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Distributed Rate Assignments for Broadband CDMA Networks Tara Javidi Electrical & Computer Engineering University of California, San Diego

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Multi-Cell Single Hop CDMA Motivation Wideband CDMA network with variable rates Mobiles communicate directly with the base station Base stations are connected directly to the traditional IP network

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Rate Assignment Problem Limited by congestion constraints in the wired network Limited by interference constraints in the wireless network Objective: Maximize the global network utility in a distributed adaptive manner

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Philosophically Related Works Wired Networks [1] F. Kelly. Mathematical Modeling of the Internet. B. Enquist and W. Schmid, editors. Mathematics Unlimited – 2001 and Beyond, pages 685-702. Springer-Verlaq, 2001. [2] J. Mo and J. Walrand. Fair End-to-End Window-Based Congestion Control. IEEE/ACM Transactions on Networking, 8(5):555-567, 2000. [3]S.H. Low and D.E. Lapsley. Optimization Flow Control I: Basic Algorithm and Convergence. IEEE/ACM Transactions on Networking, 7(6):861-874, 1999. Wireless Networks [3] T. Javidi Distributed Rate Assignment in Multi-sector CDMA. Global Telecommunications Conference, 2003. [4] M. Chiang and R. Man. Jointly Optimal Congestion Control and Power Control in Wireless Multihop Networks. Global Telecommunications Conference, 2003. [5] X. Lin and N.B. Shroff. The Impact of Imperfect Scheduling on Cross-Layer Rate Control in Wireless Networks. INFOCOM 2005.

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Cross-Layer Design: One-Shot One-shot and joint design of a rate assignment protocol (merging MAC and transport layers) Wireless and wired networks generate feedback based on their respective system constraints This feedback allows for dynamic adaptation to slowly varying network conditions

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Iterative Methods and Convergence If the Lagrange multipliers are computed using a gradient projection method, the rate assignment becomes an iterative algorithm that uses feedback from the network Theorem: Given an appropriate choice of step- size, the distributed system will converge to the solution to the primal problem (cross-layer optimal)

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Related Work [1] X. Lin and N.B. Shroff. Joint rate control and scheduling in multi- hop wireless networks. CDC04 [2] M. Neely, E. Modiano, and C. Li. Fairness and Optimal Stochastic Control for Heterogeneous Networks. Infocom05 +Due to structure of the problem, we get truly distributed solutions (little overhead comm) - Such solutions require a fundamental re-doing of the protocol stack in general and transport layer in particular

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Cross-Layer Design: Modular MAC and transport layer protocols are separate MAC chooses rate using feedback from wireless The transport layer chooses rate based on end-end feedback following a dual controller Can this be optimal in a cross-layer sense? If no wired core, the answer is yes: [1] A. Eryilmaz and R. Srikant. Fair Resource Allocation in Wireless Using Queue-based Scheduling and Congestion Control

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Outline Motivation and Overview One-Shot Rate Assignments Modular Rate Assignments The Problem with Dual Methods Practical Implementation & Cross-Layer Coordination Observations, Conclusions, & Future Work

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Notation CDMA uplink dynamic power and spreading gain control (distributed) Network Parameters M: number of nodes: N of them wireless L: number of sectors J: number of (wired) links C j : capacity of link j ij : routing function W: chip bandwidth g il : channel power gain K: acceptable interference b(i): mobile is sector Node Variables P i : transmit power for user i i : transmit rate for user i at MAC x i : transmit rate for user i at transport

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One-Shot Problem Formulation subject to Wired Link Capacity ROT-Controlled Feasible Rate Vector Bench Mark: cross-layer optimal

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Iterative Methods and Convergence If the Lagrange multipliers are computed using a gradient projection method, the rate assignment becomes an iterative algorithm that uses feedback from the network Theorem: Given an appropriate choice of step- size, the distributed system will converge to the solution to the primal problem (cross-layer optimal)

