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Scheduling for maximizing throughput

Scheduling for maximizing throughput. EECS, UC Berkeley Presented by Antonis Dimakis (dimakis@eecs). Outline. Setting, throughput optimal scheduling policies. Basic tools: Lyapunov functions and fluid limits Maximum Weight matching (MaxWeight) Varying Channel models

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Scheduling for maximizing throughput

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  1. Scheduling for maximizing throughput EECS, UC Berkeley Presented by Antonis Dimakis (dimakis@eecs)

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

  3. Scheduling in ad-hoc wireless networks • Goals: • Large throughput • Delay / Loss • Fairness • Simple protocol 2 3 4 5 1 6

  4. Scheduling in data comm. switches Q11(t) input 1 l11 output 1 l12 1 1 Q12(t) Q21(t) input 2 l21 output 2 l22 2 2 Q22(t)

  5. 1. Setting A1(t) D1(t) A2(t) D2(t) R : : Q(t) K: set of queues A(t): arrivals D(t): departures M[K]: service rate matrix R: routing matrix A|K|(t) D|K|(t) . . . m11 m12 m13 [Tassiulas’92],[McKeown et al.’95] [Andrews et al.’00],… m21 m22 m23 M[K]= m|K|1 m|K|2 m|K|3 service rate matrix

  6. Example l1 l2 l3 • K={1,2,3} • Service matrix • Routing matrix • Queueing equation: 1 2 3

  7. Feasible region & Optimal policy • Given avg. input rates l=E(A(0)), is there a static schedule that supports it? i.e., exist f¸ 0, y2 Co(M[K]) s.t. l+RTf=0, f < y. • If rates l are stable under some policy, then necessarily l is supported by a static schedule: l = limt A(t)/t = limt D(t)/t · liminfty(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 Xt is positive recurrent. • Pakes’ lemma: Assume V(x)¸ 0,8 x. If E[V(Xt+1)-V(Xt)|Xt=x]· -e<0, for all x except on a finite set C, then Xt 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 t0 s.t. Q(t)=0,8 t¸ t0 , the original queueing system is stable (pos. rec.).

  10. Fluid limit example l1 l2 1 2 • Consider sequence of systems indexed by n, with Q1n(0)+Q2n(0)=n. • Under ergodic inputs, any limit must satisfy

  11. Fluid limit example (ctd.) • If l1+l2<1, then 8 t¸ t0=1/(1-l1-l2), Q1(t)=Q2(t)=0. • This gives a Lyapunov function for Q(t).

  12. 3. Maximum Weight Matching (MaxWeight) • Choose m2 argmax{-fRQ: f2 Co(M[K])} • Example: • In this case, check max{Q1+Q3,Q2-Q3} • When Q=(2,7,3) activate {1,3}. • Basic theorem: MaxWeight is throughput optimal. l1 l2 l3 1 2 3 [Tassiulas’92]

  13. MaxWeight optimality: • V(q)=qTq is a Lyapunov function. • Recall: so, dV(Q(t))/dt=2(l+RTf(t))TQ(t) =2(lTQ(t)+f(t)TRQ(t)) =2(-fTRQ(t)+f(t)TRQ(t)) · 0, since, -f(t)TRQ(t)=max{-fTRQ(t): f2 Co(M[K])}.

  14. 4. Varying Channel model [Andrews et al.’00],… A1(t) D1(t) A2(t) D2(t) : : Q(t) A|K|(t) D|K|(t) . . . . . . channel state m11 m12 m13 m11 m12 m13 m21 m22 m23 m21 m22 m23 M2[K]= M1[K]= m|K|1 m|K|2 m|K|3 m|K|1 m|K|2 m|K|3 service rate matrix service rate matrix

  15. Varying Channel analysis l1 l2 1 2 • Consider 2 channel states (service matrices) M1, M2, w.p. pi. • Feasible region: {l¸ 0: l < p1y1+p2y2,yi2 Co(Mi)}. • MaxWeight: at state i, choose argmax{fTQ(t): f2 Co(Mi)} • Again, V(q)=q12+q22, is a Lyapunov function: • In an interval (t,t+d) channel is Mi for time pid. During this time, MaxWeight some fi2 Co(Mi) is always optimal.

  16. 5. LQF generalized switch model A1(t) D1(t) K: set of queues A(t): arrivals D(t): departures M[K]: service rate matrix Longest Queue First A2(t) D2(t) : : Q(t) A|K|(t) D|K|(t) . . . m11 m12 m13 m21 m22 m23 M[K]= m|K|1 m|K|2 m|K|3 service rate matrix

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

  18. Stability of LQF l1 l2 l3 • Necessary: l1+l2<1, l2+l3<1. • Sufficient: • Under LQF, longest queues tend to decrease: • Say, Q1¼ Q2>>Q3, for some time. • Then, Q1+Q2 decreases, and so do Q1,Q2. • Key: locally in time, service from common resource pool. 1 2 3 service vectors

  19. Local Pooling L service matrix M[L] • Assume 9 nonzero vector a¸0 s.t. af=constant C, 8f2 M[L]. • M[L], or L, is said to satisfy local pooling (LP). • Then, aQL(t)=aQL(0) + aAL(t) – C £ t has negative drift, for feasible arrival rates l. (al<ay=C) Q(t) K\L t

  20. Local Pooling • Note: If f<y for some service vectors f,y 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 l are feasible, then system is stable: Proof: • Fix time t, L:=argmaxi Qi(t). • W.l.o.g., Qi(t)=Qj(t) for all i,j2 L. • If l feasible, then lL<f2Co(M[L]). But f does not dominate DL(t)2Co(M[L]), by Local Pooling. • Thus, 9 k2 L s.t., lk<Dk(t), so Qk(t)<-e*. • maxi Qi(t) is a Lyapunov function for fluid system. . . . . .

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

  23. 2 3 1 4 6 5 Stability of LQF: Subtle Effect • Graph that does not satisfy Local Pooling: • 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}

  24. 2 3 1 4 6 5 Stability of LQF: Subtle Effect • Note: Deterministic inputs with rate close to 0.5  unstable • Assume arrival of constant 0.5-e 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 e.

  25. 2 3 1 4 6 5 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. 2 3 1 4 6 5 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 kD(n): A subset L of queues dominates the others during interval. • This subset satisfies LP  Longest queue decreases in D(n)-interval.

  28. Stability of LQF: Subtle Effect • Most of D(n)-intervals are dominated by proper subsets L of {1,…,6}  LP holds for L. • This will imply maxiQi(n(t+d))-maxiQi(nt)<-nde. Qn(nt) … Qn(n(t+d)) time n(t+d) nt D(n) D(n)=n1/6 D(n)

  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. 2 3 1 2 1 4 8 3 6 5 7 4 5 6 Stability of LQF: Subtle Effect • Examples

  31. 1 2 8 3 7 4 5 6 Example of instability • 8-cycle. Bernoulli li=l1=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), 1992. • [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. • [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, 1992. (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, 49-77 (1995). [full paper: ps file dai95a.ps (294 Kbytes) or pdf file dai95a.pdf (184 Kbytes) ] • [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 1999. 22 pages pdf

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