State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy

Abstract

This reprint differs from the original in pagination and typographic detail. 1 2 KANG, KELLY, LEE AND WILLIAMS approximation for the flow count process. In this case, the workload diffusion behaves like Brownian motion in the interior of a polyhedral cone and is confined to the cone by reflection at the boundary, where the direction of reflection is constant on any given boundary face. When all of the weights are equal (proportional fair sharing), this diffusion has a product form invariant measure. If the latter is integrable, it yields the unique stationary distribution for the diffusion which has a strikingly simple interpretation in terms of independent dual random variables, one for each of the resources of the network.We are able to extend this product form result to the case where document sizes are distributed as finite mixtures of exponentials and to models that include multi-path routing. We indicate some difficulties related to extending the diffusion approximation result to values of α = 1. We illustrate our approximation results for a few simple networks. In particular, for a two-resource linear network, the diffusion lives in a wedge that is a strict subset of the positive quadrant. This geometrically illustrates the entrainment of resources, whereby congestion at one resource may prevent another resource from working at full capacity. For a four-resource network with multi-path routing, the product form result under proportional fair sharing is expressed in terms of independent dual random variables, one for each of a set of generalized cut constraints.

Minimality of EDF networks with resource sharing

“…The literature on this topic is large. Here we mention only Gromoll and Williams (2009), Kang et al (2009), Vlasiou et al (2015), as representative examples of related asymptotic results. Stability (or the lack thereof) of the SRPT protocol in resource sharing networks was investigated by Verloop et al (2005).…”

Diffusion Approximation for an Input-queued Switch Operating under a Maximum Weight Matching Policy

“…The first property follows from Corollary II.3.1 and Corollary I.3.4 of Dantzig et al [6]. The second property follows by the same argument as in the proof of the second property in Proposition 4.1 of [12].…”

“…This is similar to an assembly or joining operation familiar in manufacturing systems. Another type of stochastic processing network that involves simultaneous actions arises from bandwidth sharing models considered recently by several authors, see e.g., [12,17,23]. However, in these models, the activities simultaneously use multiple resources or servers to process jobs; in other words, the simultaneous aspect is at the opposite end of the activity than in the case of an input-queued switch.…”

Limit Theorems for Markovian Bandwidth-Sharing Networks with Rate Constraints

“…This would imply the validity of several constraint qualifications, in particular, the Fromovitz-Mangasarian constraint qualification, see pages 71-72 of Bonnans and Shapiro (2000). Such extensions are relevant to cover more general network models involving multipath routing, see §5.5 of Kang et al (2009).…”

“…Properties of overloaded bandwidth-sharing networks have subsequently been derived by Borst et al (2014) and Egorova et al (2007). A diffusion approximation for bandwidth-sharing networks was derived in Kang et al (2009). Yao (2008, 2010) considered diffusion approximations of some bandwidth-sharing networks where the service discipline per class is FIFO rather than PS, which coincides with regular bandwidth-sharing networks in the case of exponential flow sizes.…”