International Journal of Nonlinear Analysis and Applications
On first excess level analysis of hysteretic bilevel control queue with multiple vacations
Document Type : Research Paper
Authors
Department of Mathematics, College of Education for Pure Sciences, University of Babylon, Iraq
Abstract
In this article, we consider queueing model with bilevel hysteretic control and multiple vacations. This system's mechanism depends on the queue size and the facility where the server goes to the series of vacation trips when the line is empty and returns but waiting in the system when the number of units is more than level \(M\). In this case, the server doesn't start a new busy period unless this size is more than another level \(N > M\). Furthermore, we employ N-policy and first excess level analysis to derive the probability generating function of queue size. Additionally, we assume that the vacation times are exponentially distributed random variables, and arrival batches are type 1 geometrically distributed random variables.
Keywords
ces
[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] J. R. Artalejo and A. Economou, Markovian controllable queueing systems with hysteretic policies: busy period
and waiting time analysis, Method. Comput. Appl. Prob. 7 (2005) 353–378.
[5] Y. Baba, On the MX/G/1 queue with vacation time, Oper. Res. Lett. 5 (1986) 93–98.
[6] 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.
[7] R. Bekker, Queues with L´evy input and hysteretic control, Queueing Systems, 63 (2009) 281-299.
[8] K.C. Chae and H.W. Lee, MX/G/1 vacation models with N-policy: heuristic interpretation of the mean waiting
time, J. Oper. Res. Soc. 46 (1995) 258–264.
[9] S.R. Chakravarthy and A. Rumyantsev, Analysis of a queueing model with Batch Markovian arrival process and
general distribution for group clearance, Method. Comput. Appl. Prob. (2020) 1–29.
[10] C. W. Chan, M. Armony and N. Bambos, Maximum weight matching with hysteresis in overloaded queues with
setups, Que. Syst. 82 (2016) 315–351.
[11] G. Choudhury, A batch arrival queue with a vacation time under single vacation policy, Comput. Oper. Res. 29
(2002) 1941–1955.
[12] G. Choudhury, An MX/G/1 queueing system with a setup period and a vacation period, Que. Syst. 36 (2000)
23–38.
[13] 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. Series B (2000) 303–316.
[14] V. Deart, A. Maslennikov and Y. Gaidamaka, A hysteretic model of queuing system with fuzzy logic active queue
management, Proc. 15th Conf. Open Innov. Assoc. (2014) 32–38.
[15] J.H. Dshalalow, A note on D-policy bulk queueing systems, J. Appl. Prob. 38 (2001) 280[16] J.H. Dshalalow and E.E. Dikong, On generalized hysteretic control queues with modulated input and state dependent service, Stoch. Anal. Appl. 17 (1999) 937–961.
[17] J.H. Dshalalow, S. Kim and L. Tadj, Hybrid queueing systems with hysteretic bilevel control policies, Nonlinear
Anal. Theory Meth. Appl. 65 (2006) 2153–2168.
[18] J.H. Dshalalow and A. Merie, Fluctuation analysis in queues with several operational modes and priority customers, 26(2018) 309-333.
[19] J. H. Dshalalow, A. Merie and R.T. White, Fluctuation analysis in parallel queues with hysteretic control, Method.
Comput. Appl. Prob. 22 (2020) 295–327.
[20] J.H. Dshalalow, On applications of eXcess level processes to (N, D)-policy bulk queueing systems, J. Appl. Math.
Stoch. Anal. 9 (1996) 551–562.
[21] J. H. Dshalalow, Queues with hysteretic control by vacation and post-vacation periods, Que. Syst. 29 (1998)
231–268.
[22] J. Dshalalow, Queueing processes in bulk systems under the D-policy, J. Appl. Prob. 34 (1998) 976–989.
[23] 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.
[24] J.H. Dshalalow and R.T. White, Current trends in random walks on random lattices, Math. 9 (2021) 11–48.
[25] J.H. Dshalalow and J. Yellen, Bulk input queues with quorum and multiple vacations, Math. Prob. Engin. 2 (1996)
95–106.
[26] R. F. Gebhard, A queuing process with bilevel hysteretic service-rate control, Naval Res. Log. Quart. 14 (1967)
55–67.
[27] U.C. Gupta, A.D. Banik and S.S. Pathak, Complete analysis of MAP/G/1/N queue with single (multiple) vacation
(s) under limited service discipline, J. Appl. Math. Stoch. Anal. 2005 (2005) 353–373.
[28] M. Kadi, A.A. Bouchentouf and L. Yahiaoui, On a multiserver queueing system with customers’ impatience until
the end of service under single and multiple vacation policies, Appl. Appl. Math. 15 (2020).
