Revisiting the optimality of the cμ-rule with Stochastic Flow Models

Abstract

We revisit, in the context of Stochastic Flow Models (SFMs), a classic scheduling problem for optimally allocating a resource to multiple competing users. For the two-user case, we establish the optimality of the well-known cµ-rule for arbitrary stochastic processes using calculus of variations arguments as well as an Infinitesimal Perturbation Analysis (IPA) approach. The latter allows us to derive an explicit sensitivity estimate of the cost function with respect to a controllable parameter and to further study the problem when the cost function is nonlinear, deriving simple distribution-free cost sensitivity estimates and analyzing why the cµ-rule may fail in this case.

Optimal scheduling of parallel queues using stochastic flow models

“…This result applies to any sample path, leading to the conclusion that θ * = 1 and the cμ-rule is therefore optimal. In order to extend this result to an arbitrary number of queues, we use the setting proposed in the previous section, slightly different from the one in Kebarighotbi and Cassandras (2009), which allows us to re-derive the IPA derivative and the optimality of the cμ-rule in a simpler way. Our result will then be used in the following section as a base step for an inductive argument generalizing the optimality of the cμ-rule to any N > 2 queues while also providing explicit IPA derivatives as performance gradient estimators in other problems where the cμ-rule is no longer optimal.…”

“…This was analyzed by Kebarighotbi and Cassandras (2009) where the sample performance function Q(θ) was expressed in terms of a single parameter θ, defined as the fraction of resource capacity allocated to queue 1. An explicit IPA derivative dQ/dθ was derived and it was shown that if c 1 μ 1 > c 2 μ 2 then dQ/dθ < 0.…”

“…These properties, including the unbiasedness of the estimators, were established in Cassandras et al (2002) for queueing systems with finite capacity and extended to serial networks , systems with feedback control mechanisms (Yu and Cassandras 2006), and some multi-class models Cassandras et al 2003;Panayiotou 2004). This paper starts by extending the results in Kebarighotbi and Cassandras (2009) to establish the optimality of the cμ-rule to N > 2 queues (which has proved to be a somewhat surprising challenge.) Its contributions include the following.…”

“…In Section 3, using IPA we first prove the optimality of the cμ-rule for two queues using a different approach than that of Kebarighotbi and Cassandras (2009) which subsequently allows the generalization to N > 2 queues by using this result as a base case for an induction argument. In Section 4 we provide the general IPA scheme for estimating cost gradients.…”

“…In Kebarighotbi and Cassandras (2009), we studied the basic stochastic scheduling problem above using a Stochastic Fluid (or Flow) Model (SFM). Using this model, it was shown that the cμ-rule is optimal in the two-queue case, extending previous results in the literature to a broader class of stochastic processes and without any heavy traffic conditions.…”