Departures from Many Queues in Series

We consider a series of n single-server queues, each with unlimited waiting space and the first-in first-out service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time D(k, n) required for all k customers to complete service from all n queues. In particular, we investigate the limiting behavior of D(k, n) as n → ∞ and/or k → ∞. There is a duality implying that D(k, n) is distributed the same as D(n , k) so that results for large n are equivalent to results for large k. A previous heavy-traffic limit theorem implies that D(k, n) satisfies an invariance principle as n → ∞, converging after normalization to a functional of k-dimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of D(k n , n) where k n → ∞ as n → ∞. The case of k n = xn corresponds to a hydrodynamic limit. AMS 1980 subject classifications. primary 60K25, 60F17; secondary 90B22, 60J60, 60F15.