Discrete Event Dynamic Systems

, Volume 20, Issue 3, pp 325–347 | Cite as

Performance Analysis of a Block-Structured Discrete-Time Retrial Queue with State-Dependent Arrivals

Article

Abstract

In this paper, we introduce a new discrete block state-dependent arrival (D-BSDA) distribution which provides fresh insights leading to a successful generalization of the discrete-time Markovian arrival process (D-MAP). The D-BSDA distribution is related to structured Markov chains and the method of stages. The consideration of this new discrete-time state-dependent block description gives one the ability of construct new stochastic models. The retrial queue analyzed in this paper gives an example of application of the D-BSDA distribution to construct more general and sophisticated models. We assume that the primary arrivals and the retrials follow the D-BSDA description and the service times are of discrete phase-type (PH). We study the underlying level dependent Markov chain of M/G/1-type at the epochs immediately after the slot boundaries. To this end, we employ the UL-type RG-factorization which provides an expression for the stationary probabilities. We also perform an analysis of waiting times. Numerical experiments are presented to study the system performance.

Keywords

Queueing Discrete-time Structured Markov chain Retrial LU-factorizations Waiting time 

Notes

Acknowledgement

The constructive comments of the referees on an earlier version of this paper are greatly appreciated. The authors are also grateful to Jingqi Wang (Tsinghua University) for preparing codes for the numerical implementation. J.R. Artalejo was supported by grant MTM2005-01248 from MEC. The work of Q.L. Li was supported by the National Science Foundation of China under grant No. 10671107, 10871114, 60736028 and the National Grand Fundamental Research 973 Program of China under grant No. 2006CB805901.

References

  1. Alfa AS (2006) Discrete-time analysis of the GI/G/1 system with Bernoulli retrials: an algorithmic approach. Ann Oper Res 141:51–66MATHCrossRefMathSciNetGoogle Scholar
  2. Allen LJS (2003) An introduction to stochastic processes with applications to biology. Prentice-Hall, Englewood CliffsGoogle Scholar
  3. Artalejo JR, Gomez-Corral A (2008) Retrial queueing systems: a computational approach. Springer, BerlinMATHCrossRefGoogle Scholar
  4. Artalejo JR, Lopez-Herrero MJ (2007a) On the distribution of the number of retrials. Appl Math Model 31:478–489MATHCrossRefGoogle Scholar
  5. Artalejo JR, Lopez-Herrero MJ (2007b) A simulation study of a discrete-time multiserver retrial queue with finite population. J Stat Plan Inference 137:2536–2542MATHCrossRefMathSciNetGoogle Scholar
  6. Artalejo JR, Lopez-Herrero MJ (2009) Cellular mobile networks with repeated calls operating in random environment. Comput Oper Res. doi: 10.1016/j.cor.2009.01.011 Google Scholar
  7. Artalejo JR, Atencia I, Moreno P (2005) A discrete-time Geo [X]/G/1 retrial queue with control of admission. Appl Math Model 29:1100–1120MATHCrossRefGoogle Scholar
  8. Artalejo JR, Economou A, Gomez-Corral A (2008) Algorithmic analysis of the Geo/Geo/c retrial queue. Eur J Oper Res 189:1042–1056MATHCrossRefMathSciNetGoogle Scholar
  9. Atencia I, Moreno P (2005) A single-server G-queue in discrete-time with geometrical arrival and service process. Perform Eval 59:85–97CrossRefGoogle Scholar
  10. Blondia C, Casals O (1992) Statistical multiplexing of VBR sources: a matrix-analytic approach. Perform Eval 16:5–20MATHCrossRefGoogle Scholar
  11. Bruneel H, Kim BG (1993) Discrete-time models for communication systems including ATM. Kluwer, BostonGoogle Scholar
  12. Chaudhry ML (2000) On numerical computations of some discrete-time queues. In: Grassmann WK (ed) Computational probability. Kluwer, Boston, pp 365–407Google Scholar
  13. Falin GI, Templeton JGC (1997) Retrial queues. Chapman and Hall, LondonMATHGoogle Scholar
  14. Hong X, Huang Z, Chan E (2002) A new method for evaluating the cell loss probability in an ATM multiplexer. Eur Trans Telecommun 13:197–202CrossRefGoogle Scholar
  15. Kim CS, Klimenok VI, Lee SC, Dudin AN (2007) The BMAP/PH/1 retrial queueing system operating in random environment. J Stat Plan Inference 137:3904–3916MATHCrossRefMathSciNetGoogle Scholar
  16. Latouche G, Ramaswami R (1999) Introduction to matrix analytic methods in stochastic modeling. ASA-SIAM, PhiladelphiaMATHGoogle Scholar
  17. Lee HW, Moon JM, Kim BK, Park JG, Lee SW (2005) A simple eigenvalue method for low-order D-MAP/G/1 queues. Appl Math Model 29:277–288MATHCrossRefGoogle Scholar
  18. Lenin RB (2006) Loss probability of a D-BMAP/PH/1/N queue. Am J Math Manage Sci 26:277–291MathSciNetGoogle Scholar
  19. Li QL (2009) Constructive computation in stochastic models with applications: the RG-Factorizations. Springer, Berlin and Tsinghua University Press, BeijingGoogle Scholar
  20. Li QL, Cao J (2004) Two types of RG-factorizations of quasi-birth-and-death processes and their applications to stochastic integral functionals. Stoch Models 20:299–340MATHCrossRefMathSciNetGoogle Scholar
  21. Li QL, Zhao YQ (2004) The RG-factorizations in block-structured Markov renewal processes with applications. In: Zhu X (ed) Observation, theory and modeling of atmosphere variability. World Scientific, Hackensack, pp 545–568Google Scholar
  22. Neuts MF (1981) Matrix-geometric solutions in stochastic models. The Johns Hopkins University Press, BaltimoreMATHGoogle Scholar
  23. Rom R, Sidi M (1990) Multiple access protocols. Springer, New YorkMATHGoogle Scholar
  24. Roszik J, Sztrik J, Virtamo J (2007) Performance analysis of finite-source retrial queues operating in random environments. Int J Oper Res 2:254–268MATHCrossRefMathSciNetGoogle Scholar
  25. Takagi H (1993) Queueing analysis: a foundation of performance evaluation. Discrete-Time Systems, vol. 3. North-Holland, AmsterdamGoogle Scholar
  26. Van Velthoven J, Van Houdt B, Blondia C (2005) Response time distribution in a D-MAP/PH/1 queue with general customer impatience. Stoch Models 21:745–765MATHCrossRefMathSciNetGoogle Scholar
  27. Wang J, Zhao Q (2007) A discrete-time Geo/G/1 retrial queue with starting failures and second optional service. Comput Math Appl 53:115–127MATHCrossRefMathSciNetGoogle Scholar
  28. Yang T, Li H (1995) On the steady-state queue size distribution of the discrete-time Geo/G/1 queue with repeated customers. Queueing Syst 21:199–215MATHCrossRefGoogle Scholar
  29. Zhao J-A, Li B, Cao X-R, Ahmad I (2006) A matrix-analytic solution for the DMAP/PH/1 priority queue. Queueing Syst 53:127–145MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Statistics and Operations Research, Faculty of MathematicsComplutense University of MadridMadridSpain
  2. 2.Department of Industrial EngineeringTsinghua UniversityBeijingPeople’s Republic of China

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