1/14 Introduction Motivation Objective

Problem Description

On Scheduling over Multi-server Wireless Systems

Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Sanjay Shakkottai The University of Texas at Austin

Joint work with Shreeshankar Bodas, Lei Ying, R. Srikant

February 4, 2010

2/14 Introduction Motivation Objective

Motivation Investigate scheduling in

OFDM1

downlink networks

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

1

Orthogonal Frequency Division Multiplexing

3/14

Motivation Channel allocation (to be determined)

Introduction Motivation Objective

z

Problem Description Analysis

Air interface

{

z

Channel 1 Q1 Channel 2

Why not MaxWeight? Proposed Algorithm

Conclusions

}|

Q2

Channel 3 Channel 4

Q3

Channel 5 Channel 6

Q4

Figure: System model - first glance

}|

{

4/14

Objective

Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Prime performance metric: network stability Design throughput-optimal scheduling algorithms MaxWeight-type algorithms [TasEph’92], various extensions [Sto’04], [ShaSriSto’04], [ErySriPer’05], [YinSriEry’06], [VenLin’07], . . .

4/14

Objective

Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Prime performance metric: network stability Design throughput-optimal scheduling algorithms MaxWeight-type algorithms [TasEph’92], various extensions [Sto’04], [ShaSriSto’04], [ErySriPer’05], [YinSriEry’06], [VenLin’07], . . .

Conclusions

Delay: important performance metric Real-time traffic (voice/video/. . .) Closely related to queue-lengths at base-station Comparatively, much less investigated [GanModTsi’07], [KitJav’08]

5/14

Our Contribution

Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Fact

[Sigmetrics 2009]

Longest-queues-first type algorithms ⇒ optimal small-queue performance (large deviations sense) Three important questions: LQF throughput-optimal? Small-queue performance of MaxWeight? Throughput optimality + small-queues + low-complexity?

5/14

Our Contribution

Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Fact

[Sigmetrics 2009]

Longest-queues-first type algorithms ⇒ optimal small-queue performance (large deviations sense) Three important questions: LQF throughput-optimal?

Yes

Small-queue performance of MaxWeight?

Very poor

Throughput optimality + small-queues + low-complexity?

Yes

6/14 Introduction Motivation Objective

Problem Description Multiuser, multichannel system A1 (t)

Problem Description Analysis

X11 (t)

S1

Q1 A2 (t)

Why not MaxWeight? Proposed Algorithm

X22 (t)

S2

Q2 Xn1 (t)

Conclusions

An (t)

Xnn (t)

Sn

Qn

Figure: System model

4G-systems [WiMax], [LTE] Several tens of users per base station OFDM-based slotted-time air-interface at base station

7/14 Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Problem Description Arrivals, channels: i.i.d., Bernoulli One server can serve at most one user

7/14 Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Problem Description Arrivals, channels: i.i.d., Bernoulli One server can serve at most one user Aims: Network stability Low complexity Short longest queue

7/14 Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Problem Description Arrivals, channels: i.i.d., Bernoulli One server can serve at most one user Aims: Network stability Low complexity Short longest queue Mathematically, want positive value of   −1 log P max Qi (0) > b , α(b) := lim inf n→∞ n 1≤i≤n for fixed integer b ≥ 0. α(b) is called the rate-function.

P(Qmax (0) > b) ≈ exp(−nα(b)), for n large.

8/14

MaxWeight In Action Before allocation

Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

After allocation

10

S1

5

9

S2

9

9

S3

9

Conclusions

3

S4

3

3

S5

3

Edge used for allocation Edge available for allocation

Theorem MaxWeight results in zero rate-function.

9/14

More Balanced Allocation Before allocation

Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

After allocation

10

S1

7

9

S2

8

9

S3

8

Conclusions

3

S4

3

3

S5

3

Edge used for allocation Edge available for allocation

Queue-lengths closer to each other Smaller longest queue

10/14 Introduction Motivation Objective

Problem Description

Server-Side Greedy Allocation First round of service

Second round of service

S1

S2

S2

S3

S3

S4

S4

Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Third round of service

Fourth round of service

S3

S4

S4

Final queue-lengths

11/14 Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Main Result SSG is throughput-optimal Yields strictly positive rate-function ⇒ small queues! (Compare with MaxWeight: zero rate-function)

Complexity = O(n2 ) computations per timeslot (Compare with MaxWeight: Ω(n2 ))

Intuition: MaxWeight: simultaneous server-allocations - Drains longest queues by too much - Service wastage issues (matter in small-queues regime!)

SSG: iterative resource allocation - Natural modification of MaxWeight - Uses every last drop of service

12/14 Introduction Motivation Objective

Simulations SSG much better than MaxWeight in all regimes tested

Problem Description

Performance of the SSG and MaxWeight Algorithms for p = 0.095, q = 0.75, Bursty (0 − 10) arrivals

0

10

n n n n n n

Analysis Why not MaxWeight? Proposed Algorithm

i

i

P(max Q (t) > b)

Conclusions

−1

10

−2

10

= = = = = =

50, MW 80, MW 100, MW 50, SSG 80, SSG 100, SSG

−3

10

−4

10

−5

10

−6

10

0

20

40

60 Buffer size (b)

80

Figure: Buffer overflow probabilities

100

120

13/14 Introduction Motivation Objective

Simulations MaxWeight’s performance worsens with system-size

Problem Description

Performance of the SSG and MaxWeight Algorithms for p = 0.095, q = 0.75, Bursty (0 − 10) arrivals

0

10

Analysis

Conclusions

n n n n n n

−1

10 P(Packet delay ≥ D)

Why not MaxWeight? Proposed Algorithm

−2

10

= = = = = =

50, MW 80, MW 100, MW 50, SSG 80, SSG 100, SSG

−3

10

−4

10

−5

10

−6

10

0

50

100 150 Delay D (timeslots)

Figure: Packet delay profiles

200

250

14/14

Conclusions

Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Throughput and delay are not conflicting requirements New intuition: iterative resource allocation for guaranteeing small delay Scale the number of users and bandwidth, not buffer-length or time Present a stable, low-complexity algorithm (SSG) that gives good small-queue performance

14/14

Conclusions

Introduction Motivation Objective

Problem Description Analysis Why not MaxWeight? Proposed Algorithm

Conclusions

Throughput and delay are not conflicting requirements New intuition: iterative resource allocation for guaranteeing small delay Scale the number of users and bandwidth, not buffer-length or time Present a stable, low-complexity algorithm (SSG) that gives good small-queue performance

Questions / comments ?