2019/07/10 by Hassan Hmedi, Hmedi, Hassan, Ari Arapostathis +3
Business, Management and Accounting · Computer Science · Engineering · #49L20 #60K25 #90B22 (Primary) #90B36 (Secondary) #Advanced Queuing Theory Analysis #Advanced Wireless Network Optimization #FOS: Electrical engineering #FOS: Mathematics #Network Traffic and Congestion Control #Optimization and Control (math.OC) #Probability (math.PR) #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1907.04793
openalex publication_date 2019/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the uniform stability properties of two classes of parallel server networks with multiple classes of jobs and multiple server pools of a tree topology. These include a class of networks with a single non-leaf server pool, such as the 'N' and 'M' models, and networks of any tree topology with class-dependent service rates. We show that with √(n) safety staffing, and no abandonment, in the Halfin--Whitt regime, the diffusion-scaled controlled queueing processes are exponentially ergodic and their invariant probability distributions are tight, uniformly over all stationary Markov controls. We use a unified approach in which the same Lyapunov function is used in the study of the prelimit and diffusion limit. A parameter called the spare capacity (safety staffing) of the network plays a central role in characterizing the stability results: the parameter being positive is necessary and sufficient that the limiting diffusion is uniformly exponentially ergodic over all stationary Markov controls. We introduce the concept of "system-wide work conserving policies", which are defined as policies that minimize the number of idle servers at all times. This is stronger than the so-called joint work conservation. We show that, provided the spare capacity parameter is positive, the diffusion-scaled processes are geometrically ergodic and the invariant distributions are tight, uniformly over all "system-wide work conserving policies". In addition, when the spare capacity is negative we show that the diffusion-scaled processes are transient under any stationary Markov control, and when it is zero, they cannot be positive recurrent.