This text presents a modern theory of analysis, control, and optimization for dynamic networks. Mathematical techniques of Lyapunov drift and Lyapunov optimization are developed and shown to enable constrained optimization of time averages in general stochastic systems. The focus is on communication and queueing systems, including wireless networks with time-varying channels, mobility, and randomly arriving traffic. A simple drift-plus-penalty framework is used to optimize time averages such as throughput, throughput-utility, power, and distortion. Explicit performance-delay tradeoffs are provided to illustrate the cost of approaching optimality. This theory is also applicable to problems in operations research and economics, where energy-efficient and profit-maximizing decisions must be made without knowing the future. Topics in the text include the following: - Queue stability theory - Backpressure, max-weight, and virtual queue methods - Primal-dual methods for non-convex stochastic utility maximization - Universal scheduling theory for arbitrary sample paths - Approximate and randomized scheduling theory - Optimization of renewal systems and Markov decision systems Detailed examples and numerous problem set questions are provided to reinforce the main concepts. Table of Contents: Introduction / Introduction to Queues / Dynamic Scheduling Example / Optimizing Time Averages / Optimizing Functions of Time Averages / Approximate Scheduling / Optimization of Renewal Systems / Conclusions

Stochastic Network Optimization with Application to Communication and Queueing Systems

Information flow in a telecommunication network is accomplished through the interaction of mechanisms at various design layers with the end goal of supporting the information exchange needs of the applications. In wireless networks in particular, the different layers interact in a nontrivial manner in order to support information transfer. In this text we will present abstract models that capture the cross-layer interaction from the physical to transport layer in wireless network architectures including cellular, ad-hoc and sensor networks as well as hybrid wireless-wireline. The model allows for arbitrary network topologies as well as traffic forwarding modes, including datagrams and virtual circuits. Furthermore the time varying nature of a wireless network, due either to fading channels or to changing connectivity due to mobility, is adequately captured in our model to allow for state dependent network control policies. Quantitative performance measures that capture the quality of service requirements in these systems depending on the supported applications are discussed, including throughput maximization, energy consumption minimization, rate utility function maximization as well as general performance functionals. Cross-layer control algorithms with optimal or suboptimal performance with respect to the above measures are presented and analyzed. A detailed exposition of the related analysis and design techniques is provided.

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Resource Allocation and Cross Layer Control in Wireless Networks

We consider optimal control for general networks with both wireless and wireline components and time varying channels. A dynamic strategy is developed to support all traffic whenever possible, and to make optimally fair decisions about which data to serve when inputs exceed network capacity. The strategy is decoupled into separate algorithms for flow control, routing, and resource allocation, and allows each user to make decisions independent of the actions of others. The combined strategy is shown to yield data rates that are arbitrarily close to the optimal operating point achieved when all network controllers are coordinated and have perfect knowledge of future events. The cost of approaching this fair operating point is an end-to-end delay increase for data that is served by the network.

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Fairness and optimal stochastic control for heterogeneous networks

Embedding pico/femto base-stations and relay nodes in a macro-cellular network is a promising method for achieving substantial gains in coverage and capacity compared to macro-only networks. These new types of base-stations can operate on the same wireless channel as the macro-cellular network, providing higher spatial reuse via cell splitting. However, these base-stations are deployed in an unplanned manner, can have very different transmit powers, and may not have traffic aggregation among many users. This could potentially result in much higher interference magnitude and variability. Hence, such deployments require the use of innovative cell association and inter-cell interference coordination techniques in order to realize the promised capacity and coverage gains. In this paper, we describe new paradigms for design and operation of such heterogeneous cellular networks. Specifically, we focus on cell splitting, range expansion, semi-static resource negotiation on third-party backhaul connections, and fast dynamic interference management for QoS via over-the-air signaling. Notably, our methodologies and algorithms are simple, lightweight, and incur extremely low overhead. Numerical studies show that they provide large gains over currently used methods for cellular networks.

Cell Association and Interference Coordination in Heterogeneous LTE-A Cellular Networks

In third-generation (3G) wireless data networks, mobile users experiencing poor channel quality usually have low data-rate connections with the base-station. Providing service to low data-rate users is required for maintaining fairness, but at the cost of reducing the cell's aggregate throughput. In this paper, we propose the Unified Cellular and Ad-Hoc Network (UCAN) architecture for enhancing cell throughput, while maintaining fairness. In UCAN, a mobile client has both 3G cellular link and IEEE 802.11-based peer-to-peer links. The 3G base station forwards packets for destination clients with poor channel quality to proxy clients with better channel quality. The proxy clients then use an ad-hoc network composed of other mobile clients and IEEE 802.11 wireless links to forward the packets to the appropriate destinations, thereby improving cell throughput. We refine the 3G base station scheduling algorithm so that the throughput gains of active clients are distributed proportional to their average channel rate, thereby maintaining fairness. With the UCAN architecture in place, we propose novel greedy and on-demand protocols for proxy discovery and ad-hoc routing that explicitly leverage the existence of the 3G infrastructure to reduce complexity and improve reliability. We further propose a secure crediting mechanism to motivate users to participate in relaying packets for others. Through extensive simulations with HDR and IEEE 802.11b, we show that the UCAN architecture can improve individual user's throughput by up to 310% and the aggregate throughput of the HDR downlink by up to 60%.

