Abstract.
We consider the problem of scheduling a set of pages on a single broadcast channel using time-multiplexing. In a perfectly periodic schedule, time is divided into equal size slots, and each page is transmitted in a time slot precisely every fixed interval of time (the period of the page). We study the case in which each page i has a given demand probability \(w_{i}\), and the goal is to design a perfectly periodic schedule that minimizes the average time a random client waits until its page is transmitted. We seek approximate polynomial solutions. Approximation bounds are obtained by comparing the costs of a solution provided by an algorithm and a solution to a relaxed (non-integral) version of the problem. A key quantity in our methodology is a fraction we denote by \(a_1\), that depends on the maximum demand probability: \(a_1 {\stackrel{\rm def}{=}} \sqrt{\max_i\{w_{i}\}}/ \sum\sqrt{w_i}\). The best known polynomial algorithm to date guarantees an approximation of \(\frac{3}{2}+\frac{3}{2}a_1\). In this paper, we develop a tree-based methodology for perfectly periodic scheduling, and using new techniques, we derive algorithms with better bounds. For small \(a_1\) values, our best algorithm guarantees approximation of \(1+\frac{\sqrt[3]{3a_1}}{1-\sqrt[3]{3a_1}}\). On the other hand, we show that the integrality gap between the cost of any perfectly periodic schedule and the cost of the fractional problem is at least \(1+a_1^2\). We also provide algorithms with good performance guarantees for large values of \(a_1\).
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Received: December 2001 / Accepted: September 2002
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Bar-Noy, A., Nisgav, A. & Patt-Shamir, B. Nearly optimal perfectly periodic schedules. Distrib Comput 15, 207–220 (2002). https://doi.org/10.1007/s00446-002-0085-1
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DOI: https://doi.org/10.1007/s00446-002-0085-1