American Institute of Mathematical Sciences

Advanced
July  2015, 11(3): 779-806. doi: 10.3934/jimo.2015.11.779

On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations

Shan Gao 1, and  Jinting Wang 2,

1. 

Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China
2. 

Department of Mathematics, Beijing Jiaotong University, 100044 Beijing

Received  September 2013 Revised  May 2014 Published  October 2014

This paper considers a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations, where upon arrival, a negative customer removes one positive (ordinary) customer in service if any is present and disappears immediately; otherwise, it has no effect on the system if the system is empty. As soon as the system becomes empty, the server immediately takes a working vacation. If there are no customers in the system at the end of the working vacation, the server takes another working vacation with probability $p$ or remains dormant in the system with probability $1-p$. Otherwise, the server starts to serve the customers with the normal service rate immediately if there are some customers at the end of a working vacation. This pattern does not terminate until the server has taken $J$ successive working vacations. Steady-state system length distributions at various epochs such as, pre-arrival, arbitrary and outside observer's observation epochs have been obtained. Based on the various system length distributions, we also give some important performance measures including blocking probabilities, mean queue length, probability mass function of waiting time and other performance measures along with some numerical examples. Then, we use the parabolic method to search the optimum value of the normal service rate under a established cost function.
Keywords: performance evaluation, randomized vacations, working vacations., Discrete time queues.
Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22, 68M2.
Citation: Shan Gao, Jinting Wang. On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations. Journal of Industrial and Management Optimization, 2015, 11 (3) : 779-806. doi: 10.3934/jimo.2015.11.779
References:
[1]

I. Atencia and P. Moreno, The discrete-time Geo/Geo/1 queue with negative customers and disasters, Comput. Oper. Res., 31 (2004), 1537-1548. doi: 10.1016/S0305-0548(03)00107-2.

[2]

Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Oper. Res. Lett., 33 (2005), 201-209. doi: 10.1016/j.orl.2004.05.006.

[3]

A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple working vacations-analytic analysis and computation, Appl. Math. Model., 31 (2007), 1701-1710. doi: 10.1016/j.apm.2006.05.010.

[4]

K. C. Chae, D. E. Lim and W. S. Yang, The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation, Perform. Eval., 66 (2009), 356-367. doi: 10.1016/j.peva.2009.01.005.

[5]

K. C. Chae, H. M. Park and W. S. Yang, A GI/Geo/1 queue with negative and positive customers, Appl. Math. Model., 34 (2010), 1662-1671. doi: 10.1016/j.apm.2009.09.015.

[6]

R. Chakka and P. G. Harrison, A Markov modulated multi-server queue with negative customers-the MM CPP/GE/c/L G-queue, Acta Inform., 37 (2001), 881-919. doi: 10.1007/PL00013307.

[7]

I. Dimitriou, A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations, Appl. Math. Model., 37 (2013), 1295-1309. doi: 10.1016/j.apm.2012.04.011.

[8]

T. V. Do, Bibliography on G-networks, negative customers and applications, Math. Comput. Model., 53 (2011), 205-212.

[9]

S. Gao and Z. Liu, An M/G/1 queue with single working vacation and vacation interruption under bernoulli schedule, Appl. Math. Model., 37 (2013), 1564-1579. doi: 10.1016/j.apm.2012.04.045.

[10]

S. Gao, J. Wang and D. Zhang, Discrete-time $GI^X$/Geo/1/N queue with negative customers and multiple working vacations, J. Korean. Stat. Soc., 42 (2013), 515-528.

[11]

E. Gelenbe, Random neural networks with negative and positive signals and product form solution, Neural Comput., 1 (1989), 502-510. doi: 10.1162/neco.1989.1.4.502.

[12]

E. Gelenbe, Product-form queueing networks with negative and positive customers, J. Appl. Probab., 28 (1991), 656-663. doi: 10.2307/3214499.

[13]

V. Goswami and G. B. Mund, Analysis of discrete-time batch service renewal input queue with multiple working vacations, Comput. Ind. Eng., 61 (2011), 629-636. doi: 10.1016/j.cie.2011.04.018.

[14]

P. G. Harrison, N. M. Patel and E. Pitel, Reliability modelling using G-queues, Eur. J. Oper. Res., 126 (2000), 273-287. doi: 10.1016/S0377-2217(99)00478-6.