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subject to Modular Problem Formulation Coordinate MAC and Transport Layers x i = i i i

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Dual Controller Fails OQuestion: What happens when we try to use the dual controller/gradient projection? OAnswer: The dual controller fails to converge to solution of the optimization problem OWe need to maximize a function that is strictly concave over all the primal variables

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Modular Utility Functions if i is a wired user if i is a wireless user

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A New Modular Problem Formulation subject to

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Economic Interpretation of the Dual Price for Link j Price for Sector l Individual Profit Maximization (Transport Layer) Cross-Layer Coordination Signal

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Iterative Methods and Convergence If the Lagrange multipliers are computed using a gradient projection method, the rate assignment becomes an iterative algorithm that uses feedback from the network Theorem: Given an appropriate choice of step- size, the distributed system will converge to the solution to the primal problem (cross-layer optimal)

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Wired Network Prices Individual Lagrange multipliers are generated using gradient projection This has a well known physical interpretation: queuing delay! Aggregate price q i can be interpreted as end-to-end queuing delay, which can be measured by each user if

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Wireless Network Prices Individual Lagrange multipliers are generated using gradient projection We can construct a signaling mechanism under which the aggregate price p i becomes closely related to forward link SINR on the pilot signal

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Again, individual Lagrange multipliers are generated using gradient projection These equations are similar to the equations representing delay in queues! if Cross-Layer Coordination Signal Each equality is broken into two inequalities For each inequality two multipliers computed

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Cross Layer Coordination Signal Two imaginary queues whose associated delays are i + and i - Queue 1 is our MAC- layer buffer, and Queue 2 is our token bucket Token bucket is not used to regulate service rate, but to keep track of the mismatch between transport and MAC layer rates

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The Role of the New Buffers Non-zero delay in the MAC-layer buffer corresponds to a wireless bottleneck The price from the actual link prevents the transport layer from out-running the MAC layer Non-zero delay in the token bucket corresponds to a wired bottleneck The price from the token bucket prevents the MAC layer from out-running the transport layer Generally only one of the queues is nonempty (i.e. only one of the constraints is active) at a time Without the use of a token bucket, the solution will converge but not to the desired equilibrium when wired bottle-neck

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Transport Layer Profit Maximization Information about the interference levels in the wireless network is now incorporated into the end- to-end queuing delay (q i + i + ) minus the token bucket delay ( i - ) Allows the transport layer to take interference levels into account without any major modification of current protocols add the token bucket delay to the propagation delay

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Mac Layer Profit Maximization Wireless sources now receive credit for long data queues (i.e. large i + ) and are penalized for long token buckets (i.e. large i - ) Prioritize wireless users based on their backlog (De)Prioritize wireless users based on received service so far

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Simulations

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Dynamical Behavior Convergence Since we wish to interpret the Lagrange multipliers as delay, the step size must be chosen as Δ t/C Convergence is dependent upon the step size being small enough, hence the algorithm being run fast enough Nested Feedback Loops Decoupling of the MAC and transport layer allows for the corresponding feedback loops to be run at different time scales – aid in convergence and/or robustness? Interaction of three separate feedback loops (MAC, transport, and power control) plays a significant role in dynamic situations Choice of parameters and K play an important role

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Future Work Provide a stability analysis Use the concept of Markov chain stability for queue lengths Understand the impact of realistic arrival statistics on the system How does statistical multiplexing impact the transient behavior of the system? Determine whether these results can be extended to other MAC protocols Does the addition of the MAC-layer queue and token bucket provide sufficient coordination for other MAC schemes?

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Mathematical models of the Internet Frank Kelly www.statslab.cam.ac.uk/~frank Hood Fellowship Public Lecture University of Auckland 3 April 2012.

Mathematical models of the Internet Frank Kelly www.statslab.cam.ac.uk/~frank Hood Fellowship Public Lecture University of Auckland 3 April 2012.

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