Load Balancing and Switch Scheduling Xiangheng Liu EE 384Y Final Presentation June 4, 2003

Motivation – Duality? Load Balancing Algorithms –Join the Shortest Queue (SQ) –Join a random queue (RAND) –Join the Shortest Queue of Randomly Chosen d queues (SQ(d)) –Randomization with Memory (SQ(d,m)) Bandwidth Allocation Algorithms –Longest Queue First (LQF) –Serve a random queue (RAND) –Serve the Longest Queue of Randomly chosen d queues (LQF(d)) –Randomization with Memory (LQF(d,m))

One Dimensional Case Queue Dynamics Linear Approximation Taking the negative Let we have

Intuition for the Resemblance Bandwidth allocation  Load balancing Trick: Negative dual system Cast bandwidth allocation as load balancing –Reverse the directions of all arrows –Arrivals: arrival of tokens = departure of packets –System state = Number of tokens (Note: the state is always less or equal to 0) –Service: departure of tokens = arrival of packet SQ in load balancing will translate to LQF in bandwidth allocation based on this duality. Duality only true for linear approximation.

2-D: N x N crossbar switch One arrival at a given input per time slot. No buffering at the input. Each block can only read from one input per time slot.

Dual of MWM in Load Balancing Load balancing in crossbar switches is essentially a matching problem. Switch scheduling is better studied in 2-D. Cast load balancing as switch scheduling. Design load balancing algorithm based on MWM scheduling. –At time n, if there is no packet arriving at input i, we let q ij =  for all j = 1, …, N. –Find a matching that minimizes the weighted sum of queue lengths.

Joint Load Balancing and Switch Scheduling: 1-D Dual algorithms usually give optimal solutions for linear systems. –Kalman filter and State feedback controller –Linear Modulation and Matched filter. Load Balancer Bandwidth Allocator Poisson(n ) exp(n)

Mean Field Analysis Assume the load balancer uses SQ(d) and the bandwidth allocation adopts LQF(d). s i (t) – fraction of queues with load at least i. Mean Field Analysis – differential equation In Equilibrium, Backward Recursion:

Performance Comparison

Entropy Rate of Load Balancing System Entropy rate of one dimensional load balancing system with a class of randomized algorithm. Theorem 1 [Nair, et. Al. 2001] Under mild conditions, the entropy rate of queue size process of the load balancing system with any algorithm describable by the coin toss model is equal to (H ER (A) + H(S) + H(C)). Same class of randomized algorithm exists for bandwidth allocation. If the queue dynamics were linear, the dual bandwidth allocation system should have the same entropy rate.

Dual Analysis – Entropy rate of randomized bandwidth allocation Failed Attempt – try to prove the two systems have equal entropy rates. Conjecture 1 – The dual bandwidth allocation system has an entropy rate that is upper bounded by (H ER (A) + H(S) + H(I) + H(C)) and lower bounded by (H ER (A) + H(S+I)). The bijection that proves Theorem 1 for load balancing is not true for switch scheduling. Reason – the server may choose an empty queue with the result of coin tosses and it will be tossed again in the next time slot.

Conclusions The Load Balancing and Switch Scheduling are duals of each other if we assume the linear queue dynamics approximation. We can design new algorithms in one area by taking advantage of the research in the other. Joint load balancing algorithms lead to great performance gains over the system where only load balancing is used. Failed attempt – relate the entropy rates of the dual systems.