Optimization Flow Control—I: Basic Algorithmand Convergence

Present : Li-der

Outline

• Objective of the paper• Problem Framework• The optimization problem • Synchronous Distributed Algorithm• Experimental Results• Conclusion

Objective of the paper

• Propose an optimization approach for flow control on a network whose resources are shared by a set of S of sources

• Maximization of aggregate source utility over transmission rates is aimed

• Sources select transmission rates that maximize their benefit (utility – bandwidth cost)

• Distributed algorithms for converging optimal behavior in static environment is presented

Problem Framework

The problem is formulated for • A network that consists of a set L of unidirectional links

of capacities cl, where l is element of L.

• The network is shared by a set S of sources, where source s is characterized by a utility function Us(xs) that is concave increasing in its transmission rate xs

• The goal is to calculate source rates that maximize the sum of the utilities ∑s ϵ S Us(xs) over xs subject to capacity constraints.

S

Problem Framework

s1 s2 s3 ss

..........

DESTINATION NODES

SOURCE NODES

L(s1)={l1,l2,l3,l4}

link l4 : S(l4)={s1,s3}

l1

l2

l3 l5

l6

Problem Framework

c1 c2

Sources sL(s) - links used by source sUs(xs) - utility if source rate = xs

• Network– Links l of capacities cl

x1

x2x3

121 cxx 231 cxx

The optimization problem :Primal problem

Concave utility function

TYPICAL CONCAVE UTILITY FUNCTION

The optimization problem :Lagrangian for primal problem

pl represents Lagrange multipliers utilized in standard convex optimization method By using this approach coupled link capacity constraints are integrated to the objective

function Notice separability in terms of xs so maximizing lagrangian function as aggregate of

different xs related terms means gives the same result as summing up maximum of each individual xs related term. Therefore we have

The optimization problem :Dual problem

Here pl is the price per unit bandwidth at link l. ps is the total price per unit bandwidth for all links in the path of s The dual problem has been defined as minimization of D(p) (upper bound of Lagrangian

function) for non-negative bandwidth prices. Each source can independently solve maximization problem in (3) for a given p

Source rate as demand function

The above figure depict xs(p) as a possible solution

Similar to inverse of U’ figure in previous slide the rate is obtained as a decreasing function of U’-1(rate)

This means that xs(p) acts as a demand function seen in Microeconomics.

The dual problem is solved via gradient projection method where link prices are adjusted in the opposite direction of gradient of D(p)

Synchronous Distributed Algorithmbased on gradient projection applied to dual problem

Synchronous Distributed Algorithm

S

Problem Framework

s1 s2 s3 ss

..........

DESTINATION NODES

SOURCE NODES

L(s1)={l1,l2,l3,l4}

l4

l1

l2

l3 l5

l6

Experiment Result

• Overview of Implementation– two IBM-compatible PC’s (Pentium 233 MHz)

running the FreeBSD-2.2.5 operating system. – Each PC was equipped with 64 MB of RAM and

100-MB/s PCI ether-net cards.

Experiment Result

• Each source transmitted data for a total of 120 s, with their starting times staggered by intervals of 40 s.

• source 1 started transmitting at time 0, source 2 at time 40 s, and source 3 at time 80 s.

Conclusion

• This paper have described an optimization approach to reactive flow control, and derived a simple distributed algorithm.

• The algorithm is provably convergent to the global optimal when network conditions are static and seems to track the optimum when network conditions vary slowly.