[15]

J.-H. Li and N. Tian, The discrete-time GI/Geo/1 queue with working vacations and vacation interruption, Appl. Math. Comput., 185 (2007), 1-10. doi: 10.1016/j.amc.2006.07.008.

[16]

Q.-L. Li and Y. Q. Zhao, A MAP/G/1 queue with negative customers, Queueing Syst., 47 (2004), 5-43. doi: 10.1023/B:QUES.0000032798.65858.19.

[17]

W.-Y. Liu, X.-L. Xu and N.-S. Tian, Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett., 35 (2007), 595-600. doi: 10.1016/j.orl.2006.12.007.

[18]

L. R. Ronald, Optimization in Operations Research, New Jersey, 1997.

[19]

L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (m/m/1/wv), Perform. Eval., 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3.

[20]

J. Wang, B. Liu and J. Li, Transient analysis of an M/G/1 retrial queue subject to disasters and server failures, European Journal of Operational Research, 189 (2008), 1118-1132. doi: 10.1016/j.ejor.2007.04.054.

[21]

J. Wang and P. Zhang, A discrete-time retrial queue with negative customers and unreliable server, Comput. Ind. Eng., 56 (2009), 1216-1222. doi: 10.1016/j.cie.2008.07.010.

[22]

J. Wang, Y. Huang and Z. Dai, A discrete-time on-off source queueing system with negative customers, Comput. Ind. Eng., 61 (2011), 1226-1232. doi: 10.1016/j.cie.2011.07.013.

[23]

D.-A. Wu and H. Takagi, M/G/1 queue with multiple working vacations, Perform. Eval., 63 (2006), 654-681. doi: 10.1016/j.peva.2005.05.005.

[24]

J. Wu, Z. Liu and Y. Peng, On the BMAP/G/1 G-queues with second optional service and multiple vacations, Appl. Math. Model., 33 (2009), 4314-4325. doi: 10.1016/j.apm.2009.03.013.

[25]

M. Yu, Y. Tang and Y. Fu, Steady state analysis and computation of the $GI^{[x]}$/$M^b$/1/L queue with multiple working vacations and partial batch rejection, Comput. Ind. Eng., 56 (2009), 1243-1253.

[26]

M. Yu, Y. Tang, Y. Fu and L. Pan, GI/Geom/1/N/MWV queue with changeover time and searching for the optimum service rate in working vacation period, J. Comput. Appl. Math., 235 (2011), 2170-2184. doi: 10.1016/j.cam.2010.10.013.

[27]

M. Zhang and Z. Hou, Steady state analysis of the GI/M/1/N queue with a variant of multiple working vacations, Comput. Ind. Eng., 61 (2011), 1296-1301. doi: 10.1016/j.cie.2011.08.002.

show all references

References:
[1]

I. Atencia and P. Moreno, The discrete-time Geo/Geo/1 queue with negative customers and disasters, Comput. Oper. Res., 31 (2004), 1537-1548. doi: 10.1016/S0305-0548(03)00107-2.

[2]

Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Oper. Res. Lett., 33 (2005), 201-209. doi: 10.1016/j.orl.2004.05.006.

[3]

A. D. Banik, U. C. Gupta and S. S. Pathak, On the GI/M/1/N queue with multiple working vacations-analytic analysis and computation, Appl. Math. Model., 31 (2007), 1701-1710. doi: 10.1016/j.apm.2006.05.010.

[4]

K. C. Chae, D. E. Lim and W. S. Yang, The GI/M/1 queue and the GI/Geo/1 queue both with single working vacation, Perform. Eval., 66 (2009), 356-367. doi: 10.1016/j.peva.2009.01.005.

[5]

K. C. Chae, H. M. Park and W. S. Yang, A GI/Geo/1 queue with negative and positive customers, Appl. Math. Model., 34 (2010), 1662-1671. doi: 10.1016/j.apm.2009.09.015.

[6]

R. Chakka and P. G. Harrison, A Markov modulated multi-server queue with negative customers-the MM CPP/GE/c/L G-queue, Acta Inform., 37 (2001), 881-919. doi: 10.1007/PL00013307.

[7]

I. Dimitriou, A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations, Appl. Math. Model., 37 (2013), 1295-1309. doi: 10.1016/j.apm.2012.04.011.