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UCAN: a unified cellular and ad-hoc network architecture

We are concerned with the allocation of the base station transmitter time in time-varying mobile communications with many users who are transmitting data. Time is divided into small scheduling intervals, and the channel rates for the various users are available at the start of the intervals. Since the rates vary randomly, in selecting the current user there is a conflict between full use (by selecting the user with the highest current rate) and fairness (which entails consideration for users with poor throughput to date). The proportional fair scheduler of the Qualcomm High Data Rate system and related algorithms are designed to deal with such conflicts. The aim here is to put such algorithms on a sure mathematical footing and analyze their behavior. The available analysis, while obtaining interesting information, does not address the actual convergence for arbitrarily many users under general conditions. Such algorithms are of the stochastic approximation type and results of stochastic approximation are used to analyze the long-term properties. It is shown that the limiting behavior of the sample paths of the throughputs converges to the solution of an intuitively reasonable ordinary differential equation, which is akin to a mean flow. We show that the ordinary differential equation (ODE) has a unique equilibrium and that it is characterized as optimizing a concave utility function, which shows that PFS is not ad-hoc, but actually corresponds to a reasonable maximization problem. These results may be used to analyze the performance of PFS. The results depend on the fact that the mean ODE has a special form that arises in problems with certain types of competitive behavior. There is a large set of such algorithms, each one corresponding to a concave utility function. This set allows a choice of tradeoffs between the current rate and throughout. Extensions to multiple antenna and frequency systems are given. Finally, the infinite backlog assumption is dropped and the data is allowed to arrive at random. This complicates the analysis, but the same results hold.

Convergence of proportional-fair sharing algorithms under general conditions

: We are concerned with the allocation of channel or transmitter resources for time varying mobile communications. There are many users who are competing to transmit data over the resource. Time is divided into small scheduling intervals, and information on the channel rates for the various users is available at the start of the intervals. Since the rates vary randomly, there is a conflict at any time between fully exploiting the channel (by selecting the user with the highest current rate) and being fair (giving attention to users with poor rates, to assure a fair throughput for them). The Proportional Fair Scheduler (PFS) of the Qualcomm High Data Rate (HDR) system and related algorithms are designed to deal with such conflicts. There is little analysis available for such systems and our aim is to put them on a sure mathematical footing and analyze their behavior. Such algorithms are of the stochastic approximation type and results of stochastic approximation are used to analyze the long term properties of this class. The limiting behavior of the throughputs converges to the solution of an ordinary differential equation (a mean ODE), which is akin to a mean flow. The ODE has a unique equilibrium and it is optimal in the sense that it optimizes a concave utility function. The results depend on the fact that the mean ODE has a special form that arises in problems with certain types of repeated stochastic games with competitive behavior. There are a large family of such algorithms, each member corresponding to a concave utility function. Thus, is not simply ad-hoc, but actually corresponds to a reasonable maximization problem. There are extensions to multiple antenna and frequency systems. Also, the infinite backlog assumption can be dropped and the data is allowed to arrive at random.

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Asymptotic Properties of Proportional-Fair Sharing Algorithms

: We are concerned with the allocation of transmitter time and power for randomly time varying mobile data communications. Time is divided into small scheduling intervals, called slots, and information on the channel rates for the various users is available at the start of the slot, when the user selections are made. There is a conflict between selecting the user set that can get the most immediate data through and helping users with poor average rates. The Proportional Fair Sharing method (PFS) deals with such con icts. In [5, 6] the convergence and basic qualitative properties were analyzed. Stochastic approximation results were used to analyze the long term properties. The paths of the (suitably interpolated) throughputs converge to the solution of an ODE, akin to a mean flow. The ODE has a unique equilibrium point. It is asymptotically stable and optimal in that it is the maximizer of a concave utility function. There is a large family of such algorithms, each member corresponding to a concave utility function. The basic idea of PFS extends to many systems of current importance for which it was not originally intended, and a variety of such extensions are treated here to illustrate the possibilities. One might have minimal throughput constraints, nonlinear dependence of rate on allocated power, minimal SNR requirements, etc. In some recent applications, the number of slots in a scheduling intervals is random, and the length is not known when the selection is made. The form of the PFS rule is adapted to the application. Then the basic results continue to hold. The asymptotic properties of the ODE characterize the behavior of the algorithm.

Harold J. Kushner

Papers

Convergence of proportional-fair sharing algorithms under general conditions

Asymptotic Properties of Proportional-Fair Sharing Algorithms

Asymptotic Properties of Proportional-Fair Sharing Algorithms: Extensions of the Algorithm