Fluid Limits for Bandwidth-Sharing Networks with Rate Constraints
Mathematics of Operations Research
Bandwidth-sharing networks as introduced by Roberts and Massoulié [Roberts JW, Massoulié L (1998) Bandwidth sharing and admission control for elastic traffic. Proc. ITC Specialist Seminar, Yokohama, Japan], Massoulié and Roberts [Massoulié L, Roberts JW (1999) Bandwidth sharing: Objectives and algorithms. Proc. IEEE Infocom. (Books in Statistics, New York), 1395-1403] model the dynamic interaction among an evolving population of elastic flows competing for several links. With policies based on
... ptimization procedures, such models are of interest both from a queueing theory and operations research perspective. In the present paper, we focus on bandwidth-sharing networks with capacities and arrival rates of a large order of magnitude compared to transfer rates of individual flows. This regime is standard in practice. In particular, we extend previous work by Reed and Zwart [Reed J, Zwart B (2010) Limit theorems for bandwidth-sharing networks with rate constraints. Revised, preprint http://people.stern.nyu.edu/jreed/Papers/BARevised.pdf] on fluid approximations for such networks: we allow interarrival times, flow sizes, and patient times (i.e., abandonment times measured from the arrival epochs) to be generally distributed, rather than exponentially distributed. We also develop polynomial-time computable fixed-point approximations for stationary distributions of bandwidth-sharing networks, and suggest new techniques for deriving these types of results. Downloaded from informs.org by [22.214.171.124] on 17 September 2014, at 02:53 . For personal use only, all rights reserved. Remerova, Reed, and Zwart: Bandwidth Sharing with Rate Constraints Mathematics of Operations Research 39(3), pp. 746-774, © 2014 INFORMS 747 grow large. This large capacity scaling reflects the fact that overall network capacity and individual user rate constraints may be of different orders of magnitude. For example, it is common that Internet providers set download speed limitations for individual users that are typically measured in megabits per second, whereas network capacities are measured in gigabits or terabits per second. The framework of Reed and Zwart  is rather comprehensive. In particular, it allows abandonments of flows: each flow knows how long it can stay in the system and abandons as soon as its service is finished or its patience time expires, whichever happens earlier. The present paper builds upon Reed and Zwart  by relaxing its stochastic assumptions: we assume general distribution for interarrival times and general joint distribution for the size and patience time of a flow (in particular, the flow size and patience time are allowed to be dependent), whereas Reed and Zwart  assume a Markovian setting with independent arrivals, flow sizes, and patience times. We study the behavior of bandwidth-sharing networks in terms of measure-valued processes that are called state descriptors and that keep track of residual flow sizes and residual patience times. The first main result of the paper is a fluid limit theorem (it generalizes the fluid limit result of Reed and Zwart  to non-Markovian stochastic assumptions). We propose a fluid model, or a formal deterministic approximation of the stochastic bandwidth-sharing model, and show that the scaled state descriptors are tight with all weak limit points a.s. solving the fluid model equation. We provide a sufficient condition for the fluid model to have a unique solution, which converts tightness of the scaled state descriptors into convergence to this fluid model solution (FMS). In the sense of techniques used in the proofs, this part of the paper is closely related to previous work on bandwidth sharing (Gromoll and Williams ), processor sharing with impatience (Gromoll et al. ), and bandwidth sharing in overload (Borst et al. , Egorova et al. ). The rate constraints play a crucial role in adopting these techniques. For example, the proof of convergence to fluid model solutions in Gromoll et al.  requires an additional assumption of overload to eliminate problems at zero. However, in our case, because of the rate constraints, the network never empties, and the load conditions become irrelevant. Our second main result, which is a new type of result for bandwidth-sharing networks, is convergence of the scaled network stationary distribution to the fixed point of the fluid model, provided the fixed point is unique. There is a similar result by Kang and Ramanan  for a call center model, but the techniques of Kang and Ramanan  are different than ours. Applying the approach of Borst et al.  , we prove that in many cases the fixed point can be found by solving an optimization problem with a strictly concave objective function and a polyhedral constraint set, and thus is unique and computable in polynomial time. We also construct an example with multiple fixed points, which is a feature that is distinctive from earlier cited works. Besides proving new results for the particular model of bandwidth sharing, we also suggest new ideas and believe that they can be adjusted to other models, too. In particular, we derive equations for asymptotic bounds for fluid model solutions (see Theorem 3) that can be solved for a wide class of networks, and then asymptotic stability of the fixed point can be shown. Another interesting idea is that, in the stationary regime, the properties of a network depend on newly arriving flows only, since all initial flows are gone after some point (see Lemma 2). Throughout this part of the paper, we assume Poisson arrivals, since that guarantees existence of a unique stationary distribution. Poisson arrivals also imply M/G/ bounds that are exploited heavily in the proofs. The structure of the paper is as follows. Section 2 describes the stochastic bandwidth-sharing model, and §3 introduces its deterministic analogue, the fluid model. Also §3 states sufficient conditions for a fluid model solution to be unique, and for a fixed fluid model solution to be unique and asymptotically stable. Sections 4 and 5 discuss convergence of the scaled state descriptor and its stationary distribution to the fluid model and its fixed point, respectively. Sections 6, 7, and 8 contain the proofs of the statements from § §3, 4, and 5. The appendix proves auxiliary results. In the remainder of this section, we list the notation we use throughout the paper. Notation. To introduce the notation, we use the signs = and =. The standard sets are denoted as follows: the reals = − , the nonnegative reals + = 0 , the positive reals 0 , the nonnegative integers + = 0 1 2 , and the natural numbers = 1 2 . The signs ∧ and ∨ stand for minimum and maximum, respectively. For x ∈ , x + = x ∨ 0. The signs lim and lim denote the lower and upper limits of a sequence of numbers. The coordinates of a vector from a set S I are denoted by the same symbol as the vector with lower indices 1 I added. If a vector has a superscript, tilde sign, or overlining, they remain in its coordinates. For examplex 0 ∈ S I ,x 0 = x 0 1 x 0 I . The space I is endowed with the supremum norm x = max 1≤i≤I x i . Vector inequalities hold coordinate-wise. The coordinate-wise product of vectors of the same dimensionality I is x * y = x 1 y 1 x I y I . Downloaded from informs.org by [126.96.36.199] on 17 September 2014, at 02:53 . For personal use only, all rights reserved.