1. Bandwidth sharing: objectives and algorithms Jim Roberts France Télécom - CNET Laurent Massoulié Microsoft Research

2. Motivation Optimize TCP for Web-like traffic transport  adequate rate sharing principles?  control laws realising such rate shares?

3. Outline Rate sharing objectives – max-min fairness [ Bertsekas & Gallager ] – proportional fairness [ Kelly ] – “minimum delay” Distributed implementation (users / network) – fixed window control / scheduling + buffering – stochastic algorithms and adaptive sending rates

4. Network model – links l, capacity C l – users r (for route), r  subset of links –feasible rate allocations: rate r to route r so that  l,  r  l r  C l –e.g., a naïve approach: maximize  r r (e.g., 0 =0, r =1, r>0) route 1 route 2 route L route 0 link 1 link 2 link L linear network :

5. Rate sharing objectives  max-min fairness:  r,  bottleneck l  r so that  r’  l r’ =C l and r =max r’  l r’ (e.g., r =1/2  r)  proportional fairness: choose r so as to maximise  r log( r ) (e.g., 0 =1/(L+1) and r =L/(L+1)  r>0)  minimum delay criterion: choose r so as to minimise  r 1/ r (e.g., 0 =1/(1+  L) and r =  L/(1+  L)  r>0)

6. Fixed window control route r  window size B r, round trip time T r –fluid model (packet size  0) –greedy sources –buffering at links ’ access –no congestion on return path some relevant work: [Hahne&Gallager], [Mitra&Seery], [Mo&Walrand] source destination packets ack’s

7. Fixed window control (ctd) Theorem: assume scheduling among different routes at each link is FIFO (resp., Longest Queue First, Fair Queueing,  Q-proportional). Then system admits unique static regime, with r characterized as: –argmax  r B r log( r ) - r T r (FIFO) –argmax  r B r r - (1/2)T r ( r ) 2 (LQF) –max-min fair allocation with bound (B r / T r ) on r (FQ) –argmin  r (B r / r ) + T r log( r )(  Q- prop.)

8. s1s1 s2s2 d2d2 d1d1 Example FIFO scheduling if B 1 /T 1 +B 2 /T 2  1 then i =B i /T i, i=1,2 otherwise 1 + 2 = 1, where 1 as a function of T 1, T 2 (B 1 = B 2 =1)

9. Stochastic algorithms for rates updates  route r  sending rate  r, for integer-valued r  user r adapts r randomly according to r : n  n+1 at rate b(n) if compatible with capacity constraints, 0 otherwise, r : n  n -1 at rate d(n)  distributed mechanism based on binary information “rate increment by  feasible or not”  simplifying assumptions: – no queueing at links – feedback information instantaneously available

10. Stochastic algorithms (ctd)  Stationary distribution:  ({ r })   r [b(0)…b( r -1)] / [d(1)…d( r )] 1 {constraints satisfied}  e.g., b(n)  b, d(n)  d:   ({ r })  exp [ log(b/d)  r r ] 1 {constraints satisfied} as b/d increases,  concentrates on argmax  r r  e.g., b(n)=(n+1)a, d(n)=(n-1)a:   ({ r })  exp [ a  r log( r )] 1 {constraints satisfied} as b/d increases,  concentrates on argmax  r r

11. Deterministic increase/decrease rules  Smooth deterministic sending rates updates, based on same binary information: d r /dt = f r ( r ) if capacity available, = - g r ( r ) otherwise. e.g., TCP-like additive increase/multiplicative decrease: f r , g r ( r )= r Theorem: these dynamics admit as stable points argmax

12. Deterministic increase/decrease rules (ctd)  e.g., additive increase-multiplicative decrease mechanism: f r  , g r ( r ) = r stable points at argmax  argmax  r log( r )  “additive increase / multiplicative decrease achieves proportional fairness” (cf. [Kelly et al.], [Leboudec et al.])  e.g., f r ( r )=1/ r, g r ( r )= r stable points near argmin  r 1/ r  “logarithmic increase / multiplicative decrease achieves minimum delay”

13. Conclusions Introduction of minimum delay rate sharing principle Analysis of fixed window control  impact of scheduling and round trip delays New class of stochastic algorithms  impact of end user reaction to congestion For further study: –consideration of dynamic users population –combination of end users reaction, scheduling and round trip delays