SYSTEMS WITH LARGE FLEXIBLE SERVER POOLS: INSTABILITY OF NATURAL LOAD BALANCING

• 成果类型: Article
• 署名作者: Stolyar, Alexander L.; Yudovina, Elena
• 署名单位: AT&T; Alcatel-Lucent; University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP895
• 发表日期: 2013
• 页码: 2099-2138
• 关键词: multiserver pools; QUEUE

摘要

We consider general large-scale service systems with multiple customer classes and multiple server (agent) pools, mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a natural (load balancing) routing/scheduling rule, Longest-Queue Freest-Server (LQFS-LB), in the many-server asymptotic regime, such that the exogenous arrival rates of the customer classes, as well as the number of agents in each pool, grow to infinity in proportion to some scaling parameter r. Equilibrium point of the system under LQBS-LB is the desired operating point, with server pool loads minimized and perfectly balanced. Our main results are as follows. (a) We show that, quite surprisingly (given the tree assumption), for certain parameter ranges, the fluid limit of the system may be unstable in the vicinity of the equilibrium point; such instability may occur if the activity graph is not too small. (b) Using (a), we demonstrate that the sequence of stationary distributions of diffusion-scaled processes [measuring O(root r) deviations from the equilibrium point] may be nontight, and in fact may escape to infinity. (c) In one special case of interest, however, we show that the sequence of stationary distributions of diffusion-scaled processes is tight, and the limit of stationary distributions is the stationary distribution of the limiting diffusion process.