1. Optimization Flow Control—I: Basic Algorithm and Convergence Present : Li-der

2. Outline Objective of the paper Problem Framework The optimization problem Synchronous Distributed Algorithm Experimental Results Conclusion

3. 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

4. Problem Framework The problem is formulated for A network that consists of a set L of unidirectional links of capacities c l, 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 U s (x s ) that is concave increasing in its transmission rate x s The goal is to calculate source rates that maximize the sum of the utilities ∑ s S U s (x s ) over x s subject to capacity constraints.

5. S Problem Framework s1s1 s2s2 s3s3s.......... DESTINATION NODES SOURCE NODES L(s 1 )={l 1,l 2,l 3,l 4 } link l 4 : S(l 4 )={s 1,s 3 } l1l1 l2l2 l3l3 l5l5 l6l6

6. Problem Framework c1c1 c2c2 Sources s L(s) - links used by source s U s (x s ) - utility if source rate = x s Network – Links l of capacities c l x1x1 x2x2 x3x3

7. The optimization problem : Primal problem

8. Concave utility function TYPICAL CONCAVE UTILITY FUNCTION

9. The optimization problem : Lagrangian for primal problem  p l 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 x s so maximizing lagrangian function as aggregate of different x s related terms means gives the same result as summing up maximum of each individual x s related term. Therefore we have

10. The optimization problem : Dual problem  Here p l is the price per unit bandwidth at link l.  p s 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

11. Source rate as demand function  The above figure depict x s (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 x s (p) acts as a demand function seen in Microeconomics.

12.  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 Algorithm based on gradient projection applied to dual problem

13. Synchronous Distributed Algorithm

14. S Problem Framework s1s1 s2s2 s3s3s.......... DESTINATION NODES SOURCE NODES L(s 1 )={l 1,l 2,l 3,l 4 } l4l4 l1l1 l2l2 l3l3 l5l5 l6l6

15. 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.

16. 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.

18. 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.