1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic ...
-
date post
15-Jan-2016 -
Category
Documents
-
view
227 -
download
0
Transcript of 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic ...
![Page 1: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/1.jpg)
1
Achieving 100% throughputWhere we are in the course…
1. Switch model2. Uniform traffic
Technique: Uniform schedule (easy)
3. Non-uniform traffic, but known traffic matrix Technique: Non-uniform schedule (Birkhoff-von Neumann)
4. Unknown traffic matrix Technique: Lyapunov functions (MWM)
5. Faster scheduling algorithms Technique: Speedup (maximal matchings) Technique: Memory and randomization (Tassiulas) Technique: Twist architecture (buffered crossbar)
6. Accelerate scheduling algorithm Technique: Pipelining Technique: Envelopes Technique: Slicing
7. No scheduling algorithm Technique: Load-balanced router
![Page 2: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/2.jpg)
Buffered CrossbarsWith Performance Guarantees
Taken from the 2004 Ph.D. defense of:
Shang-Tse (Da) ChuangDepartment of Electrical Engineering,Stanford University, http://yuba.stanford.edu/~stchuang
![Page 3: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/3.jpg)
3
Motivation
Network operators want performance guarantees Throughput guarantee Delay guarantee
High performance routers use crossbars
Hard to build crossbar-based routers with guarantees
My talk: How a crossbar with a small amount of internal
buffering can give guarantees
![Page 4: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/4.jpg)
4
Contents
Throughput Guarantees Buffered Crossbar - 100% Throughput Buffered Crossbar - Work Conservation
![Page 5: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/5.jpg)
5
Generic Crossbar-Based Architecture
Speedup of S
Scheduler
VOQs
![Page 6: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/6.jpg)
6
Admissible Traffic
1 , , j
iji
ij
Traffic Matrix
Traffic is admissible if
![Page 7: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/7.jpg)
7
100% Throughput An algorithm delivers 100% throughput if for any
admissible traffic the average backlog is finite
Throughput Guarantee
Speedup of S
Scheduler
![Page 8: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/8.jpg)
8
Previous Work
1985 1990 1995 2000 2005
Wave Front Arbiter [Tamir]
Parallel Iterative Matching [Anderson et al.]
iSLIP [McKeown]
Longest Port First [Mekkittikul et al.]
Maximum Weight Matching [McKeown et al.]
Maximal Matching S=2[Dai,Prabhakar]
Heuristics
TheoreticallyProven
![Page 9: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/9.jpg)
9
Maximal Matching Has Become Hard
TTX Switch Fabric Uses maximal matching Speedup less than 2 Consumes up to 8kW Limited to ~2.5Tb/s No 100% throughput guarantee
![Page 10: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/10.jpg)
10
Traditional Crossbar
Crossbar Requirements An input can send at most one cell An output can receive at most one cell
Scheduling Problem Must overcome two constraints simultaneously
New Crossbar Relieve contention Remove dependency between inputs and outputs
![Page 11: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/11.jpg)
11
Contents
Throughput Guarantees Buffered Crossbar - 100% Throughput Buffered Crossbar - Work Conservation
Delay Guarantees Traditional Crossbar – Emulating an OQ Switch Buffered Crossbar – Emulating an OQ Switch
![Page 12: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/12.jpg)
12
Buffered Crossbar
Arrival Phase Scheduling Phases – Speedup of 2 Departure Phase
![Page 13: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/13.jpg)
13
Scheduling Phase
Input Schedule Each input selects in parallel a cell for an empty crosspoint
Output Schedule Each output selects in parallel a cell from a full crosspoint
![Page 14: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/14.jpg)
14
Example of Input/Output Scheduling
Round-robin Policy Each input schedules in a round-robin order Each output schedules in a round-robin order
![Page 15: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/15.jpg)
15
Previous Work
Buffered Crossbar Simulations [Rojas-Cessa et al. 2001] 32x32 switch, Uniform Bernoulli Traffic, Round-Robin, S=1
0.01
0.1
1
10
100
1000
0.025 0.125 0.225 0.325 0.425 0.525 0.625 0.725 0.825 0.925
Offered Load p
Ave
rag
e D
elay
(C
ell
Tim
e)
1-SLIP
4-SLIP
Buffered Crossbar
Ideal Router
![Page 16: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/16.jpg)
16
Theorem 1 A buffered crossbar with speedup of 2 delivers 100%
throughput for any admissible Bernoulli iid traffic using any work-conserving input/output schedules.
