By Eitan Altman
Opening new instructions in study in either discrete occasion dynamic platforms in addition to in stochastic keep watch over, this quantity makes a speciality of a large category of regulate and of optimization difficulties over sequences of integer numbers. this can be a counterpart of convex optimization within the surroundings of discrete optimization. the idea built is utilized to the keep watch over of stochastic discrete-event dynamic platforms. a few functions are admission, routing, provider allocation and holiday regulate in queuing networks. natural and utilized mathematicians will take pleasure in interpreting the publication because it brings jointly many disciplines in arithmetic: combinatorics, stochastic strategies, stochastic keep an eye on and optimization, discrete occasion dynamic platforms, algebra.
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Extra resources for Discrete-Event Control of Stochastic Networks: Multimodularity and Regularity
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Proof. For each letter a, 1a (G) is periodic with period pa . The period of G is lcm(pa , a ∈ A). In the next lemma, we give a characterization of constant gap sequences that stresses the fact that constant gap is some kind of strong balance. Proposition 4. G is constant gap if and only if, for any two finite words, W and W included in G with ||W | − |W || ≤ 1, then for each letter a, ||W |a − |W |a | ≤ 1. Proof. Let a be a letter in the alphabet. First, assume that G is constant gap. If |W |a − |W |a ≥ 2, then, necessarily, |W | − |W | ≥ 2.
Also note that B(N, α, p) is lower semi-continuous in α and in p. 9), we derive the following lower bounds Lemma 8. Under assumptions < 1 >, < 2 > and < 3 >, m 1 lim fn (a1 , · · · , an ) ≥ inf fN (q, · · · , q), m→∞ m q∈L n=1 where L is the set of all limit points of pα as α ↑ 1. Proof. 22) The Lemma follows since for any given p, by definition of B, B(N, 1, p) = fN (p, p, · · · , p). 3 Optimality of the Bracket Sequences Theorem 6. Under assumptions < 1 >,< 2 > and < 3 >, and given some p ∈ [0, 1], if the functions fk (a1 , · · · , ak ) are increasing in all ai , then the bracket sequence ap (θ) for any θ ∈ [0, 1), minimizes the average cost g(a) over all sequences that satisfy the constraint: 1 lim N N →∞ N an ≥ p.
Assume {(θi , qi ), i = 1 · · · K} is an exact covering sequence and that P = maxi qi . Then P appears at least twice in the set q1 , · · · , qK . Proof. The proof given here is similar to the discussion in [115] on exact def covering sequences. Let w = e2iπ/r for some integer r > 1. By definition, w is a primitive r-th root of one. We have: K (w − x)S(x) = i=1 Let x → w. This yields i:r|qi (w − x)xθi w−x . = q i 1−x 1−x −wθi = 0. 1) Now, take r = P . The set {i : P |qi } is exactly the set {i : P = qi }.