.

Numerical data are presented concerning the mean and the standard deviation of the waiting-time distribution for multiserver systems with queues in parallel, in which customers choose one of the shortest queues upon arrival. Moreover, a new numerical method is outlined for calculating state probabilities and moments of queue-length distributions. This method is based on power series expansions and recursion. It is applicable to many systems with more than one waiting line.

Previously derived sample path upper bounds for multi-channel work in system and work in queue are shown to be false. A new proof is given for the corresponding stochastic bounds on these quantities.