TY - JOUR
T1 - Stationary distribution convergence of the offered waiting processes for GI/ GI/ 1 + GI queues in heavy traffic
AU - Lee, Chihoon
AU - Ward, Amy R.
AU - Ye, Heng Qing
N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/2/1
Y1 - 2020/2/1
N2 - 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 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 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.
AB - 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 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 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.
KW - Customer abandonment
KW - Heavy traffic
KW - Stationary distribution convergence
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U2 - 10.1007/s11134-019-09641-y
DO - 10.1007/s11134-019-09641-y
M3 - Article
AN - SCOPUS:85075952610
SN - 0257-0130
VL - 94
SP - 147
EP - 173
JO - Queueing Systems
JF - Queueing Systems
IS - 1-2
ER -