Limit theorems for workload input models

General models for workload input into service systems are considered. Scaling limit theorems appropriate for the formulation of fluid and heavy traffic approximations for systems driven by these inputs are given. Under appropriate assumptions, it is shown that fractional Brownian motion can be obtained as the limiting workload input process. Motivation for these results comes from data on communication network traffic exhibiting scaling properties similar to those for fractional Brownian motion. 1 Discrete source models. We consider models for the input of work into a system from a large number of sources. Each source “turns on” at a random time and inputs work into the system for some period of time. Associated with each active period of a source is a cumulative input process, that is, a nondecreasing stochastic process X such that X(t) is the cumulative work input into the system during the first t units of time during the active period. One representation of the process is as follows. Let N(t) denote the number of source activations up to time t, and for the ith activation, let Xi(s) denote the cumulative work input into the system during the first s units of time that the source is on. We will refer to N as the source activation process. The total work input into the system up to time t is then given by

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