Stationary distribution convergence of the offered waiting processes for GI/GI/1+GI queues in heavy traffic

• Published in: Queueing systems Vol. 94; no. 1-2; pp. 147 - 173
• Main Authors: Lee, Chihoon, Ward, Amy R., Ye, Heng-Qing
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2020; Springer Nature B.V
• Subjects: Abandonment; Business and Management; Computer Communication Networks; Control; Convergence; Markov analysis; Operations Research/Decision Theory; Probability distribution functions; Probability Theory and Stochastic Processes; Queues; Queuing theory; Supply Chain Management; Systems Theory; 90B22; Customer abandonment; 93E15; Heavy traffic; Primary: 60K25; Stationary distribution convergence; secondary: 60J25
• ISSN: 0257-0130; 1572-9443
• DOI: 10.1007/s11134-019-09641-y

A result of Ward and Glynn (Queueing Syst 50(4):371–400, 2005) asserts that the sequence of scaled offered waiting time processes of the G I / G I / 1 + G I queue converges weakly to a reflected Ornstein–Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate the scaled stationary distribution of the offered waiting time in a G I / G I / 1 + G I queue; however, no such result has been proved. We prove the aforementioned convergence, and the convergence of the moments, in heavy traffic, thus resolving a question left open in 2005. In comparison with Kingman’s classical result (Kingman in Proc Camb Philos Soc 57:902–904, 1961) showing that an exponential distribution approximates the scaled stationary offered waiting time distribution in a GI  /  GI  / 1 queue in heavy traffic, our result confirms that the addition of customer abandonment has a non-trivial effect on the queue’s stationary behavior.