Abstract
Many-server queuing networks with general service and abandonment times have proven to be a realistic model for scenarios such as call centers and health-care systems. The presence of abandonment makes analytical treatment difficult for general topologies. Hence, such networks are usually studied by means of fluid limits. The current state of the art, however, suffers from two drawbacks. First, convergence to a fluid limit has been established only for the transient, but not for the steady state regime. Second, in the case of general distributed service and abandonment times, convergence to a fluid limit has been either established only for a single queue, or has been given by means of a system of coupled integral equations which does not allow for a numerical solution. By making the mild assumption of Coxian-distributed service and abandonment times, in this paper we address both drawbacks by establishing convergence in probability to a system of coupled ordinary differential equations (ODEs) using the theory of Kurtz. The presence of abandonments leads in many cases to ODE systems with a global attractor, which is known to be a sufficient condition for the fluid and the stochastic steady state to coincide in the limiting regime. The fact that our ODE systems are piecewise affine enables a computational method for establishing the presence of a global attractor, based on a solution of a system of linear matrix inequalities.
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References
Anselmi, J., & Verloop, I. (2011). Energy-aware capacity scaling in virtualized environments with performance guarantees. Performance Evaluation, 68(11), 1207–1221.
Ascher, U. M., & Petzold, L. R. (1988). Computer methods for ordinary differential equations and differential-algebraic equations. Philadelphia: SIAM.
Baskett, F., Chandy, K. M., Muntz, R. R., & Palacios, F. G. (1975). Open, closed, and mixed networks of queues with different classes of customers. Journal of the ACM, 22(2), 248–260.
Benaim, M. (1998). Recursive algorithms, urn processes and chaining number of chain recurrent sets. Ergodic Theory and Dynamical Systems, 18, 53–87.
Billingsley, P. (1999). Convergence of probability measures (2nd ed.)., Wiley series in probability and statistics: Probability and statistics New York: Wiley. (A Wiley-Interscience Publication).
Bolch, G., Greiner, S., de Meer, H., & Trivedi, K. (2005). Queueing networks and Markov chains: Modeling and performance evaluation with computer science applications. New York: Wiley.
Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory, studies in applied mathematics (Vol. 15). Philadelphia, PA: SIAM.
Chen, H., & Yao, D. D. (2001). Fundamentals of queueing networks. Berlin: Springer.
Cumani, A. (1982). On the canonical representation of homogeneous Markov processes modelling failure-time distributions. Microelectronics Reliability, 22(3), 583–602.
Dai, J., Dieker, A., & Gao, X. (2014). Validity of heavy-traffic steady-state approximations in many-server queues with abandonment. Queueing Systems, 78(1), 1–29. doi:10.1007/s11134-014-9394-x.
Dai, J., & He, S. (2012). Many-server queues with customer abandonment: A survey of diffusion and fluid approximations. Journal of Systems Science and Systems Engineering, 21(1), 1–36.
Dayar, T., Hermanns, H., Spieler, D., & Wolf, V. (2011). Bounding the equilibrium distribution of Markov population models. Numerical Linear Algebra with Applications, 18(6), 931–946.
Gans, N., Koole, G., & Mandelbaum, A. (2003). Telephone call centers: Tutorial, review, and research prospects. Manufacturing and Service Operations Management, 5(2), 79–141.
Gast, N., & Gaujal, B. (2010). A mean field model of work stealing in large-scale systems. In V. Misra, P. Barford & MS. Squillante (Eds.), SIGMETRICS 2010, Proceedings of the 2010 ACM SIGMETRICS InternationalConference on Measurement and Modeling of Computer Systems, New York, USA, 14–18 June 2010. ACM, New York. doi:10.1145/1811039.
Halfin, S., & Whitt, W. (1981). Heavy-traffic limits theorem for queues with many exponential servers. Operations Research, 29, 567–588.
Hamza, K., & Klebaner, F. C. (1995). Conditions for integrability of Markov chains. Journal of Applied Probability, 32(2), 541–547.
Hayden, R. (2010). Convergence of ODE approximations and bounds on performance models in the steady-state. In: Ninth workshop on process algebra and stochastically timed activities (PASTA). http://aesop.doc.ic.ac.uk/conferences/pasta/2010.
Jennings, O. B., & Puha, A. L. (2013). Fluid limits for overloaded multiclass FIFO single-server queues with general abandonment. Stochastic Systems, 3(1), 262–321.
Kang, W., & Pang, G. (2013). Fluid limit of a many-server queueing network with abandonment. (submitted).
