Steady-State Queue Length Analysis of a Batch Arrival Queue under N-Policy with Single Vacation and Setup Times

ABSTRACT

This paper investigates the steady state property of queue length for a batch arrival queue under N-policy with single vacation and setup times. When the system becomes empty, the server is turned off at once and takes a single vacation of random length . When he returns, if the queue length reaches or exceeds threshold , the server is immediately turned on but is temporarily unavailable due to a random setup time before offering service. If not, the server stays in the system until the queue length at least being . We derive the system size distribution and confirm the stochastic decomposition property. We also derive the recursion expressions of queue length distribution and other performance measures. Finally, we present some numerical examples to show the analytical results obtained. Sensitivity analysis is also performed.

This paper investigates the steady state property of queue length for a batch arrival queue under N-policy with single vacation and setup times. When the system becomes empty, the server is turned off at once and takes a single vacation of random length . When he returns, if the queue length reaches or exceeds threshold , the server is immediately turned on but is temporarily unavailable due to a random setup time before offering service. If not, the server stays in the system until the queue length at least being . We derive the system size distribution and confirm the stochastic decomposition property. We also derive the recursion expressions of queue length distribution and other performance measures. Finally, we present some numerical examples to show the analytical results obtained. Sensitivity analysis is also performed.

Cite this paper

nullZ. Yu, M. Liu and Y. Ma, "Steady-State Queue Length Analysis of a Batch Arrival Queue under N-Policy with Single Vacation and Setup Times,"*Intelligent Information Management*, Vol. 2 No. 6, 2010, pp. 365-374. doi: 10.4236/iim.2010.26044.

nullZ. Yu, M. Liu and Y. Ma, "Steady-State Queue Length Analysis of a Batch Arrival Queue under N-Policy with Single Vacation and Setup Times,"

References

[1] B. T. Doshi, “Queueing Systems with Vacations–a Survey,” Queueing System, Vol. 1, No. 1, 1986, pp. 29-66.

[2] H. Takagi, “Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority System,” Elsevier, Amsterdam, Vol. 1, 1991.

[3] S. W. 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.

[4] M. Yadin and P. Naor, “Queueing Systems with a Removable Service Station,” Operational Research Quarterly, Vol. 14, No. 3, 1963, pp. 393-405.

[5] H. W. Lee, S. S. Lee and K. C. Chae, “Operating Characteristics of Mx/G/1 Queue with N-Policy,” Queueing Systems, Vol. 15, No. 1-4, 1994, pp. 387-399.

[6] 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.

[7] 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. 2, 1995, pp. 173-189.

[8] G. Choudhury and M. Paul, “A Batch Arrival Queue with an Additional Service Channel under N-Policy,” Applied Mathematics and Computation, Vol. 156, No. 1, 2004, pp. 115-130.

[9] G. Choudhury and K. C. Madan, “A Two-Stage Batch Arrival Queueing System with a Modified Bernoulli Schedule Vacation under N-Policy,” Mathematical and Computer Modelling, Vol. 42, No. 1-2, 2005, pp. 71-85.

[10] G. Choudhury and M. Paul, “A Batch Arrival Queue with a Second Optional Service Channel under N-Policy,” Stochastic Analysis and Applications, Vol. 24, No. 1, 2006, pp. 1-21.

[11] J.-C. Ke, “The Control Policy of an Mx/G/1 Queueing System with Server Startup and Two Vacation Types,” Mathematical Methods of Operations Research, Vol. 54, No. 3, 2001, pp. 471-490.

[12] J.-C. Ke, “Optimal Strategy Policy in Batch Arrival Queue with Server Breakdowns and Multiple Vacations,” Ma- thematical Methods of Operations Research, Vol. 58, No. 1, 2003, pp. 41-56.

[13] R. G. V. Krishna, R. Nadarajan and R. Arumuganathan, “Analysis of a Bulk Queue with N-Policy Multiple Vacations and Setup Times,” Computers and Operations Research, Vol. 25, No. 11, 1998, pp. 957-967.

[14] 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.

[15] J. E. Shore, “Information Theoretic Approximations for M/G/1 and G/G/1 Queueing Systems,” Acta Informatica, 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 Informatica, Vol. 19, No. 4, 1983, pp. 339-355.

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

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

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

[20] 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.

[21] J.-C. Ke, “Batch Arrival Queues under Vacation Policies with Server Breakdowns and Startup/Closedown Times,” Applied Mathematical Modelling, Vol. 31, No. 7, 2007, pp. 1282-1292.

[1] B. T. Doshi, “Queueing Systems with Vacations–a Survey,” Queueing System, Vol. 1, No. 1, 1986, pp. 29-66.

[2] H. Takagi, “Queueing Analysis: A Foundation of Performance Evaluation, Vacation and Priority System,” Elsevier, Amsterdam, Vol. 1, 1991.

[3] S. W. 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.

[4] M. Yadin and P. Naor, “Queueing Systems with a Removable Service Station,” Operational Research Quarterly, Vol. 14, No. 3, 1963, pp. 393-405.

[5] H. W. Lee, S. S. Lee and K. C. Chae, “Operating Characteristics of Mx/G/1 Queue with N-Policy,” Queueing Systems, Vol. 15, No. 1-4, 1994, pp. 387-399.

[6] 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.

[7] 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. 2, 1995, pp. 173-189.

[8] G. Choudhury and M. Paul, “A Batch Arrival Queue with an Additional Service Channel under N-Policy,” Applied Mathematics and Computation, Vol. 156, No. 1, 2004, pp. 115-130.

[9] G. Choudhury and K. C. Madan, “A Two-Stage Batch Arrival Queueing System with a Modified Bernoulli Schedule Vacation under N-Policy,” Mathematical and Computer Modelling, Vol. 42, No. 1-2, 2005, pp. 71-85.

[10] G. Choudhury and M. Paul, “A Batch Arrival Queue with a Second Optional Service Channel under N-Policy,” Stochastic Analysis and Applications, Vol. 24, No. 1, 2006, pp. 1-21.

[11] J.-C. Ke, “The Control Policy of an Mx/G/1 Queueing System with Server Startup and Two Vacation Types,” Mathematical Methods of Operations Research, Vol. 54, No. 3, 2001, pp. 471-490.

[12] J.-C. Ke, “Optimal Strategy Policy in Batch Arrival Queue with Server Breakdowns and Multiple Vacations,” Ma- thematical Methods of Operations Research, Vol. 58, No. 1, 2003, pp. 41-56.

[13] R. G. V. Krishna, R. Nadarajan and R. Arumuganathan, “Analysis of a Bulk Queue with N-Policy Multiple Vacations and Setup Times,” Computers and Operations Research, Vol. 25, No. 11, 1998, pp. 957-967.

[14] 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.

[15] J. E. Shore, “Information Theoretic Approximations for M/G/1 and G/G/1 Queueing Systems,” Acta Informatica, 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 Informatica, Vol. 19, No. 4, 1983, pp. 339-355.

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

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

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

[20] 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.

[21] J.-C. Ke, “Batch Arrival Queues under Vacation Policies with Server Breakdowns and Startup/Closedown Times,” Applied Mathematical Modelling, Vol. 31, No. 7, 2007, pp. 1282-1292.