On Universal Scaling of Distributed Queues under Load Balancing

This paper considers the steady-state performance of load balancing algorithms in a many-server system with distributed queues. The system has $N$ servers, and each server maintains a local queue with buffer size $b-1,$ i.e. a server can hold at most one job in service and $b-1$ jobs in the queue. Jobs in the same queue are served according to the first-in-first-out (FIFO) order. The system is operated in a heavy-traffic regime such that the workload per server is $\lambda = 1 - N^{-\alpha}$ for $0.5\leq \alpha<1.$ We identify a set of algorithms such that the steady-state queues have the following universal scaling, where {\em universal} means that it holds for any $\alpha\in[0.5,1)$: (i) the number of of busy servers is $\lambda N-o(1);$ and (ii) the number of servers with two jobs (one in service and one in queue) is $O(N^{\alpha}\log N);$ and (iii) the number of servers with more than two jobs is $O\left(\frac{1}{N^{r(1-\alpha)-1}}\right),$ where $r$ can be any positive integer independent of $N.$ The set of load balancing algorithms that satisfy the sufficient condition includes join-the-shortest-queue (JSQ), idle-one-first (I1F), and power-of-$d$-choices (Po$d$) with $d\geq N^\alpha\log^2 N.$ We further argue that the waiting time of such an algorithm is near optimal order-wise.