International Journal of Nonlinear Analysis and Applications
On fluctuation analysis of different kinds of n-policy queues with single vacation
Document Type : Research Paper
Authors
Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Iraq
Abstract
In this paper, we debate queueing systems with N-policy and single vacation. We consider these systems when the vacation times have Erlang distribution. Moreover, we adapted the input by studying two different kinds: first, an ordinary Poisson input, and second, type 2 geometric batch input. We derive the probability generating function of the number of units in the system in two cases by using fluctuation analysis.
Keywords
[1] L. Abolnikov and A. Dukhovny, Markov chains with transition delta-matrix: ergodicity conditions, invariant
probability measures and applications, J. Appl. Math. Stoch. Anal. 4 (1991) 333-355.
[2] L. Abolnikov, J.H. Dshalalow and A. Treerattrakoon, On a dual hybrid queueing system, Nonlinear Anal. Hybrid
Syst. 2 (2008) 96–109.
[3] R.P. Agarwal and J.H. Dshalalow, New fluctuation analysis of D-policy bulk queues with multiple vacations, Math.
Comput. Model. 41 (2005) 253–269.
[4] Y. Baba, On the MX/G/1 queue with vacation time, Operat. Res. Let. 5 (1986) 93–98.
[5] J.B. Bacot and J.H. Dshalalow, A bulk input queueing system with batch gated service and multiple vacation
policy, Math. Comput. Model. 34 (2001) 873–886.
[6] K.C. Chae and H.W. Lee, MX/G/1 vacation models with N-policy: heuristic interpretation of the mean waiting
time, J. Operat. Res. Soc. 46 (1995) 258-264.
[7] G. Choudhury, A batch arrival queue with a vacation time under single vacation policy, Comput. Operat. Res. 29
(2002) 1941–1955.
[8] G. Choudhury, An MX/G/1 queueing system with a setup period and a vacation period, Que. Syst. 36 (2000)
23-38.
[9] G. Choudhury and H.K. Baruah, Analysis of a Poisson queue with a threshold policy and a grand vacation process:
an analytic approach, Sankhy¯a: Indian J. Stat. Ser. B (2000) 303–316.
[10] G. Choudhury and M. Paul, A batch arrival queue with an additional service channel under N-policy, Appl. Math.
Comput. 156 (2004) 115–130.
[11] J.H. Dshalalow, A note on D-policy bulk queueing systems, J. Appl. Prob. 38 (2001) 280–283.
[12] J. H. Dshalalow and A. Merie, Fluctuation analysis in queues with several operational modes and priority customers, Top 26 (2018) 309–333.
[13] J. H. Dshalalow and J. Yellen, Bulk input queues with quorum and multiple vacations, Math. Prob. Engin. 2
(1996) 95–106.
[14] J.H. Dshalalow and L. Tadj, A queueing system with a fixed accumulation level, random server capacity and
capacity dependent service time, Int. J. Math. Math. Sci. 15 (1992) 189–194.
[15] J.H. Dshalalow and R.T. White, Current Trends in Random Walks on Random Lattices, Math. 9 (2021) 11–48.
[16] J. H. Dshalalow, Queueing processes in bulk systems under the D-policy, J. Appl. Prob. 34 (1998) 976–989.
[17] S. Kalita and G. Choudhury, Some aspects of a batch arrival Poisson queue with N-policy, Stoch. Model. Appl.
5 (2002) 21–32.
[18] H. W. Lee, S. L. Soon and C. C. Kyung, Operating characteristics of MX/G/1 queue with N-policy, Que. Syst.
15 (1994) 387–399.[19] H.W. Lee and M.M. Srinivasan, Control policies for the MX/G/1 queueing system, Manag. Sci. 35 (1989) 708–721.
[20] H. W. Lee, S. S. Lee, J.O. Park and K.C. Chae, Analysis of the M x/G/1 queue by N-policy and multiple vacations,
J. Appl. Prob. 31 (1994) 476–496.
[21] S.S. Lee, H. W. Lee, S.H. Yoon and K.C. Chae, Batch arrival queue with N-policy and single vacation, Comput.
Operat. Res. 22 (1995) 173–189.
[22] K. C. Madan and W. Abu–Dayyeh, Restricted admissibility of batches into an M/G/1 type bulk queue with
modified Bernoulli schedule server vacations, ESAIM: Prob. Stat. 6 (2002) 113–125.
[23] J. Medhi, Single server queueing system with Poisson input: a review of some recent developments, Adv. Combin.
Meth. Appl. Prob. Stat. 5 (1997) 317–338.
[24] E. Rosenberg and U. Yechiali, The Mx/G/1 queue with single and multiple vacations under the LIFO service
regime, Operat. Res. Let. 14 (1993) 171–179.
