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Internet Networking Spring 2003

Internet Networking Spring 2003. Tutorial 9 Max-Min Fairness. Motivation. Given a network and sessions we would like to find maximal flow that it is fair We will see different definitions for max-min fairness and will learn a flow control algorithm

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Internet Networking Spring 2003

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  1. Internet Networking Spring 2003 Tutorial 9 Max-Min Fairness

  2. Motivation • Given a network and sessions we would like to find maximal flow that it is fair • We will see different definitions for max-min fairness and will learn a flow control algorithm • The tutorial will give understanding what is max-min fairness Session 2 Session 1 Session 3 Session 0 קיבול = 1 קיבול = 1

  3. Intuition of Fairness Intuitive notion of fairness is that any session is entitled to as much network use as is any other session

  4. Example of max-min fair flow control Session 2 Session 3 Session 1 קיבול = 1 קיבול = 1 Session 0 Maximal fair flow division will be to give for the sessions 0,1,2 a flow rate of 1/3 and for the session 3 a flow rate of 2/3

  5. Max-Min Flow Control Rule • The rule is maximizing the network use allocated to the sessions with the minimum allocation • An alternative definition: is to maximize the allocation of each session i under constrain that an increase in i’s allocation doesn’t cause a decrease in some other session allocation with the same or smaller rate than i

  6. Max-Min Notation • Directed network graph G=(N,A) (N is set of vertexes and A is set of edges) • Ca – is the capacity of a link a • Fa – is the flow on a link a • P – is a set of the sessions • r – is some flow in the network • rp – is the rate of a session p • We assume a fixed, single-path routing method

  7. Max-Min Definitions We have following constraints on the vector r = {rp | p Є P} A vector r satisfying these constraints is said to be feasible

  8. Max-Min Definitions • A vector of rates r is said to be max-min fair if is a feasible and for each p Є P, rp can not be increased while maintaining feasibility without decreasing rp’ for some session p’ for which rp’ ≤ rp • We want to find a rate vector that is max-min fair

  9. Bottleneck Link for a Session • Given some feasible flow r, we say that a is a bottleneck linkwith respect to r, for a session p crossing a if Fa = Caand rp ≥ rp’for all sessions p’ crossing link a d 4:1 b 1:2/3 1 4 5 c 5:1/3 2 3 3:1/3 2:1/3 a All link capacity is 1. Bottlenecks for 1,2,3,4,5 respectively are: c,a,a,d,a Put attention: c is not a bottleneck for 5 and b is not a bottleneck for 1

  10. Max-Min Fairness Definition Using Bottleneck Theorem: A feasible rate vector r is max-min fair if and only if each session has a bottleneck link with respect to r Proof: Exercise

  11. Algorithm for Computing Max-Min Fair Rate Vectors The idea of the algorithm: • Bring all the sessions to the state that they have a bottleneck link and then according to theorem it will be the maximal fair flow • We start with all-zero rate vector and to increase rates on all paths together until Fa = Cafor one or more links a. • At this point, each session using a saturated link has the same rate as every other session using this link. Thus, these saturated links serve as bottleneck links for all sessions using them

  12. Algorithm for Computing Max-Min Fair Rate Vectors • At the next step, all sessions not using the saturated links are incremented equally in rate until one or more new links become saturated • Note that the sessions using the previously saturated links might also be using these newly saturated links (at a lower rate) • The algorithm continues from step to step, always equally incrementing all sessions not passing through any saturated link until all session pass through at least one such link

  13. Algorithm for Computing Max-Min Fair Rate Vectors Init: k=1, Fa0=0, rp0=0, P1=P and A1=A • For all aA, nak:= num of sessions pPk crossing link a • Δr=minaAk(Ca-Fak-1)/nak (find inc size) • For all p  Pk, rpk:=rpk-1+ Δr, (increment)for other p, rpk:=rpk-1 • Fak:=Σp crossing arpk (Update flow) • Ak+1:=קבוצת הקשתות הלא רוויות כעת • Pk+1:=all p’s, such that p doesn’t cross links in Ak+1 • k:=k+1 • If Pk is empty then stop, else goto 1

  14. Example of Algorithm Running d 4:1 b 1:2/3 1 4 5 c 5:1/3 2 3 3:1/3 2:1/3 a All link capacity is 1 Step 1: All sessions get a rate of 1/3, because of a and the link a is saturated. Step 2: Sessions 1 and 4 get an additional rate increment of 1/3 for a total of 2/3. Link c is saturated now. Step 3: Session 4 gets an additional rate increment of 1/3 for a total of 1. Link d is saturated. End

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