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Charge-Sensitive TCP and Rate Control . Richard J. La Department of EECS UC Berkeley November 22, 1999. Motivation. Network users have a great deal of freedom as to how they can share the available bandwidth in the network

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Charge sensitive tcp and rate control l.jpg

Charge-Sensitive TCP and Rate Control

Richard J. La

Department of EECS

UC Berkeley

November 22, 1999


Motivation l.jpg
Motivation

  • Network users have a great deal of freedom as to how they can share the available bandwidth in the network

  • The increasing complexity and size of the Internet renders centralized rate allocation impractical

    • distributed algorithm is desired

  • Two classes of flow/congestion control mechanisms

    • rate-based : directly controls the transmission rate based on feedback

    • window-based : controls the congestion window size to adjust the transmission rate and backlog


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Motivation

  • Transmission Control Protocol (TCP) does not necessarily results in a fair or efficient allocation of the available bandwidth

  • Many algorithms have been proposed to achieve fairness among the connections

  • Fairness alone may not be a suitable objective

    • most algorithms do not reflect the user utilities or preferences

    • good rate allocation should not only be fair, but should also maximize the overall utility of the users


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Model

  • Network with a set J of links and a set I of users


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Model (Kelly)

  • system is not likely to know

  • impractical for a centralized system to compute and allocate the user rates



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Background (Kelly’s work)

  • One can always find vectors and such that

    1) solves for all

    2) solves

    3)

    4) is the unique solution to


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Fairness

  • Max-min fairness :

    • a user’s rate cannot be increased without decreasing the rate of another user who is already receiving a smaller rate

    • gives an absolute priority to the users with smaller rates

  • (weighted) proportional fairness :

    • is weighted proportionally fair with weight vector if is feasible and for any other feasible vector



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Fluid Model (Mo & Walrand)

  • Theorem 1 (Mo & Walrand) : For all w there exists a uniquex that satisfies the constraints (1)-(4)

    • this theorem tells us that the rate vector is a well defined function of the window sizes w.

    • denote the function by x(w)

    • x(w) is continuous and differentiable at an interior point

    • q(w) may not be unique, but the sum of the queuing delay along any route is well defined


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Mo & Walrand’s Algorithm

  • (p, 1)-proportionally fair algorithm :

    where


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Mo & Walrand’s Algorithm

  • Theorem 2 (Mo & Walrand) :The window sizes converge to a unique point w*such that for all

    Further, the resulting rate at the unique stable point w*is weighted proportionally fair that solves NETWOKR(A, C ; p).


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Pricing Scheme

  • Price per unit flow at a switch is the queuing delay at the switch, i.e.,

    • the total price per unit flow of user i is given by

      where is connection i’s queue size at resource j


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User Optimization & Assumption

  • User optimization problem :

    where is the price per unit flow, which is the queuing delay

  • Assumption 1 : The optimal price

    is a decreasing function of .



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Price Updating Rule

  • At time t, each user i updates its price according to


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Price Updating Rule

  • Define a mapping to be

  • Fixed point of the mapping T is a vector p such that T(p) = p.

  • Theorem : There exists a unique fixed point p* of the mapping T, and the resulting rate allocation from p* is the optimal rate allocation x* that solves SYSTEM(U,A,C).


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Algorithm I

  • Suppose that users update their prices according to

  • Assumption 2 : There exists M > 0 such that

    (a) for all p such that

    (b) for allpsuch that


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Convergence in Single Bottleneck Case

  • Theorem : Under the assumptions 1 and 2, the user prices p(n) converges to the unique fixed point of the mapping T under both Jacobi and the totally asynchronous update schemes as .


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Algorithm II

  • Suppose that users update their window sizes according to

    where


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Assumption & Convergence

  • Assumption 3: The utility functions satisfy

    where

  • Theorem : Under assumption 3, the window sizes converge to a unique stable point of the algorithm II, where the resulting rates solve SYSTEM(U,A,C).