Charge-Sensitive TCP and Rate Control

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Charge-Sensitive TCP and Rate Control

• Richard J. La
• Department of EECS
• UC Berkeley
• November 22, 1999

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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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Model (Kelly)

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Background (Kellys 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 users 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)

• where

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Fluid Model (Mo Walrand)

• Theorem 1 (Mo Walrand) For all w there exists
  a unique x 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 Walrands Algorithm

• (p, 1)-proportionally fair algorithm
• where

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Mo Walrands 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 is 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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Examples of Utility Functions

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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 gt 0 such that
• (a) for all p such that
• (b) for all p such 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).