Heavy-traffic limits for many-server queues with service interruptions

• Published in: Queueing systems Vol. 61; no. 2-3; pp. 167 - 202
• Main Authors: Pang, Guodong, Whitt, Ward
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2009; Springer Nature B.V
• Subjects: Approximation; Business and Management; Computer Communication Networks; Control; Customer services; Decomposition; Integral equations; Operations Research/Decision Theory; Ordinary differential equations; Probability Theory and Stochastic Processes; Queuing theory; Random variables; Servers; Stochastic models; Studies; Supply Chain Management; Systems Theory; Traffic flow; 60J75; 60G44; 54A20; Service interruptions; Jump-diffusion process; topology; Continuous mapping theorem; Skorohod; Levy-driven O–U process; Heavy-traffic limits; 90B22; Many-server queues; Stochastic-decomposition property; 90B15; 60K25; 60F17
• ISSN: 0257-0130; 1572-9443
• DOI: 10.1007/s11134-009-9104-2

We establish many-server heavy-traffic limits for G / M / n + M queueing models, allowing customer abandonment (the + M ), subject to exogenous regenerative service interruptions. With unscaled service interruption times, we obtain a FWLLN for the queue-length process, where the limit is an ordinary differential equation in a two-state random environment. With asymptotically negligible service interruptions, we obtain a FCLT for the queue-length process, where the limit is characterized as the pathwise unique solution to a stochastic integral equation with jumps. When the arrivals are renewal and the interruption cycle time is exponential, the limit is a Markov process, being a jump-diffusion process in the QED regime and an O–U process driven by a Levy process in the ED regime (and for infinite-server queues). A stochastic-decomposition property of the steady-state distribution of the limit process in the ED regime (and for infinite-server queues) is obtained.