Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford...
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![Page 1: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/1.jpg)
Scheduling in Delay Graphswith Applications to Optical Networks
Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis (Bell-Labs)
![Page 2: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/2.jpg)
Problem Motivation
MAN WDM optical ring with N nodes
For each node i, one tunable transmitter, and one fixed receiver at wavelength λi.
1 23
i
Nλ1 λ2
λ3
λi
λN
When and how can we guarantee 100% throughput?
![Page 3: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/3.jpg)
Outline
Introduction: Scheduling with no Delays
Bipartite Delay Graph
TSS Algorithm
Theorems on Separable Architectures
Non-Separable Architectures
![Page 4: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/4.jpg)
Assume no propagation delays (each packet transmitted is immediately received)
A single transmitter and receiver per node => when i sends to j, i cannot send to j’≠j and j cannot receive from i’≠i
Slotted time, fixed-size packets
Scheduling with no Delays
![Page 5: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/5.jpg)
Input:
Birkhoff-von Neumann (BvN) schedule: A frame of F matrices S1,…,SF such that Arrivals ≤ Services: R’ ≤ S1 + .. + SF
{Si}’s are permutation matrices: any node sends and receives at most one packet per time-slot
Known result: decomposition always exists
Frame-Based Scheduling
.j,
i,
matrix,integer integer,
: with,1
matrix rate Arrival
iij
jij
FR'
F,R'
R'F
RF
R
![Page 6: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/6.jpg)
Example of BvN Schedule
,...
001
100
010
,
010
001
100
,
010
001
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,
001
100
010
,
010
001
100
,
010
001
100
:Schedule
001
100
010
010
001
100
010
001
100
':ionDecomposit
021
102
210
3
1'
3
1:matrix Rate
R
RR
No transmitter conflicts
No receiver conflicts
Frame Frame
![Page 7: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/7.jpg)
Propagation delays << time-slot ?
Example: MAN WDM ring 30km ring, 10Gbps, 1kb packets Time-slot = 1kb/(10Gb/s) = 100ns Max propagation delay = 30km/(3.108 m/s)
= 100μs
Clearly impossible to neglect delays
Neglecting Delays?
![Page 8: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/8.jpg)
Outline
Introduction: Scheduling with no Delays
Bipartite Delay Graph
TSS Algorithm
Theorems on Separable Architectures
Non-Separable Architectures
![Page 9: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/9.jpg)
Question: Can we extend Birkhoff-von Neumann (BvN) to general case of WDM mesh with delays
Method:
1. Provide simple model for mesh
2. Use model to extend BvN
Question
![Page 10: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/10.jpg)
General WDM Mesh Architecture
2
λ2
λ11
N λN
i λi
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Star Coupler
Examples of WDM Architectures
1
2
N
λ1
λ2
λN
i
λi
1 23
i
Nλ1 λ2
λ3
λi
λN
Ring
![Page 12: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/12.jpg)
Arbitrary mesh with constant delays Arbitrary routing policy such that all paths to a
given node form a spanning tree
Mesh Model
di
sλd
![Page 13: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/13.jpg)
Property: if packets collide on the path, they would also have collided at the receiver
Mesh Model
di
sdd
dd λd
![Page 14: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/14.jpg)
Property: if packets collide on the path, they would also have collided at the receiver
No collision at receiver no collision on path
We need to prevent only two types of collision: At the transmitter At the receiver
Mesh Model
![Page 15: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/15.jpg)
Bipartite delay graph: bipartite graph with weights ij (delay from i to j)
Bipartite Delay Graph
i
j
ij
![Page 16: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/16.jpg)
Example of Bipartite Delay Graph
1
3 2
λ1
λ3 λ2
12=1
23=1
31=12
3
1
2
3
1312
23
1
1 23
![Page 17: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/17.jpg)
Using the Bipartite Delay Graph in the Schedule
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:Schedule
transmitter conflicts
receiver conflicts
2
3
1
2
3
1312
23
1123
1
3 2
λ1
λ3 λ2
12=1
23=1
31=1
Conflict
![Page 18: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/18.jpg)
Delay Graph of a Star Coupler
i
N
1
j
N
1u1
ui
uN
v1
vj
vN
jiij vu
Delay in a star coupler:
![Page 19: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/19.jpg)
Delay Graph of a Ring
Tvu jiij mod
Delay in a ring:
1 2k
i
N
j TvuT
vu
vu
T
ijiij
ikiik
iii
j11
k11
1i1
,
of RTT
ui
vi
![Page 20: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/20.jpg)
Outline
Introduction: Scheduling with no Delays
Bipartite Delay Graph
TSS Algorithm
