1. BSnetworks.pptTKK/ComNet Research Seminar, 7.3.2008 1 SRPT Applied to Bandwidth Sharing Networks (to appear in Annals of Operations Research) Samuli Aalto TKK Helsinki University of Technology, Finland Urtzi Ayesta LAAS-CNRS, France

2. 2 Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations

3. 3 Resource sharing Single resource –loss system (e.g. Erlang model M/G/n/n) one customer per server, no buffer model for telephone traffic at the connection level –queueing system (e.g. M/G/1-FCFS) customers served one-by-one, infinite buffer model for data traffic at the packet level –sharing system (e.g. M/G/1-PS) customers share the server, infinite buffer model for (elastic) data traffic at the flow level Multiple resources –loss networks (e.g. Ross) –queueing networks (e.g. Jackson, BCMP, Kelly,...) –bandwidth sharing networks (Massoulié & Roberts)

4. 4 Bandwidth sharing network Characterization: –Flow-level model of a data network loaded with elastic flows (such as file transfers using TCP) Resources: –Links l with bandwidth c l (bps) –Routes r (route = collection of consecutive links) Traffic: –Elastic flows i with arrival time (possibly random), size (bits to be transferred), and route (fixed) Control: –Bandwidth allocation to flows inter-route bw allocation intra-route bw allocation

5. 5 Example: Symmetric linear network with L = 2 link 1link 2 route 0 route 1 route 2 Symmetric with unit capacities:

6. 6 Bandwidth allocation Inter-route bandwidth allocation –  r = total bandwidth allocated to the flows on route r – feasible allocation satisfies the capacity constraints: Intra-route bandwidth allocation –tells how the total bandwidth  r is shared among the flows using route r –an example: PS = Processor-Sharing = bandwidth is shared evenly among the flows with the same route

7. 7 Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations

8. 8 Static setting Characterization: –Fixed number of saturated flows, n  (n r ; r  R) Problem: –Fair bandwidth sharing Solutions (with PS intra-route discipline): –MMF: max-min fairness [  ]; trad., Jaffe (1981) –PF: proportional fairness [  ]; Kelly (1997) –TM: throughput maximization [  ]; Massoulié & Roberts (1999) –PDM: potential delay minimization [  ]; Massoulié & Roberts (1999) –alpha-fairness [  ]; Mo & Walrand (1998,2000) –U-utility maximization; Ye et al. (2003,2005) –BF: balanced fairness; Bonald & Proutière (2003)

9. 9 Fairness in symmetric linear network with L = 2 alpha-fairness [  ] TM [  ] PF [  ] = BF (in this case, not generally) MMF [  ]

10. 10 Fairness in symmetric linear network with L = 2 Note: Throughput maximization does not specify a unique bandwidth allocation when n   or n   TM as a limit  TM* with preemptive priority to routes 1 and 2

11. 11 Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations

12. 12 Dynamic setting Randomly varying number of flows –Poisson arrivals, generally distributed flow sizes Necessary stability conditions: Definition: –Bandwidth allocation policy is (maximally) stable if the necessary stability conditions are also sufficient Primary concern: –stable bandwidth sharing Secondary concern: –(mean) delay optimization among stable bandwidth allocations

13. 13 Single (bottleneck) link M/G/1 queue Fair bandwidth sharing: PS (Processor-Sharing) Stability: WC (Work-Conserving disciplines) Anticipating mean delay optimization: SRPT (Shortest- Remaining-Processing-Time); Schrage (1968) Non-anticipating mean delay optimization for DHR (Decreasing Hazard Rate) service times: FB (Foregroud- Background) = LAS (Least-Attained-Service); Yashkov (1987)

14. 14 Stable bw allocations for multilink networks Exponential flow sizes –PF stable in linear networks; Massoulié & Roberts (1998) –MMF stable in linear networks; De Veciana et al. (1999) –alpha-fair bw allocations stable for any topology; Bonald and Massoulié (2001) –U-utility maximization bw allocations stable for any topology; Ye et al. (2003,2005): General flow sizes –BF stable for any topology; Bonald and Proutière (2003) –MMF stable for any topology; Bramson (2005) –PF stable for any topology; Massoulié (2005) –alpha-fair bw allocation stable for tree topologies; Gromoll and Williams (2006) Note: Above, intra-route discipline always PS

