Abstract
One of the key performance measures in queueing systems is the decay rate of the steady-state tail probabilities of the queue lengths. It is known that if a corresponding fluid model is stable and the stochastic primitives have finite moments, then the queue lengths also have finite moments, so that the tail probability ℙ(⋅>s) decays faster than s −n for any n. It is natural to conjecture that the decay rate is in fact exponential.
In this paper an example is constructed to demonstrate that this conjecture is false. For a specific stationary policy applied to a network with exponentially distributed interarrival and service times, it is shown that the corresponding fluid limit model is stable, but the tail probability for the buffer length decays slower than s −log s.
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Dai, J.G.: On the positive Harris recurrence for multiclass queueing networks: A unified approach via fluid models. Ann. Appl. Probab. 5, 49–77 (1995)
Dai, J.G., Meyn, S.P.: Stability and convergence of moments for multiclass queueing networks via fluid limit models. IEEE Trans. Autom. Controls 40, 1889–1904 (1995)
Douc, R., Fort, G., Moulines, E., Soulier, P.: Practical drift conditions for subgeometric rates of convergence. Adv. Appl. Probab. 14(3), 1353–1377 (2004)
Down, D., Meyn, S.P., Tweedie, R.L.: Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23(4), 1671–1691 (1995)
Duffield, N.G., O’Connell, N.: Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Camb. Philos. Soc. 118(2), 363–374 (1995)
Gamarnik, D., Zeevi, A.: Validity of heavy traffic steady-state approximations in open queueing networks. Ann. Appl. Probab. 16(1), 56–90 (2006)
Ganesh, A., O’Connell, N., Wischik, D.: Big Queues. Lecture Notes in Mathematics, vol. 1838. Springer, Berlin (2004)
Glynn, P.W., Whitt, W.: Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. J. Appl. Probab. 31, 131–156 (1994)
Kumar, P.R., Meyn, S.P.: Duality and linear programs for stability and performance analysis queueing networks and scheduling policies. IEEE Trans. Autom. Control 41(1), 4–17 (1996)
Kumar, P.R., Seidman, T.I.: Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems. IEEE Trans. Autom. Control AC-35(3), 289–298 (1990)
Malyshev, V.A., Men’shikov, M.V.: Ergodicity, continuity and analyticity of countable Markov chains. Tr. Mosk. Mat. Obs. 39, 3–48 (1979). Trans. Mosc. Math. Soc. 1, 1–48 (1981)
Meyn, S.P.: Sequencing and routing in multiclass queueing networks. Part I: Feedback regulation. SIAM J. Control Optim. 40(3), 741–776 (2001)
Meyn, S.P.: Workload models for stochastic networks: Value functions and performance evaluation. IEEE Trans. Autom. Control 50(8), 1106–1122 (2005)
Meyn, S.P.: Control Techniques for Complex Networks. Cambridge University Press, Cambridge (2007)
Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press, London (2009). Published in the Cambridge Mathematical Library. 1993 edition online: http://black.csl.uiuc.edu/~meyn/pages/book.html
Resnick, S.: Adventures in Stochastic Processes. Birkhäuser, Boston (1992)
Rybko, A., Stolyar, A.: On the ergodicity of stochastic processes describing open queueing networks. Probl. Pered. Inf. 28, 3–26 (1992)
Shwartz, A., Weiss, A.: Large Deviations for Performance Analysis. Chapman and Hall, London (1995)
Stolyar, A.: On the stability of multiclass queueing networks: A relaxed sufficient condition via limiting fluid processes. Markov Process. Relat. Fields 491–512 (1995)
Tuominen, P., Tweedie, R.L.: Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Probab. 26, 775–798 (1994)
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Gamarnik, D., Meyn, S. On exponential ergodicity of multiclass queueing networks. Queueing Syst 65, 109–133 (2010). https://doi.org/10.1007/s11134-010-9173-2
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DOI: https://doi.org/10.1007/s11134-010-9173-2