Scheduling in Delay Graphs with 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 λ i i N λ1λ1 λ2λ2 λ3λ3 λiλi λNλN  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 S 1,…,S F such that  Arrivals ≤ Services: R’ ≤ S S F  {S i }’s are permutation matrices: any node sends and receives at most one packet per time-slot  Known result: decomposition always exists Frame-Based Scheduling

Example of BvN Schedule No transmitter conflicts No receiver conflicts Frame

 Propagation delays << time-slot ?  Example: MAN WDM ring  30km ring, 10Gbps, 1kb packets  Time-slot = 1kb/(10Gb/s) = 100ns  Max propagation delay = 30km/( 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 λ2λ2 λ1λ1 1 N λNλN i λiλi

 Star Coupler Examples of WDM Architectures 1 2 N λ1λ1 λ2λ2 λNλN i λiλi 12 3 i N λ1λ1 λ2λ2 λ3λ3 λiλi λNλN  Ring

 Arbitrary mesh with constant delays  Arbitrary routing policy such that all paths to a given node form a spanning tree Mesh Model d i s λdλd

 Property: if packets collide on the path, they would also have collided at the receiver Mesh Model d i s d d d d λdλd

 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 λ1λ1 λ3λ3 λ2λ2  12 =1  23 =1  31 =

Using the Bipartite Delay Graph in the Schedule transmitter conflicts receiver conflicts λ1λ1 λ3λ3 λ2λ2  12 =1  23 =1  31 =1 Conflict

Delay Graph of a Star Coupler i N 1 j N 1 u1u1 uiui uNuN v1v1 vjvj vNvN  Delay in a star coupler:

Delay Graph of a Ring  Delay in a ring: 12 k i N j uiui vivi

Outline  Introduction: Scheduling with no Delays  Bipartite Delay Graph  TSS Algorithm  Theorems on Separable Architectures  Non-Separable Architectures

Birkhoff-von Neumann Schedule Example with 3 nodes Frame Sender 1 Sender 2 Sender 3 time Frame

u1u1 u2u2 u3u3 v1v1 v2v2 v3v3 Time-Shifted Scheduling (TSS) in a Star Coupler

u1u1 u2u2 u3u3 v1v1 v2v2 v3v3 Sender 1 Sender 2 Sender 3 time (at senders) u1u1 u2u2 u3u time (at star coupler) Time-Shifted Scheduling (TSS) in a Star Coupler

u1u1 u2u2 u3u3 v1v1 v2v2 v3v3 Sender 1 Sender 2 Sender time (at star coupler) Time-Shifted Scheduling (TSS) in a Star Coupler

u1u1 u2u2 u3u3 v1v1 v2v2 v3v3 Sender 1 Sender 2 Sender time (at star coupler) v1v1 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

 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)

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

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?

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

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.