100% Throughput
![Page 17: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/17.jpg)
17
Intuition of Proof
ε
<1-ε
<1-ε
1 2
1-ε 1-ε+ + ε = 2- ε
When a flow is backed up, the services for this backlog exceeds the arrivals
![Page 18: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/18.jpg)
18
Contents
Throughput Guarantees Buffered Crossbar - 100% Throughput Buffered Crossbar - Work Conservation
Delay Guarantees Traditional Crossbar – Emulating an OQ Switch Buffered Crossbar – Emulating an OQ Switch
![Page 19: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/19.jpg)
19
Work-conserving Property If there is a cell for a given output in the system, that
output is busy.
Work Conservation
Output Queued (OQ) Switch
![Page 20: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/20.jpg)
20
?
Emulating an OQ switch
Under identical inputs, the departure time of every cell from both switches is identical
![Page 21: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/21.jpg)
21
4
Input Priority List
57 6
56
1
1
2
9
2
3
8 3
1
Label each cell with their corresponding departure times Arrange input cells into an input priority list Output selects crosspoint with earliest departure time
4
![Page 22: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/22.jpg)
22
Input Priority List
57 6
56
4
132
9
4
2
13
1
8
2
Good guy
Bad guysBad guy
Label each cell with their corresponding departure times Arrange input cells into an input priority list Output selects crosspoint with earliest departure time
![Page 23: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/23.jpg)
23
Definitions
57 6
56
2
4
132
9
4
2
13
Output Margin – cells at its output with earlier departure time Input Margin – cells ahead in input priority list destined to
different outputs Total Margin – Output Margin minus Input Margin
1
8
2 good guys2 bad guys
![Page 24: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/24.jpg)
24
Emulation of FIFO OQ Switch
57 6
56
2
4
12
9
4
2
13
Scheduling Phase Crosspoint is full – Output Margin will increase by one Crosspoint is empty – Input Margin will decrease by one
Total Margin increases by two
1
8 3
![Page 25: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/25.jpg)
25
Emulation of FIFO OQ Switch
57 6
56
2
4
12
9
4
2
13
Arrival Phase Input Margin might increase by one
Departure Phase Output Margin will decrease by one
Total Margin decreases by at most two
1
8 3
3
![Page 26: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/26.jpg)
26
Emulation of FIFO OQ Switch
57 6
56
2
4
2
9
4
2
3
8 33
Lemma 1 For every time slot, total margin does not decrease
![Page 27: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/27.jpg)
27
FIFO Insertion Policy
56
4
2
9
4
2
3
857 6 323
47
Arrival Phase Cell for non-empty VOQ, insert behind cells for same
output Cell for empty VOQ, insert at head of input priority list
![Page 28: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/28.jpg)
28
FIFO Insertion Policy
57 6
56
2
4
2
9
4
2
3
8 33
Lemma 2 An arriving cell will have a non-negative total margin
4 7
![Page 29: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/29.jpg)
29
Theorem 2 A buffered crossbar with speedup of 2 can exactly emulate a
FIFO OQ switch.
Result was shown independently B. Magill, C. Rohrs, R. Stevenson, “Output-Queued Switch
Emulation by Fabrics With Limited Memory”, in IEEE Journal on Selected Areas in Communications, pp.606-615, May. 2003.
Theorem 3 A buffered crossbar with speedup of 2 can be work-conserving
with a distributed algorithm.
Emulation of FIFO OQ Switch
![Page 30: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/30.jpg)
30
Summary
Buffered crossbars Uses crosspoints to relieve contention Inputs and outputs schedule independently and in
parallel
Performance guarantees Throughput – any work-conserving input/output
schedule Work Conservation – simple insertion policy
![Page 31: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/31.jpg)
31
Relevant Papers
Crossbars Shang-Tse Chuang, Ashish Goel, Nick McKeown,
Balaji Prabhakar, “Matching Output Queuing with a Combined Input Output Queued Switch,” IEEE Journal on Selected Areas in Communications, vol.17, n.6, pp.1030-1039, Dec.1999.
Buffered Crossbars Shang-Tse Chuang, Sundar Iyer, Nick McKeown,
“Practical Algorithms for Performance Guarantees in Buffered Crossbars,” in preparation for IEEE/ACM Transactions on Networking.
![Page 32: 1 Achieving 100% throughput Where we are in the course… 1. Switch model 2. Uniform traffic Technique: Uniform schedule (easy) 3. Non-uniform traffic,](https://reader036.fdocuments.net/reader036/viewer/2022081504/56649d6d5503460f94a4d6af/html5/thumbnails/32.jpg)
32
Thank you!