Kang, W., & Ramanan, K. (2010). Fluid limits of many-server queues with reneging. The Annals of Applied Probability, 200(6), 2204–2260.
Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure Markov processes. Journal of applied Probability, 7(1), 49–58.
Liu, Y., & Whitt, W. (2011). A network of time-varying many-server fluid queues with customer abandonment. Operations Research, 59(4), 835–846.
Liu, Y., & Whitt, W. (2011). Large-time asymptotics for the Gt/Mt/st+GIt many-server fluid queue with abandonment. Queueing Systems: Theory and Applications, 67(2), 145–182. doi:10.1007/s11134-010-9208-8.
Liu, Y., & Whitt, W. (2011). Nearly periodic behavior in the overloaded \(G/D/s+GI\) queue. Stochastic Systems, 1(2), 340–410.
Liu, Y., & Whitt, W. (2012). A many-server fluid limit for the queueing model experiencing periods of overloading. Operations Research Letters, 40(5), 307–312.
Liu, Y., & Whitt, W. (2014). Algorithms for time-varying networks of many-server fluid queues. INFORMS Journal on Computing, 26(1), 59–73.
Liu, Y., & Whitt, W. (2014). Many-server heavy-traffic limit for queues with time-varying parameters. The Annals of Applied Probability, 24(1), 378–421.
Long, Z., & Zhang, J. (2014). Convergence to equilibrium states for fluid models of many-server queues with abandonment. Operations Research Letters, 42(6), 388–393.
Mandelbaum, A., Massey, W. A., & Reiman, M. I. (1998). Strong approximations for Markovian service networks. Queueing Systems, 30(1–2), 149–201.
Mandelbaum, A., Massey, W. A., Reiman, M. I., Stolyar, A. L., & Rider, B. (2002). Queue lengths and waiting times for multiserver queues with abandonment and retrials. Telecommunication Systems, 21(2), 149–171.
Mandelbaum, A., & Momčilović, P. (2012). Queues with many servers and impatient customers. Mathematics of Operations Research, 37(1), 41–65. doi:10.1287/moor.1110.0530.
Meyn, S. P., & Tweedie, R. L. (1993). Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Advances in Applied Probability, 25(3), 518–548.
Nelson, Barry L., & Taaffe, Michael R. (2004). The [Pht/Pht/8]K queueing system: Part II-the multiclass network. INFORMS Journal on Computing, 16(3), 275–283. doi:10.1287/ijoc.1040.0071.
Pavlov, A., Wouw, N. V. D., & Nijmeijer, H. (2005). Convergent piecewise affine systems: Analysis and design Part I: Continuous case. In: 44th IEEE Conference on decision and control and European control conference ECC (2005).
Stewart, W. J. (2009). Probability, Markov chains, queues, and simulation. Princeton: Princeton University Press.
Urgaonkar, B., Pacifici, G., Shenoy, P., Spreitzer, M., & Tantawi, A. (2005). An analytical model for multi-tier internet services and its applications. In DL. Eager, CL. Williamson, SC. Borst & JCS Lui (Eds.), Proceedings of the International Conference on Measurements and Modeling of Computer Systems, SIGMETRICS 2005, June 6–10, 2005, Banff, Alberta, Canada. ACM. doi:10.1145/1064212.
Van Houdt, B. (2013). A mean field model for a class of garbage collection algorithms in flash-based solid state drives. In M. Harchol-Balter & JR. Douceur, J. Xu (Eds.), ACMSIGMETRICS / International Conference on Measurement and Modeling of Computer Systems, Pittsburgh, PA, USA, June 17–21, 2013. ACM, New York. doi:10.1145/2465529.
Van Houdt, B., & Bortolussi, L. (2012). Fluid limit of an asynchronous optical packet switch with shared per link full range wavelength conversion. In PG. Harrison, MF. Arlitt & G. Casale (Eds.), ACM SIGMETRICS/PERFORMANCE Joint International Conference on Measurement and Modeling of Computer Systems London, United Kingdom, June 11–15, 2012. ACM, New York. doi:10.1145/2254756.
Whitt, W. (2006). Fluid models for multiserver queues with abandonments. Operations Research, 54(1), 37–54.
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This work was partially supported by the EU project QUANTICOL, 600708.
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Tschaikowski, M., Tribastone, M. A computational approach to steady-state convergence of fluid limits for Coxian queuing networks with abandonment. Ann Oper Res 252, 101–120 (2017). https://doi.org/10.1007/s10479-016-2193-5
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DOI: https://doi.org/10.1007/s10479-016-2193-5