15. 15 Stable bw allocations for multilink networks Verloop et al. (2006): –PR0 and PR12 stable in symm. linear network PR0 gives preemptive priority to class 0 whenever nonempty PR12 gives preemptive priority to classes 1 and 2 whenever both of them are nonempty; otherwise preemptive priority is given to class 0 –Intuitive argument: Both policies ensure that the whole capacity of each link is used whenever there are flows loading the link Note: PR12  TM* which gives preemptive priority to classes 1 and 2 whenever at least one of them is nonempty

16. 16 Unstable bw allocations for multiple link networks Bonald and Massoulié (2001): –TM* not maximally stable in linear network –TM* stable in symmetric linear network only if Verloop et al. (2005): –global SRPT not maximally stable in linear network –global LAS not maximally stable in linear network Note: In all these cases, the whole capacity of a link is not necessarily used while there are flows loading the link

17. 17 Delay optimization among stable bw allocations Yang & de Veciana (2002,2004): –optimal allocation:  r  n  or  (depending on state n ) in symmetric linear network Verloop et al. (2006): –determined optimal non-anticipating allocation in symmetric linear network with exponential flow sizes –if  1  2  0, then PR0 optimal –if  1  2  0 and  1  2  0, then PR12 optimal Bonald and Proutière (2003): –BF insensitive for any topology

18. 18 Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations

19. 19 Theoretical setup Consider a BS network with –a general topology, –Poisson arrivals, and –generally distributed flow sizes  = family of stable bw allocation policies 

20. 20 Delay reduction by local SRPT’  n = family of stable bw allocation policies  for which where – Z r (t) = total bw allocated to class r at time t – N r (t) = number of flows on route r at time t – N(t)  (N r (t); r  R) Note: All fair bw allocation policies mentioned above   n

21. 21 Delay reduction by local SRPT’ Let  n be fixed.  = a modified policy –with the same inter-route bw allocation process, –but the intra-route disciplines may be different from the original ones  ’ = the modified policy –that applies SRPT as the intra-route discipline  * = the policy –for which the inter-route bw allocation process is – and that applies SRPT as the intra-route discipline 

22. 22 Delay reduction by local SRPT’ Note the difference between  ’ and  * Proposition 1: –Let  n, r  R and t . For any modification  (including  ), Theorem 1: –Let  n. For any modification  (including  ), Here T refers to delay (= total transfer time)  

23. 23 Delay reduction by local SRPT*  w = family of stable bw allocation policies  for which where – Z r (t) = total bw allocated to class r at time t – W r (t) = total workload on route r at time t – W(t)  (W r (t); r  R) Note: Policies PR0 and PR12   w

24. 24 Now there is no difference between  ’ and  * Proposition 2: –Let  w, r  R and t . For any modification  (including  ), Theorem 2: –Let  w. For any modification  (including  ), Delay reduction by local SRPT*  

25. 25 Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations

26. 26 Simulation setup Symmetric linear network with L = 2 and unit capacities –Poisson arrivals with constant total rate –Flow size distribution with mean b  deterministic exponential:  b hyperexponential: p 1 ,  1  b ; p 2 ,  2   b Now, PR12 is the optimal non-anticipating allocation policy Mean delay comparison between basic policy  and its SRPT- modifications  ’ and  * using basic policies –BF = PF, PR0, PR12

27. 27 Deterministic flow sizes PR12   ’  * PR12 BF

28. 28 Exponential flow sizes   ’  * PR12 BF PR12

29. 29 Hyperexponential flow sizes   ’  * PR12 BF

30. 30 Increasing variability BFPR0 PR12

31. 31 Observations Mean delay improved by  ’ for each class as Prop 1 predicts As Prop 2 implies, for the basic policies PR0 and PR12 In all numerical cases, for the basic policy BF In all numerical cases, for all basic policies (BF, PR0, PR12) Basic policy PR12 is better than BF for deterministic and exponential flow sizes but worse for hyperexpontial Delay reduction of BF very similar for exponential and hyperexponential flow sizes

32. 32 THE END