Convergence of a queueing system in heavy traffic with general patience-time distributions

• Published in: Stochastic processes and their applications Vol. 121; no. 11; pp. 2507 - 2552
• Main Authors: Lee, Chihoon, Weerasinghe, Ananda
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.11.2011; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Controlled queueing systems; Customer abandonment; Customer impatience; Diffusion approximations; Distribution theory; Exact sciences and technology; Heavy traffic theory; Mathematics; Probability and statistics; Probability theory and stochastic processes; Reneging; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stochastic analysis; Stochastic control; Stochastic control Controlled queueing systems Heavy traffic theory Diffusion approximations Customer abandonment Customer impatience Reneging; Stochastic processes; secondary; Heavy traffic theory; Customer abandonment; Stochastic control; Customer impatience; Diffusion approximations; primary; Controlled queueing systems; Reneging; primary 60K25; Stochastic process; Waiting time; Convergence; Drift coefficient; Stochastic control(mathematics); Queue length; secondary 90B18; Heavy traffic; Distribution function; Diffusion process; 68M20; Infinite horizon; Diffusion coefficient; Approximation; Functional; Arrival time; Time distribution; Service time; 90B22; Queueing system; Dependent process
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2011.07.003

We analyze a sequence of single-server queueing systems with impatient customers in heavy traffic. Our state process is the offered waiting time, and the customer arrival process has a state dependent intensity. Service times and customer patient-times are independent; i.i.d. with general distributions subject to mild constraints. We establish the heavy traffic approximation for the scaled offered waiting time process and obtain a diffusion process as the heavy traffic limit. The drift coefficient of this limiting diffusion is influenced by the sequence of patience-time distributions in a non-linear fashion. We also establish an asymptotic relationship between the scaled version of offered waiting time and queue-length. As a consequence, we obtain the heavy traffic limit of the scaled queue-length. We introduce an infinite-horizon discounted cost functional whose running cost depends on the offered waiting time and server idle time processes. Under mild assumptions, we show that the expected value of this cost functional for the n -th system converges to that of the limiting diffusion process as n tends to infinity.