[29] S. Kalita, G. Choudhury, S. Kalita and G. Choudhury, Some aspects of a batch arrival Poisson queue with
N-policy, Stoch. Model. Appl. 5 (2002) 21–32
[30] J.C. Ke, An M/G/1 queue under hysteretic vacation policy with an early startup and un-reliable server, Math.
Meth. Oper. Res. 63 (2006) 357.
[31] M.Y. Kitaev and R.F. Serfozo, M/M/1 queues with switching costs and hysteretic optimal control, Oper. Res. 47
(1999) 310–312.
[32] 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.
[33] S.S. Lee, H.W. Lee, S.H. Yoon and K.C. Chae, Batch arrival queue with N-policy and single vacation, Comput.
Oper. Res. 22 (1995) 173–189.
[34] 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.
[35] H.S. Lee and M.M. Srinivasan, Control policies for the MX/G/1 queueing system, Manag. Sci. 35 (1989) 708–721.
[36] F.V. Lu and R.F. Serfozo, M/M/1 queueing decision processes with monotone hysteretic optimal policies, Oper.
Res. 32 (1984) 1116–1132.
[37] 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.
[38] J. Medhi, Single server queueing system with Poisson input: a review of some recent developments, Adv. Combin.
Meth. Appl. Prob. Stat. (1997) 317-338.
[39] A.V. Pechinkin, R.R. Razumchik and I.S. Zaryadov, First passage times in M2[X]| G|1|R queue with hysteretic
overload control policy, AIP Conf. Proc. 1738 (2016) 220007.
[40] E. Rosenberg and U. Yechiali, The MX/G/1 queue with single and multiple vacations under the LIFO service
regime, Oper. Res. Lett. 14 (1993) 171–179.
[41] C. Shekhar, A. Gupta, N. Kumar, A. Kumar and S. Varshney, Transient Solution of Multiple Vacation Queue
with Discouragement and Feedback, Scientia Iranica, 2020.
[42] L. Tadj, L. Benkherouf and L. Aggoun, A hysteretic queueing system with random server capacity, Comput.
Math. Appl. 38 (1999) 51–61.
[43] L. Tadj and J.C. Ke, A hysteretic bulk quorum queue with a choice of service and optional re-service, Qual. Tech.
Quant. Manag. 5 (2008) 161–178.
[44] L. Tadj and J.C. Ke, Control policy of a hysteretic queueing system, Math. Meth. Oper. Res. 57 (2003) 367–376.
[45] J. Teghem, Control of the service process in a queueing system, Euro. J. Oper. Res. 23 (1986) 141–158.
[46] J. Teghem, On a decomposition result for a class of vacation queueing systems, J. Appl. Prob. 27 (1990) 227–231.
[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] J. R. Artalejo and A. Economou, Markovian controllable queueing systems with hysteretic policies: busy period
and waiting time analysis, Method. Comput. Appl. Prob. 7 (2005) 353–378.
[5] Y. Baba, On the MX/G/1 queue with vacation time, Oper. Res. Lett. 5 (1986) 93–98.
[6] 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.
[7] R. Bekker, Queues with L´evy input and hysteretic control, Queueing Systems, 63 (2009) 281-299.
[8] K.C. Chae and H.W. Lee, MX/G/1 vacation models with N-policy: heuristic interpretation of the mean waiting
time, J. Oper. Res. Soc. 46 (1995) 258–264.
[9] S.R. Chakravarthy and A. Rumyantsev, Analysis of a queueing model with Batch Markovian arrival process and
general distribution for group clearance, Method. Comput. Appl. Prob. (2020) 1–29.
[10] C. W. Chan, M. Armony and N. Bambos, Maximum weight matching with hysteresis in overloaded queues with
setups, Que. Syst. 82 (2016) 315–351.
[11] G. Choudhury, A batch arrival queue with a vacation time under single vacation policy, Comput. Oper. Res. 29
(2002) 1941–1955.
[12] G. Choudhury, An MX/G/1 queueing system with a setup period and a vacation period, Que. Syst. 36 (2000)
23–38.
[13] 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. Series B (2000) 303–316.
[14] V. Deart, A. Maslennikov and Y. Gaidamaka, A hysteretic model of queuing system with fuzzy logic active queue
management, Proc. 15th Conf. Open Innov. Assoc. (2014) 32–38.
[15] J.H. Dshalalow, A note on D-policy bulk queueing systems, J. Appl. Prob. 38 (2001) 280[16] J.H. Dshalalow and E.E. Dikong, On generalized hysteretic control queues with modulated input and state dependent service, Stoch. Anal. Appl. 17 (1999) 937–961.
[17] J.H. Dshalalow, S. Kim and L. Tadj, Hybrid queueing systems with hysteretic bilevel control policies, Nonlinear
Anal. Theory Meth. Appl. 65 (2006) 2153–2168.
