On an M/G/1 queue in random environment with Min(N, V) policy

Li, Jianjun; Liu, Liwei

RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 1, pp. 61-77.

envIn this paper, we analyze an $M∕G∕1$ queue operating in multi-phase random environment with $Min(N,V)$ vacation policy. In operative phase $i,i=1,2,...,n$ customers are served according to the discipline of First Come First Served (FCFS). When the system becomes empty, the server takes a vacation under the $\mathrm{𝑀𝑖𝑛}(𝑁,𝑉)$ policy, causing the system to move to vacation phase $0$. At the end of a vacation, if the server finds no customer waiting, another vacation begins. Otherwise, the system jumps from the phase $0$ to some operative phase $i$ with probability $q_i,i=1,2,...,n$. And whenever the number of the waiting customers in the system reaches $N$, the server interrupts its vacation immediately and the system jumps from the phase $0$ to some operative phase $i$ with probability $q_i,i=1,2,...,n$, too. Using the method of supplementary variable, we derive the distribution for the stationary system size at arbitrary epoch. We also obtain mean system size, the results of the cycle analysis and the sojourn time distribution. In addition, some special cases and numerical examples are presented.

Classification : 60K25, 68M20
Mots clés : M∕G∕1 queue, vacation, sojourn time, Min(N, V) policy, random environment, queueing theory

[1] M. Baykal-Gursoy, W. Xiao and K. Ozbay, Modeling traffic flow interruped by incidents. Eur. J. Oper. Res. 195 (2009) 127–138. | DOI | MR | Zbl

[2] C.E. Bell, Characterization and computation of optimal policies for operating an M/G/1 queueing system with removable server. Oper. Res. 19 (1971) 208–218. | DOI | Zbl

[3] J. Blom, O. Kella and M. Mandjes, Markov-modulated infinite-server queues with general service times. Queue. Syst. 76 (2014) 403–424. | DOI | MR | Zbl

[4] O. Boxma and I. Kurkova, The M/G/1 queue with two service speeds. Adv. Appl. Probab. 33 (2001) 520–540. | MR | Zbl

[5] J. Cordeiro and J. Kharoufeh, The unreliable M/M/1 retrial queue in a random environment. Stoch. Model. 28 (2012) 29–48. | DOI | MR | Zbl

[6] B. Doshi, Queueing systems with vacations – a survey. Queue. Syst. 1 (1986) 29–66. | DOI | MR | Zbl

[7] G. Falin, The M/M/∞ queue in random environment. Queue. Syst. 58 (2008) 65–76. | DOI | MR | Zbl

[8] S.W. Fuhrmann and R.B. Cooper, Stochastic decomposition in the M/G/1 queue with generalized vacations. Oper. Res. 33 (1985) 1117–1129. | DOI | MR | Zbl

[9] D.P. Heyman, Optimal operating policies for M/G/1 queueing system. Oper. Res. 16 (1968) 362–382. | DOI | Zbl

[10] L. Huang and T. Lee, Generalized Pollaczek-Khinchin formula for Markov channels. IEEE Trans. Commun. 61 (2013) 3530–3540. | DOI

[11] C. Jing, C. Ying and N. Tian, Analysis of the M/G/1 queueing systme with Min(N, V)-policy. Oper. Res. Manag. Sci. (Chin.) 15 (2006) 53–58.

[12] J.-C. Ke,The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns. Comput. Ind. Eng. 44 (2003) 567–579. | DOI

[13] J.C. Ke and K.H. Wang, A recursive method for the N policy G/M/1 queueing system with fnite capacity. Eur. J. Oper. Res. 142 (2002) 577–594. | DOI | MR | Zbl

[14] J.C. Ke, C.H. Wu and Z.G. Zhang, Recent developments in vacation queueing models: a short survey. Int. J. Oper. Res. 7 (2010) 3–8.

[15] O. Kella, The threshold policy in the M/G/1 queue with server vacations. Nav. Res. Logist. 36 (1989) 111–123. | DOI | MR | Zbl

[16] B. Kim and J. Kim, A single server queue with Markov modulated service rates and impatient customers. Perform. Eval. 83–84 (2015) 1–15. | DOI

[17] 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. | DOI | Zbl

[18] J. Li and L. Liu, Performance analysis of a complex queueing system with vacations in random environment. Adv. Mech. Eng. 9 (2017) 1–9.

[19] J. Li, L. Liu and T. Jiang, Analysis of the M/G/1 queue with vacations and multiple phases of operation. Math. Methods Oper. Res. 28 (2018) 51–72. | DOI | MR | Zbl

[20] Z. Liu and S. Yu, The M/M/C queueing system in a random environment. J. Math. Anal. Appl. 436 (2016) 556–567. | DOI | MR | Zbl

[21] M.F. Neuts, A queue subject to extraneous phase changes. Adv. Appl. Probab. 3 (1971) 78–119. | DOI | MR | Zbl

[22] B. Sengupta, A queue with service interruptions in an alternating random environment. Oper. Res. 38 (1990) 308–318. | DOI | MR | Zbl

[23] J.G. Shanthikumar, On stochastic decomposition in M/G/1 tpye queues with generilized server vacations. Oper. Res. 36 (1988) 566–569. | DOI | MR | Zbl

[24] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation Vol. 1. North-Holland, Amsterdam (1991). | MR | Zbl

[25] N. Tian and Z. Zhang, Vacation Queueing Models–Theory and Applications. Springer-Verlag, New York (2006). | DOI | MR | Zbl

[26] H.C. Tijms, Stochastic Modelling and Analysis. Wiley, New York (1986). | MR

[27] K.H. Wang and J.C. Ke, A recursive method to the optimal control of an M/G/1 queueing system with finite capacity and infinite capacity. Appl. Math. Model. 24 (2000) 899–914. | DOI | Zbl

[28] K.H. Wang, T.Y. Wang and W.L. Pearn, Optimal control of the N policy M/G/1 queueing system with server breakdowns and general startup times. Appl. Math. Model. 31 (2007) 2199–2212. | DOI | Zbl

[29] W. Wu, Y. Tang and M. Yu, Analysis of an M/G/1 queue with N-policy, single vacation, unreliable service station and replaceable repair facility. Opsearch 52 (2015) 670–691. | DOI | MR | Zbl

[30] M. Yadin and P. Naor, Queueing systems with a removable service station. Opl. Res. Q. 14 (1963) 393–405. | DOI

[31] U. Yechiali and P. Naor, Queueing problems with heterogeneous arrivals and service. Oper. Res. 19 (1971) 722–734. | DOI | MR | Zbl

[32] Z.G. Zhang and N. Tian, The N threshold policy for the GI/M/1 queue. Oper. Res. Lett. 32 (2004) 77–84. | DOI | MR | Zbl