American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

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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. Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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