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'Proportional Fairness: Dynamics, Stability and Pathology'

'Proportional Fairness: Dynamics, Stability and Pathology'. John Carroll Cambridge University Engineering Department. JEC; CUED; NORTEL. Acknowledgements. (Nortel) Paul Kirkby Sabesan Subramaniam. Martin Biddiscombe John Hudson Radhakrishnan Kadengal (Cambridge) Frank Kelly.

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'Proportional Fairness: Dynamics, Stability and Pathology'

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  1. 'Proportional Fairness: Dynamics, Stability and Pathology' John Carroll Cambridge University Engineering Department JEC; CUED; NORTEL

  2. Acknowledgements • (Nortel) • Paul Kirkby • Sabesan Subramaniam. • Martin Biddiscombe • John Hudson • Radhakrishnan Kadengal • (Cambridge) • Frank Kelly JEC; CUED; NORTEL

  3. Background • Internet traffic needs differentiation of services • Premium traffic (perhaps different grades) • ‘Guaranteed’ delivery at a price - different delays • Best efforts - may be lost • ‘Proportional Fairness’ is one scheme being studied for dynamic pricing to control access to network JEC; CUED; NORTEL

  4. Outline • 1. Proportional Fairness - key concepts • 2. Form of Network • 3. Routes, Resources, Capacities • 4. Theory of Small Changes • 5. Stability (frequency domain) • 6. Stability (time domain) • 7. Price Limited Proportional Fairness • 8 Nil-Change and Max-Change Offers • 9. Conclusions JEC; CUED; NORTEL

  5. Proportional Fairness - key concepts • 1. Each resource on network has price/bandwidth/ unit time specific to that resource. • This price varies with time. No limits. • 2. All customers pay the price (Lp) for using the resource p • 3. Resource p has a limited capacity (Cp). • 4. Customers bandwidth-allocation along each route (covering many resources) is determined from the amount they are willing to pay for the route : ‘WtP’ or ‘bid’ • 5. All resource prices are continually adjusted so as to fill the capacity of all resources, given customers bid. JEC; CUED; NORTEL

  6. Outline • 1. Proportional Fairness - key concepts • 2. Form of Network • any network with pre-assigned routes of limited capacity. • simple example of 4 node ring to illustrate algebra JEC; CUED; NORTEL

  7. Network formed with 4 nodes servicing 16 routes Node 1 Node 2 Node 4 Node 3 JEC; CUED; NORTEL

  8. Outline • 1. Proportional Fairness - key concepts • 2. Form of Network • 3. Routes, Resources, Capacities JEC; CUED; NORTEL

  9. Routes & prices x11 x12 x13 x14 x21 x22 x23 x24 = x x31 x32 x33 x34 x41 x42 x43 x44 ordered routes m11 m12m13m14 m21m22m23m24 = mcolumn vector m31m32m33m34 m41m42m43 m44 Willingness-to-pay or bid price JEC; CUED; NORTEL

  10. Node 1 Resource 1 price L1capacity C1 Resource 4 price L4capacity C4 Node 2 Node 4 Resource 3 price L3capacity C3 Resource 2 price L2capacity C2 Node3 JEC; CUED; NORTEL

  11. Capacities & price/bw. for resources • x11m11 x12 m12 C1 L1 x13 m13 x14 m14 • x21 m21 x22 m22 C2 L2 x23 m23 =C=L x24 m24 • x31 m31 x32 m32 C3 L3 x33 m33 x34 m34 • x41 m41 x42 m42 C4 L4 x43 m43 x44 m44 JEC; CUED; NORTEL

  12. 10 Routes filling Resource 1 - C1 Node 1 • x11 started at t x12 x13 x14 x21 x22 x23 x24started at t -3T x31 x32 x33 started at t - 2T x34 x41 x42 x43 started at t - Tx44 0 Node 2 -3T -T Node 4 Node 3 -2T JEC; CUED; NORTEL JEC; CUED; NORTEL

  13. Delay Operator • All delays in unit of hop-time T (all hops taken of equal length) • Delay operator z-1 f( t- T) = z-1 f • For single frequency w then z-1 = exp( -jw T) JEC; CUED; NORTEL JEC; CUED; NORTEL

  14. If Capacity C1 Fully Used:- • C1= 1 xx11 + 1xx12 + 1 xx13 + 1 xx14 + 0xx21 + 0 xx22 + 0xx23+ z-3 xx24 + 0 xx31 + 0 xx32 + z-2xx33+ z-2 xx34 + 0 xx41 + z-1xx42 + z-1xx43+ z-1 xx44 • Similar calculations for capacities C2 , C3 and C4 • C =Scapx global matrix form JEC; CUED; NORTEL JEC; CUED; NORTEL

  15. Price/bandwidth for Routes • x11 uses L1 x12 uses L1 + L2 x13 uses L1 + L2 + L3 x14 uses L1 + L2 + L3+ L4 Node 1 as typical JEC; CUED; NORTEL JEC; CUED; NORTEL

