Optimal control of fluid limits of queuing networks and stochasticity corrections. (English) Zbl 0894.60095

Avram, Florin

Yin, G. George (ed.) et al., Mathematics of stochastic manufacturing systems. AMS-SIAM summer seminar in applied mathematics. June 17–22, 1996. Williamsburg, VA, USA. Providence, RI: AMS, American Mathematical Society. Lect. Appl. Math. 33, 1-36 (1997).

Summary: On one hand we present some examples of fluid networks control which illustrate the general method for optimally emptying networks with linear holding costs proposed by F. Avram, D. Bertsimas and M. Ricard [in: Stochastic networks. IMA Vol. Math. Appl. 71, 199-234 (1995; Zbl 0837.60083)]. On the other hand we attempt to study the changes in the optimal policies for fluid models caused by stochasticity effects. In the case of the usual expected total emptying cost objective, we found out numerically in the particular case of the tandem that the optimal switch curve is very close to being an upward shift of the optimal fluid switch line. We give a conjecture for the asymptotical value of this shift, obtained by the perturbation techniques promoted by C. Knessl, B. J. Matkowsky, Z. Schuss and C. Tier [IEEE Trans. Commun. COM-34, 1170-1175 (1986; Zbl 0612.60086)].
We turn then to another objective which is much easier to work with: the so called “totally risk averse” objective. This is a limiting approach promoted by Whittle and Fleming which replaces stochastic control problems by deterministic differential games. For a tandem with “Gaussian stochasticity”, this method yields a quadratic switch curve depending on some “worry” parameters in the objective. The switch curve approaches the line obtained for the fluid model when the “worry” parameters tend to 0. We also indicate how the method proposed in the first quoted paper may be used to develop an algorithm for finding explicit solutions for the “totally risk averse” optimal scheduling of general Gaussian networks.
For the entire collection see [Zbl 0869.00058].

MSC:

60K25	Queueing theory (aspects of probability theory)
93E20	Optimal stochastic control
90B15	Stochastic network models in operations research

Keywords:

totally risk averse objective; expected total emptying cost; optimal switch curve; quadratic switch curve; optimal scheduling

Citations:

Zbl 0837.60083; Zbl 0612.60086