Advanced Search
Article Contents
Article Contents

Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

  • * Corresponding author: Jinbiao Wu

    * Corresponding author: Jinbiao Wu

The first author is supported by Provincial Natural Science Foundation of Hunan under Grant 2019JJ50677 and the Program of Hehua Excellent Young Talents of Changsha Normal University

Abstract Full Text(HTML) Figure(4) Related Papers Cited by
  • We consider an $ M^X/G/1 $ retrial queue with impatient customers subject to disastrous failures at which times all customers in the system are lost. When the server finishes serving a customer and finds the orbit empty, the server becomes dormant until $ N $ or more customers accumulate. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest join the orbit to repeat their request later. Otherwise, if the server is dormant or busy or down, all customers of the coming batch enter the orbit. When the server is under repair, customers in the orbit can become impatient after waiting a random amount of time and leave the system. By using the characteristic method for partial differential equations, the steady-state distributions of the server state and the number of customers in the orbit are obtained along with various performance measures. In addition, the reliability of the system is analyzed detailed. Finally, an application to telecommunication networks is provided and the effects of various parameters on the system performance are demonstrated numerically.

    Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The mean system size versus $ p $ with $ (\alpha, \delta, \theta) = (0.1, 3, 0.7) $

    Figure 2.  The mean system size versus $ \delta $ with $ (\alpha, p, \theta) = (0.1, 0.1, 0.7) $

    Figure 3.  The mean system size versus $ \theta $ with $ (\alpha, p, \delta) = (0.1, 0.1, 3) $

    Figure 4.  The mean system size versus $ \alpha $ with $ (\theta, p, \delta) = (0.7, 0.1, 3) $

  • [1] J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 51 (20100), 1071-1081.  doi: 10.1016/j.mcm.2009.12.011.
    [2] J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7 (1999), 187-211.  doi: 10.1007/BF02564721.
    [3] A. Aissani, On the M/G/1 queueing system with repeated orders and unreliable server, Journal of Technology, 6 (1988), 93-123. 
    [4] E. Altman and U. Yechiali, Analysis of customers'impatience in queues with server vacations, Queueing Systems, 52 (2006), 261-279.  doi: 10.1007/s11134-006-6134-x.
    [5] F. M. ChangT. H. Liu and J. C. Ke, On an unreliable-server retrial queue with customer feedback and impatience, Applied Mathematical Modelling, 55 (2018), 171-182.  doi: 10.1016/j.apm.2017.10.025.
    [6] J. Chang and J. Wang, Unreliable M/M/1/1 retrial queues with set-up time, Quality Technology & Quantitative Management, 15 (2018), 589-601.  doi: 10.1080/16843703.2017.1320459.
    [7] G. ChoudhuryJ. Ke and L. Tadj, The $N$-policy for an unreliable server with delaying repair and two phases of service, Journal of Computational and Applied Mathematics, 231 (2009), 349-364.  doi: 10.1016/j.cam.2009.02.101.
    [8] R. B. Cooper, Introduction to Queueing theory, North-Holland, New York, 1981.
    [9] G. I. Falin, A survey of retrial queues, Queueing Systems, 7 (1990), 127-167.  doi: 10.1007/BF01158472.
    [10] G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997.
    [11] G. Jain and K. Sigman, A Pollaczek-Khintchine formula for M/G/1 queues with disasters, Journal of Applied Probability, 33 (1996), 1191-1200.  doi: 10.2307/3214996.
    [12] V. G. Kulkarni and B. D. Choi, Retrial queues with server subject to breakdowns and repairs, Queueing Systems, 7 (1990), 191-208.  doi: 10.1007/BF01158474.
    [13] Q. L. LiY. DuG. Dai and M. Wang, On a doubly dynamically controlled supermarket model with impatient customers, Computers & Operations Research, 55 (2015), 76-87.  doi: 10.1016/j.cor.2014.10.004.
    [14] H. S. Lee and M. M. Srinivasan, Control policies for $M^X/G/1$ queueing system, Management Science, 35 (1989), 708-721.  doi: 10.1287/mnsc.35.6.708.
    [15] H. W. LeeS. S. Lee and K. C. Chae, Operating characteristics of $M^X/G/1$ queue with $N$-policy, Queueing Systems, 15 (1994), 387-399.  doi: 10.1007/BF01189247.
    [16] S. S. LeeH. W. LeeS. H. Yoon and K. C. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189. 
    [17] Z. LiuY. Chu and J. Wu, Heavy-traffic asymptotics of a priority polling system with threshold service policy, Computers & Operations Research, 65 (2016), 19-28.  doi: 10.1016/j.cor.2015.06.013.
    [18] G. C. Mytalas and M. A. Zazanis, An $M^X/G/1$ queueing system with disasters and repairs under a multiple adapted vacation policy, Naval Research Logistics, 62 (2015), 171-189.  doi: 10.1002/nav.21621.
    [19] T. Phung-Duc, Multiserver retrial queues with two types of nonpersistent customers, Asia-Pacific Journal of Operational Research, 31 (2014), 1440009, 27pp. doi: 10.1142/S0217595914400090.
    [20] N. Perel and U. Yechiali, Queues with slow servers and impatient customers, European Journal of Operational Research, 201 (2010), 247-258.  doi: 10.1016/j.ejor.2009.02.024.
    [21] A. G. Pakes, Some conditions for ergodicity and recurrence of Markov chains, Operations Research, 17 (1969), 1058-1061.  doi: 10.1287/opre.17.6.1058.
    [22] T. Phung-Duc, Retrial queueing models: A survey on theory and applications, preprint, arXiv: 1906.09560, 2019.
    [23] L. I. SennotP. A. Humblet and R. L. Tweedie, Mean drifts and the non-ergodicity of Markov chains, Operations Research, 31 (1983), 785-789.  doi: 10.1287/opre.31.4.783.
    [24] J. WangJ. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380.  doi: 10.1023/A:1010918926884.
    [25] J. WangB. 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.
    [26] J. Wu and Z. Lian, A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule, Computers & Industrial Engineering, 64 (2013), 84-93.  doi: 10.1016/j.cie.2012.08.015.
    [27] R. W. Wolff, Poisson arrival see time average, Operations Research, 30 (1982), 223-231.  doi: 10.1287/opre.30.2.223.
    [28] T. YangM. J. M. Posner and J. G. C. Templeton, The M/G/1 retrial queue with nonpersistent customers, Queueing Systems, 7 (1990), 209-218.  doi: 10.1007/BF01158475.
    [29] T. Yang and J. G. C. Templeton, A survey on retrial queues, Queueing Systems, 2 (1987), 201–233. doi: 10.1007/BF01158899.
    [30] U. Yechiali, Queues with system disasters and impatient customers when system is down, Queueing Systems, 56 (2007), 195-202.  doi: 10.1007/s11134-007-9031-z.
  • 加载中



Article Metrics

HTML views(2515) PDF downloads(398) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint