1/18 Introduction Motivation Objective

Problem Description

Low-complexity Scheduling Algorithms for Multi-channel Downlink Wireless Networks

Why not MaxWeight? Proposed Algorithm Throughput

Shreeshankar Bodas The University of Texas at Austin

Simulations Conclusions

Joint work with Sanjay Shakkottai, Lei Ying, R. Srikant

March 18, 2010

2/18 Introduction Motivation Objective

Motivation Investigate scheduling in

OFDM1

downlink networks

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations Conclusions

1

Orthogonal Frequency Division Multiplexing

3/18

Motivation Channel allocation (to be determined)

Introduction Motivation Objective

z

Problem Description Why not MaxWeight? Proposed Algorithm Throughput

Air interface z

{

Channel 1 Q1 Channel 2 Q2

Channel 3 Channel 4

Simulations Conclusions

}|

Q3

Channel 5 Channel 6

Q4

Figure: System model - first glance

}|

{

4/18

Motivation

Introduction Motivation Objective

Problem Description Why not MaxWeight?

Typical parameters for WiMax-like systems:

Proposed Algorithm

20 MHz downlink bandwidth

Throughput

50 sub-bands (channels)

Simulations

Each channel can support 400 kbps

Conclusions

Timeslot duration: 5 ms

5/18

Objective

Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations 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], . . .

5/18

Objective

Introduction Motivation Objective

Problem Description 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], . . .

Throughput Simulations 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] “Large queues” regime primarily studied

6/18

Our Contribution

Introduction Motivation Objective

Problem Description Why not MaxWeight?

Fact

Longest-queues-first type algorithms ⇒ optimal small-queue performance (large deviations sense)

Proposed Algorithm Throughput

Three important questions:

Simulations Conclusions

[Sigmetrics’09]

LQF throughput-optimal? Small-queue performance of MaxWeight? Throughput optimality + small-queues + low-complexity?

6/18

Our Contribution

Introduction Motivation Objective

Problem Description Why not MaxWeight?

Fact

[Sigmetrics’09]

Longest-queues-first type algorithms ⇒ optimal small-queue performance (large deviations sense)

Proposed Algorithm Throughput

Three important questions:

Simulations Conclusions

LQF throughput-optimal?

Yes

Small-queue performance of MaxWeight?

Very poor

Throughput optimality + small-queues + low-complexity?

Yes

7/18 Introduction Motivation Objective

Problem Description Multiuser, multichannel system A1 (t)

Problem Description Why not MaxWeight?

X11 (t) Q1

A2 (t)

X22 (t)

S2

Q2

Proposed Algorithm

Xn1 (t)

Throughput Simulations

S1

An (t)

Xnn (t)

Sn

Conclusions

Qn

Figure: System model

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

8/18 Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations Conclusions

Problem Description Arrivals, channels: - Stationary, ergodic (For throughput) - I.i.d., Bernoulli (For small-queues)

One server can serve at most one user

8/18 Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations Conclusions

Problem Description Arrivals, channels: - Stationary, ergodic (For throughput) - I.i.d., Bernoulli (For small-queues)

One server can serve at most one user Aims: Network stability Low complexity Short longest queue

8/18 Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations Conclusions

Problem Description Arrivals, channels: - Stationary, ergodic (For throughput) - I.i.d., Bernoulli (For small-queues)

One server can serve at most one user Aims: Network stability Low complexity Short longest queue Mathematically, want positive value of   −1 α(b) := lim inf log P max Qi (0) > b , 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.

9/18

Why not MaxWeight?

Introduction Motivation Objective

Before allocation

After allocation

10

S1

5

9

S2

9

9

S3

9

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations Conclusions

3

S4

3

3

S5

3

Edge used for allocation Edge available for allocation

Theorem MaxWeight results in zero rate-function.

10/18

More Balanced Allocation Before allocation

Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations Conclusions

After allocation

10

S1

7

9

S2

8

9

S3

8

3

S4

3

3

S5

3

Edge used for allocation Edge available for allocation

Queue-lengths closer to each other Smaller longest queue

11/18 Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm

Server-Side Greedy Allocation First round of service

Second round of service

S1

S2

S2

S3

S3

S4

S4

Throughput Simulations Conclusions

Third round of service

Fourth round of service

S3

S4

S4

Final queue-lengths

12/18 Introduction Motivation Objective

Problem Description Why not MaxWeight?

Theorem SSG gives strictly positive rate-function ⇒ small queues! (Compare with MaxWeight: zero rate-function)

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

Proposed Algorithm Throughput Simulations Conclusions

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

13/18

Throughput-optimality

Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput

MaxWeight is T.O. under very general system models - Stationary, ergodic channel process - Arrival process with finite variance .. .

Simulations Conclusions

Symmetric, ON-OFF arrivals, channels: too restrictive Are SSG, iLQF T.O.?

14/18

Weight of a Schedule

Introduction Motivation Objective

Problem Description Why not MaxWeight?

10

S1

5 9

15

S2 4

Proposed Algorithm Throughput

2

12

3

Weight of schedule = 2 × 10 + 9 × 15 + 4 × 10

S3

Simulations Conclusions

Theorem Weight(SSG ) ≥ Weight(MW ) − const. Weight(iLQF ) ≥ Weight(MW ) − const. ⇒ SSG, iLQF throughput-optimal!

15/18 Introduction Motivation Objective

SSG, iLQF Comparison SSG and iLQF give very similar performance

Problem Description

0

Why not MaxWeight?

10

i

i

P(max Q (t) > b)

Simulations Conclusions

p p p p p p

−1

Proposed Algorithm Throughput

Performance of the SSG and Modified iLQF with PullUp Algorithms for n = 20, q = 0.5, Bursty (0 − 4) arrivals

10

−2

10

= = = = = =

0.2, iLQF 0.22, iLQF 0.24, iLQF 0.2, SSG 0.22, SSG 0.24, SSG

−3

10

−4

10

−5

10

−6

10

0

2

4

6

8 10 Buffer size (b)

12

Figure: Buffer overflow probabilities

14

16

16/18 Introduction Motivation Objective

SSG, MaxWeight Comparison SSG much better than MaxWeight in all regimes tested

Problem Description Why not MaxWeight?

i

i

P(max Q (t) > b)

Simulations Conclusions

n n n n n n

−1

10

Proposed Algorithm Throughput

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

0

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

17/18 Introduction Motivation Objective

SSG, MaxWeight Comparison MaxWeight’s performance worsens with system-size

Problem Description Why not MaxWeight?

Throughput Simulations Conclusions

n n n n n n

−1

10 P(Packet delay ≥ D)

Proposed Algorithm

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

0

10

−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

18/18

Conclusions

Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations 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 throughput-optimal algorithms (SSG, iLQF) that give good small-queue performance

18/18

Conclusions

Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput Simulations 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 throughput-optimal algorithms (SSG, iLQF) that give good small-queue performance

Questions / comments ?

19/18

Large bipartite graphs

Introduction Motivation Objective

Problem Description Why not MaxWeight? Proposed Algorithm Throughput

Consider balanced bipartite graphs Matching: set of disjoint edges Each edge present with probability q, i.i.d. u1

v1

u2

v2

u3

v3

These graphs have perfect matchings with very high probability, for n large.

Simulations Conclusions

Lemma: For n large, (1−q)n ≤ P(No PM) ≤ 3n(1−q)n .

un

vn

Figure: Perfect matching

Take-away: no perfect matching, “because” isolated node.