Near optimal control of queueing networks over a finite time horizon

Abstract

We propose a method for the control of multi-class queueing networks over a finite time horizon. We approximate the multi-class queueing network by a fluid network and formulate a fluid optimization problem which we solve as a separated continuous linear program. The optimal fluid solution partitions the time horizon to intervals in which constant fluid flow rates are maintained. We then use a policy by which the queueing network tracks the fluid solution. To that end we model the deviations between the queuing and the fluid network in each of the intervals by a multi-class queueing network with some infinite virtual queues. We then keep these deviations stable by an adaptation of a maximum pressure policy. We show that this method is asymptotically optimal when the number of items that is processed and the processing speed increase. We illustrate these results through a simple example of a three stage re-entrant line.

A Push–Pull Network with Infinite Supply of Work

“…In [26] we have proposed an adaptation of the max pressure policy of Dai and Lin to multiclass queueing networks (MCQN) with infinite virtual queues (IVQs). Here the classes are K = K 0 ∪ K ∞ , where k ∈ K 0 are standard queues, Q k (t) ≥ 0, while k ∈ K ∞ are IVQs with infinite supply of work.…”

Control of Continuum Models of Production Systems

“…Near optimal control of queueing networks over a finite time horizon has been achieved by [10] for a fluid model (not re-entrant) using continuous linear programming.…”

“…We define a cost functional over a fixed time period from (10) where the outflux is generated by our usual PDE model (2a). The cost functional (10) penalizes overproduction and underproduction equally.…”

Positive Harris Recurrence and Diffusion Scale Analysis of a Push Pull Queueing Network

“…The push pull network was introduced by Kopzon et al [19,20] who assumed exponential processing times. Infinite supply of work and infinite virtual queues are discussed in [1,2,14,15,30,34]. A brief survey of these results is in Chapter 2 of [28].…”

“…We can choose 0 ≤ θ ≤ 1, and specify θ 1,1 = θ 1,2 = θ, θ 2,1 = θ 2,2 = 1 − θ and use ν 1 = µ 1 θ as nominal rate for type 1 and ν 2 = µ 2 (1 − θ) as nominal rate for type 2. As shown in [30], we can use an adaptation of the maximum pressure policy of Dai and Lin [9] to serve jobs of types 1 and 2 at these rates, under full utilization. However, the network will become congested, with expected O( √ T ) jobs in the network at time T .…”