  16. JEC; CUED; NORTEL Price/route That Must Be Offered • x11x(L1 ) = m11 x12x(L1 + z-a L2) = m12x13x(L1 + z-a L2 + z-b L3) = m13 x14x(L1 + z-a L2 + z-b L3+ z-c L4) = m14 • Yellow terms calculated directly (globally or locally) from previous capacity equations • Delay operators depend on how prices become known to manager at node 1 • Additional smoothing can be introduced • Similar calculations for nodes 2 , 3 & 4

  17. Offer/allocation = STRUCTURE* Price/resource • m11 / x11 1 0 0 0 m12 / x12 1 z-a 0 0 m13 / x13 1 z-a z-b 0 m14 / x14 1 z-a z-b z-cm21 / x21 0 1 0 0 m22 / x22 0 1 z-a 0 L1 m23 / x23 0 1 z-a z-b L2 m24 / x24 = z-c 1 z-a z-b L3 m31 / x31 0 0 1 0 L4 m32 / x32 0 0 1 z-a m33 / x33 z-b 0 1 z-a m34 / x34 z-b z-c 1 z-a m41 / x41 0 0 0 1 m42 / x42 z-a 0 0 1 m43 / x43 z-a z-b 0 1 m44 / x44 z-a z-b z-c 1 JEC; CUED; NORTEL JEC; CUED; NORTEL

  18. Offer/allocation = STRUCTURE* Price/resource • m11 / x11 1 0 0 0 m12 / x12 1 Da 0 0 m13 / x13 1 Da Db 0 m14 / x14 1 Da Db dcm21 / x21 0 1 0 0 m22 / x22 0 1 Da 0 L1 m23 / x23 0 1 Da Db L2 m24 / x24 = Dc 1 Da Db L3 m31 / x31 0 0 1 0 L4 m32 / x32 0 0 1 Da m33 / x33 Db 0 1 Da m34 / x34 Db Dc 1 Da m41 / x41 0 0 0 1 m42 / x42 Da 0 0 1 m43 / x43 Da Db 0 1 m44 / x44 Da Db dc 1 Sres L (m./x) MATLAB Notation JEC; CUED; NORTEL JEC; CUED; NORTEL

  19. Matrix Formulation of Network • (m./x) = SresL resources • C = Scapx capacities • z-1 = 1 in steady state whenScap = Srestr:incidence matrix • Compact but non-linear because of 1/x JEC; CUED; NORTEL JEC; CUED; NORTEL

  20. Outline • 1. Proportional Fairness - key concepts • 2. Form of Network • 3. Routes, Resources, Capacities • 4. Theory of Small Changes JEC; CUED; NORTEL

  21. m®m + Dm bids x®x + Dx allocations of bandwidth C®C + DC resource capacities L®L + DL resource prices D indicates small changes from steady state. Neglect Dx. Dx , Dx.Dm etc. Matrix Formulation: Small Changes JEC; CUED; NORTEL JEC; CUED; NORTEL

  22. Define: D %m=Dm. /m: fractional bid changes; D %x=Dx. /x : fractional alloc. changes; Xd as a diagonal matrix with diagonal elementsx (allocations); steady stateinformation. Md as a diagonal matrix with diagonal elementsm (willingness to pay); steady state information. Matrix Formulation: Small Changes JEC; CUED; NORTEL JEC; CUED; NORTEL

  23. Matrix Formulation: Small Changes • Using the constraint on Dx it is possible to show that changes in bids m and resource prices L are linked • ScapXd Dm = RDL where • R = ScapXdMd-1XdSres • R must have inverse to find DL • Det|R| ¹ 0 at all real w JEC; CUED; NORTEL JEC; CUED; NORTEL

  24. Matrix Formulation: Small Changes • R matrix (Mnodex Mnode) encapsulates steady state values and information delays. • Det|R| always has zeros at some complex frequency wreal-jwimag • ‘Zeros’ give outputs with no inputs - transients • If wimag> 0 then transients grow • ® Instability JEC; CUED; NORTEL JEC; CUED; NORTEL

  25. Small changes allow one to integratefrom approximate to ‘exact’ solutions. Allow integration from one exact solution to another with different bids Examine effects of ‘coalitions’ where customers combine in cartels. Here we concentrate on stability of allocations in response to varying bids JEC; CUED; NORTEL Matrix Formulation: Small Changes

  26. Outline • 1. Proportional Fairness - key concepts • 2. Form of Network • 3. Routes, Resources, Capacities • 4. Theory of Small Changes • 5. Stability (frequency domain) JEC; CUED; NORTEL

  27. Stability Criterion • Calculate a typical steady state; • Allow for small changes • Calculate y= log(abs(det|R|)) over a grid of complex frequencies; note (det|R|)) periodic in wreal • Seek minima in y ; • Check if minima deepen for increasing wimag> 0 in wreal-jwimag JEC; CUED; NORTEL JEC; CUED; NORTEL

  28. Example 1 reverse price propagation JEC; CUED; NORTEL JEC; CUED; NORTEL

  29. Propagation of Information about Price/bandwidth on resources Node 1 Resource 4: known to manager at node 1 from update at t= -3T Node 2 Node 4 Node 3 JEC; CUED; NORTEL JEC; CUED; NORTEL

