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.