Theorems on Separable Architectures
Non-Separable Architectures
![Page 21: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/21.jpg)
Birkhoff-von Neumann ScheduleExample with 3 nodes
3
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1
Frame
Sender 1
Sender 2
Sender 3
time
Frame Frame Frame
![Page 22: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/22.jpg)
2
3
1
2
3
1u1u2
u3
v1
v2
v3
Time-Shifted Scheduling (TSS) in a Star Coupler
![Page 23: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/23.jpg)
2
3
1
2
3
1u1u2
u3
v1
v2
v3
Sender 1
Sender 2
Sender 3
time (at senders)
u1
u2
u3
3 3 2 3 3 2 3 3 2 3 3 2
1 1 3 1 1 3 1 1 3 1 1 3
2 2 1 2 2 1 2 2 1 2 2 1
time (at star coupler)
Time-Shifted Scheduling (TSS) in a Star Coupler
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2
3
1
2
3
1u1u2
u3
v1
v2
v3
Sender 1
Sender 2
Sender 3
3 3 2 3 3 2 3 3 2 3 3 2
1 1 3 1 1 3 1 1 3 1 1 3
2 2 1 2 2 1 2 2 1 2 2 1
time (at star coupler)
Time-Shifted Scheduling (TSS) in a Star Coupler
![Page 25: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/25.jpg)
2
3
1
2
3
1u1u2
u3
v1
v2
v3
Sender 1
Sender 2
Sender 3
1 1 1 1 1 1 1 1
1 1 1 1
time (at star coupler)
v1
time (at node 1)
Time-Shifted Scheduling (TSS) in a Star Coupler
![Page 26: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/26.jpg)
In a star coupler, TSS works:
In a ring with RTT T, and a schedule of frame length F=T, TSS also works (shifting time by T doesn’t matter):
and the schedule is modulo F=T.
TSS in a Star Coupler and in a Ring
Tvu jiij mod
jiij vu
![Page 27: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/27.jpg)
Separable architecture:
T-Separable architecture:
A separable architecture is T-separable for all T
F-rate matrix: Rate matrix for which (optimal) BvN decomposition has frame length F
Definitions (more general setting)
Tvu jiij mod
jiij vu
![Page 28: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/28.jpg)
Properties
Property 1: Using the TSS algorithm, an F-separable architecture can schedule any F-rate matrix. Example: ring of RTT F
Property 2: Using the TSS algorithm, a separable architecture can schedule any rate matrix. Example: star coupler
![Page 29: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/29.jpg)
Outline
Introduction: Scheduling with no Delays
Bipartite Delay Graph
TSS Algorithm
Theorems on Separable Architectures
Non-Separable Architectures
![Page 30: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/30.jpg)
Can we always extend BvN?
No! Even for simple matrices… Example: ring
With cyclical scheduling of two matrices, each of the 3 pairs has to be associated to either matrix, but there are at most 3 elements ( one pair) per matrix BvN impossible here
1
3 2
1
1
1
011
101
110
'R ???
010
100
010
001
001
100
011
101
110
'R
![Page 31: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/31.jpg)
Theorems (Necessity and Sufficiency)
Theorem 1: An architecture can schedule any F-rate matrix iff the architecture is F-separable. Proof: if not F-separable, exhibit counter-example
Theorem 2: An architecture can schedule any rate matrix iff the architecture is separable. Proof: needs to be F-separable for all F
Corollary (Negative result): Guaranteed frame-based scheduling cannot be achieved in non-separable architectures.
![Page 32: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/32.jpg)
Outline
Introduction: Scheduling with no Delays
Bipartite Delay Graph
TSS Algorithm
Theorems on Separable Architectures
Non-Separable Architectures
![Page 33: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/33.jpg)
Non-Separable Delay Graphs
Guaranteed schedule in non-separable architecture? need to make it separable
Assume we can add delay lines ij between nodes.
How to minimize the sum of these delay lines?
0
ˆ
:s.t. min,
ij
jiijijij
jiij
vu
![Page 34: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/34.jpg)
Non-Separable Delay Graphs
Dual formulation of Maximum Weight Matching Problem in Bipartite Delay Graph
Separable architecture: all matches are MWM Non-separable architecture: solving MWM
gives minimum amount of additional delay lines
ijji
jj
ii
vu
vu
:s.t. min
![Page 35: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/35.jpg)
Summary
The bipartite delay graph can model any mesh architecture
An architecture can schedule any F-rate matrix iff it is F-separable (e.g. ring of RTT=F)
An architecture can schedule any rate matrix iff it is separable (e.g. star coupler)
Non-separable architectures can schedule any rate matrix at minimum cost by adding delay lines and using maximum weight matching
![Page 36: Scheduling in Delay Graphs with Applications to Optical Networks Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis.](https://reader030.fdocument.org/reader030/viewer/2022032521/56649d5f5503460f94a3fbb9/html5/thumbnails/36.jpg)
Thank you.