Competition-Free Instances, The Fraenkel's Conjecture, and Optimal Admission Sequences
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 . The multimodular functions were first studied by Hajek , see Sect. 6.9 for their definition, and later by Altman et al.  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 , and Altman et al.  prove that the expected queue sizes and more generally expected travel times in queuing networks represented by stochastic event graphs are multimodular functions.
KeywordsIdeal Position Regular Sequence Regular Word Asymptotic Rate Exact Cover
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