Fluid Models with Burst Arrivals: A Sample Path Analysis

Abstract

Motivated by packetwise transmission of data in telecommunications, we introduce a fluid model with a continuous burst arrival process ψ = {(tn, Sn):n ≥ 0}. Each epoch, tn, begins a burst of length Sn that is, a continuous flow of fluid (work, information, etc.) at rate 1 to a system. The system processes fluid at rate 1. The model more generally can be used to approximate a storage system where fluid arrives over time due to many different (unrelated) sources. We analyze the model using sample path techniques (ψ is assumed deterministic, having arisen as a sample path from some underlying probability space) and by doing so obtain a variety of expressions for such quantities as average work inc system and average work in service as well as the empirical distribution for work in service. The expressions are given in terms of familiar quantities (such as customer delay) from the corresponding classic single- and infinite-channel queues that we construct from the same arrival sample path. In particular, we obtain an interesting decomposition of work in terms of the work in these two wellknown queueing models. In our final remarks, we also point out the optimality features of this model when compared to other models having different rules regulating the flow of burst work. In addition, we also give some stochastic ordering results when comparing two burst systems of the M/GI/1 or GI/M/1 type.