Matrix analytic method and working vacation queues - A survey

International Journal of Information and Management Sciences

Volumn 20, Issue 4, 2009, Pages 603-633

  Tian, Nai Shuo ^b   Li, Ji Hong ^c   Zhang, Zhe George ^a,d  

^a WESTERN WASHINGTON UNIVERSITY   (United States)

^b YANSHAN UNIVERSITY   (China)

^c SHANXI UNIVERSITY   (China)

^d SIMON FRASER UNIVERSITY   (Canada)

Author keywords

GI M 1 type model; M G 1 type model; Matrix analytic approach; QBD process; Queueing systems; Stochastic decomposition; Working vacation queue

Indexed keywords

MODEL MATRICES; QBD PROCESS; QUEUEING SYSTEM; STOCHASTIC DECOMPOSITION; WORKING VACATION;

OPTICAL COMMUNICATION; QUEUEING NETWORKS; QUEUEING THEORY; STOCHASTIC SYSTEMS; SURVEYS;

STOCHASTIC MODELS;

EID: 75149158446     PISSN: 10171819     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (73)

1. Alfa, A., Vacation models in discrete time, Queueing Syst., Vol.44, pp.5-30, 2003.
2. Baba, Y., Analysis of a GI/M/1 queue with multiple working vacations, Oper. Res. Letters, Vol.33, pp.201-209, 2005.
3. Banik, A., Gupta, U. and Pathak, S., On the GI/M/1/N queue with multiple working vacationsanalytic analysis and computation, Appl. Math. Modelling, Vol.31, pp.1701-1710, 2007.
4. Blondia, C., A discrete time batch Markovian arrival process as B-ISDN traffic model, Belgior J. Oper. Res. Statist. Comput. Sci., Vol.32, pp.13-23, 1993.
5. Blondia, C. and Cosal, O., Statistical multiplexing of VBR sources: a matrix-analytic approach, Perform. Evaluation, Vol.16, pp.5-20, 1992.
6. Boxma, O., Peudo-conservation laws in cyclic service systems, J. Appl. Prob., Vol.24, pp.949-964, 1987.
7. Bruneel, H. and Kim. B., Discrete-time models for communication Systems Including ATM, Kluwer Academic Publishers, Boston, 1993.
8. Chao, X. and Zhao, Y., Analysis of multi-server queues with station and server vacation, Eur. J. Oper. Res., Vol.110, pp.392-406, 1998.
9. Chatterjee, U. and Mukherjee, S., GI/M/1 queue with server vacation, J. Oper. Res. Soc., Vol.41, pp.83-87, 1990.
10. Doshi, B., Queueing systems with vacations-a survey, Queueing Syst., Vol.1, pp.29-66, 1986.
11. x/1 systems and Riemann boundary value problems, Queueing Syst., Vol.27, pp.351-366, 1997.
12. Erlang, A. K., The theory of probabilities and telephone conversations, Nyt. Tidsskrift Matematik, Vol.B20, pp.33-39, 1909.
13. Fuhrman, S., A note on the M/G/1 queue with server vacations, Oper. Res., Vol.32, pp.1368-1373, 1984.
14. Fuhrmann, S. and Cooper, R., Stochastic decompositions in the M/G/1 queue with generalized server vacations, Oper. Res., Vol.33, pp.1117-1129, 1985.
15. Gravey, A. and Hebuterne, G., Simultaneity in discrete time single server queue with Bernoulli inputs, Perform. Evaluation, Vol.14, pp.123-131, 1992.
16. Gross, D. and Harris, C., Foundamentals of Queueing Theory, second ed, Johns Wiley & Sons, New York, 1985.
17. Harris, C., Marchai, W., State dependence in M/G/1 server vacation models, Oper. Res., Vol.36, pp.560-565, 1988.
18. Hunter, J. J., Mathematical techniques of applied probability, Vol.2 Discrete Time Models: Techniques and Applications. Academic Press. New York, 1983.
19. Ke, J. C., The analysis of a general input queue with N-policy and exponential vacations, Queueing Syst., Vol.45, pp.135-160, 2003.
20. Keilson, J. and Rammaswami, R., The backlog and depletion-time process for M/G/1 vacation models with exhaustive service discipline, J. Appl. Prob., Vol.25, pp.404-412, 1988.
21. Keilson, J. and Servi, L., Dynamics of the M/G/1 vacation model, Oper. Res., Vol.35, pp.575-582, 1987.
