Reducing the Variance of Sojourn Times in Multiclass Queueing Systems

Cover Reducing the Variance of Sojourn Times in Multiclass Queueing Systems
Reducing the Variance of Sojourn Times in Multiclass Queueing Systems
Lawrence M Wein
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42 customers when QafO > Qii^) ^'^^ serve 52 class customers when Qiit) > Q2{t)- For the general case where < c < oo, the optimal dual objective function h{\V) in (2. 25) is given by r(6±H:)2 if iVe [-11. -6]; h{\V)= I if Vi'6 [-6, 0]; (4. 5) [^ ifU'€[0, 3], and thus f{a. B} is given by , , 1 /ca2 63 q3 jgQ2 ;^08a 216n, ^^ /(a, 6) = - (_ + _-_-_ _ _ — . (4. 6) — a \ 2 2t (0 lO to to / Therefore, our optimal interval endpoints can be found by minimizing f{a. B) over a € [-11, -6] and be [0, 3]. Now let us consider the numerical example in the context of a network with con- trollable inputs. If we choose a long run average throughput rate of . 1286 customers per unit of time and choose the scaling parameter n = 100, then p\ = P2 = -9. The solution {Z'[t). -*{t)) to the dual quadratic program (3. 11)-(3. 12) is given in Table I, and the dy- namic reduced costs (3. 31) are given in Table II. In each of these tables, the solution and costs have been broken down into three regions (corresponding to the columns in the two tables), depending upon the value of the workload imbalance process Vr(f).

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