Scheduling for maximizing throughput EECS, UC Berkeley Presented by Antonis Dimakis

2 Outline 1.Setting, throughput optimal scheduling policies. 2.Basic tools: Lyapunov functions and fluid limits 3.Maximum Weight matching (MaxWeight) 4.Varying Channel models 5.Longest Queue First (LQF) 6.Summary & Open problems

3 Scheduling in ad-hoc wireless networks Goals: –Large throughput –Delay / Loss –Fairness –Simple protocol

4 Scheduling in data comm. switches input 1 input 2 output 2 output 1 Q 11 (t) Q 12 (t) Q 21 (t) Q 22 (t)

5 1. Setting...  11  21  |K|1  12  22  |K|2  13  23  |K|3 service rate matrix Q(t) M[K]= [Tassiulas’92],[McKeown et al.’95] [Andrews et al.’00],… K: set of queues A(t): arrivals D(t): departures M[K]: service rate matrix R: routing matrix A 1 (t) A 2 (t) A |K| (t) D 1 (t) D 2 (t) D |K| (t) :: R

6 Example K={1,2,3} Service matrix Routing matrix Queueing equation:

7 Feasible region & Optimal policy Given avg. input rates  is there a static schedule that supports it? i.e., exist f¸ 0,  2 Co(M[K]) s.t. +R T f=0, f < . If rates are stable under some policy, then necessarily is supported by a static schedule:  = lim t A(t)/t = lim t D(t)/t · liminf t  (t) 2 Co(M[K]). Feasible rates = rates supported by static schedules. Optimal policy = stabilizes all feasible rates.

8 2. Basic tools: Lyapunov functions Goal: show irreducible Markov chain X t is positive recurrent. Pakes’ lemma: Assume V(x)¸ 0,8 x. If E[V(X t+1 )-V(X t )|X t =x]· - , for all x except on a finite set C, then X t is positive recurrent.

9 2. Basic tools: fluid limits Goal: show a queueing system is stable Queueing equation Consider deterministic fluid model theorem: If 9 t 0 s.t. Q(t)=0,8 t¸ t 0, the original queueing system is stable (pos. rec.).

10 Fluid limit example Consider sequence of systems indexed by n, with Q 1 n (0)+Q 2 n (0)=n. Under ergodic inputs, any limit must satisfy

11 Fluid limit example (ctd.) If <1, then 8 t¸ t 0 =1/( ), Q 1 (t)=Q 2 (t)=0. This gives a Lyapunov function for Q(t).

12 3. Maximum Weight Matching (MaxWeight) Choose  2 argmax{-  RQ:  2 Co(M[K])} Example: In this case, check max{Q 1 +Q 3,Q 2 -Q 3 } When Q=(2,7,3) activate {1,3}. Basic theorem: MaxWeight is throughput optimal [Tassiulas’92]

13 MaxWeight optimality: V(q)=q T q is a Lyapunov function. Recall: so, dV(Q(t))/dt=2( +R T  (t)) T Q(t) =2( T Q(t)+  (t) T RQ(t)) =2(-f T RQ(t)+  (t) T RQ(t)) · 0, since, -  (t) T RQ(t)=max{-  T RQ(t):  2 Co(M[K])}.

Varying Channel model...  11  21  |K|1  12  22  |K|2  13  23  |K|3 service rate matrix Q(t) M 1 [K]= [Andrews et al.’00],… A 1 (t) A 2 (t) A |K| (t) D 1 (t) D 2 (t) D |K| (t) ::  11  21  |K|1  12  22  |K|2  13  23  |K|3 service rate matrix M 2 [K]= channel state

15 Varying Channel analysis Consider 2 channel states (service matrices) M 1, M 2, w.p. p i. Feasible region: { ¸ 0: < p 1  1 +p 2  2,  i 2 Co(M i )}. MaxWeight: at state i, choose argmax{  T Q(t):  2 Co(M i )} Again, V(q)=q 1 2 +q 2 2, is a Lyapunov function: In an interval (t,t+  ) channel is M i for time p i . During this time, MaxWeight some  i 2 Co(M i ) is always optimal

16 5. LQF generalized switch model...  11  21  |K|1  12  22  |K|2  13  23  |K|3 service rate matrix Q(t) M[K]= K: set of queues A(t): arrivals D(t): departures M[K]: service rate matrix Longest Queue First A 1 (t) A 2 (t) A |K| (t) D 1 (t) D 2 (t) D |K| (t) ::

17 Longest Queue First (LQF) 1.Easy case: local pooling ) stability. 2. Subtle effect: fluctuations can stabilize. rank condition and non-deterministic arrivals ) stability.