22. Keilson, J., Servi, L., A distributional form of Little's law, Oper. Res. Letters, Vol.7, pp.223-227, 1988.
23. Kim, J., Choi, D. and Chae, K., Analysis of queue-length distribution of the M/G/1 queue with working vacations. International Conference on Statistics and Related Fields. Hawaii, 2003.
24. Kobayashi, H. and Konhein, A., Queueing models for computer communications system analysis, IEEE Tran. Comm. COM-25, pp. 1-29, 1977.
25. Latouche, G. and Ramaswami, V., Introduction to Matrix-Analysis Methods in Statistics Modeling, ASA-SIAM Series on Applied Probability. New York, 1999.
26. Levy, H. and Kleinrock, L., A queue with starter and a queue with vacations: Delay analysis by decomposition, Oper. Res., Vol.34, pp.426-436, 1986.
27. Levy, Y. and Yechiali, U., Utilization of idle time in an M/G/1 queueing system, Manag. Sci., Vol.22, pp.202-211, 1975.
28. Li, J., Liu, W. and Tian, N., Steady state analysis of a discrete time batch arrival queue with working vacations, Perform. Evaluations (2008, in press).
29. Li, J. and Tian, N., The discrete time GI/Geo/1 queue with working vacations and vacation interruption, Appl. Math, and Comput., Vol.185, pp.1-10, 2007.
30. Li, J. and Tian, N., The M/M/1 queue with working vacations and vacation interruptions, J. Syst. Sci. Syst. Engin., Vol.16, pp.121-127, 2007.
31. Li, J. and Tian, N., Analysis of the discrete time Geo/Geo/1 queue with single working vacation, Quality Technology & Quantitative Management, Vol.5, pp.77-89, 2008.
32. Li, J. and Tian, N., The waiting time of the GI/M/1 queue with working vacations, The research report (in press).
33. Li, J. and Tian, N., The GI/M/1 queue with phase-type working vacations and vacation interruption, J. Appl. Math, and Comput., Vol.30, pp.121-141, 2009.
34. Li, J., Tian, N. and Liu, W., Discrete time GI/Geo/1 queue with multiple working vacations, Queueing Syst., Vol.56, pp.53-63, 2007.
35. Li, J., Tian, N. and Ma, Z., Performance analysis of GI/M1 queue with working vacations and vacation interruption, Appl. Math. Modelling, Vol.32, pp.2715-2730, 2008.
36. Li, J., Tian, N., Zhang, Z.G. and Lu, H. P., Analysis of the M/G/1 queue with exponential working vacations-A matrix analytic approach, Queueing Syst., Vol.61, pp.139-166, 2009.
37. Liu, W., Xu, X. and Tian, N., Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Letters., Vol.35, pp.595-600, 2007.
38. Lucantoni, D., The BMAP/G/1 Queue, in Models and Techniques for Performance Evaluation of Computer and Communication System, Springer-Verlag, pp.330-358, 1993.
39. Ma, Z., Tian, N. and Liu, M., A Geom/G/1 Gate Service System with Multiple Adaptive Vacation, Inter. J. Infor. Manag. Sciences, Vol.18, pp.209-221, 2007.
40. Meisling, T., Discrete time queueing theory, Oper. Res., Vol.6, pp.96-105, 1958.
41. Miyazawa, M., Decomposition formulas for single server queue with vacation: An unified approach, Stoch. Models, Vol.12, pp.455-472, 1994.
42. Narula-Tam, A., Finn, S. and Medard, M., Analysis of reconfiguration in IP over WPM access network, In: Proceeding of the Optical Fiber Communication Conference(OFC). pp. MN4.1-MN4.3, 2001.
43. Neuts, M., Markov chain with applications in queueing theory which have a matrix-geometric invariant vector, Adv. Appl. Prob., Vol.10, pp.185-211, 1978.
44. Neuts, M., Matrix-Geometric Solutions in Stochastic models, Johns Hopkins University Press, Baltimore, 1981.
45. Neuts, M., Matrix-analytic methods in queueing theory. In advances in queueing. (ed.) CRC press, Dshalaow, 1995.
46. Neuts, M., tructured Stochastic Matrices of M/G/1 type and Their Applications, Marcel Dekker, New York, 1999.
47. Ross, D., Stochastic Processes. John Wiley & Sons. New York, 1983.
