belief-propagation assisted scheduling in input-queued switches s. atalla 1, d. cuda 2, p. giaccone...

Belief-Propagation Assisted Scheduling in Input-Queued

Switches

S. Atalla1, D. Cuda2, P. Giaccone1, M. Pretti2

1Politecnico di Torino2Italian National Research Council

Hot Interconnects 2010August 2010

Outline Background motivations System model Basic belief-propagation algorithm for MWM Assisted scheduling Belief-propagation for assisted scheduling Performance evaluation Hardware implementation Conclusions

2 Hot Interconnects 2010

Background motivations

Internet traffic is steadily increasing Routers and switches require to process growing

amount of data faster and faster Input Queued (IQ) switches can be

considered as a reference architecture Memory speed = line rate

IQ switches require suitable scheduling algorithms that Ensure good performance (throughput, delay,) Run fast (few ns to take each scheduling decision) Are implementable in hardware (HW)


System model NxN crossbar with Virtual Output Queuing

one FIFO queue for each input output pair total of N2 queues

Synchronous architecture: time is slotted fixed sized packets

Hot Interconnects 20104

Scheduling algorithm At each timeslot, the scheduler selects a set of

head-of-line packets compatible with the crossbar constraint: At the most one packet can be transferred to/from

each output/input port equivalent to choose a matching in a bipartite graph

Inputs: lengths of the VOQ Outputs: matching described through binary variable:

xij=1 iff input i transfer packet to output j )]([ txtX ij

)]([ tqtQ ij

qij

Scheduler(MWM, iSLIP, iLQF,

…)

tQ tX

x00=1

x33=05 Hot Interconnects 2010

3 33 3

0 00 0

Scheduling algorithm dichotomy Maximum Weight Matching (MWM) is

Optimal in terms of performance Difficult to implement in HW

O(N3) operations, difficult to be parallelized

Heuristic algorithms mimicking MWM E.g., iSLIP, iLQF, WFA (and many others) Efficient to be implemented in HW

e.g., iSLIP was implemented in CISCO 12000 serie Possible traffic losses under critical traffic patterns


Basic belief-propagation for MWM Recently, Belief-Propagation (BP) algorithm has been

proposed to solve MWM problem [1,2] BP algorithms are message passing algorithms firstly

conceived to study Graphical Models (GMs) GMs combine graphic theory and probability theory

BP is exact for MWM over bipartite graph (see [1]), but To ensure convergence, MWM must be unique

Small random noise can be added to queue length It takes O(N3/ε) to converge

ε: difference in weight between the first two heaviest matchings not known a priori


[1]M. Bayati, D. Shah, and M. Sharma, “Max-product for maximum weight matching: Convergence, correctness, and LP duality,” Information Theory, IEEE Transactions on, vol. 54, no. 3, pp. 1241–1251, Mar. 2008.

[2]M. Bayati, B. Prabhakar, D. Shah, and M. Sharma, “Iterative scheduling algorithms,” in INFOCOM 2007, IEEE, 6-12 2007, pp. 445 –453.

Basic belief-propagation for MWM

0 0


3 3


0 0


3 3


0 0


3 3

00030

01030

02030

130 ,,,0max fqfqfqb


0 0

After convergence, each output it is matched to the input associated with the largest message.


Assisted scheduling

Our major contribution is the introduction of the concept of assisted scheduling: Instead of the queue length, scheduling

algorithms are modified to use messages computed by BP as weights We show that BP assisted scheduling boosts

performance of existing schedulers while keeping backward compatibility


Assisted scheduling We introduce the Belief-Propagation Message-Processing

module between the VOQs and the Scheduler

BP-MP computes message values as a function of the queue length Q(t), based on a BP algorithm

The scheduler works in the usual way, but scheduling decisions are based on the messages F(t) computed by the BP-MP module instead that on Q(t) F(t) can be see as a correction of the VOQ lengths Q(t)

tQBP-MPfew I

tF

tF

Scheduler

tX


Assisted scheduling BP propagation has been improved with:

Relaxation of the MWM uniqueness constraint We do not need BP to converge anymore No random noise

Finite (and small) number of iterations Integer number representation Memory Self-Asynchronous update


Messages for assisted scheduling It runs for a fixed (and small) number of iterations I


Messages are bounded

Messages represented through

integer numbersSame numerical

range of the queue length(around log2 Qmax bits)

Memory for assisted schedulingQueues exhibit a strong correlation that is reflected in the message dynamics

Queue length can change at the most by 1 at each timeslot

Memory: messages are initialized to the last computed messages

Memory speeds up convergence


Self-asynchronous update for assisted schedulingStudies in BP showed that messages updated in a random sequential order are beneficial for the convergence (asynchronous update) Not easy to implement in HW

Self-asynchronous update:•exploits randomness of the arrival process•updates only messages associated with queues which have changed from the previous timeslot•mimics asynchronous update


