The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown

ABSTRACT

This paper develops a new method for calculating the system size distribution on two different M/G/1 queueing system under N-policy with general startup/closedown. Firstly, the stochastic decomposition property is used to derive the p.g.f. of the system size distribution. By the Leibniz formula of derivation, we investigate the additional system size distribution, and then, we get the recursion expression of system sizes distribution. Finally, several examples are given for illustrating the application of the recursion expression and sensitivity analysis is also performed.

This paper develops a new method for calculating the system size distribution on two different M/G/1 queueing system under N-policy with general startup/closedown. Firstly, the stochastic decomposition property is used to derive the p.g.f. of the system size distribution. By the Leibniz formula of derivation, we investigate the additional system size distribution, and then, we get the recursion expression of system sizes distribution. Finally, several examples are given for illustrating the application of the recursion expression and sensitivity analysis is also performed.

Cite this paper

nullM. Liu, Y. Ma and B. Deng, "The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown,"*iBusiness*, Vol. 2 No. 4, 2010, pp. 363-369. doi: 10.4236/ib.2010.24047.

nullM. Liu, Y. Ma and B. Deng, "The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown,"

References

[1] M. Yadin and P. Naor, “Queueing Systems with a Removable Service Station,” Operational Research Quarterly, Vol. 14, No. 4, 1963, pp. 393-405.

[2] O. Kella, “The Threshold Policy in the M/G/1 Queue with Server Vacations,” Naval Research Logistics, Vol. 36, No. 2, 1989, pp. 111-123.

[3] A. Borthakur, J. Medhi and R. Gohain, “Poisson Input Queueing Systems with Startup Time and under Control Operating Policy,” Computers and Operations Research, Vol. 14, No. 1, 1987, pp. 33-40.

[4] K. R. Baker, “A Note on Operating Policies for the Queue M/M/1 with Exponential Startup,” INFOR, Vol. 11, No. 1, 1973, pp. 71-72.

[5] H. W. Lee, S. S. Lee, J. O. Park and K. C. Chae, “Analysis of MX/G/1 Queue with N Policy and Multiple Vacations,” Journal of Applied Probability, Vol. 31, No. 2, 1994, pp. 467-496.

[6] S. S. Lee, H. W. Lee, S. H. Yoon and K. C. Chae, “Batch Arrival Queue with N Policy and Single Vacation,” Computers and Operations Research, Vol. 22, No. 1, 1995, pp. 173-189.

[7] H. S. Lee and M. M. Srinivasan, “Control Policies for the MX/G/1 queueing System,” Management Science, Vol. 35, No. 6, 1989, pp. 708-721.

[8] S. Fuhrmann and R. B. Cooper, “Stochastic Decompositions in the M/G/1 Queue with Generalized Vacations,” Operations Research, Vol. 33, No. 5, 1985, pp. 1117- 1129.

[9] H. Takagi, “Vacation and Priority Systems,” Part I. In: Queueing Analysis: A Foundation of Performance Eva- luation. Vol. I, Amsterdam, 1991.

[10] J.-C. Ke, “On M/G/1 System under NT Policies with Breakdowns, Startup and Closedown,” Applied Mathematical Modelling, Vol. 30, No. 1, 2006, pp. 49-66.

[11] K. H. Wang, “Optimal Control of a Removable and Non- Reliable Server in an M/M/1 Queueing System with Exponential Startup Time,” Mathematical Methods of Operations Research, Vol. 58, No. 1, 2003, pp. 29-39.

[12] K.-H. Wang and K.-L. Yen, “Optimal Control of an M/ Ek/1 Queueing System with a Removable Server,” Mathematical Methods of Operations Research, Vol. 57, No. 2, 2003, pp. 255-262.

[13] K. H. Wang, “Optimal Control of an M/Ek/1 Queueing System with Removable Service Station Subject to Breakdowns,” Journal of the Operational Research Society, Vol. 48, No. 9, 1997, pp. 936-946.

[14] K. H. Wang, K.-W. Chang and B. D. Sivazlian, “Optimal Control of a Removable and Non-Reliable Server in an Infinite and a Finite M/H2/1 Queueing System,” Applied Mathematical Modelling, Vol. 23, No. 8, 1999, pp. 651- 666.

