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- Title
Stationary distribution convergence of the offered waiting processes for GI/GI/1+GI queues in heavy traffic.
- Authors
Lee, Chihoon; Ward, Amy R.; Ye, Heng-Qing
- Abstract
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.
- Subjects
URUGUAY; ORNSTEIN-Uhlenbeck process; OPEN-ended questions
- Publication
Queueing Systems, 2020, Vol 94, Issue 1/2, p147
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-019-09641-y