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

Chihoon Lee (), Amy R. Ward () and Heng-Qing Ye ()
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Chihoon Lee: Stevens Institute of Technology
Amy R. Ward: The University of Chicago
Heng-Qing Ye: Hong Kong Polytechnic University

Queueing Systems: Theory and Applications, 2020, vol. 94, issue 1, No 7, 147-173

Abstract: 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 $$GI/GI/1+GI$$GI/GI/1+GI 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 $$GI/GI/1+GI$$GI/GI/1+GI 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.

Keywords: Customer abandonment; Heavy traffic; Stationary distribution convergence; Primary: 60K25; secondary: 60J25; 93E15; 90B22 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11134-019-09641-y

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