Broadcast disks with polynomial cost functions

Abstract

In broadcast disk systems, information is broadcast in a shared medium. When a client needs an item from the disk, it waits until that item is broadcast. The fundamental algorithmic problem for such systems is to determine the broadcast schedule based on the demand probability of items, and the cost incurred to the system by clients waiting. The goal is to minimize the mean access cost of a random client. Typically, it was assumed that the access cost is proportional to the waiting time. In this paper, we ask what are the best broadcast schedules for access costs which are arbitrary polynomials in the waiting time. These may serve as reasonable representations of reality in many cases, where the patience of a client is not necessarily proportional to its waiting time. We present an asymptotically optimal algorithm for a fluid model, where the bandwidth may be divided to allow for fractional concurrent broadcasting. This algorithm, besides being justified in its own right, also serves as a lower bound against which we test known discrete algorithms. We show that the Greedy algorithm has the best performance in most cases. Then we show that the performance of other algorithms deteriorate exponentially with the degree of the cost polynomial and approach the fractional solution for sub-linear cost. Finally, we study the quality of approximating the greedy schedule by a finite schedule.