1. D. AriflerCMPE 548 Fall 20051 CMPE 548 Routing and Congestion Control

2. D. AriflerCMPE 548 Fall 20052 Flow models and optimal routing Origin for OD pair w 1 Origin for OD pair w 2 Destination for OD pair w 1 r w1 r w2 x1x1 x2x2 x3x3 W: Set of all OD pairs w 1, w 2, … P W : Set of all directed paths connecting OD pair w x p : Flow (data units/s) of path p r w : Traffic input of OD pair w Path 1 Path 2 Path 3

3. D. AriflerCMPE 548 Fall 20053 Formulation of optimal routing problem Define the following: –C ij : Transmission capacity of link (i,j) –F ij : Traffic arrival rate on link (i,j) –d ij : Processing and propagation delay Cost function: –Average no. of packets on link (i,j): Optimal routing problem: Note: D ij is defined on [0,C ij )

4. D. AriflerCMPE 548 Fall 20054 Congestion/flow control Congestion/flow control: Restricting input rates to “reasonable” levels Such restriction must be done fairly! –Fairness characterizes how competing users should share bottleneck resources (See “Max-Min Fairness” lecture) Assume we are given “utility functions” U w –Utility functions map service delivered (rate) into performance We will consider the problem that combines optimal routing and congestion control: –Select x p and r w to maximize sum of utilities of flows minus the sum of link costs

5. D. AriflerCMPE 548 Fall 20055 Utility functions Suppose an amount x of commodity has utility (or value) U(x) to a consumer –U(x) is called the utility function Utility functions are non-decreasing, continuously differentiable, concave functions –e.g. U(x)=βlog(x) for β>0 Marginal utility U’(x): –If x is increased by a small amount ε, the utility U(x) is increased by εU’(x)+o(x) –Since U is concave, U’ is monotone non-increasing in x

6. D. AriflerCMPE 548 Fall 20056 Fair allocation (1) Consider a vector of allocations x=(x 1, …, x n ) –x i : Amount allocated to i th user Suppose x is constrained to lie in a set S assumed to be a closed, bounded and convex subset of –e.g. S={ Ax<c } for matrix A and vector c –A set is convex if the line segment joining any two points in the set is also in the set Suppose U i measures the “value” of allocation to user i

7. D. AriflerCMPE 548 Fall 20057 Fair allocation (2) A reasonable allocation criteria is to select x to solve –If x* is optimal and x is any other vector in S and 0<ε≤1, then the vector x*+ε(x-x*) lies on the line segment joining x* and x, hence such vector is in S –By optimality of x*

8. D. AriflerCMPE 548 Fall 20058 Fair allocation (3) Letting ε→0, the following is a necessary and sufficient condition for x* to be optimal: When we have logarithmic utility functions, U i (x i )=β i log(x i ):

9. D. AriflerCMPE 548 Fall 20059 Proportionally fair allocation This states that no vector x can offer a positive weighted sum of normalized improvement over x* A vector x* that satisfies the inequality above is said to be proportionally fair (with weight β i )

10. D. AriflerCMPE 548 Fall 200510 Optimal routing and congestion control Origin for OD pair w 1 Destination for OD pair w 1 r w1 x1x1 x2x2 x3x3 r w : Desired input by OD pair w x p : Flow (data units/s) of path p r w : Traffic input of OD pair w y w : Overflow (portion blocked out) Overflow link: y w =r w -r w

11. D. AriflerCMPE 548 Fall 200511 Combined problem Combined optimal routing and congestion control problem:

12. D. AriflerCMPE 548 Fall 200512 Additional note on the constraints Note that in optimal routing problem (and in the one combined with congestion control), one additional obvious constraint is the capacity constraint F ij <C ij This is normally ignored since D ij →∞ as F ij →C ij, and its effect is automatically brought into play

13. D. AriflerCMPE 548 Fall 200513 Additional note on S={ Ax<c } The constraint xεS ensures that the flow on link m does not exceed its capacity –A is called the incidence matrix –A mn =1 if path n crosses link m, and A mn =0, otherwise –c is a vector whose elements are the capacities of corresponding links 12 34 x1x1 x2x2 Example:

14. D. AriflerCMPE 548 Fall 200514 Additional note on the derivative Question: Apply the definition of the derivative to the expression on slide 7 to obtain the necessary and sufficient condition on slide 8.

15. D. AriflerCMPE 548 Fall 200515 Additional notes: (p,α)-proportionally fair Let p=(p 1, …, p n ) and α be positive numbers x* is (p,α)-proportionally fair if it is feasible and for any other x –This reduces to proportionally fair when p=(1, …, 1) and α=1 –(p,α)-proportionally fair rate vector approaches max-min fair rate vector as α→∞ –Max-min fairness is the most common notion of fairness: x is max-min fair if any rate x i cannot be increased without decreasing some x j which is smaller than or equal to x i

16. D. AriflerCMPE 548 Fall 200516 Additional notes on max-min fairness Resources are allocated in order of increasing demand No source gets a resource share larger than its demand Sources with unsatisfied demands get an equal share of the resource For more, see lecture on “Max-Min Fairness”