Interchange arguments in stochastic scheduling

Abstract

Interchange arguments are applied to establish the optimality of priority list policies in three problems. First, we prove that in a multiclass tandem of two ·/M/1 queues it is always optimal in the second node to serve according to the cµ rule. The result holds more generally if the first node is replaced by a multiclass network consisting of ·/M/1 queues with Bernoulli routing. Next, for scheduling a single server in a multiclass node with feedback, a simplified proof of Klimov's result is given. From it follows the optimality of the index rule among idling policies for general service time distributions, and among pre-emptive policies when the service time distributions are exponential. Lastly, we consider the problem of minimizing the blocking in a communication link with lossy channels and exponential holding times.

A survey of Markov decision models for control of networks of queues

“…Whittle [71] shows that the optimality of this index rule can be derived from the general theory of Gittins indices (Gittins [27]). Other references on the optimality of index rules for preemptive as well as non-preemptive scheduling of a single-server facility include Whittle [72,73], Baras et al [3], Baras et al [4], Varaiya et al [61], Walrand [65], Nain [50], Nain et al [51], and Liu and Nain [47].…”

Stochastic scheduling of parallel queues with set-up costs

“…The great majority of these models, however, have considered the case with zero switchover costs. It is well known, for example, that for an M/G/1 queue with multiple customer classes, if jobs of type i are charged holding costs at rate ci and are processes at rate #i, the c# rule minimizes the average holding cost per unit time (see Baras et al [1], Buyukkoc et al [3], Cox and Smith [4], Gittins [7], Nain [21], Nain et al [22], and Walrand [30]). Other stochastic scheduling problems in the literature for which there are no costs for switching from one type of job to another may be found in Baras et al [1], Dempster et al [5], Gittins [7], Harrison [10,11], Klimov [13,14], Lai and Ying [16], Nain [21], Nain et al [22], Varaiya et al [29], and Walrand [30].…”

The µc-rule is not optimal in the second node of the tandem queue: a counterexample

“…The relevant passages are [1], p. 183, line 4 of the proof of Theorem 4.1: 'node 1 has the same behavior under both policies', and [2], p. 818, line 4: 'in the above proof, policies 1'C and ft result in identical arrivals for node 2'. Therefore we consider two cases.…”

“…In case 2 this idle time is equal in distribution but independent of this service time. Before we proceed to this case, we remark that the definition of ft on p. 817 of [2] generates in our example a definitely different arrival process in node 2.…”

“…The proof in Section 2 of Nain et al [2] and Section 4 of Nain [1] of the optimality of the JlC-rule in the second node uses an interchange argument for the order of service in the second node. They compare the optimal policy, say R*, with a policy, say R o , which is constructed such that the service process in the first node generates the same arrival process for node 2 as policy R*.…”