scheduling in delay graphs with applications to optical networks isaac keslassy (stanford...

Scheduling in Delay Graphswith Applications to Optical Networks

Isaac Keslassy (Stanford University), Murali Kodialam, T.V. Lakshman, Dimitri Stiliadis (Bell-Labs)

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?

Outline

Introduction: Scheduling with no Delays

Bipartite Delay Graph

TSS Algorithm

Theorems on Separable Architectures

Non-Separable Architectures

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

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

Example of BvN Schedule

,...

001

100

010

,

010

001

100

,

010

001

100

,

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

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?

Outline

Introduction: Scheduling with no Delays

Bipartite Delay Graph

TSS Algorithm

Theorems on Separable Architectures

Non-Separable Architectures

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

General WDM Mesh Architecture

2

λ2

λ11

N λN

i λi

Star Coupler

Examples of WDM Architectures

1

2

N

λ1

λ2

λN

i

λi

1 23

i

Nλ1 λ2

λ3

λi

λN

Ring

Arbitrary mesh with constant delays Arbitrary routing policy such that all paths to a

given node form a spanning tree

Mesh Model

di

sλd

Property: if packets collide on the path, they would also have collided at the receiver

Mesh Model

di

sdd

dd λd

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

Bipartite delay graph: bipartite graph with weights ij (delay from i to j)

Bipartite Delay Graph

i

j

ij

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

Using the Bipartite Delay Graph in the Schedule

,...

001

100

010

,

010

001

100

,

010

001

100

,

001

100

010

,

010

001

100

,

010

001

100

: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

Delay Graph of a Star Coupler

i

N

1

j

N

1u1

ui

uN

v1

vj

vN

jiij vu

Delay in a star coupler:

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

Outline

Introduction: Scheduling with no Delays

Bipartite Delay Graph

TSS Algorithm

Theorems on Separable Architectures

Non-Separable Architectures

Birkhoff-von Neumann ScheduleExample with 3 nodes

3

1

2

3

1

2

2

3

1

3

1

2

3

1

2

2

3

1

3

1

2

3

1

2

2

3

1

3

1

2

3

1

2

2

3

1

Frame

Sender 1

Sender 2

Sender 3

time

Frame Frame Frame

2

3

1

2

3

1u1u2

u3

v1

v2

v3

Time-Shifted Scheduling (TSS) in a Star Coupler

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

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

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

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

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

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

Outline

Introduction: Scheduling with no Delays

Bipartite Delay Graph

TSS Algorithm

Theorems on Separable Architectures

Non-Separable Architectures

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

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.

Outline

Introduction: Scheduling with no Delays

Bipartite Delay Graph

TSS Algorithm

Theorems on Separable Architectures

Non-Separable Architectures

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

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

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

Thank you.