Storage-Limited Queues in Heavy Traffic

Abstract

This paper models primary computer storage in the context of a general (GI/GI/l) queueing system. Queued items are described by sizes, or storage requirements, as well as by arrival and service times; the sum of the sizes of the items in the system is the occupied storage. Capacity constraints are represented by two different protocols for determining whether an arriving item is admitted to the system: (1) an item is accepted if and only if at its arrival time the currently occupied storage does not exceed a given constantC> 0, and (2) an item is accepted if and only if at its arrival time the occupied storage is at mostC, and the occupied storage plus the item's size is at mostC(l + ε) for some given ε > 0. We prove for both systems that in heavy traffic the occupied storage, suitably normalized, converges weakly to reflected Brownian motion with boundaries at 0 and atC. A distinctive feature of the proof is the characterization of reflected Brownian motion as a limit of unrestricted penalized processes.These results make more plausible an earlier conjecture of the authors, i.e., that one obtains the same heavy traffic limit when the admission rule is: accept an item if and only if at its arrival time the occupied storage plus the item's size is no greater thanC.