A gated service single vacation M/G/1 queue with setup and closedown periods, and different customer arrival rates, is studied in this paper. The probability generating function of the number of systems for customers who are at the initial moment of service period is analyzed by using a total probability theorem, and the stability condition of the system is obtained. The stationary distribution of the queue length is solved by the regeneration cycle method. The stochastic decomposition of queue length in the steady state is calculated, and the service cycle is obtained. Moreover, classified discussions are established in order to solve the steady-state distribution for the waiting time. The variation of system performance indicators with parameters is analyzed by performing numerical experiments.
Ying SUN
,
Zhanyou MA*
,
Tongyu XU
. A GATED SERVICE SINGLE VACATION M/G/1 QUEUE SYSTEM WITH SETUP AND CLOSEDOWN TIMES AND DIFFERENT CUSTOMER ARRIVAL RATES[J]. Acta mathematica scientia, Series B, 2026
, 46(1)
: 407
-426
.
DOI: 10.1007/s10473-026-0122-5
[1] Cheng Y S, Zhu Y J. A simplified algorithm for stochastic decomposition of queueing systems for M/G/1 non-exhaustive service vacations. Qual Technol Quant M, 2004, $\textbf{13}$(2): 8-12
[2] Cheng Y S, Zhu Y J. Analysis of the equilibrium conditions of the M/G/1 non-exhaustive service vacation queueing system. Operations Research Transactions, 2005, $\textbf{24}$(3): 83-88
[3] Choudhury G, Kalita C R. An M/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server's breakdown and delayed repair. Qual Technol Quant M, 2018, $\textbf{15}$(5): 622-654
[4] Gao S, Liu Z M. An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Appl Math Model, 2013, $\textbf{37}$(3): 1564-1579
[5] He L Q, Tian R L, Han Y N. Optimal joining strategies in a repairable retrial queue with reserved time and $N$-policy. Oper Res-Ger, 2014, $\textbf{24}$(1): Art 3
[6] Hu R, Tang Y H.Transient queue length distributions for M/G/1 queueing with delayed vacation and Min(${{\textit{N}}}$, ${{\textit{V}}}$)-policy controls (in Chinese). Mathematics in Practice and Theory, 2019, $\textbf{49}$(13): 145-155
[7] Jain M, Kumar A. Unreliable server ${\rm{M}}^{[{X}]}$/G/1 retrial feedback queue with balking, working vacation and vacation interruption. Proceeding of the National Academy of Sciences, India Section A: Physical Sciences, 2023, $\textbf{93}$(1): 57-73
[8] Jiang T. Analysis of a discrete-time Geo/G/1 queue in a multi-phase service environment with disasters. Journal of System Science Information, 2018, $\textbf{6}$(13): 145-155
[9] Kumar M S, Dadlani A, Kim K. Performance analysis of an unreliable M/G/1 retrial queue with two-way communication. Oper Res-Ger, 2020, $\textbf{20}$: 2267-2280
[10] Lee H W, Won J S, Se W L. Analysis of the M/G/1 queueing system under the triadic(${{\textit{N}}}$, ${{\textit{D}}}$, ${{\textit{J}}}$)-policies. Qual Technol Quant M, 2011, $\textbf{8}$(3): 333-357
[11] Levy Y, Yechiali U. Utilization of idle time in an M/G/1 queueing system. Manage Sci, 1975, $\textbf{22}$(2): 202-211
[12] Li J J, Liu L W, Jiang T. Analysis of an M/G/1 queue with vacations and multiple phases of operation. Math Method Oper Res, 2018, $\textbf{87}$(1): 51-72
[13] Li T, Zhang L Y, Gao S. An M/G/1 retrial queue with balking customers and Bernoulli working vacation interruption. Qual Technol Quant M, 2019, $\textbf{16}$(9): 511-530
[14] Lim D E, Lee D H, Yang W S, Chae K C. Analysis of the GI/Geo/1 queue with $N$-policy. Appl Math Model, 2013, $\textbf{37}$(7): 4643-4652
[15] Luo L, Tang Y H. Optimal design and optimal control strategy for the capacity of M/G/1 queueing systems with $p$-entry rules and Min(${{\textit{N}}}$, ${{\textit{D}}}$, ${{\textit{V}}}$)-policies. Acta Math Sci, 2019, $\textbf{39B}$(5): 1228-1246
[16] Ma Z Y.Steady State Theory of M/G/1 Queueing Systems with Multiple Adaptive Vacation[PhD thesis]. Qinhuangdao: Yanshan University, 2006
[17] Mao B W, Wang F W, Tian N S. Fluid model driven by an M/G/1 queue with multiple exponential vacations. Appl Math and Comput, 2011, ${\bf 218}$ (8): 4041-4048
[18] Qi J, Yu J, Jin S. Nash equilibrium and social optimization of transactions in blockchain system based on discrete-time queue. IEEE Access, 2020, $\textbf{8}$: 2314-2329
[19] Qin X P, Tang Y H.Analysis of an M/G/1 queueing system with setup time and multiple adaptive vacation under Min(${{\textit{N}}}$, ${{\textit{V}}}$)-policy control (in Chinese). Mathematics in Practice and Theory, 2021, $\textbf{51}$(17): 130-141
[20] Sun K, Wang J T. Game-theoretic analysis of the single vacation queue with negative customers. Qual Technol Quant M, 2022, ${\bf 19}$(4): 403-427
[21] Tang B L, Tang Y H. Optimal control policies for two classes of M/G/1 single vacation queueing systems with ${{\textit{N}}}$-policy. Operations Research Transactions, 2021, $\textbf{25}$(4): 15-30
[22] Tang Y H, Liu M W. Queue length distribution of M/G/1 single vacation with $N$-policy. Mathematica Applicata, 2008, $\textbf{88}$(1): 21-27
[23] Tian N S, Zhang Z G.Vacation Queueing Models: Theory and Applications. New York: Springer Science Business Medis, 2006
[24] Tuan P D. Single server retrial queues with setup time. J Ind Manag Optim, 2017, $\textbf{13}$(3): 1329-1345
[25] Wang K H, Huanh K B. A maximum entropy approach for the $\langle$ ${{\textit{p}}}$, ${{\textit{N}}}$ $\rangle$-policy M/G/1 queue with a removable and unreliable server. Appl Math Model, 2009, $\textbf{33}$(4): 2024-2034
[26] Wang Z, Liu L, Shao Y F, et al. Equilibrium joining strategy in a batch transfer queuing system with gated policy. Methodol Comput Appl, 2022, $\textbf{22}$(1): 75-99
[27] Wu J B, Yin X L. An M/G/1 retrial G-queue with non-exhaustive random vacations and an unreliable server. Comput Math Appl, 2011, $\textbf{62}$(5): 2314-2329