[18] J.H. Dshalalow and A. Merie, Fluctuation analysis in queues with several operational modes and priority customers, 26(2018) 309-333.
[19] J. H. Dshalalow, A. Merie and R.T. White, Fluctuation analysis in parallel queues with hysteretic control, Method.
Comput. Appl. Prob. 22 (2020) 295–327.
[20] J.H. Dshalalow, On applications of eXcess level processes to (N, D)-policy bulk queueing systems, J. Appl. Math.
Stoch. Anal. 9 (1996) 551–562.
[21] J. H. Dshalalow, Queues with hysteretic control by vacation and post-vacation periods, Que. Syst. 29 (1998)
231–268.
[22] J. Dshalalow, Queueing processes in bulk systems under the D-policy, J. Appl. Prob. 34 (1998) 976–989.
[23] 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.
[24] J.H. Dshalalow and R.T. White, Current trends in random walks on random lattices, Math. 9 (2021) 11–48.
[25] J.H. Dshalalow and J. Yellen, Bulk input queues with quorum and multiple vacations, Math. Prob. Engin. 2 (1996)
95–106.
[26] R. F. Gebhard, A queuing process with bilevel hysteretic service-rate control, Naval Res. Log. Quart. 14 (1967)
55–67.
[27] U.C. Gupta, A.D. Banik and S.S. Pathak, Complete analysis of MAP/G/1/N queue with single (multiple) vacation
(s) under limited service discipline, J. Appl. Math. Stoch. Anal. 2005 (2005) 353–373.
[28] M. Kadi, A.A. Bouchentouf and L. Yahiaoui, On a multiserver queueing system with customers’ impatience until
the end of service under single and multiple vacation policies, Appl. Appl. Math. 15 (2020).
[29] S. Kalita, G. Choudhury, S. Kalita and G. Choudhury, Some aspects of a batch arrival Poisson queue with
N-policy, Stoch. Model. Appl. 5 (2002) 21–32
[30] J.C. Ke, An M/G/1 queue under hysteretic vacation policy with an early startup and un-reliable server, Math.
Meth. Oper. Res. 63 (2006) 357.
[31] M.Y. Kitaev and R.F. Serfozo, M/M/1 queues with switching costs and hysteretic optimal control, Oper. Res. 47
(1999) 310–312.
[32] 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.
[33] S.S. Lee, H.W. Lee, S.H. Yoon and K.C. Chae, Batch arrival queue with N-policy and single vacation, Comput.
Oper. Res. 22 (1995) 173–189.
[34] 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.
[35] H.S. Lee and M.M. Srinivasan, Control policies for the MX/G/1 queueing system, Manag. Sci. 35 (1989) 708–721.
[36] F.V. Lu and R.F. Serfozo, M/M/1 queueing decision processes with monotone hysteretic optimal policies, Oper.
Res. 32 (1984) 1116–1132.
[37] 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.
[38] J. Medhi, Single server queueing system with Poisson input: a review of some recent developments, Adv. Combin.
Meth. Appl. Prob. Stat. (1997) 317-338.
[39] A.V. Pechinkin, R.R. Razumchik and I.S. Zaryadov, First passage times in M2[X]| G|1|R queue with hysteretic
overload control policy, AIP Conf. Proc. 1738 (2016) 220007.
[40] E. Rosenberg and U. Yechiali, The MX/G/1 queue with single and multiple vacations under the LIFO service
regime, Oper. Res. Lett. 14 (1993) 171–179.
[41] C. Shekhar, A. Gupta, N. Kumar, A. Kumar and S. Varshney, Transient Solution of Multiple Vacation Queue
with Discouragement and Feedback, Scientia Iranica, 2020.
[42] L. Tadj, L. Benkherouf and L. Aggoun, A hysteretic queueing system with random server capacity, Comput.
Math. Appl. 38 (1999) 51–61.
[43] L. Tadj and J.C. Ke, A hysteretic bulk quorum queue with a choice of service and optional re-service, Qual. Tech.
Quant. Manag. 5 (2008) 161–178.
[44] L. Tadj and J.C. Ke, Control policy of a hysteretic queueing system, Math. Meth. Oper. Res. 57 (2003) 367–376.
[45] J. Teghem, Control of the service process in a queueing system, Euro. J. Oper. Res. 23 (1986) 141–158.
[46] J. Teghem, On a decomposition result for a class of vacation queueing systems, J. Appl. Prob. 27 (1990) 227–231.
)
Volume 12, Issue 2
November 2021Pages 2131-2144
- Receive Date: 11 March 2021
- Revise Date: 18 May 2021
- Accept Date: 04 July 2021