  30. 4 3 2 1 0 Loge[abs(det R)] as a function of wr T and wi T (pricing information travels by reverse route) Linefor wi T = 0 increasing wi T steps of 0.025, (decreasing depth of minima indicating stability) 0 p/2 wr T JEC; CUED; NORTEL JEC; CUED; NORTEL

  31. Example 2 forward price propagation JEC; CUED; NORTEL JEC; CUED; NORTEL

  32. Propagation of Information about Price/bandwidth on resources Node 1 Resource 4: known to manager at node 1 from update at t= -T Node 2 Node 4 Node 3 JEC; CUED; NORTEL JEC; CUED; NORTEL

  33. 4 3 2 1 0 Loge[abs(det R)] as a function of wr T and wi T (pricing information travels by completing ring) Linefor w i T = 0 increasing wi T; steps of 0.0125, (initial increase in minima indicating location of instability) wr T 0 p/2 JEC; CUED; NORTEL JEC; CUED; NORTEL

  34. Stability Theorem • If feedback system is small signal stable/unstable in one steady state, then it will be stable/unstable in neighbouring steady states. • Margin of stability needs criteria - given from magnitude of wimag • Stability of ‘a calculation system’ (w = 0) does not guarantee stability of dynamic system.

  35. Outline • 1. Proportional Fairness - key concepts • 2. Form of Network • 3. Routes, Resources, Capacities • 4. Theory of Small Changes • 5. Stability (frequency domain) • 6. Stability (time domain) JEC; CUED; NORTEL

  36. JEC; CUED; NORTEL Local Price DeterminationReverse Propagation of Price • Local manager at each node controlling ingress of traffic to network • Local manager controls local resource prices to match supply and demand ( bids) • Local Manager informed of distant resource prices (delayed information) • Has sufficient information to determine local resource prices and to determine a locally proportionally fair allocation

  37. Example : reverse price propagation with local price control JEC; CUED; NORTEL

  38. Group 1 Group 2 Group 3 Group 4 Normalised offers - - - - dynamic allocations _____o 28 initial start 5 Timestep 1 4 8 12 16 Route no. JEC; CUED; NORTEL JEC; CUED; NORTEL

  39. Normalised offers - - - - dynamic allocations _____o 29 - step change 5 1 4 12 16 8 Route no. JEC; CUED; NORTEL JEC; CUED; NORTEL

  40. Normalised offers - - - - dynamic allocations _____o 31 5 1 4 12 16 8 Route no. JEC; CUED; NORTEL JEC; CUED; NORTEL

  41. Normalised offers - - - - dynamic allocations _____o 34 5 1 4 12 16 8 Route no. JEC; CUED; NORTEL JEC; CUED; NORTEL

  42. Normalised offers - - - - dynamic allocations _____o 36 5 Steady state allocation 1 4 12 16 8 Route no. JEC; CUED; NORTEL JEC; CUED; NORTEL

  43. System stabilises for even quite large changes in 4 - 12 time steps (1 to 3 circuits) JEC; CUED; NORTEL JEC; CUED; NORTEL

  44. Outline • 1. Proportional Fairness - key concepts • 2. Form of Network • 3. Routes, Resources, Capacities • 4. Theory of Small Changes • 5. Stability (frequency domain) • 6. Stability (time domain) • 7. Price Limited Proportional Fairness JEC; CUED; NORTEL

  45. ‘Pathology’ of Proportional fairness • All may not seem fair in a proportionally fair system • One set of bids m gives bandwidth allocations x then km any another set where k is the same for all routes gives same allocations! • even if k =0.001 or 10there is still no change in allocation to any one route JEC; CUED; NORTEL

  46. Price Limited Proportional Fairness • 1. Each resource on network has price/bandwidth/ unit time specific to that resource. • This price varies with time. Not allowed to fall below a specified level at each resource. • 2. All customers pay the price (Lp) for using the resource p • 3. Resource p has a limited capacity (Cp). • 4. Customers bandwidth-allocation along each route (covering many resources) is determined from the amount they are willing to pay for the route : ‘WtP’ or ‘bid’ • 5. Capacity of all resources is only filled if customers pay sufficient. JEC; CUED; NORTEL

  47. Dynamic Example: reverse price propagation with local price control and Price-Limited Proportional Fairness (PLPF) JEC; CUED; NORTEL JEC; CUED; NORTEL

  48. S P F Offer PLP F 28 initial start 5 Timestep Group 1 Group 2 Group 3 Group 4 1 4 8 12 16 Route no. JEC; CUED; NORTEL JEC; CUED; NORTEL

  49. S P F Offer PLP F 30 5 Better PLPFcontrol 1 4 12 16 8 Route no. JEC; CUED; NORTEL JEC; CUED; NORTEL

  50. S P F Offer PLP F 32 5 1 4 12 16 8 Route no. JEC; CUED; NORTEL JEC; CUED; NORTEL

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