Abstract
We study a sequence of single server queues with customer abandonment (\(GI/GI/1+GI\)) under heavy traffic. The patience time distributions vary with the sequence, which allows for a wider scope of applications. It is known Lee and Weerasinghe (Stochastic Process Appl 121(11):2507–2552, 2011) and Reed and Ward (Math Oper Res 33(3):606–644, 2008) that the sequence of scaled offered waiting time processes converges weakly to a reflecting diffusion process with nonlinear drift, as the traffic intensity approaches one. In this paper, we further show that the sequence of stationary distributions and moments of the offered waiting times, with diffusion scaling, converge to those of the limit diffusion process. This justifies the stationary performance of the diffusion limit as a valid approximation for the stationary performance of the \(GI/GI/1+GI\) queue. Consequently, we also derive the approximation for the abandonment probability for the \(GI/GI/1+GI\) queue in the stationary state.
Similar content being viewed by others
References
Asmussen, S.: Applied Probability and Queues. Applications of Mathematics, vol. 51, 2nd edn. Springer, New York (2003)
Baccelli, F., Boyer, P., Hébuterne, G.: Single-server queues with impatient customers. Adv. Appl. Probab. 16(4), 887–905 (1984)
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)
Bramson, M.: State space collapse with application to heavy traffic limits for multiclass queueing networks. Queu. Syst. 30(1–2), 89–148 (1998)
Bramson, M.: Stability of Queueing Networks, volume 1950 of Lecture Notes in Mathematics. Springer, Berlin (2008)
Braverman, A., Dai, J.G., Miyazawa, M.: Heavy traffic approximation for the stationary distribution of a generalized Jackson network: the BAR approach. Stochastic Syst. 7(1), 143–196 (2017)
Budhiraja, A., Lee, C.: Stationary distribution convergence for generalized Jackson networks in heavy traffic. Math. Oper. Res. 34(1), 45–56 (2009)
Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. Applications of Mathematics, vol. 46. Springer, New York (2001)
Chen, H., Ye, H.-Q.: Asymptotic optimality of balanced routing. Oper. Res. 60(1), 163–179 (2012)
Echeverria, P.: A criterion for invariant measures of Markov processes. Z. Wahrsch. Verw. Gebiete 61, 1–16 (1982)
Eryilmaz, A., Srikant, R.: Asymptotically tight steady-state queue length bounds implied by drift conditions. Queu. Syst. 72(3–4), 311–359 (2012)
Gamarnik, D., Zeevi, A.: Validity of heavy traffic steady-state approximations in open queueing networks. Ann. Appl. Probab. 16(1), 56–90 (2006)
Huang, J., Gurvich, I.: Beyond heavy-traffic regimes: Universal bounds and controls for the single-server queue. Oper. Res. 66(4), 1168–1188 (2018)
Huang, J., Zhang, H., Zhang, J.: A unified approach to diffusion analysis of queues with general patience-time distributions. Math. Oper. Res. 41(3), 1135–1160 (2016)
Hurtado-Lange, D., Maguluri, S.T.: Transform methods for heavy-traffic analysis. Stochastic Syst. 10(4), 275–309 (2020)
Lee, C., Ward, A.R.: Pricing and capacity sizing of a service facility: customer abandonment effects. Prod. Oper. Manag. 28(8), 2031–2043 (2019)
Lee, C., Ward, A.R., Ye, H.-Q.: Stationary distribution convergence of the offered waiting processes for \(GI/GI/1+GI\) queues in heavy traffic. Queu. Syst. 94(7), 147–173 (2020)
Lee, C., Weerasinghe, A.: Convergence of a queueing system in heavy traffic with general patience-time distributions. Stochastic Process. Appl. 121(11), 2507–2552 (2011)
Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25, 518–548 (1993)
Reed, J.E., Ward, A.R.: Approximating the \(GI/GI/1+GI\) queue with a nonlinear drift diffusion: hazard rate scaling in heavy traffic. Math. Oper. Res. 33(3), 606–644 (2008)
Ward, A.R., Glynn, P.W.: A diffusion approximation for a \(GI/GI/1\) queue with balking or reneging. Queu. Syst. 50(4), 371–400 (2005)
Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)
Ye, H.-Q., Yao, D.D.: Diffusion limit of fair resource control–stationarity and interchange of limits. Math. Oper. Res. 41(4), 1161–1207 (2016)
Ye, H.-Q., Yao, D.D.: Justifying diffusion approximations for stochastic processing networks under a moment condition. Ann. Appl. Probab. 28(6), 3652–3697 (2018)
Acknowledgements
Amy Ward is supported as Charles M. Harper Faculty Fellow at the University of Chicago Booth School of Business, and Heng-Qing Ye is supported by the HK/RGC Grant 15503519.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Lee, C., Ward, A.R. & Ye, HQ. Stationary distribution convergence of the offered waiting processes in heavy traffic under general patience time scaling. Queueing Syst 99, 283–303 (2021). https://doi.org/10.1007/s11134-021-09716-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-021-09716-9