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Accuracy of fluid approximations to controlled birth-and-death processes: absorbing case

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Abstract

We consider the fluid model of a controlled birth-and-death process with an absorbing state. Instead of analyzing the trajectories, we investigate the performance functionals of the underlying process by considering algebraic equations of the dynamic programming type. We provide the accuracy of such fluid approximations and give illustrative examples.

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References

  • Avrachenkov K, Ayesta U, Piunovskiy A (2005) Optimal Choice of Buffer Size of an Internet Router. In: Proceedings of IEEE CDC/ECC, Spain

  • Avrachenkov K, Piunovskiy A, Zhang Y (2011) Asymptotic Fluid Optimality and Efficiency of Tracking Policies for Bandwidth-Sharing Networks. J Appl Prob (accepted)

  • Ayesta U, Piunovskiy A, Zhang Y (2008) Fluid Model of An Internet Router Under the MIMD Control Scheme. In: Raghavan S, Golden B, Wasil E (eds) Telecommunications Modeling Policy and Technology. Springer-Verlag, New York, pp 239–251

    Chapter  Google Scholar 

  • Bassamboo A, Randhawa R (2010) On the accuracy of fluid models for capacity sizing in queueing systems with impatient customers. Oper Res 58(5): 1398–1413

    Article  MathSciNet  Google Scholar 

  • Bäuerle N (2000) Asymptotic Optimality of Tracking Policies in Stochastic Networks. Ann Appl Prob 10(4): 1065–1083

    Article  MATH  Google Scholar 

  • Bäuerle N (2002) Optimal Control of Queueing Networks: an Approach via Fluid Models. Adv Appl Prob 34: 313–328

    Article  MATH  Google Scholar 

  • Bertsekas D, Shreve S (1978) Stochastic Optimal Control: the Discrete-Time Case. Academic Press, NY

    MATH  Google Scholar 

  • Chen H (1995) Fluid Approximations and Stability of Multiclass Queueing Networks: Work-Conserving Disciplines. Ann Appl Prob 5(3): 637–665

    Article  MATH  Google Scholar 

  • Chen H (1996) Rate of Convergence of the Fluid Approximation for Generalized Jackson Networks. J Appl Prob 33: 804–814

    Article  MATH  Google Scholar 

  • Chen H, Mandelbaum A (1991) Discrete Flow Networks: Bottleneck Analysis and Fluid Approximations. Math Oper Res 16(2): 408–446

    Article  MathSciNet  MATH  Google Scholar 

  • Chen H, Mandelbaum A (1994) Hierachical Modeling of Stochastic Networks, Part I: Fluid Models. In: Yao D (ed) Stochastic Modeling and Analysis of Manufacturing Systems. Springer, New York, pp 47–105

    Google Scholar 

  • Clancy D, Piunovskiy A (2005) An Explicit Optimal Isolation Policy for a Determinisitc Epidemic Model. Applied Mathematics and Computation 163: 1109–1121

    Article  MathSciNet  MATH  Google Scholar 

  • Dai J (1995) On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach via Fluid Limit Models. Ann Appl Prob 5: 49–77

    Article  MATH  Google Scholar 

  • Ethier S, Kurtz T (1986) Markov Processes: Chracterization and Convergence. Wiley, NY

    Book  Google Scholar 

  • Foss S, Kovalevskii A (1999) A Stability Criterion via Fluid Limits and its Application to a Polling System. Queueing Syst 32: 131–168

    Article  MathSciNet  MATH  Google Scholar 

  • Gajrat AS, Hordijk A (2005) On the Structure of the Optimal Server Control for Fluid Networks. Math Meth Oper Res 62: 55–75

    Article  MathSciNet  MATH  Google Scholar 

  • Gajrat AS, Hordijk A, Malyshev V, Spieksma F (1997) Fluid Approximations of Markov Decision Chains. Markov Process Related Fields 3: 129–150

    MathSciNet  MATH  Google Scholar 

  • Gajrat AS, Hordijk A, Ridder A (2003) Large-Deviations Analysis of the Fluid Approximations for a Controllable Tandem Queue. Ann Appl Prob 13(4): 1423–1448

    Article  MathSciNet  MATH  Google Scholar 

  • Guo X, Hernández-Lerma O, Prieto-Rumeau T (2006) A Survey of Recent Results on Continuous-Time Markov Decision Processes. Top 14: 177–257

    Article  MathSciNet  MATH  Google Scholar 

  • Kitaev M (1986) Semi-Markov and Jump Markov Controlled Models: Average Cost Criterion. Theory Probab Appl 30(2): 272–288

    Article  MathSciNet  MATH  Google Scholar 

  • Kitaev M, Rykov V (1995) Controlled Queueing Systems. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Mandelbaum A, Pats G (1995) State-dependent Queues: Approximations and Applications. In: Kelly F, Williams R (eds) Proceedings of the IMA. Stochastic networks, Springer, NY, pp 239–282

  • Mandelbaum A, Massey W, Reiman M (1998) Strong Approximations for Markovian Service Networks. Queueing Syst 30: 149–201

    Article  MathSciNet  MATH  Google Scholar 

  • Pang G, Day M (2007) Fluid Limits of Optimally Controlled Queueing Networks. J Appl Math Stoch Anal. Article ID 68958. doi:10.I155/2007/68958

  • Piunovskiy A (1998) A Controlled Jump Discounted Model with Constraints. Theory Probab Appl 42(1): 51–71

    Article  Google Scholar 

  • Piunovskiy A (2004) Multicriteria Impulsive Control of Jump Markov Processes. Math Meth Oper Res 60: 125–144

    Article  MathSciNet  MATH  Google Scholar 

  • Piunovskiy A (2009a) Controlled Jump Markov Processes with Local Transitions and Their Fluid Approximation. WSEAS Trans on Systems and Control 4(8): 399–412

    MathSciNet  Google Scholar 

  • Piunovskiy A (2009b) Random Walk, Birth-and-Death Process and Their Fluid Approximations: ‘Absorbing Case’. Math Meth Oper Res 70(2): 285–312

    Article  MathSciNet  MATH  Google Scholar 

  • Piunovskiy A, Clancy D (2008) An Explicit Optimal Intervention Policy for a Determinisitc Epidemic Model. Optim Control Appl Meth 29: 413–428

    Article  MathSciNet  Google Scholar 

  • Piunovskiy A, Zhang Y (2011) On the Fluid Approximations of a class of General Inventory Level-Dependent EOQ and EPQ Models. Adv Oper Res (submitted)

  • Puterman M (1994) Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley Series in Probability and Mathematical Statistics. Wiley, NY, USA

    MATH  Google Scholar 

  • Verloop IM (2009) Scheduling in Stochastic Resource-sharing Systems. PhD thesis, Eindhoven University of Technology

  • Zhang Y, Piunovskiy A, Ayesta U, Avrachenkov K (2010) Convergence of Trajectories and Optimal Buffer Sizing for MIMD Congestion Control. Com Com 33(2): 149–159

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK

    Alexey Piunovskiy & Yi Zhang

Authors
  1. Alexey Piunovskiy
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  2. Yi Zhang
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Correspondence to Alexey Piunovskiy.

Additional information

This research was partially supported by the Alliance: Franco-British Research Partnership Programme, project PN08.021, British Council.

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Piunovskiy, A., Zhang, Y. Accuracy of fluid approximations to controlled birth-and-death processes: absorbing case. Math Meth Oper Res 73, 159–187 (2011). https://doi.org/10.1007/s00186-010-0340-3

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  • DOI: https://doi.org/10.1007/s00186-010-0340-3

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