Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

DIMENSION REDUCTION FOR APPROXIMATION OF ADVANCED RETRIAL QUEUES : TUTORIAL AND REVIEW

  • SHIN, YANG WOO (Department of Statistics, Changwon National University)
  • Received : 2017.05.06
  • Accepted : 2017.08.22
  • Published : 2017.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Retrial queues have been widely used to model the many practical situations arising from telephone systems, telecommunication networks and call centers. An approximation method for a simple Markovian retrial queue by reducing the two dimensional problem to one dimensional problem was presented by Fredericks and Reisner in 1979. The method seems to be a promising approach to approximate the retrial queues with complex structure, but the method has not been attracted a lot of attention for about thirty years. In this paper, we exposit the method in detail and show the usefulness of the method by presenting the recent results for approximating the retrial queues with complex structure such as multi-server retrial queues with phase type distribution of retrial time, impatient customers with general persistent function and/or multiclass customers, etc.

Keywords

Acknowledgement

Supported by : Changwon National University

References

  1. J.R. Artalejo, Analysis of an M/G/1 queue with constant repeated attempts and server vacations, Computers & Operations Research 24 (1997), 493-504. https://doi.org/10.1016/S0305-0548(96)00076-7
  2. J.R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, 7 (1999), 187-211.
  3. J.R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling 30 (1999), 1-6.
  4. J.R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009, Mathematical and Computer Modelling 51 (2000) 1071-1081.
  5. J.R. Artalejo, A. Gomez-Corral, Retrial Queueing Systems, A Computational Approach, Springer-Verlag, Hidelberg, 2008.
  6. S. Asmussen, Applied Probability and Queues, 2nd Ed., Springer-Verlag, New York, 2003.
  7. A. Bobbio, A. Horvath and M. Telek, Matching three moments with minimal acyclic phase type distributions, Stochastic Models 21 (2005), 303-326. https://doi.org/10.1081/STM-200056210
  8. M. Boualem, N. Djellab and D. Aissani, Stochastic inequality for M/G/1 retrial queues with vacations and constant retrial policy, Mathematical and Computer Modelling 50 (2009), 207-212. https://doi.org/10.1016/j.mcm.2009.03.009
  9. L. Breuer, A. Dudin and V. Klimenok, A Retrial BMAP/PH/N system, Queueing Systems 40 (2002), 433-457. https://doi.org/10.1023/A:1015041602946
  10. B.D. Choi and Y. Chang, Single server retrial queues with probability calls, Mathematical and Computer Modelling 30 (1999), 7-32.
  11. B.D. Choi, K.B. Choi and Y.W. Lee, M/G/1 retrial queueing systems with two types of calls and finite capacity, Queueing Systems 19 (1995), 215-229. https://doi.org/10.1007/BF01148947
  12. J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecommunication Review 18 (1957), 49-100.
  13. G. Choudhury, Steady state analysis of an M/G/1 queue with linear retrial policy and two phase service under Bernoulli vacation schedule, Applied Mathematical Modelling 32 (2008), 2480-2489. https://doi.org/10.1016/j.apm.2007.09.020
  14. G. Choudhry and J.C. Ke, A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delayed repair, Applied Mathematical Modelling 36 (2012), 255-269. https://doi.org/10.1016/j.apm.2011.05.047
  15. J.E. Diamond, A.S. Alfa, Approximation method for M/PH/1 retrial queues with phase type inter-retrial times, European Journal of Operational Research 113 (1999), 620-631. https://doi.org/10.1016/S0377-2217(98)00004-6
  16. G.I. Falin, On a multiclass batch arrival retrial queue, Advances in Applied Probability 20 (1988), 483-487. https://doi.org/10.1017/S0001867800017109
  17. G.I. Falin, A survey of retrial queues, Queueing Systems 7 (1990), 127-168. https://doi.org/10.1007/BF01158472
  18. G.I. Falin, J.R. Artalejo and M. Martin, On the single server retrial queue with priority customers, Queueing Systems 14 (1993), 439-455. https://doi.org/10.1007/BF01158878
  19. G.I. Falin and J.G.C. Templeton, Retrial Queues, Chapman and Hall, London, 1997.
