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The Decay Parameter and Invariant Measures for Markovian Bulk-Arrival Queues with Control at Idle Time

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Abstract

We consider properties regarding decay parameter and invariant measures for unstable Markovian bulk-arrival Queues with control at idle time. The exact value of the decay parameter, denoted by λ Z , is firstly revealed. An elegant criterion regarding λ Z -recurrence and λ Z -positive is obtained. The corresponding subinvariant and invariant measures are considered and the structure of all the subinvariant and invariant measures are presented.

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References

  • Anderson W (1991) Continuous-time markov chains: an applications-oriented approach. Springer, New York

    Book  MATH  Google Scholar 

  • Asmussen S (2003) Applied probability and queues, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Chaudhry ML, Templeton JGC (1983) A first course in bulk queues. Wiley, New York

    MATH  Google Scholar 

  • Chen AY, Renshaw E (1997) The M/M/1 queue with mass exodus and mass arrivals when empty. J Appl Probab 34:192–207

    Article  MathSciNet  MATH  Google Scholar 

  • Chen AY, Renshaw E (2004) Markovian bulk-arriving queues with state-dependent control at idle time. Adv Appl Probab 36:499–524

    Article  MathSciNet  MATH  Google Scholar 

  • Chen AY, Li JP, Ramesh NI (2005) Uniqueness and extinction of weighted Markov branching processes. Methodol Comput Appl Probab 7:489–516

    Article  MathSciNet  MATH  Google Scholar 

  • Chen AY, Pollett P, Li J, Zhang H (2010) Markovian bulk-arrival and bulk-service queues with state-dependent control. Queueing Syst 64:267–304

    Article  MathSciNet  MATH  Google Scholar 

  • Chung KL (1967) Markov chains with stationary transition probabilities, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Darroch JN, Seneta E (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J Appl Probab 4:192–196

    Article  MathSciNet  MATH  Google Scholar 

  • Flaspohler DC (1974) Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann Inst Stat Math 26:351–356

    Article  MathSciNet  MATH  Google Scholar 

  • Gross D, Harris CM (1985) Fundamentals of queueing theory. Wiley, New York

    MATH  Google Scholar 

  • Kelly FP (1983) Invariant measures and the generator. In: Kingman JFC, Reuter GE (eds) Probability, statistics, and analysis. London Math. Soc. Lecture notes series 79. Cambridge University Press, Cambridge, pp 143–160

    Google Scholar 

  • Kijima M (1963) Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous-time. J Appl Probab 30:509–517

    Article  MathSciNet  Google Scholar 

  • Kingman JFC (1963) The exponential decay of Markov transition probability. Proc Lond Math Soc 13:337–358

    Article  MathSciNet  MATH  Google Scholar 

  • Kleinrock I (1975) Queueing systems, vol 1. Wiley, New York

    Google Scholar 

  • Li JP, Chen AY (2008) Decay properties of stopped Markovian bulk-arriving queues. Adv Appl Probab 40:95–121

    Article  MATH  Google Scholar 

  • Medhi J (1991) Stochastic models in queuing theory. Academic Press, San Diego

    Google Scholar 

  • Nair MG, Pollett PK (1993) On the relationship between μ-invariant measures and quasistationary distributions for continuous-time Markov chains. Adv Appl Probab 25:82–102

    Article  MathSciNet  MATH  Google Scholar 

  • Pollett PK (1988) Reversibility, invariance and mu-invariance. Adv Appl Probab 20:600–621

    Article  MathSciNet  MATH  Google Scholar 

  • Tweedie RL (1974) Some ergodic properties of the Feller minimal process. Q J Math (Oxford) 25(2):485–493

    Article  MathSciNet  MATH  Google Scholar 

  • Van Doorn EA (1985) Conditions for exponential ergodicity and bounds for the decay parameter of a birth–death process. Adv Aappl Probab 17:514–530

    Article  MATH  Google Scholar 

  • Van Doorn EA (1991) Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv Appl Probab 23:683–700

    Article  MATH  Google Scholar 

  • Vere-Jones D (1962) Geometric ergidicity in denumerable Markov chains. Q J Math (Oxford) 13(2):7–28

    Article  MathSciNet  MATH  Google Scholar 

  • Yang XQ (1990) The construction theory of denumerable Markov processes. Wiley, New York

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Science and Computing Technology, Central South University, Changsha, 410075, China

    Li Junping

  2. Division of Statistics and Probability, Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK

    Chen Anyue

  3. Department of Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China

    Chen Anyue

Authors
  1. Li Junping
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  2. Chen Anyue
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Correspondence to Li Junping.

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Junping, L., Anyue, C. The Decay Parameter and Invariant Measures for Markovian Bulk-Arrival Queues with Control at Idle Time. Methodol Comput Appl Probab 15, 467–484 (2013). https://doi.org/10.1007/s11009-011-9252-9

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  • DOI: https://doi.org/10.1007/s11009-011-9252-9

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