Scheduling for maximizing throughput

EECS, UC Berkeley

Presented by Antonis Dimakis (dimakis@eecs)

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

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Scheduling in ad-hoc wireless networks

• Goals: – Large throughput– Delay / Loss– Fairness– Simple protocol

2 3 4 5 1 6

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Scheduling in data comm. switches

1112

22

21

1 1

2 2

input 1

input 2output 2

output 1

Q11(t)

Q12(t)

Q21(t)

Q22(t)

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1. Setting

. . .1121

|K|1

1222

|K|2

1323

|K|3

service rate matrix

Q(t)

M[K]=

[Tassiulas’92],[McKeown et al.’95][Andrews et al.’00],…

K: set of queuesA(t): arrivalsD(t): departuresM[K]: service rate matrixR: routing matrix

A1(t)A2(t)

A|K|(t)

D1(t)D2(t)

D|K|(t): : R

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Example

• K={1,2,3}• Service matrix

• Routing matrix

• Queueing equation:

1 2 3

1 2 3

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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. +RTf=0, f < .

• If rates are stable under some policy, then necessarily is supported by a static schedule: = limt A(t)/t = limt D(t)/t · liminft (t) 2 Co(M[K]).

• Feasible rates = rates supported by static schedules.• Optimal policy = stabilizes all feasible rates.

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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]· -, for all x except on a finite set C,

then Xt is positive recurrent.

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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.).

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Fluid limit example

• Consider sequence of systems indexed by n, with Q1n(0)+Q2

n(0)=n.

• Under ergodic inputs, any limit must satisfy

1 2

1 2

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Fluid limit example (ctd.)

• If 1+2<1, then 8 t¸ t0=1/(1-1-2), Q1(t)=Q2(t)=0.

• This gives a Lyapunov function for Q(t).

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3. Maximum Weight Matching (MaxWeight)

• Choose 2 argmax{-RQ: 2 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.

1 2 3

1 2 3

[Tassiulas’92]

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MaxWeight optimality:• V(q)=qTq is a Lyapunov function.

• Recall:

so, dV(Q(t))/dt=2(+RT(t))TQ(t)

=2(TQ(t)+(t)TRQ(t))

=2(-fTRQ(t)+(t)TRQ(t))

· 0,

since, -(t)TRQ(t)=max{-TRQ(t): 2 Co(M[K])}.

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. . .

4. Varying Channel model

. . .1121

|K|1

1222

|K|2

1323

|K|3

service rate matrix

Q(t)

M1[K]=

[Andrews et al.’00],…

A1(t)A2(t)

A|K|(t)

D1(t)D2(t)

D|K|(t): :

1121

|K|1

1222

|K|2

1323

|K|3

service rate matrix

M2[K]=

channel state

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Varying Channel analysis

• Consider 2 channel states (service matrices) M1, M2, w.p. pi.

• Feasible region: { ¸ 0: < p1 1+p2 2,i2 Co(Mi)}.

• MaxWeight: at state i, choose argmax{TQ(t): 2 Co(Mi)}

• Again, V(q)=q12+q2

2, is a Lyapunov function:

• In an interval (t,t+) channel is Mi for time pi. During this time, MaxWeight some i2 Co(Mi) is always optimal.

1 2

1 2

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5. LQF generalized switch model

. . .1121

|K|1

1222

|K|2

1323

|K|3

service rate matrix

Q(t)

M[K]=

K: set of queuesA(t): arrivalsD(t): departuresM[K]: service rate matrix

Longest Queue First

A1(t)A2(t)

A|K|(t)

D1(t)D2(t)

D|K|(t): :

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Longest Queue First (LQF)

1. Easy case: local pooling ) stability.

2. Subtle effect: fluctuations can stabilize. rank condition and non-deterministic arrivals ) stability.

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Stability of LQF

• Necessary: 1+2<1, 2+3<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

1 2 3

service vectors

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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,

QL(t)=QL(0) + AL(t) – C £ t

has negative drift, for feasible arrival rates . (<=C)

L

K\L

Q(t)

t

service matrixM[L]

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Local Pooling

• Note:

If < for some service vectors , of the subsystem L, then Local Pooling cannot hold for L.

• Characterization:

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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:=argmaxi Qi(t).

2. W.l.o.g., Qi(t)=Qj(t) for all i,j2 L.

3. If feasible, then L<2Co(M[L]). But does not dominate DL(t)2Co(M[L]), by Local Pooling.

4. Thus, 9 k2 L s.t., k<Dk(t), so Qk(t)<-*.

5. maxi Qi(t) is a Lyapunov function for fluid system.

. .

...

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Stability of LQF

Trees3, 4, 5, 7 Cycles

Combinations

• Graphs that satisfy Local Pooling:

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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.

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6 5

1 4

2 • Service Vectors: {1, 3, 5}, {2, 4, 6} {1, 4}, {2, 5}, {3, 6}

• Graph that does not satisfy Local Pooling:

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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 .

3

6 5

1 4

2

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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.

3

6 5

1 4

2

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Stability of LQF: Subtle Effect

Assume all queues are longest for a while {2, 3} and {5, 6} served at same rate

3

6 5

1 4

2

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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.

(n)

(k + 1)(n)k(n)

L

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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 maxiQi(n(t+))-maxiQi(nt)<-n .

nt n(t+)

…

n) n)=n1/6 n)

time

Qn(nt)Qn(n(t+))

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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.

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Stability of LQF: Subtle Effect

• Examples1 2

3

4

56

7

83

6 5

1 4

2

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Example of instability

• 8-cycle. Bernoulli i=1=0.4984<1/2, uniform tie-breaking policy.

1 2

3

4

56

7

8

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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.

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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.

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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