Scheduling algorithms iLQF vs. BP assisted iLQF (BP-iLQF)

Distributed greedy algorithm Each input (each output) is equipped with an arbiter which selects

output (input) associated with the longest queue Greedy MWM (GMWM) vs. BP assisted GMWM (BP-GMWM)

centralized scheduling, iterating N times at each iteration it selects the unmatched input/output couple

associated with the longest queue iSLIP

as iLQF, but sending only a binary information (queue empty/not-empty)


Performance evaluation settings Simulation settings:

Traffic patterns:

Critical traffic pattern


Performance evaluation results

BP assisted scheduling improves performance (I=3)

Memory

No Memory


Self-asynchronous

Synchronous

Asynchronous

Hardware design: General overview

2N modules running in parallel

BP-MPBackward messages

Forward messages

• When n=I, IM sends F(t) to the scheduler

• IM and OM perform the same operations

VOQ tQ

tFScheduler


Hardware design: IM details

Self-asynchronous:if wij(t)≠ wij(t-1) eij=1else eij=0

Flags associated with VOQ at input i

Memory: registers storingmessages computed during the previous timeslot

Max operation

Tournament implementationlog2 (N-1) stages and (N-2) comparisons

c used to select between 0 and the result of the subtraction operation

Subtraction operation

When n=I messages are sent to the scheduler


N registers of size log2 Qmax

Conclusion We proposed BP assisted scheduling to boost

performance of existing scheduling algorithms keeping backward compatibility BP runs for few iterations

We simplified and improved basic BP algorithm: Relaxation of MWM uniqueness constraint Integer messages (backward compatibility) Message memory Self-asynchronous update

We provided a high-level description of a possible HW implementation of the BP-MP: BP-MP can be efficiently implemented in HW and it is

compatible with existing implementations


Belief-Propagation Assisted Scheduling in Input-Queued

Switches

S. Atalla1, D. Cuda2, P. Giaccone1, M. Pretti2

1Politecnico di Torino2Italian National Research Council

Hot Interconnects 2010August 2010

Any questions?

Thank you for your attention!


Example: MWM computation over a tree Node “1” must decide to add or not edge (1,2) to

the matching Node “1” takes its decision based on the information

provided only by nodes belonging to its neighborhood

E.g., Node “2” sends to “1” two messages: : MWM of the sub-tree rooted at “2” comprising (2,1)

given that (2,1) is part of the MWM rooted at “1” : MWM of the sub-tree rooted at 2 comprehending (2,1)

given that (2,1) is part of the MWM rooted at “1”

12W

12W

1

2

34

5 6

7

Take or not to take (2,1)?

w 32

w21

w42

w42

w61

w71


Example: MWM computation over a tree

Message definitions:

If (2,1) is part of the MWM, then (3,2), (4,2), (5,2) can not be in the MWM

if (2,1) is not the MWM, then at the most one (or none) among (3,2), (4,2), (5,2) can part of the MWM

It is possible to reduce the number of exchanged messages combining into a single message

2524232112 WWWwW

12

34 5

w21

25W

},

,,max{

252423252423

25242325242312

WWWWWW

WWWWWWW1

2

34

5 25W

,12W


Example: MWM computation over a tree

Node “1” decision: Node “1” adds edge (1,2) to the MWM if:

or equivalently

},

,max{

171612171612

171612171512

WWWWWW

WWWWWW

}0,,max{ 161512 1

2

34

5 6

7

Take or not to take (2,1)?

w 32

w21

w42

w42

w61

w71


Graphical models BP algorithms are message passing algorithms

conceived firstly to study Graphical Models (GMs) GMs are a “marriage” between probability theory and

graph theory lo direi solo a voce, non significa niente qui GMs are becoming a powerful tool in several fields

of science (AI, speech recognition, coding/decoding, bioinformatics) to compute marginal probabilities and maximum a posteriori probability (max-product algorithm) “BP” and “max-product “ are usually simply referred as

“BP” since computing the maximum a posteriori probability requires first to compute the marginal distributions io questa frase non l’ho capita e mi pare rischiosissima!!!


VOQ BP-MP Scheduler


Scheduler: iLQF If the MWM is unique, BP assisted iLQF, running

with weights computes exactly the MWM


Performance evaluation: results

BP assisted scheduling improves performance(I=3)

Average delays : delays BP-iLQF/GWM are at the most 1.37 times delays of iLQF/GWM.

Memory

No Memory


Self-asynchronous

Synchronous

Asynchronous


0 0


3 3

belief-propagation assisted scheduling in input-queued switches s. atalla 1, d. cuda 2, p. giaccone...

Documents

scheduling beliefpropagation

mwm problem

suitable scheduling

scheduling decision

bp algorithms

j q ij scheduler mwm

inputqueued switches

heuristic algorithms