Optimization Flow Control, I : Basic Algorithm and Convergence

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Optimization Flow Control, I Basic Algorithm
and Convergence

• Steven Low and David Lapsley

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The purpose of flow control

• Consider file transfer
• Sender sends a stream of packets
• Cant send too slow or too fast
• Too slow wastes time
• Too fast packet losses
• How to find the correct rate?

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In the following

• Formulate the flow control problem
• Derive the solution
• Explain experimental results

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Network Model

• A set of links L
• cl link capacity
• S(l) sources that passed link l.
• A set of sources S
• xs source transmission rate
• L(s) links passed by source s.
• Us(xs) utility function that reflect user
  satisfaction of xs, e.g. the video quality
  perceived by users.
• I(s) range of transmission rate

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Formulate the flow control problem
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Difficulties with the primal problem

• It is HARD to implement !!! It requires that
• Utility functions for all sources must be known.
• Transmission rates are coupled by link
  capacities
• What if we look at its Dual Problem?

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Lets take a look at the Dual Problem

• Define the Lagrangian
• Dual objective function

Consider pl as the link price for every unit
bandwidth
Since Slater's condition holds and the original
problem is concave, we have strong duality.
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Decentralized nature of the Dual Problem
Dual problem

• Note the decisions of links and sources are
  separable.

• Note decision of sources are also separable.

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How to solve the problem?

• Gradient Projection

• Illustration
• Its like steepest decent
• But we can not get out of
• the feasible solution set.

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Interpretation of the solution

• Solve this equation

• We have

• Given aggregate source rate at link l, link l can
  calculate its price separately.
• It is like the Law of Supply and Demand
• If the demand for bandwidth at a link exceeds
  the supply, raise the price otherwise reduce
  price.

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A quick summary

• Given price of the path, individual source can
  calculate its transmission rate separately.
• Given aggregate source rate at link l, link l can
  calculate its price separately.
• They are the processors of a distributed system!

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Synchronous Algorithm

• Source s At time t1,2,3
• Receives link prices on its path
• Chooses a rate to transmit until next update
• Communicates new rate to links on
  its path
• Link l At time t1,2,3
• Receives rates from sources using link
  l
• Computes new price
• Communicates to sources using
  link l

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Convergence

• Theorem 1
• If
• suppose the utility function is strictly
  increasing and concave with
• step size satisfies
• then
• from any initial rates, the solution generated by
  Algorithm A1 are primal-dual optimal.

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Synchronous solution is not practical

• The synchronous model assumes that updates at the
  sources and links are synchronized to occur at
  time t1,2,3...
• In the real network,
• sources may be located at different distance from
  links. Feedbacks may reach the sources after
  different delays.
• Some processors may compute faster, and some may
  communicate more frequently than others.
• Next we are going to show the asynchronous
  algorithm A2
• We update link ls price pl and source rate xs at
  time set T1l and T2s respectively.
• In the time in the interval of the update time we
  just receive the state and store it to replace
  the oldest one.

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The key of the Asyn. Solution

• Recall the original link price update equation

• Now link will make an estimation of the source
  rate at update time

• Also recall the rate update equation

• Source will make an estimation of the link price

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Asynchronous Algorithm

• Source s
• Receives link prices from time to time
• At update time computes new rate based on prices
  currently in local memory
• Communicates current rate to links on its path
• Link l
• Receives source rates from time to time
• At update time computes new price based on rates
  currently in local memory
• Communicate current prices to sources using link l

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Convergence

• Theorem 1
• If
• suppose the utility function is strictly
  increasing and concave with
• step size satisfies
• And the time between consecutive updates is
  bounded
• then
• from any feasible initial rates and price, the
  solution generated by asynchronous distributed
  Algorithm converges to primal-dual optimal.

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Fairness

• If user utilities are all equal but not
  necessarily logarithmic, the following properties
  follow
• If the sources share the same path then
• If the path of s1 is a subset of s2, then
• The proposed algorithm is biased against flow
  that travels a long path.

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Experiments

• The experimental setting is as follows

S1 goes into the network at time 0s120s S2 goes
into the network at time 40s160s S3 goes into
the network at time 80s200s stepsize is equal
to 0.015 The links and sources update every
500ms The utility functions are identical
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Results
Theoretical rate for each source

• Theoretical price for path price
• The sum of measured link prices should roughly
  equal to the theoretical price

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Conclusion

• This paper considers an optimization approach to
  distributed flow control.
• Specifically, the dual problem suggest that we
  can treat sources and links as processors of a
  distributed system.
• A high level description of the algorithm is as
  follows
• Each processor can use only local information to
  make and advertise the decision.
• By communication between processors, the system
  can achieve global optimality.

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• Thank you!

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Motivation

• The Transmission Control Protocol of the Internet
  is distributed and is working very well,
• But
• Previous flow control protocols are built by
  intuitions. Can we justify them from a
  theoretical angle?
• They work well in current networks, are they
  still suitable as the networks evolve? Can we
  prove their generality?
• This paper considered distributed flow control
  from a theoretical angle !!!

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Example

• To provide efficiency

• To provide fairness

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Why do people prefer decentralized flow control?

• Real examples of different flow control method
• Centralized ATM
• network explicitly imposes the transmission rate
  to sources.
• Decentralized Internet
• sources adjust their transmission rate according
  to network conditions.