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Layering As Optimization Decomposition: a Mathematical INVITED PAPER Layering as Optimization Decomposition: A Mathematical Theory of Network Architectures There are various ways that network functionalities can be allocated to different layers and to different network elements, some being more desirable than others. The intellectual goal of the research surveyed by this article is to provide a theoretical foundation for these architectural decisions in networking. By Mung Chiang, Member IEEE, Steven H. Low, Senior Member IEEE, A. Robert Calderbank, Fellow IEEE, and John C. Doyle ABSTRACT | Network protocols in layered architectures have layering architectures. This paper surveys the current status of historically been obtained on an ad hoc basis, and many of the horizontal decomposition into distributed computation, and recent cross-layer designs are also conducted through piece- vertical decomposition into functional modules such as con- meal approaches. Network protocol stacks may instead be gestion control, routing, scheduling, random access, power holistically analyzed and systematically designed as distributed control, and channel coding. Key messages and methods solutions to some global optimization problems. This paper arising from many recent works are summarized, and open presents a survey of the recent efforts towards a systematic issues discussed. Through case studies, it is illustrated how understanding of Blayering[ as Boptimization decomposition,[ BLayering as Optimization Decomposition[ provides a common where the overall communication network is modeled by a language to think about modularization in the face of complex, generalized network utility maximization problem, each layer networked interactions, a unifying, top-down approach to corresponds to a decomposed subproblem, and the interfaces design protocol stacks, and a mathematical theory of network among layers are quantified as functions of the optimization architectures. variables coordinating the subproblems. There can be many alternative decompositions, leading to a choice of different KEYWORDS | Ad hoc network; channel coding; computer net- work; congestion control; cross-layer design; distributed algo- rithm; feedback control; game theory; Internet; Lagrange duality; Manuscript received July 24, 2005; revised September 6, 2006. The works at medium access control (MAC); network utility maximization Princeton University and Caltech that are summarized in this paper were supported by the National Science Foundation (NSF) Grants ANI-0230967, EIA-0303620, (NUM); optimization; power control; reverse-engineering; CNS-0417607, CNS-0435520, CCF-0440443, CCF-0448012, CNS-0417607, routing; scheduling; stochastic networks; transmission control CNS-0427677, CNS-0430487, CCF-0635034, and CNS-0519880, by the Air Force Office of Scientific Research (AFOSR) Grants F49620-03-1-0119 protocol (TCP)/Internet protocol (IP); wireless communications and FA9550-06-1-0297, by the ARO Grant DAAD19-02-1-0283, by the Defense Advanced Research Projects Agency (DARPA) Grant HR0011-06-1-0008 and CBMANET program, and by the Cisco Grant GH072605. M. Chiang is with the Electrical Engineering Department, Princeton University, I. INTRODUCTION Princeton, NJ 08544 USA (e-mail: [email protected]). S. H. Low is with the Computer Science and Electrical Engineering Departments, California Institute of Technology, Pasadena, CA 91125 USA (e-mail: [email protected]). A. Overview A. R. Calderbank is with the Electrical Engineering and Mathematics Departments, Princeton University, Princeton, NJ 08544 USA (e-mail: [email protected]). J. C. Doyle is with the Control and Dynamic Systems, California Institute of Technology, 1) Structures of the Layered Protocol Stack: Network Pasadena, CA 91125 USA (e-mail: [email protected]). architecture determines functionality allocation: Bwho does Digital Object Identifier: 10.1109/JPROC.2006.887322 what[ and Bhow to connect them,[ rather than just resource 0018-9219/$25.00 Ó2007 IEEE Vol. 95, No. 1, January 2007 | Proceedings of the IEEE 255 Chiang et al.: Layering as Optimization Decomposition: A Mathematical Theory of Network Architectures allocation. It is often more influential, harder to change, quantitatively understood through the mathematical lan- and less understood than any specific resource allocation guage of decomposition theory for constrained optimization scheme. Functionality allocations can happen, for example, [104]. This framework of BLayering as Optimization between the network management system and network Decomposition[ exposes the interconnections between elements, between end-users and intermediate routers, and protocol layers as different ways to modularize and dis- between source control and in-network control such as tribute a centralized