[25] J. Teghem, Control of the service process in a queueing system, Euro. J. Operat. Res. 23 (1986) 141–158.
[26] J. Teghem, On a decomposition result for a class of vacation queueing systems, J. Appl. Prob.27 (1990) 227–231.
probability measures and applications, J. Appl. Math. Stoch. Anal. 4 (1991) 333-355.
[2] L. Abolnikov, J.H. Dshalalow and A. Treerattrakoon, On a dual hybrid queueing system, Nonlinear Anal. Hybrid
Syst. 2 (2008) 96–109.
[3] R.P. Agarwal and J.H. Dshalalow, New fluctuation analysis of D-policy bulk queues with multiple vacations, Math.
Comput. Model. 41 (2005) 253–269.
[4] Y. Baba, On the MX/G/1 queue with vacation time, Operat. Res. Let. 5 (1986) 93–98.
[5] J.B. Bacot and J.H. Dshalalow, A bulk input queueing system with batch gated service and multiple vacation
policy, Math. Comput. Model. 34 (2001) 873–886.
[6] K.C. Chae and H.W. Lee, MX/G/1 vacation models with N-policy: heuristic interpretation of the mean waiting
time, J. Operat. Res. Soc. 46 (1995) 258-264.
[7] G. Choudhury, A batch arrival queue with a vacation time under single vacation policy, Comput. Operat. Res. 29
(2002) 1941–1955.
[8] G. Choudhury, An MX/G/1 queueing system with a setup period and a vacation period, Que. Syst. 36 (2000)
23-38.
[9] G. Choudhury and H.K. Baruah, Analysis of a Poisson queue with a threshold policy and a grand vacation process:
an analytic approach, Sankhy¯a: Indian J. Stat. Ser. B (2000) 303–316.
[10] G. Choudhury and M. Paul, A batch arrival queue with an additional service channel under N-policy, Appl. Math.
Comput. 156 (2004) 115–130.
[11] J.H. Dshalalow, A note on D-policy bulk queueing systems, J. Appl. Prob. 38 (2001) 280–283.
[12] J. H. Dshalalow and A. Merie, Fluctuation analysis in queues with several operational modes and priority customers, Top 26 (2018) 309–333.
[13] J. H. Dshalalow and J. Yellen, Bulk input queues with quorum and multiple vacations, Math. Prob. Engin. 2
(1996) 95–106.
[14] J.H. Dshalalow and L. Tadj, A queueing system with a fixed accumulation level, random server capacity and
capacity dependent service time, Int. J. Math. Math. Sci. 15 (1992) 189–194.
[15] J.H. Dshalalow and R.T. White, Current Trends in Random Walks on Random Lattices, Math. 9 (2021) 11–48.
[16] J. H. Dshalalow, Queueing processes in bulk systems under the D-policy, J. Appl. Prob. 34 (1998) 976–989.
[17] S. Kalita and G. Choudhury, Some aspects of a batch arrival Poisson queue with N-policy, Stoch. Model. Appl.
5 (2002) 21–32.
[18] H. W. Lee, S. L. Soon and C. C. Kyung, Operating characteristics of MX/G/1 queue with N-policy, Que. Syst.
15 (1994) 387–399.[19] H.W. Lee and M.M. Srinivasan, Control policies for the MX/G/1 queueing system, Manag. Sci. 35 (1989) 708–721.
[20] H. W. Lee, S. S. Lee, J.O. Park and K.C. Chae, Analysis of the M x/G/1 queue by N-policy and multiple vacations,
J. Appl. Prob. 31 (1994) 476–496.
[21] S.S. Lee, H. W. Lee, S.H. Yoon and K.C. Chae, Batch arrival queue with N-policy and single vacation, Comput.
Operat. Res. 22 (1995) 173–189.
[22] K. C. Madan and W. Abu–Dayyeh, Restricted admissibility of batches into an M/G/1 type bulk queue with
modified Bernoulli schedule server vacations, ESAIM: Prob. Stat. 6 (2002) 113–125.
[23] J. Medhi, Single server queueing system with Poisson input: a review of some recent developments, Adv. Combin.
Meth. Appl. Prob. Stat. 5 (1997) 317–338.
[24] E. Rosenberg and U. Yechiali, The Mx/G/1 queue with single and multiple vacations under the LIFO service
regime, Operat. Res. Let. 14 (1993) 171–179.
[25] J. Teghem, Control of the service process in a queueing system, Euro. J. Operat. Res. 23 (1986) 141–158.
[26] J. Teghem, On a decomposition result for a class of vacation queueing systems, J. Appl. Prob.27 (1990) 227–231.
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Volume 12, Issue 2
November 2021Pages 2029-2040
- Receive Date: 04 March 2021
- Revise Date: 16 May 2021
- Accept Date: 02 July 2021