Large-System Insensitivity of Zero-Waiting Load Balancing Algorithms

This paper studies the sensitivity (or insensitivity) of a class of load balancing algorithms that achieve asymptotic zero-waiting in the sub-Halfin-Whitt regime, named LB-zero. Most existing results on zero-waiting load balancing algorithms assume the service time distribution is exponential. This paper establishes the large-system insensitivity of LB-zero for jobs whose service time follows a Coxian distribution with a finite number of phases. This result justifies that LB-zero achieves asymptotic zero-waiting for a large class of service time distributions as the Coxian family is dense in the class of positive-valued distributions. To prove this result, this paper develops a new technique, called “iterative state-space peeling” (ISSP). ISSP first identifies an iterative relation between the upper and lower bounds on the queue states and then proves that the system lives near the fixed point of the iterative bounds with a high probability. Based on ISSP, the steady-state distribution of the queue length is further analyzed by applying Stein’s method in the neighborhood of the fixed point. ISSP, like state-space collapse in heavy-traffic analysis, is a general approach that may be used to study other complex stochastic systems.