Poly-symmetry in processor-sharing systems

Abstract

We consider a system of processor-sharing queues with state-dependent service rates. These are allocated according to balanced fairness within a polymatroid capacity set. Balanced fairness is known to be both insensitive and Pareto-efficient in such systems, which ensures that the performance metrics, when computable, will provide robust insights into the real performance of the system considered. We first show that these performance metrics can be evaluated with a complexity that is polynomial in the system size if the system is partitioned into a finite number of parts, so that queues are exchangeable within each part and asymmetric across different parts. This in turn allows us to derive stochastic bounds for a larger class of systems which satisfy less restrictive symmetry assumptions. These results are applied to practical examples of tree data networks, such as backhaul networks of Internet service providers, and computer clusters.

Stochastic non-bipartite matching models and order-independent loss queues

“…Sketch of proof. The proof consists of showing that (3) is satisfied if and only if the arrival rates satisfy (8). The key idea is to observe that the denominators in (3) only depend on the set of classes in the sequence ðc 1 , :::, c p Þ: Guided by this observation, we can define a new partition of the state space C, that is a refinement of the partition ðC I , I 2 I 0 Þ, such that the restriction of the sum in (3) to each part of this new partition can be written as a finite product of geometric series that are convergent if and only if the conditions (8) are satisfied.…”

“…The only difference is that the set of customer classes that can be in the queue at the same time is an independent set, so that the stability condition is also restricted to independent sets. The stability condition (8) means that, for each independent set I, there are enough items that can be matched with the classes in I to prevent items of these classes from building up in the buffer. As proved in Ref.…”

“…Therefore, we can first compute wðIÞ for each I 2 I by applying recursion (10) with the base case wð;Þ ¼ 1, and then derive pð;Þ by normalization, that is, pð;Þ ¼ ð P I2I 0 wðIÞÞ À1 : Since there is a finite number of independent sets, this provides us with a closed-form expression for the normalization constant. Note that the denominator in (10) is always positive thanks to the stability condition (8).…”

“…Consequently, both formulas can be applied to make calculations only when there are few classes, when symmetries between classes can be exploited to reduce complexity (see for instance Ref. [8]), or when the graph structure guarantees that the number of independent sets is small. The same remark holds for the formulas derived in Propositions 6 to 10 below.…”

Performance of Balanced Fairness in Resource Pools

“…This is why existing results consider either small systems (e.g., 3 servers) or symmetric systems (e.g., all servers have the same service rate and each job can be processed by k servers chosen uniformly at random) [10,9,8]. The notion of poly-symmetry has recently been introduced to enlarge the class of tractable models but it still relies on some specific (poly-)symmetric properties of the system [4].…”