Equilibrium allocation and pricing of variable resources among user-suppliers☆
Introduction
This paper is motivated by two emerging trends in packet switched networks in the past decade. The first is the increasing popularity of resource reservation as a means to guarantee quality of service (QoS), as exemplified by the recent standards on the mechanism (but not the algorithm) of resource reservation, e.g., [25], [33]. However, unlike circuit switched or leased line networks where reservation takes the form of a fixed amount of dedicated bandwidth, in packet switched networks, it may also include a variable component, as in the available bit rate (ABR) service of an ATM network where a user can receive a minimum cell rate (MCR) plus some random extra bandwidth. In response to the first phenomenon is the design of a large number of packet scheduling policies to provide the reserved bandwidth to competing users; some recent examples include, e.g., generalized process sharing [22], a family of fair queueing algorithms [7], [28], and the virtual partitioning policy [18], etc; also see [32] for a survey of several earlier schemes. The common theme across these algorithms is that a user is guaranteed a minimum share of resources and gets random extra amounts depending on network condition. For elastic traffics [27] that can tolerate some degree of delay or loss, buffer is also a scarce resource to be traded-off in network resource allocation, e.g., [5], [6], [15], [23]. Again buffer allocation can be implemented by schemes ranging from complete partitioning, in which all users are guaranteed fixed amounts of buffer, to complete sharing, in which no user is guaranteed any fixed amounts of buffer, e.g., [2].
In this paper we describe a novel model for such bandwidth and buffer allocation schemes and study the equilibrium allocation that would result when users interact under such schemes.
We consider the idealized situation where users can freely choose their shares of fixed and variable bandwidth or buffer to maximize their benefit. The network coordinates their choices through resource pricing. This approach seems more desirable than one in which the network decides the allocation to all users without regard to their individual valuation of resources. A critical issue in such an approach is whether the coordinated interaction of competing users leads to a stable equilibrium and whether the equilibrium has desirable properties, e.g., fairness. In this paper, we provide conditions under which the first question can be answered in the affirmative, and will partially answer the second question by characterizing equilibrium allocations and prices. We present an example to illustrate how this characterization might be exploited to speed up convergence to an equilibrium.
We make two remarks before proceeding further. First the prices in our model may or may not form part of the monetary tariff a user faces. In addition to encouraging efficient sharing of resources, pricing for network services also serves other critical business needs. If congestion pricing interferes too much with other functions of pricing, then the ‘prices’ discussed in this paper should be treated simply as a control signal to guide users’ decision. Second the model considered in this paper is motivated by a possible future scenario of the deregulated telecommunication environment where a large number of resellers may purchase network services (in terms of reserved resources) and retail them to end users. Some of these resellers (referred to as ‘users’ in the sequel) may themselves own resources and can therefore either purchase services from the network or sell services to other resellers, depending for example on the current prices. In [16] we consider the interaction among end users who can only purchase services from the network (see remarks after Eq. (1) in Section 2).
In the next section, we describe a model of allocating variable bandwidth and buffer, summarize and interpret our main results, and comment on related works. In the sections following, we present our assumptions and proofs.
Section snippets
Model
For our purposes a network provides two types of resources, bandwidth and buffer, in two flavors, fixed and variable. A bandwidth allocation is specified by a pair x=(x0,x1)≥0, with the interpretation that a fixed amount x0R0 and a random amount x1R1 of bandwidth will be made available to a user that is granted the allocation x. Similarly a buffer allocation is specified by a pair y=(y0,y1)≥0, with the interpretation that, if granted, a fixed amount y0B0 and a random amount y1B1 of buffer will
Assumptions, notations and preliminaries
We make the following assumptions: Assumption 1 ER1>0,EB1>0,cov(R1,B1)≠0, and that R1 and B1 are linearly independent, i.e., there exists no constant a such that R1=aB1 almost surely. Assumption 2 is concave in its arguments and satisfies Assumption 3 All prices (p,q)>0 are strictly positive.
As discussed in Section 2 by Assumption 2 we assume that user n’s utility increases with the mean allocation and decreases with its variance.
Let be the mean of R1 and B1, respectively. Let σR
Convergence
In this section we will show that an equilibrium can indeed be approached, at least locally. Specifically, we will provide (implicitly) a rationing rule on the resources and a simple price adjustment rule, and show that if every source requests an allocation to maximize its own utility un given the current prices, then the requested allocations and the prices approach an equilibrium in the sense of Definition 1, provided that certain conditions are satisfied.
We model the prices and the
Equilibrium allocations and prices
In this section we present some interesting properties of equilibrium and conclude with an example application of the results.
Conclusion
We have described a model for resource allocation schemes that provide a user with a fixed and a variable share of bandwidth and buffer. When these users interact by exchanging their resource allocations in order to maximize their utility an equilibrium may result at which the marginal increase in utility due to the higher expected return of variable resources is balanced by the marginal decrease in utility due to their variability. We have shown that an equilibrium could be approached under a
Acknowledgements
The author would like to thank Albert Greenberg of AT&T Laboratories for his hospitality where part of this work was done, and Australian Research Council for grant S49813050. He also thanks David Lapsley for help with Matlab script that generated Fig. 1, Fig. 2.
Steven Low recieved his B.S. degree from Cornell University in 1987 and Ph.D. from the University of California, Berkeley in 1992, both in Electrical Engineering. He was with AT&T Bell Laboratories, Murray Hill, from 1992 to 1996, and joined the University of Melbourne in 1996 as a Senior Lecturer. He has held visiting positions at Rutgers University, NJ, in 1995 and University of Science and Technology, Hong Kong, in 1996. He has consulted for NEC, AT&T, Lucent, the Australian Taxation Office,
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Steven Low recieved his B.S. degree from Cornell University in 1987 and Ph.D. from the University of California, Berkeley in 1992, both in Electrical Engineering. He was with AT&T Bell Laboratories, Murray Hill, from 1992 to 1996, and joined the University of Melbourne in 1996 as a Senior Lecturer. He has held visiting positions at Rutgers University, NJ, in 1995 and University of Science and Technology, Hong Kong, in 1996. He has consulted for NEC, AT&T, Lucent, the Australian Taxation Office, the Public Transportation Corporation, Victoria, and other organizations in the US and Australia. He was the co-recipient of the IEEE William R. Bennett Prize paper award in 1996 and the 1996 R&D 100 Award. He is on the editorial board of IEEE/ACM Transactions on Networking, and has been a guest editor for the IEEE Journal on Selected Area in Communications. He has been on the programme committee of the Workshop on Information Hiding from 1996 to 1999, of the SPIE Conference 1998, and of the Infocom 1999. His research interests are in the control and optimization of communications networks and protocols, and network security and privacy.
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An earlier version of this paper was presented at the Performance and Control of Network Systems Conference, Wai Sum Lai, Hisashi Kobayashi (Eds.), Proceedings of SPIE (The International Society for Optical Engineering), vol. 3231, Dallas, TX, 3–5 November 1997.