# Competition-Free Instances, The Fraenkel's Conjecture, and Optimal Admission Sequences

Part of the International Series in Operations Research & Management Science book series (ISOR, volume 127)
A number of applications, we have seen some of them already in Chap. 3, deal with sequences over a finite alphabet where each letter of the alphabet is required to occur with a pre-specified rate r. The sequences are modeled generally as infinite however for the rational rates the sequences become cyclic and then we can limit ourselves to studying finite cycles. The sequence projection on a particular letter results in an isomorphic zero-one valued sequence with the ones in the positions occupied by the letter in the original sequence and zeros elsewhere. Thus, for each letter we can consider the zero-one valued sequences and search for an optimal one for a given letter regardless of all other letters. It turns out that the objective functions for a single letter are often minimized by sequences with the letter being in positions defined be the following formula
$$\left[ {{j \over r} - {\theta \over r}} \right]$$
(6.1)

where j=1,2,... for some phaseθ, which may be letter-dependent, and 0 ≥θ < 1. This was the case for the just-in-time sequences minimizing the total and maximum deviation, see Theorem 3.1. There,θ= ½. It also holds for a class of multimodular functions as proven by Hajek [6]. The multimodular functions were first studied by Hajek [6], see Sect. 6.9 for their definition, and later by Altman et al. [7] as discrete counterparts of continuos convex functions. Their most prominent application thus far is to the load balancing problem in queueing networks, where the sequences are designed to implement an admission policy, see Sect. 6.10 for details of this policy. Hajek [6], and Altman et al. [7] prove that the expected queue sizes and more generally expected travel times in queuing networks represented by stochastic event graphs are multimodular functions.

## Keywords

Ideal Position Regular Sequence Regular Word Asymptotic Rate Exact Cover
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