48. Servi, L. and Finn, S., M/M/1 queue with working vacations (M/M/1/WV). Perform. Evaluation, Vol.50, pp.41-52, 2002.
49. Shanthinkumar, J., On stochastic decomposition in M/G/1 type queue with generalized server vacations, Oper. Res., Vol.36, pp.566-569, 1988.
50. Takagi, H., Queueing Analysis, Vol.1, vacation and priority systems, North-Holland, Amsterdar, 1991.
51. Takagi, H., Queueing Analysis. Vol.3, Discrete Time Systems. North-Holland / Eiservier, Amsterdam, 1993.
52. Takagi, H., Queueing analysis of polling models, Dshalalow ed. Fronties in Queueing. CRC Press, pp.119-146, 1997.
53. Takine, T. and Hasekawa, T., A generalization of the decomposition property in the M/G/1 queue with server vacations, Oper. Res. Letters, Vol.12, pp.97-99, 1992.
54. Teghem, J., Control of the service process in a queueing system, European J. Oper. Res., Vol.23, pp.141-158, 1986.
55. Tian, N., Ma, Z. and Liu, M., The discrete time Geo/Geo/1 queue with multiple working vacations, Appl. Math. Modelling, Vol.32, pp.2941-2953, 2008.
56. Tian, N., Xu, X. and Ma, Z.: Discrete Time Queueing Theory. Science Press, Beijing (2008), in Chinese.
57. Tian, N., Zhang, D. and Cao, C., The GI/M/1 queue with exponential vacations, Queueing Syst., Vol.5, pp.331-344, 1989.
58. Tian, N. and Zhang, Z. G., The discrete time GI/Geo/1 queue with multiple vacations, Queueing Syst., Vol.40, pp.283-294, 2002.
59. Tian, N. and Zhang, Z. G., Stationary distributions on GI/M/c queue with PH type vacations, Queueing Syst., Vol.44, pp.183-202, 2003.
60. Tian, N., Zhang and Z. G., A note on GI/M/1 queues with phase-type setup times or server vacations, INFOR, Vol.41, pp.341-351, 2003.
61. Tian, N. and Zhang, Z. G., Vacation Queueing Models-Theory and Applications, Springer, New York, 2006.
62. Tian, N. and Zhang, Z. G., Quantitying the performance effects of idle time utilization in multiserver systems, Naval Research Logistics, Vol.54, pp.189-199, 2007.
63. Tian, N., Zhao, X. and Wang, K., The M/M/1 queue with single working vacation. Inter. J. Infor. Manag. Sciences, Vol.19, pp.621-634, 2008.
64. Wallace, P., The solution of quasi birth and death processes arising from multiple access computer systems. Ph. D. diss. Systems Engineering Laboratory, University of Michigan. Tech. rept. 077426-T, 1969.
65. Wu, D. and Takagi, H., M/G/1 queue with multiple working vacations, Perform. Evaluation, Vol.63, pp.654-681, 2006.
66. Xu, X. and Zhang, Z. G., The analysis of multi-server Queue with a single vacation (e,d)-policy, Perform. Evaluation, Vol.63, pp.625-638, 2006.
67. Xu, X., Zhang, Z. and Tian, N., The M/M/1 queue with single working vacation and setup times, Inter. J. Oper. Res., Vol.6, pp.420-434, 2009.
68. Yi, X., Kim, J., Choi, D. and Chae, K., The Geo/G/1 queue with disaster and multiple working vacations, Stoch. Models, Vol.23, pp.537-549, 2007.
69. Zhang, Z. G. and Tian, N., Geo/G/1 queue with multiple adaptive vacations, Queueing Syst., Vol.38, pp.419-430, 2001.
70. Zhang, Z. G. and Tian, N., Analysis of queueing systems with synchronous single vacation for some servers, Queueing Syst., Vol.45, pp.161-174, 2003.
71. Zhang, Z. G. and Tian, N., Analysis on queueing systems with synchronous vacations of partial Server, Perform. Evaluation, Vol.52, pp.269-282, 2003.
72. Zhang, Z. G. and Tian, N., A two threshold vacation policy in multi-server queueing systems, Eur. J. Oper. Res., Vol.168, pp.153-163, 2006.
73. Zhang, Z. J. and Xu, X., Analysis for the M/M/1 queue with multiple working vacations and NPolicy, Inter. J. Infor. Manag. Sciences, Vol.19, pp.495-506, 2008.