Abstract
In this paper we study two transient characteristics of a Markov-fluid-driven queue, viz., the busy period and the covariance function of the workload process. Both metrics are captured in terms of their Laplace transforms. Relying on sample-path large deviations, we also identify the logarithmic asymptotics of the probability that the busy period lasts longer than t, as t→∞. Examples illustrating the theory are included.
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References
Ahn S, Ramaswami V (2005) Efficient algorithms for transient analysis of stochastic fluid flow models. J Appl Probab 42:531–549
Anick D, Mitra D, Sondhi M (1982) Stochastic theory of data-handling system with multiple sources. Bell Syst Tech J 61:1871–1894
Asmussen S (1994) Busy period analysis, rare events and transient behavior in fluid models. J Appl Math Stoch Anal 7(3):269–299
Asmussen S, Rolski T (1994) Risk theory in a periodic environment: the Cramer–Lundberg approximation and Lundberg’s inequality. Math Oper Res 19:410–433
Barbot N, Sericola B, Telek M (2001) Distribution of the busy period in stochastic fluid models. Stoch Models 17:407–427
Chang C-S (1995) Sample path large deviations and intree networks. Queueing Syst 20:7–36
da Silva Soares A, Latouche G (2006) Matrix-analytic methods for fluid queues with finite buffers. Perform Eval 63:295–314
Dembo A, Zeitouni O (1998) Large deviations techniques and applications, 2nd edn. Springer, New York
Es-Saghouani A, Mandjes M (2008) On the correlation structure of a Lévy-driven queue. J Appl Probab 45:940–952
Geršgorin S (1931) Über die Abgrenzung der Eigenwerte einer Matrix. Izv Akad Nauk SSSR Ser Mat 1:749–454
Kesidis G, Walrand J, Chang C-S (1993) Effective bandwidths for multiclass Markov fluids and other ATM sources. IEEE/ACM Trans Netw 1:424–428
Kosten L (1974) Stochastic theory of a multi-entry buffer (I). Delft Progr Rep, Ser F 1:10–18
Kosten L (1984) Stochastic theory of data-handling systems with groups of multiple sources. In: Rudin H, Bux W (eds) Performance of computer-communication systems. Elsevier, Amsterdam, pp 321–331
Kulkarni V (1997) Fluid models for single buffer systems. In: Frontiers in queueing. CRC Press, Boca Raton, pp 321–338
Mandjes M, Scheinhardt W (2008) A fluid model for a relay node in an ad hoc network: evaluation of resource sharing policies. J Appl Math Stoch Anal. doi:10.1155/2008/518214. Article ID 518214, 25 pages
Narayanan A, Kulkarni VG (1996) First passage times in fluid models with an application to two priority fluid systems. In Proceedings of the 2nd IPDS ’96, pp 166–175
Ott T (1977) The covariance function of the virtual waiting-time process in an M/G/1 queue. Adv Appl Probab 9:158–168
Prabhu N (1998) Stochastic storage processes: queues, insurance risk, dams and data communication, 2nd edn. Springer, New York
Ren Q, Kobayashi H (1995) Transient solution for the buffer behavior in statistical multiplexing. Perform Eval 21:65–87
Scheinhardt W (1998) Markov-modulated and feedback fluid queues. PhD thesis, University of Twente, The Netherlands
Sonneveld P (2004) Some properties of the generalized eigenvalue problem M x=λ(Γ−c I)x, where M is the infinitesimal generator of a Markov process, and Γ is a real diagonal matrix. Delft University of Technology Report 04-02
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Part of this work was done while M. Mandjes was at Stanford University, Stanford, CA 94305, USA.
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Es-Saghouani, A., Mandjes, M. Transient analysis of Markov-fluid-driven queues. TOP 19, 35–53 (2011). https://doi.org/10.1007/s11750-009-0078-3
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DOI: https://doi.org/10.1007/s11750-009-0078-3