Abstract
Understanding how delayed information impacts queueing systems is an important area of research. However, much of the current literature neglects one important feature of many queueing systems, namely non-stationary arrivals. Non-stationary arrivals model the fact that customers tend to access services during certain times of the day and not at a constant rate. In this paper, we analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information with time-varying arrivals. In the first model, customers receive queue length information that is delayed by a constant \(\Delta \). In the second model, customers receive information about the queue length through a moving average of the queue length where the moving average window is \(\Delta \). We analyze the impact of a time-varying arrival rate and show using asymptotic analysis that the time-varying arrival rate does not impact the critical delay unless the frequency of the time-varying arrival rate is twice that of the critical delay. When the frequency of the arrival rate is twice that of the critical delay, then the stability is enlarged by a wedge that is determined by the model parameters. As a result, this problem allows us to combine the theory of nonlinear dynamics, parametric excitation, delays, and time-varying queues together to provide insight into the impact of information on queueing systems.
Similar content being viewed by others
References
Allon, G., Bassamboo, A.: The impact of delaying the delay announcements. Oper. Res. 59(5), 1198–1210 (2011)
Allon, G., Bassamboo, A., Gurvich, I.: “We will be right with you”: managing customer expectations with vague promises and cheap talk. Oper. Res. 59(6), 1382–1394 (2011)
Armony, M., Maglaras, C.: On customer contact centers with a call-back option: customer decisions, routing rules, and system design. Oper. Res. 52(2), 271–292 (2004)
Armony, M., Shimkin, N., Whitt, W.: The impact of delay announcements in many-server queues with abandonment. Oper. Res. 57(1), 66–81 (2009)
Armony, M., Israelit, S., Mandelbaum, A., Marmor, Y.N., Tseytlin, Y., Yom-Tov, G.B., et al.: On patient flow in hospitals: a data-based queueing-science perspective. Stoch. Syst. 5(1), 146–194 (2015)
Ben-Akiva, M., Bierlaire, M.: Discrete choice methods and their applications to short term travel decisions. Handb. Transp. Sci. 23, 5–33 (1999)
Dong, J., Yom-Tov, E., Yom-Tov, G.B.: The impact of delay announcements on hospital network coordination and waiting times. Technical report, Working Paper (2015)
Eick, S.G., Massey, W.A., Whitt, W.: Mt/G/\(\infty \) queues with sinusoidal arrival rates. Manag. Sci. 39(2), 241–252 (1993a)
Eick, S.G., Massey, W.A., Whitt, W.: The physics of the Mt/G/\(\infty \) queue. Oper. Res. 41(4), 731–742 (1993b)
Guo, P., Zipkin, P.: Analysis and comparison of queues with different levels of delay information. Manag. Sci. 53(6), 962–970 (2007)
Guo, P., Zipkin, P.: The impacts of customers’delay-risk sensitivities on a queue with balking. Probab. Eng. Inf. Sci. 23(03), 409–432 (2009)
Hassin, R.: Information and uncertainty in a queuing system. Probab. Eng. Inf. Sci. 21(03), 361–380 (2007)
Hui, M.K., Tse, D.K.: What to tell consumers in waits of different lengths: an integrative model of service evaluation. J. Mark. 60, 81–90 (1996)
Hul, M.K., Dube, L., Chebat, J.C.: The impact of music on consumers’ reactions to waiting for services. J. Retail. 73(1), 87–104 (1997)
Ibrahim, R., Whitt, W.: Real-time delay estimation in call centers. In Proceedings of the 40th Conference on Winter Simulation. Winter Simulation Conference, pp. 2876–2883 (2008)
Ibrahim, R., Whitt, W.: Real-time delay estimation in overloaded multiserver queues with abandonments. Manag. Sci. 55(10), 1729–1742 (2009)
Ibrahim, R., Whitt, W.: Real-time delay estimation based on delay history in many-server service systems with time-varying arrivals. Prod. Oper. Manag. 20(5), 654–667 (2011a)
Ibrahim, R., Whitt, W.: Wait-time predictors for customer service systems with time-varying demand and capacity. Oper. Res. 59(5), 1106–1118 (2011b)
Ibrahim, R., Armony, M., Bassamboo, A.: Does the past predict the future? the case of delay announcements in service systems (2015)
Jennings, O.B., Pender, J.: Comparisons of ticket and standard queues. Queueing Syst. 84, 145–202 (2016)
Jouini, O., Dallery, Y., Akşin, Z.: Queueing models for full-flexible multi-class call centers with real-time anticipated delays. Int. J. Prod. Econ. 120(2), 389–399 (2009)
Jouini, O., Aksin, Z., Dallery, Y.: Call centers with delay information: models and insights. Manuf. Serv. Oper. Manag. 13(4), 534–548 (2011)
Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics, vol. 34. Springer, Berlin (2013)
Ko, Y.M., Pender, J.: Strong Approximations for Time-Varying Infinite-Server Queues with Non-Renewal Arrival and Service Processes. Cornell University, Ithaca (2016)
Ko, Y.M., Pender, J.: Diffusion limits for the (MAPt/Pht/\(\infty \)) N queueing network. Oper. Res. Lett. 45(3), 248–253 (2017)
Munichor, N., Rafaeli, A.: Numbers or apologies? Customer reactions to telephone waiting time fillers. J. Appl. Psychol. 92(2), 511 (2007)
Ng, L., Rand, R.: Nonlinear effects on coexistence phenomenon in parametric excitation. Nonlinear Dyn. 31(1), 73–89 (2003)
Pender, J.: A poisson-charlier approximation for nonstationary queues. Oper. Res. Lett. 42(4), 293–298 (2014)
Pender, J., Ko, Y.M.: Approximations for the queue length distributions of time-varying many-server queues. INFORMS J. Comput 29(4), 688–704 (2017)
Pender, J., Rand, R.H., Wesson, E.: Managing information in queues: the impact of giving delayed information to customers. arXiv preprint arXiv:1610.01972 (2016)
Pender, J., Rand, R.H., Wesson, E.: Queues with choice via delay differential equations. Int. J. Bifurc. Chaos 27(04), 1730016 (2017)
Pruyn, A., Smidts, A.: Effects of waiting on the satisfaction with the service: beyond objective time measures. Int. J. Res. Mark. 15(4), 321–334 (1998)
Ruelas, R.E., Rand, D.G., Rand, R.H.: Nonlinear parametric excitation of an evolutionary dynamical system. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 226(8), 1912–1920 (2012)
Sarel, D., Marmorstein, H.: Managing the delayed service encounter: the role of employee action and customer prior experience. J. Serv. Mark. 12(3), 195–208 (1998)
Taylor, S.: Waiting for service: the relationship between delays and evaluations of service. J. Mark. pp 56–69 (1994)
Train, K.E.: Discrete Choice Methods with Simulation. Cambridge University Press, Cambridge (2009)
Whitt, W.: Improving service by informing customers about anticipated delays. Manag. Sci. 45(2), 192–207 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pender, J., Rand, R.H. & Wesson, E. An analysis of queues with delayed information and time-varying arrival rates. Nonlinear Dyn 91, 2411–2427 (2018). https://doi.org/10.1007/s11071-017-4021-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-017-4021-0