  20. A.A. Fredericks and G.A. Reisner, Approximations to stochastic service systems with an application to a retrial model, Bell Systems Technical Journal 58 (1979), 557-576. https://doi.org/10.1002/j.1538-7305.1979.tb02235.x
  21. N. Gharbi, C. Dutheillet and M. Ioualalen, Colored stochastic Petri nets for modelling and analysis of multiclass retrial systems, Mathematical and Computer Modelling 49 (2009), 1436-1448. https://doi.org/10.1016/j.mcm.2008.11.006
  22. A. Gomez-Corral, A bibliographical guid to the analysis of retrial queues through matrix analytic techniques, Annals of Operations Research 141 (2006), 163-191. https://doi.org/10.1007/s10479-006-5298-4
  23. B.S. Greenberg and R.W. Wolf, An upper bound on the performance of queues with returning customer, Journal of Applied Probbaility 24 (1987), 466-475. https://doi.org/10.1017/S0021900200031107
  24. S.A. Grishechkin, Multiclass batch arrival retrial queues analyzed as branching process with immigration, Queueing Systems 11 (1992), 395-418. https://doi.org/10.1007/BF01163863
  25. Q.M. He, H. Li, Y.Q. Zhao, Ergodicity of the BMAP/PH/s/s + K retrial queue with PH-retrial times, Queueing Systems 35 (2000), 323-347. https://doi.org/10.1023/A:1019110631467
  26. M.A. Johnson, M.R. Taaffe, Matching moments to phase distributions : mixture of Erlang distributions of common order, Stochastic Models 5 (1989), 711-743. https://doi.org/10.1080/15326348908807131
  27. J.C. Ke and F.M. Chang, Modified vacation policy for M/G/1 retrial queue with balking and feedback, Computers & Industrial Engineering 57 (2009), 433-443. https://doi.org/10.1016/j.cie.2009.01.002
  28. Z. Khalil, G.I. Falin and T. Yang, Some analytical results for congestion in subscriber line modules, Queueing Systems 10 (1992), 381-402. https://doi.org/10.1007/BF01193327
  29. J. Kim and B. Kim, A survey of retrial queueing systems, Annals of Operations Research 247 (2016), 3-36. https://doi.org/10.1007/s10479-015-2038-7
  30. J. Kim, J. Kim J and B. Kim, Analysis of the M/G/1 queue with discriminatory random order service policy, Performance Evaluation 68 (2011), 256-270. https://doi.org/10.1016/j.peva.2010.12.001
  31. V.G. Kulkarni, On queueing systems with retrials, Journal of Applied Probability 20 (1983), 380-389. https://doi.org/10.2307/3213810
  32. V.G. Kulkarni, Expected waiting times in a multiclass batch arrival retrial queue, Journal of Applied Probability 23 (1986), 144-154. https://doi.org/10.1017/S0021900200106345
  33. V.G. Kulkarni and H.M. Liang, Retrial queues revisited, In Frontiers in Queueing: Models and Applications in Science and Engineering (J.H. Dshalalow, ed.), CRC Press, Boca Raton, 19-34.
  34. B.K. Kummar, R. Rukmani and V. Thangaraj, An M/M/c retrial queueing system with Bernoulli vacations, Journal of Systems Science and Systems Engineering 18 (2009), 222-242. https://doi.org/10.1007/s11518-009-5106-1
  35. H.M. Liang, Retrial queues, Ph,D Thesis, University of North Carolina at Chapel Hill, 1991.
  36. H.M. Liang, Service station factors in monotonicity of retrial queues, Mathematical and Computer Models 30 (1999), 189-196. https://doi.org/10.1016/S0895-7177(99)00141-7
  37. H.M. Liang, V.G. Kulkarni, Monotonicity properties of single server retrial queues, Stochastic Models 9 (1993), 373-400. https://doi.org/10.1080/15326349308807271
  38. M. Martin and J.R. Artalejo, Analysis of an M/G/1 queue with two types of impatient units, Advances in Applied Probability 27 (1995), 840-861. https://doi.org/10.1017/S0001867800027178
  39. E. Morozov, A multiserver retrial queue: regenerative stability analysis, Queueing Systems 56 (2007), 157-168. https://doi.org/10.1007/s11134-007-9024-y
  40. M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models - An Algorithmic Approach, Johns Hopkins University Press, Baltimore, 1981.
  41. M.F. Neuts and B.M. Rao, Numerical investigation of a multiserver retrial model, Queueing Systems 7 (1990), 169-190. https://doi.org/10.1007/BF01158473
  42. T. Osogami and M. Harchol-Balter, Closed form solutions for mapping general distributions to quasiminimal PH distributions, Performance Evaluation 62 (2006), 524-552.
  43. T. Phung-Duc and K. Kawanishi, An efficient method for performance analysis of blended call centers with retrial, Asia Pacific Journal of Operational Research 31 (2014), 1440008 (33pages). https://doi.org/10.1142/S0217595914400089
