Optimal Load Balancing in Heterogeneous Server Systems

A fundamental problem in large-scale data centers is to reduce the average response time of jobs. The Join-the-Shortest-Queue (JSQ) load balancing scheme is known to minimise the average response time of jobs for homogeneous systems consisting of identical servers. However, heterogeneous systems consisting of servers with different speeds, JSQ performs poorly. Furthermore, JSQ suffers from high communication overhead as it requires knowledge of the queue length of all servers while assigning incoming jobs to one of the destination servers. Therefore, the Join-the-Idle-Queue (JIQ) scheme was introduced which not only reduces the communication overhead but also minimises the average response time of jobs under homogeneous systems. Despite these advantages, JIQ is still known to be inefficient in minimising the average response time of jobs under heterogeneous systems. In this paper, we consider a speed-aware version of JIQ for heterogeneous systems and show that it achieves delay optimality in the fluid limit. One of the technical challenges to establishing this optimality is to show the tightness of the sequence of steady-state distributions indexed by system size. We show this tightness result by evaluating the drift of appropriate Lyapunov functions. This approach to proving tightness is different from the usual coupling approach used for homogeneous systems. Another important challenge in proving the optimality result is to establish the fluid limit which is done using the time-scale separation technique. Finally, using the monotonicity of the fluid process we have shown that the fluid limit has a unique and globally attractive fixed point.