computation. Even though the design routing and physical resource sharing. The study of network of a complex system will always be broken down into architectures involves the exploration and comparison of simpler modules, this theory will allow us to systematically alternatives in functionality allocation. This paper presents carry out this layering process and explicitly tradeoff a set of conceptual frameworks and mathematical languages design objectives. for a foundation of network architectures. The core ideas in BLayering as Optimization Decom- Architectures have been quantified in fields such as position[ are as follows. Different vertical decompositions information theory, control theory, and computation of an optimization problem, in the form of a generalized theory. For example, the source-channel separation prin- network utility maximization (NUM), are mapped to dif- ciple is a fundamental result on architecture in informa- ferent layering schemes in a communication network. Each tion theory. The choices of architectural decisions are even decomposed subproblem in a given decomposition cor- more complicated in networking. For example, the responds to a layer, and certain functions of primal or functionality of rate allocation among competing users Lagrange dual variables (coordinating the subproblems) may be implemented through various combinations of the correspond to the interfaces among the layers. Horizontal following controls: end-to-end congestion control, local decompositions can be further carried out within one scheduling, per-hop adaptive resource allocation, and functionality module into distributed computation and routing based on end-to-end or per-hop actions. However, control over geographically disparate network elements. we do not yet have a mature theoretical foundation of Since different decompositions lead to alternative layering network architectures. architectures, we can also tackle the question of Bhow and Layered architectures form one of the most fundamen- how not to layer[ by investigating the pros and cons of tal structures of network design. They adopt a modularized decomposition methods. Furthermore, by comparing the and often distributed approach to network coordination. objective function values under various forms of optimal Each module, called layer, controls a subset of the decision decompositions and suboptimal decompositions, we can variables, and observes a subset of constant parameters and seek Bseparation theorems[ among layers: conditions the variables from other layers. Each layer in the protocol under which layering incurs no loss of optimality. Robust- stack hides the complexity of the layer below and provides ness of these separation theorems can be further char- a service to the layer above. Intuitively, layered architec- acterized by sensitivity analysis in optimization theory: tures enable a scalable, evolvable, and implementable net- how much will the differences in the objective value work design, while introducing limitations to efficiency (between different layering schemes) fluctuate as constant and fairness and potential risks to manageability of the parameters in the generalized NUM formulation are network. There is clearly more than one way to Bdivide perturbed. and conquer[ the network design problem. From a data- There are two intellectually fresh cornerstones behind plane performance point of view, some layering schemes BLayering as Optimization Decomposition.[ The first is may be more efficient or fairer than others. Examining Bnetwork as an optimizer.[ The idea of viewing protocols these choices of modularized design of networks, we as a distributed solution (to some global optimization would like to tackle the question of Bhow to[ and Bhow problem in the form of the basic NUM) has been suc- not to[ layer. cessfully tested in the trials for transmission control While the general principle of layering is widely rec- protocol (TCP) [56]. The key innovation from this line of ognized as one of the key reasons for the enormous success work (e.g., [64], [72], [73], [87], [89], [90], [96], [116], of data networks, there is little quantitative understanding [125],and[161])istoviewtheTCP/IPnetworkasan to guide a systematic, rather than an ad hoc, process of optimization solver, and each variant of congestion control designing layered protocol stack for wired and wireless protocol as a distributed algorithm solving a specified basic networks. One possible perspective to understand layering NUM with a particular utility function. The exact shape of is to integrate the various protocol layers into a single the utility function can be reverse-engineered from the theory, by regarding them as carrying out an asynchronous given protocol. In the basic NUM, the objective is to
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