  44. Y.W. Shin, Monotonicity properties in various retrial queues and their applications, Queueing Systems 53 (2006), 147-157. https://doi.org/10.1007/s11134-006-6702-0
  45. Y.W. Shin, Fundamental matrix of transient QBD generator with finite states and level dependent transitions, Asia-Pacific Journal of Operational Research 26 (2009), 697-714. https://doi.org/10.1142/S0217595909002407
  46. Y.W. Shin, Algorithmic solutions for M/M/c retrial queue with $PH_2$ retrial times, Journal of Applied Mathematics and Informatics 29 (2011), 803-811.
  47. Y.W. Shin, Interpolation approximation of M/G/c/K retrial queues with ordinary queues, Journal of Applied Mathematics and Informatics 30 (2012), 531-540.
  48. Y.W. Shin, Algorithmic approach to Markovian multi-server retrial queue with vacations, Applied Mathematics and Computations 250 (2015), 287-279. https://doi.org/10.1016/j.amc.2014.10.079
  49. Y.W. Shin, Stability of MAP/PH/c/K retrial queue with customer retrials and server vacations, Bulletin of the Korean Mathematical Society 53 (2016), 985-1004. https://doi.org/10.4134/BKMS.b150337
  50. Y.W. Shin, T.S. Choo, M/M/s queue with impatient customers and retrials, Applied Mathematical Modelling 33 (2009), 2596-2606. https://doi.org/10.1016/j.apm.2008.07.018
  51. Y.W. Shin and Y.C. Kim, Stochastic comparisons of Markovian retrial queues, Journal of the Korean Statistical Society 29 (2000), 473-488.
  52. Y.W. Shin and D.H. Moon, Approximations of retrial queu with limited number of retrials, Computers and Operations Research 37 (2010), 1262-1270. https://doi.org/10.1016/j.cor.2009.03.025
  53. Y.W. Shin and D.H. Moon, Approximation of M/M/c retrial queu with PH-retrial times, European Journal of Operational Research 213 (2011), 205-209. https://doi.org/10.1016/j.ejor.2011.03.024
  54. Y.W. Shin and D.H. Moon, Sensitivity of M/M/c retrial queu with respect to retrial times : experimental investigation, Journal of the Korean Institute of Industrial Engineers 37 (2011), 83-87. https://doi.org/10.7232/JKIIE.2011.37.2.083
  55. Y.W. Shin and D.H. Moon, Approximation of M/G/c retrial queue with M/PH/c retrial queue, Communications of the Korean Stistical Society 19 (2012), 169-175.
  56. Y.W. Shin and D.H. Moon, Approximation of M/M/s/K retrial queu with nonpersistent customers, Applied Mathematical Modelling 37 (2013), 753-761. https://doi.org/10.1016/j.apm.2012.02.029
  57. Y.W. Shin and D.H. Moon, On approximations for GI/G/c retrial queues, Journal of Applied Mathematics and Informatics 31 (2013), 311-325. https://doi.org/10.14317/jami.2013.311
  58. Y.W. Shin and D.H. Moon, Approximation of PH/PH/c retrial queu with PH-retrial times, Asia-Pacific Journal of Operational Research 31 (2014), 1440010 (21 pages). https://doi.org/10.1142/S0217595914400107
  59. Y.W. Shin and D.H. Moon, M/M/c retrial queue with multiclass of customers, Merhodology and Computing in Applied Probability 16 (2014), 931-949. https://doi.org/10.1007/s11009-013-9340-0
  60. Y.W. Shin and D.H. Moon, Approximate analysis of M/M/c retrial queue with server vacations, Journal of the Korean Society for Industrial and Applied Mathematics 19 (2015), 443-457. https://doi.org/10.12941/jksiam.2015.19.443
  61. S.N. Stepanov, Generalized model with repeated calls in case of extreme load, Queueing Systems 27 (1997), 131-151. https://doi.org/10.1023/A:1019110030674
  62. H. Tijms, A First Course in Stochastic Models, Wiley, 2003.
  63. W. Whitt, Approximating a point process by a renewal process, I: two basic methods, Operations Research 30 (1982), 125-147. https://doi.org/10.1287/opre.30.1.125
  64. R.W. Wolff, Stochastic Modeling and The Theory of Queues, New Jersey, Prentice Hall, 1989.
  65. X. Xu and Z.G. Zhang, Analysis of multiple-server queue with a single vacation (e, d)-policy, Performance Evaluation 63 (2006), 825-838. https://doi.org/10.1016/j.peva.2005.09.003
  66. T. Yang and J.G.C. Templeton, A survey on retrial queues, Queueing Systems 2 (1987), 201-233. https://doi.org/10.1007/BF01158899
  67. T. Yang, M.J.M. Posner, J.G.C. Templeton and H. Li, An approximation method for the M/G/1 retrial queues with general retrial times, European Journal of Operational Research 76 (1994), 552-562. https://doi.org/10.1016/0377-2217(94)90286-0