[8]

T. V. Do, Bibliography on G-networks, negative customers and applications, Math. Comput. Model., 53 (2011), 205-212.

[9]

S. Gao and Z. Liu, An M/G/1 queue with single working vacation and vacation interruption under bernoulli schedule, Appl. Math. Model., 37 (2013), 1564-1579. doi: 10.1016/j.apm.2012.04.045.

[10]

S. Gao, J. Wang and D. Zhang, Discrete-time $GI^X$/Geo/1/N queue with negative customers and multiple working vacations, J. Korean. Stat. Soc., 42 (2013), 515-528.

[11]

E. Gelenbe, Random neural networks with negative and positive signals and product form solution, Neural Comput., 1 (1989), 502-510. doi: 10.1162/neco.1989.1.4.502.

[12]

E. Gelenbe, Product-form queueing networks with negative and positive customers, J. Appl. Probab., 28 (1991), 656-663. doi: 10.2307/3214499.

[13]

V. Goswami and G. B. Mund, Analysis of discrete-time batch service renewal input queue with multiple working vacations, Comput. Ind. Eng., 61 (2011), 629-636. doi: 10.1016/j.cie.2011.04.018.

[14]

P. G. Harrison, N. M. Patel and E. Pitel, Reliability modelling using G-queues, Eur. J. Oper. Res., 126 (2000), 273-287. doi: 10.1016/S0377-2217(99)00478-6.

[15]

J.-H. Li and N. Tian, The discrete-time GI/Geo/1 queue with working vacations and vacation interruption, Appl. Math. Comput., 185 (2007), 1-10. doi: 10.1016/j.amc.2006.07.008.

[16]

Q.-L. Li and Y. Q. Zhao, A MAP/G/1 queue with negative customers, Queueing Syst., 47 (2004), 5-43. doi: 10.1023/B:QUES.0000032798.65858.19.

[17]

W.-Y. Liu, X.-L. Xu and N.-S. Tian, Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett., 35 (2007), 595-600. doi: 10.1016/j.orl.2006.12.007.

[18]

L. R. Ronald, Optimization in Operations Research, New Jersey, 1997.

[19]

L. D. Servi and S. G. Finn, M/M/1 queues with working vacations (m/m/1/wv), Perform. Eval., 50 (2002), 41-52. doi: 10.1016/S0166-5316(02)00057-3.

[20]

J. Wang, B. Liu and J. Li, Transient analysis of an M/G/1 retrial queue subject to disasters and server failures, European Journal of Operational Research, 189 (2008), 1118-1132. doi: 10.1016/j.ejor.2007.04.054.

[21]

J. Wang and P. Zhang, A discrete-time retrial queue with negative customers and unreliable server, Comput. Ind. Eng., 56 (2009), 1216-1222. doi: 10.1016/j.cie.2008.07.010.

[22]

J. Wang, Y. Huang and Z. Dai, A discrete-time on-off source queueing system with negative customers, Comput. Ind. Eng., 61 (2011), 1226-1232. doi: 10.1016/j.cie.2011.07.013.

[23]

D.-A. Wu and H. Takagi, M/G/1 queue with multiple working vacations, Perform. Eval., 63 (2006), 654-681. doi: 10.1016/j.peva.2005.05.005.

[24]

J. Wu, Z. Liu and Y. Peng, On the BMAP/G/1 G-queues with second optional service and multiple vacations, Appl. Math. Model., 33 (2009), 4314-4325. doi: 10.1016/j.apm.2009.03.013.

[25]

M. Yu, Y. Tang and Y. Fu, Steady state analysis and computation of the $GI^{[x]}$/$M^b$/1/L queue with multiple working vacations and partial batch rejection, Comput. Ind. Eng., 56 (2009), 1243-1253.

[26]

M. Yu, Y. Tang, Y. Fu and L. Pan, GI/Geom/1/N/MWV queue with changeover time and searching for the optimum service rate in working vacation period, J. Comput. Appl. Math., 235 (2011), 2170-2184. doi: 10.1016/j.cam.2010.10.013.

[27]

M. Zhang and Z. Hou, Steady state analysis of the GI/M/1/N queue with a variant of multiple working vacations, Comput. Ind. Eng., 61 (2011), 1296-1301. doi: 10.1016/j.cie.2011.08.002.

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