Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T00:09:02.391Z Has data issue: false hasContentIssue false
  • English
  • Français

Heavy-Traffic Limits for a Many-Server Queueing Network with Switchover

Part of: Operations research and management science Special processes Limit theorems

Published online by Cambridge University Press:  04 January 2016

Guodong Pang  and
David D. Yao
Guodong Pang*
Affiliation:
Pennsylvania State University
David D. Yao*
Affiliation:
Columbia University
*
∗ Postal address: The Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, PA 16802, USA. Email address: [email protected]
∗∗ Postal address: Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027-6699, USA. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a multiclass Markovian queueing network with switchover across a set of many-server stations. New arrivals to each station follow a nonstationary Poisson process. Each job waiting in queue may, after some exponentially distributed patience time, switch over to another station or leave the network following a probabilistic and state-dependent mechanism. We analyze the performance of such networks under the many-server heavy-traffic limiting regimes, including the critically loaded quality-and-efficiency-driven (QED) regime, and the overloaded efficiency-driven (ED) regime. We also study the limits corresponding to mixing the underloaded quality-driven (QD) regime with the QED and ED regimes. We establish fluid and diffusion limits of the queue-length processes in all regimes. The fluid limits are characterized by ordinary differential equations. The diffusion limits are characterized by stochastic differential equations, with a piecewise-linear drift term and a constant (QED) or time-varying (ED) covariance matrix. We investigate the load balancing effect of switchover in the mixed regimes, demonstrating the migration of workload from overloaded stations to underloaded stations and quantifying the load balancing impact of switchover probabilities.

Keywords

Many-server queuemulticlassMarkoviantime-varying arrivalstate-dependent switchovercustomer abandonmentheavy trafficQED regimeED regimemixed regimefluid limitdiffusion limitmultidimensional (piecewise-linear) Ornstein‒Uhlenbeck process

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F17: Functional limit theorems; invariance principles 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 645 - 672
Copyright
© Applied Probability Trust 

References

Berman, A. and Plemmons, R. J. (1979). Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Bonald, T. et al. (2009). Flow-level performance and capacity of wireless networks with user mobility. Queueing Systems 63, 131–164.Google Scholar
Borst, S., Mandelbaum, A. and Reiman, M. I. (2004). Dimensioning large call centers. Operat. Res. 52, 17–34.Google Scholar
Borst, S., Proutiere, A. and Hegde, N. (2006). Capacity of wireless data networks with intra- and inter-cell mobility. In Proc. IEEE INFOCOM 2006 (Barcelona, Spain), 12pp.CrossRefGoogle Scholar
Bramson, M., Lu, Y. and Prabhakar, B. (2010). Randomized load balancing with general service time distributions. ACM SIGMETRICS Performance Evaluation Rev. 38, 275–286.Google Scholar
Chen, H. and Yao, D. D. (2001). Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization (Appl. Math. 46). Springer, New York.Google Scholar
Dieker, A. B. and X, Gao.. (2013). Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov function. Ann. Appl. Prob. 23, 1291–1317.Google Scholar
Erlang, A. K. (1948). On the rational determination of the number of circuits. In The Life and Works of A. K. Erlang, eds Brockmeyer, E., Halstrom, H. L. and Jensen, A., The Copenhagen Telephone Company, Copenhagen, Denmark.Google Scholar
Fleming, P. J., Stolyar, A. and Simon, B. (1995). Heavy traffic limit for a mobile phone system loss model. In Proc. Nashville ACM Telecommunications Conf.Google Scholar
Ganesh, A. et al. (2010). Load balancing via random local research in closed and open systems. ACM SIGMETRICS Performance Evaluation Rev. 38, 287–298.Google Scholar
Garnett, O., Mandelbaum, A. and Reiman, M. I. (2002). Designing a call center with impatient customers. Manufacturing Service Operat. Manag. 4, 208–227.Google Scholar
Grossglauser, M. and Tse, D. N. C. (2002). Mobility increases the capacity of ad hoc wireless networks. IEEE/ACM Trans. Networking 10, 477–486.Google Scholar
Gurvich, I. and Whitt, W. (2009). Queue-and-idleness-ratio controls in many-server service systems. Math. Operat. Res. 34, 363–396.Google Scholar
Halfin, S. and Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Operat. Res. 29, 567–588.Google Scholar
He, S. and Dai, J. (2013). Many-server queues with customer abandonment: numerical analysis of their diffusion models. To appear in Stoch. Systems. Google Scholar
Kang, W. and Pang, G. (2013). Fluid limit of a multiclass many-server queueing network with abandonment and feedback. Submitted.Google Scholar
Kang, W. and Ramanan, K. (2010). Fluid limits of many-server queues with reneging. Ann. Appl. Prob. 20, 2204–2260.Google Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Mandelbaum, A., Massey, W. A. and Reiman, M. I. (1998). Strong approximations for Markovian service networks. Queueing Systems 30, 149–201.Google Scholar
Pang, G., Talreja, R. and Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Prob. Surveys 4, 193–267.Google Scholar
Perry, O. and Whitt, W. (2013). A fluid limit for an overloaded X model via a stochastic averaging principle. Math. Operat. Res. 38, 294–349.Google Scholar
Puhalskii, A. (1994). On the invariance principle for the first passage time. Math. Operat. Res. 19, 946–954.Google Scholar
Saure, D., Glynn, P. W. and Zeevi, A. (2009). A linear programming algorithm for computing the stationary distribution of semimartingale reflected Brownian motion. Working paper.Google Scholar
Simatos, F. and Tibi, D. (2010). Spatial homogenization in a stochastic network with mobility. Ann. Appl. Prob. 20, 312–355.Google Scholar
Talreja, R. and Whitt, W. (2009). Heavy-traffic limits for waiting times in many-server queues with abandonment. Ann. Appl. Prob. 19, 2137–2175.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.Google Scholar
Whitt, W. (2004). Efficiency-driven heavy-traffic approximations for many-server queues with abandonments. Manag. Sci. 50, 1449–1461.Google Scholar