Abstract
We previously introduced and analyzed the G t /M t /s t +GI t many-server fluid queue with time-varying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper, we establish an asymptotic loss of memory (ALOM) property for that fluid model, i.e., we show that there is asymptotic independence from the initial conditions as time t evolves, under regularity conditions. We show that the difference in the performance functions dissipates over time exponentially fast, again under the regularity conditions. We apply ALOM to show that the stationary G/M/s+GI fluid queue converges to steady state and the periodic G t /M t /s t +GI t fluid queue converges to a periodic steady state as time evolves, for all finite initial conditions.
Similar content being viewed by others
References
Eick, S.G., Massey, W.A., Whitt, W.: M t /G/∞ queues with sinusoidal arrival rates. Manag. Sci. 39(2), 241–252 (1993)
Garnett, O., Mandelbaum, A., Reiman, M.I.: Designing a call center with impatient customers. Manuf. Serv. Oper. Manag. 4, 208–227 (2002)
Granovsky, B.L., Zeifman, A.: Nonstationary queues: estimating the rate of convergence. Queueing Syst. 46, 363–388 (2004)
Hall, R.W.: Queueing Methods for Services and Manufacturing. Prentice-Hall, Englewood Cliffs (1991)
Heyman, D., Whitt, W.: The asymptotic behavior of queues with time-varying arrival rates. J. Appl. Probab. 21, 143–156 (1984)
Isaacson, D., Madsen, R.: Markov Chains: Theory and Applications. Wiley, New York (1976)
Krichagina, E.V., Puhalskii, A.A.: A heavy-traffic analysis of a closed queueing system with a GI/∞ service center. Queueing Syst. 25, 235–280 (1997)
Liu, Y., Whitt, W.: A fluid approximation for the G t /GI/s t +GI queue. Columbia University, NY (2010). http://www.columbia.edu/~ww2040/allpapers.html
Liu, Y., Whitt, W.: A network of time-varying many-server fluid queues with customer abandonment. Columbia University, NY (2010). http://www.columbia.edu/~ww2040/allpapers.html
Liu, Y., Whitt, W.: The heavily loaded many-server queue with abandonment and deterministic service times, Columbia University, NY (2010). http://www.columbia.edu/~ww2040/allpapers.html
Mandelbaum, A., Massey, W.A., Reiman, M.I.: Strong approximations for Markovian service networks. Queueing Syst. 30, 149–201 (1998)
Mandelbaum, A., Massey, W.A., Reiman, M.I., Rider, B.: Time varying multiserver queues with abandonments and retrials. In: Key, P., Smith, D. (eds.) Proceedings of the 16th International Teletraffic Congress (1999)
Mandelbaum, A., Massey, W.A., Reiman, M.I., Stolyar, A.: Waiting time asymptotics for time varying multiserver queues with abandonment and retrials. In: Proceedings of the Thirty-Seventh Annual Allerton Conference on Communication, Control and Computing, Allerton, IL, pp. 1095–1104 (1999)
Newell, G.F.: Applications of Queueing Theory, 2nd edn. Chapman and Hall, London (1982)
Pang, G., Whitt, W.: Two-parameter heavy-traffic limits for infinite-server queues. Queueing Syst. 65, 325–364 (2010)
Whitt, W.: Fluid models for multiserver queues with abandonments. Oper. Res. 54, 37–54 (2006)
Wilie, H.: Periodic steady state of loss systems. Adv. Appl. Probab. 30, 152–166 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, Y., Whitt, W. Large-time asymptotics for the G t /M t /s t +GI t many-server fluid queue with abandonment. Queueing Syst 67, 145–182 (2011). https://doi.org/10.1007/s11134-010-9208-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-010-9208-8
Keywords
- Nonstationary queues
- Queues with time-varying arrivals
- Many-server queues
- Deterministic fluid model
- Customer abandonment
- Loss of memory
- Weakly ergodic
- Periodic steady state
- Transient behavior