Effect of Router Buffers on Stability of Internet Conges8on Control Algorithms

Transcription

1 Effect of Router Buffers on Stability of Internet Conges8on Control Algorithms Somayeh Sojoudi Steven Low John Doyle Oct 27,

2 Resource alloca+on problem Objec8ve Fair assignment of rates to the users to maximize the u8lity (Kelly, 1997): 2

3 Conges+on control: A decentralized solu+on to the resource alloca+on problem Conges8on control algorithms: Primal Dual Primal- dual 3

4 Primal- dual: Dual: 4

5 Communica8on from links to sources & vice versa: " Queueing delay (implicit) Communica8on from sources to links: " Queue size (implicit)

6 Queue dynamics: Link price dynamics: 6

7 Pricing mechanisms: i. Queueing delay ii. Queue size Conges8on control algorithms are globally asympto8cally stable! 7

8 Pricing mechanisms: i. Queueing delay ii. Queue size Conges8on control algorithms are globally asympto8cally stable! Unrealis+c assump+on: input rate at every link = source rate x s (t) x s (t) x s (t) x s (t) 8

9 Pricing mechanisms: i. Queueing delay ii. Queue size Conges8on control algorithms are globally asympto8cally stable! Unrealis+c assump+on: input rate at every link = source rate x s (t) x s (t) x s (t) x s (t) True model: input rate at link = output rate at previous link x s (t) 9

10 Goal: Derive a more accurate model, which takes into account the effect of buffering on output flows. Study the stability of the dual and primal- dual algorithms. 10

11 New model of queue dynamics Source 1 Source 2 11

12 Define: for different service disciplines: Generalized Weighted Fair Queuing (WFQ) First in First out (FIFO) 12

13 Generalized WFQ Source 1 Source 2 If, then 13

14 First in First out: Source 1 Source 2 14

15 i s Diagonal entries are equal to 1 Nonsingular matrix by assump8on 15

16 Queue dynamics 16

17 Stability of primal- dual and dual Dual: Primal- dual: Queue dynamics Common model: More accurate model: 17

18 Stability of primal- dual 18

19 Stability of primal- dual Lineariza8on around the equilibrium point: Unstable eigenvalues : 19

20 Stability of primal- dual Simula8on for ini8al values 20

21 Stability of primal- dual Sketch of the proof: Part i) Lyapunov func8on Part ii) Decompose the lineariza8on matrix 21

22 Stability of primal- dual 22

23 Stability of dual No unstable example. Why? i. Θ is highly structured. ii. Dual has less dynamics involved compare to primal- dual. 23

24 Stability of dual and primal- dual 24

25 Remarks: i. Explicit feedback and stability or Implicit communica8on and gebng close to instability? ii. General ideas w.r.t control and queueing (TCP, an illustra8ve example). iii. The issues will presumably be present in any flow- based control (may be of increasing interest as we try to design new architectures). 25

26 Summary Derived a new model which takes into account the effect of buffering on output flows. Studied the stability of the conges8on control algorithms under the new model (these algorithms might no longer be stable!). Provided sufficient condi8ons for the stability of dual and primal- dual algorithms. Proposed a new pricing mechanism for the stability of these algorithms. 26

27 GEOGRAPHICAL LOAD BALANCING Zhenhua Liu, Minghong Lin Adam Wierman, Steven Low Caltech Lachlan Andrew Swinburne, Australia

28 Energy optimization Dynamic voltage scaling Spin down or park the disk Dynamic transmission power Applica8on level power manager Dynamically turn on/off servers Adam Wierman, Caltech

29 Geographical load balancing 8 6 workload 4 Tradeoff between energy & delay Optimize over request routing and server capacity Balance propagation delay (which DC?) queueing delay (how much server capacity at DC?) energy cost hour Exploit fluctuations of price & traffic over time & space 2

30 30 λ 2j (t) d λ 1j (t) λ ij (t) Source j Σ i λ ij (t)=l j (t) Mean arrival rate at time t is L j (t) Data Center i # of active servers is m i (t) Total # of servers is M i

31 31 λ 2j (t) d λ 1j (t) λ ij (t) Source j Σ i λ ij (t)=l j (t) Mean arrival rate at time t is L j (t) Data Center i # of active servers is m i (t) Total # of servers is M i Goal energy cost delay costs Dynamic Dynamic capacity routing provision

32 energy cost delay costs Goal Structural implications of optimality condition Sources only choose data centers with the same, lowest marginal cost All datacenter utilizations are equalized Simplest (sparse) routing structure DC Liu, Lin, Wierman, L., Andrew, Sigmetrics 2011 Req

33 33 Setup: Workload d ij Workload: I/O Microsoft Cambridge from 8 servers over a 48-hour period, starting at midnight PDT on Monday August 4, one source per state workload scaled by population with Internet accesses shift by time zones propagation delay proportional to distance

34 Setup Hotmail traffic trace Aug 4, hr dura8on 48- hr of traffic traces for Hotmail 20 data centers (Google- like) Industrial electricity price of each state in May 2010 Compare op8mal strategy and min- delay strategy rela8ve to min- load strategy

35 Op8mizing delivery Global load balancing tradi8onally op8mizes for server load or distance, not energy Route jobs to data centers with min load min distance from client

36 Op8mizing delivery Our proposal: jointly minimize delay and energy consump8on must op8mally balance load, distance, electricity cost Control ac8ons available Rou8ng: which data center to serve a job/request Server: how many servers to ac8vate (vs in sleep mode) Exploit fluctua8ons in traffic & electricity prices Both fluctuate across 8me and space Op8mally match traffic, requirements, prices

37 Example: optimizing delivery Conclusion Optimal strategy Min-delay strategy Min-load strategy Overall performance Electricity cost Best Good Poor Best Poor Poor Delay Good Best Poor prevalent strategy

38 Op8mal strategy is 25% to 50% beler overall rela8ve to min- load strategy % improvement in total performance rela8ve to min- load scheme op8mal strategy is 25% - 50% beler overall min- delay strategy is 25% beler overall hour

39 Op8mal strategy incurs 25% - 65% lower cost rela8ve to min- load strategy op8mal strategy incurs 25% - 65% lower electricity cost min- delay strategy incurs same electricity cost hour

40 Op8mal strategy incurs 30% smaller delay rela8ve to min- load strategy min- delay strategy incurs 35% - 50% smaller delay op8mal strategy incurs 30% smaller delay hour