18 Stability of LQF Necessary: <1, <1. Sufficient: Under LQF, longest queues tend to decrease: –Say, Q 1 ¼ Q 2 >>Q 3, for some time. –Then, Q 1 +Q 2 decreases, and so do Q 1,Q 2. Key: locally in time, service from common resource pool service vectors

19 Local Pooling Assume 9 nonzero vector  ¸0 s.t.  =constant C, 8  2 M[L]. M[L], or L, is said to satisfy local pooling (LP). Then,  Q L (t)=  Q L (0) +  A L (t) – C £ t has negative drift, for feasible arrival rates. (  <  =C) L K\L Q(t) t service matrix M[L]

20 Local Pooling Note: If  <  for some service vectors ,  of the subsystem L, then Local Pooling cannot hold for L. Characterization:

21 Stability of LQF  If every L½ K satisfies Local Pooling and arrival rates are feasible, then system is stable: Proof: 1.Fix time t, L:=argmax i Q i (t). 2.W.l.o.g., Q i (t)=Q j (t) for all i,j2 L. 3.If feasible, then L <  2Co(M[L]). But  does not dominate D L (t)2Co(M[L]), by Local Pooling. 4.Thus, 9 k2 L s.t., k <D k (t), so Q k (t)<-  *. 5.max i Q i (t) is a Lyapunov function for fluid system......

22 Stability of LQF Trees 3, 4, 5, 7 Cycles Combinations Graphs that satisfy Local Pooling:

23 Stability of LQF: Subtle Effect ½{1,3,5}+ ½{2,4,6} > (1/3){1,4}+(1/3){2,5}+(1/3){3,6}.  {1,…,6} does not satisfy Local Pooling. Every proper subset satisfies Local Pooling Service Vectors: {1, 3, 5}, {2, 4, 6} {1, 4}, {2, 5}, {3, 6} Graph that does not satisfy Local Pooling:

24 Stability of LQF: Subtle Effect Note: Deterministic inputs with rate close to 0.5  unstable Assume arrival of constant 0.5-  work to each queue. Initial state: all queues are equal. Tie breaking rule: with >0 prob. a size-2 service vector is selected. For any sequence of service vectors, all-equal state is reached again. Sequence of service vectors does not depend on 

25 Stability of LQF: Subtle Effect theorem: LQF stable for i.i.d. arrivals with nonzero variance. key idea: {1,…,6} cannot be set of longest queues for a positive fraction of time  Local Pooling holds most of the time.  Longest queue decreases

26 Stability of LQF: Subtle Effect Assume all queues are longest for a while  {2, 3} and {5, 6} served at same rate 

27 Stability of LQF: Subtle Effect Max-min large at k  (n): A subset L of queues dominates the others during interval.  This subset satisfies LP  Longest queue decreases in  (n)- interval.

28 Stability of LQF: Subtle Effect 1.Most of  (n)-intervals are dominated by proper subsets L of {1,…,6}  LP holds for L. 2.This will imply max i Q i (n(t+  ))-max i Q i (nt)<-n  . nt n(t+  ) …  n)  n)=n 1/6  n) time Q n (nt) Q n (n(t+  ))

29 Stability of LQF: Subtle Effect Theorem: Assume that whenever a set L does not satisfy LP, the corresponding service vectors have rank · |L|-2. Assume also the arrivals are i.i.d. with positive variance (and satisfy a large deviation bound). Then LQF is stable for any feasible arrival rates.

30 Stability of LQF: Subtle Effect Examples

31 Example of instability 8-cycle. Bernoulli i = 1 =0.4984<1/2, uniform tie- breaking policy

32 Summary Lyapunov functions & fluid limits. MaxWeight throughput optimal –No need to know arrival rates. –Works under varying channel conditions. –Must know independent sets. LQF is not always optimal –No need to know arrival rates or independent sets. –Stability depends on variance, not only average rates.

33 Open problems How suboptimal LQF is in reality? Optimal policy that does not use knowledge of independent sets? Fair scheduling? Merits of using load-aware scheduling? –Ethernet works “suboptimally”, but only ~10 nodes.

34 References [Tassiulas’92] Tassiulas & Ephremides, “Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks”, IEEE Trans. On Aut.Con., 37(12), [Andrews et al.’00] M. Andrews, K. Kumaran, K. Ramanan, A.L. Stolyar, R. Vijayakumar, P. Whiting, “Scheduling in a Queueing System with Asynchronously Varying Service Rates”, Probability in the Engineering and Informational Sciences, 2004, Vol.18.Scheduling in a Queueing System with Asynchronously Varying Service Rates [Rybko & Stolyar’92] A.N. Rybko and A.L.Stolyar, “Ergodicity of stochastic processes describing the operation of open queueing networks,” Problems of Information Transmission, vol. 28, (Translated from Problemy Peredachi Informatsii, vol. 28, no. 3, pp. 3-26, 1992.) [Dai’95] J. G. Dai, "On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models", Annals of Applied Probability, Vol 5, (1995). [full paper: ps file dai95a.ps (294 Kbytes) or pdf file dai95a.pdf (184 Kbytes) ]dai95a.ps dai95a.pdf [Dimakis & Walrand’06] “Sufficient conditions for stability of longest queue first scheduling: second order properties using fluid limits" to appear in Advances in Applied Probability 38.2 (June 2006). [McKeown et al.’95] Nick McKeown, Adisak Mekkittikul, Venkat Anantharam and Jean Walrand "Achieving 100% Throughput in an Input-Queued Switch (Extended Version)" IEEE Transactions on Communications, Vol.47, No.8, August pages pdfpdf