[15] J. E. Shore, “Information Theoretic Approximations for M/G/1 and G/G/1 Queueing Systems,” Acta Information, Vol. 17, No. 1, 1982, pp. 43-61.

[16] M. A. El-Affendi and D. D. Kouvatsos, “A Maximum Entropy Analysis of the M/G/1 and G/M/1 Queueing Systems at Equilibrium,” Acta Information, Vol. 19, No. 4, 1983, pp. 339-355.

[17] K.-H. Wang, S.-L. Shuang and W. L. Pearn, “Maximum Entropy Analysis to the N Policy M/G/1 Queueing System with a Removable Server,” Applied Mathematical Modelling, Vol. 26, No. 12, 2002, pp. 1151-1162.

[18] K.-H. Wang, L.-P. Wang, J.-C. Ke and G. Chen, “Comparative Analysis for the N Policy M/G/1 Queueing System with a Removable and Unreliable Server,” Mathematical Methods of Operations Research, Vol. 61, No. 3, 2005, pp. 505-520.

[19] K.-H. Wang, T.-Y. Wang and W. L. Pearn, “Maximum Entropy Analysis to the N Policy M/G/1 Queueing System with Server Breakdowns and General Startup Times,” Applied Mathematics and Computation, Vol. 165, No. 1, 2005, pp. 45-61,.

[20] K.-H. Wang and K.-B. Huang, “A Maximum Entropy Approach for the

[1] M. Yadin and P. Naor, “Queueing Systems with a Removable Service Station,” Operational Research Quarterly, Vol. 14, No. 4, 1963, pp. 393-405.

[2] O. Kella, “The Threshold Policy in the M/G/1 Queue with Server Vacations,” Naval Research Logistics, Vol. 36, No. 2, 1989, pp. 111-123.

[3] A. Borthakur, J. Medhi and R. Gohain, “Poisson Input Queueing Systems with Startup Time and under Control Operating Policy,” Computers and Operations Research, Vol. 14, No. 1, 1987, pp. 33-40.

[4] K. R. Baker, “A Note on Operating Policies for the Queue M/M/1 with Exponential Startup,” INFOR, Vol. 11, No. 1, 1973, pp. 71-72.

[5] H. W. Lee, S. S. Lee, J. O. Park and K. C. Chae, “Analysis of MX/G/1 Queue with N Policy and Multiple Vacations,” Journal of Applied Probability, Vol. 31, No. 2, 1994, pp. 467-496.

[6] S. S. Lee, H. W. Lee, S. H. Yoon and K. C. Chae, “Batch Arrival Queue with N Policy and Single Vacation,” Computers and Operations Research, Vol. 22, No. 1, 1995, pp. 173-189.

[7] H. S. Lee and M. M. Srinivasan, “Control Policies for the MX/G/1 queueing System,” Management Science, Vol. 35, No. 6, 1989, pp. 708-721.

[8] S. Fuhrmann and R. B. Cooper, “Stochastic Decompositions in the M/G/1 Queue with Generalized Vacations,” Operations Research, Vol. 33, No. 5, 1985, pp. 1117- 1129.

[9] H. Takagi, “Vacation and Priority Systems,” Part I. In: Queueing Analysis: A Foundation of Performance Eva- luation. Vol. I, Amsterdam, 1991.

[10] J.-C. Ke, “On M/G/1 System under NT Policies with Breakdowns, Startup and Closedown,” Applied Mathematical Modelling, Vol. 30, No. 1, 2006, pp. 49-66.

[11] K. H. Wang, “Optimal Control of a Removable and Non- Reliable Server in an M/M/1 Queueing System with Exponential Startup Time,” Mathematical Methods of Operations Research, Vol. 58, No. 1, 2003, pp. 29-39.

[12] K.-H. Wang and K.-L. Yen, “Optimal Control of an M/ Ek/1 Queueing System with a Removable Server,” Mathematical Methods of Operations Research, Vol. 57, No. 2, 2003, pp. 255-262.

[13] K. H. Wang, “Optimal Control of an M/Ek/1 Queueing System with Removable Service Station Subject to Breakdowns,” Journal of the Operational Research Society, Vol. 48, No. 9, 1997, pp. 936-946.

[14] K. H. Wang, K.-W. Chang and B. D. Sivazlian, “Optimal Control of a Removable and Non-Reliable Server in an Infinite and a Finite M/H2/1 Queueing System,” Applied Mathematical Modelling, Vol. 23, No. 8, 1999, pp. 651- 666.

[15] J. E. Shore, “Information Theoretic Approximations for M/G/1 and G/G/1 Queueing Systems,” Acta Information, Vol. 17, No. 1, 1982, pp. 43-61.

[16] M. A. El-Affendi and D. D. Kouvatsos, “A Maximum Entropy Analysis of the M/G/1 and G/M/1 Queueing Systems at Equilibrium,” Acta Information, Vol. 19, No. 4, 1983, pp. 339-355.

[17] K.-H. Wang, S.-L. Shuang and W. L. Pearn, “Maximum Entropy Analysis to the N Policy M/G/1 Queueing System with a Removable Server,” Applied Mathematical Modelling, Vol. 26, No. 12, 2002, pp. 1151-1162.

[18] K.-H. Wang, L.-P. Wang, J.-C. Ke and G. Chen, “Comparative Analysis for the N Policy M/G/1 Queueing System with a Removable and Unreliable Server,” Mathematical Methods of Operations Research, Vol. 61, No. 3, 2005, pp. 505-520.

[19] K.-H. Wang, T.-Y. Wang and W. L. Pearn, “Maximum Entropy Analysis to the N Policy M/G/1 Queueing System with Server Breakdowns and General Startup Times,” Applied Mathematics and Computation, Vol. 165, No. 1, 2005, pp. 45-61,.

[20] K.-H. Wang and K.-B. Huang, “A Maximum Entropy Approach for the

-Policy M/G/1 Queue with a Removable and Unreliable Server,” Applied Mathematical Modelling, Vo1. 33, No. 4, 2009, pp. 2024-2034.

[21] K.-H. Wang, M.-C. Chan and J.-C. Ke, “Maximum Entropy Analysis of the Mx/M/1 Queueing System with Multiple Vacations and Server Breakdowns,” Computers & Industrial Engineering, Vol. 52, No. 2, 2007, pp. 192- 202.

[22] J.-C. Ke and C.-H. Lin, “Maximum Entropy Solutions for Batch Arrival Queue with an Un-Reliable Server and Delaying Vacations,” Applied Mathematics and Computation, Vol. 183, No. 2, 2006, pp. 1328-1340.

[23] J.-C. Ke and C.-H. Lin, “Maximum Entropy Approach for Batch-Arrival Queue under N Policy with an Un-Reliable Server and Single Vacation,” Journal of Computational and Applied Mathematics, Vol. 221, No. 1, 2008, pp. 1-15.

[24] Y. H. Tang, “The Transient Solution for M/G/1 Queue with Server Vacations,” Acta Math Scientia, Vol. 17(B), No. 3, 1997, pp. 276-282.

[25] Y. H. Tang and X. W. Tang, “The Queue-Length Distribution for Mx/G/1 Queue with Single Server Vacation,” Acta Mathematical Scientia, Vol. 20(B), No.3, 2000, pp. 397-408.

[26] Y.H. Tang, X. Yun and S. J. Huang, “Discrete-Time Geox/G/1 Queue with Unreliable Server and Multiple Adaptive Delayed Vacations,” Journal of Computational and Applied Mathematics, Vol. 220, No. 1-2, 2008, pp. 439-455.

[27] J. Medhi and J. G. C. Templeton, “A Poisson Input Queue under N-Policy and with a General Start up Time,” Computers and Operations Research, Vol. 19, No. 1, 1992, pp. 35-41.

[28] Y. H. Tang, “The Transient and Equilibrium Distributions of the Queue-Length for M/G/1 Queue with Delayed N-Policy,” System Engineering - Theory & Practice, Vol. 27, No. 11, 2007, pp. 130-134 (in Chinese).