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Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking

  • * Corresponding author: Ikuo Arizono

    * Corresponding author: Ikuo Arizono 

The authors are supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 18K04611: "Evaluation of system performance and reliability under incomplete information environment"

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  • Behavior that a customer who has just arrived at a crowded queueing system leaves without joining the queue is known as the phenomenon of balking. Queueing systems with balking have been studied continually as one of significant subjects. In this paper, the theoretical approach for the steady-state analysis of the Markovian queueing systems with balking is considered based on the concept of the statistical mechanics. Here, it can be easily seen that the strength of balking is not constant but various in each queueing systems. Note that the strength of balking means how degree a customer who has just arrived at a crowded queueing system leaves without joining the queue. In our approach, under considering the difference of the strength of balking for each queueing systems, we have proposed a statistical mechanics model for analyzing the M/M/$ s $ queueing system with balking by introducing a parameter influencing the strength of balking. Further, we define a procedure for estimating the model parameter influencing the strength of balking. In addition, we consider a method of improving the performance of the M/M/$ s $ queueing system with balking by utilizing the statistical mechanics approach.

    Mathematics Subject Classification: Primary: 60K25; Secondary: 60K20, 60K35.

    Citation:

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  • Figure 1.  The relationships between the number of customers $ n $ and the arrival rate $ \lambda_{n} $ in the case of $ s = 1 $

    Figure 2.  The steady-state probability distributions in the case of $ s = 1 $ and $ r = 0.5 $

    Figure 3.  The relationships between the number of customers $ n $ and the arrival rate $ \lambda_{n} $ in the case of $ s = 3 $

    Figure 4.  The steady-state probability distributions in the case of $ s = 3 $ and $ r = 0.5 $

    Figure 5.  The relationship between the cost $ c $ and the profit $ T $

    Table 1.  The arrival rate $ \lambda_{n} $ against various $ r $ in the case of $ s = 1 $

    $ n $ $ r = 0.00 $ $ r = 0.25 $ $ r = 0.50 $ $ r=0.75 $ $ r = 1.00 $ $ r = 1.25 $
    $ 0 $ 20.00 20.00 20.00 20.00 20.00 20.00
    $ 1 $ 20.00 16.82 14.14 11.89 10.00 8.41
    $ 2 $ 20.00 15.20 11.55 8.77 6.67 5.07
    $ 3 $ 20.00 14.14 10.00 7.07 5.00 3.54
    $ 4 $ 20.00 13.37 8.94 5.98 4.00 2.67
    $ 5 $ 20.00 12.78 8.16 5.22 3.33 2.13
    $ 6 $ 20.00 12.30 7.56 4.65 2.86 1.76
    $ 7 $ 20.00 11.89 7.07 4.20 2.50 1.49
    $ 8 $ 20.00 11.55 6.67 3.85 2.22 1.28
    $ 9 $ 20.00 11.25 6.32 3.56 2.00 1.12
    $ 10 $ 20.00 10.98 6.03 3.31 1.82 1.00
     | Show Table
    DownLoad: CSV

    Table 2.  The steady-state probability $ P_{n} $ under some $ \rho $ in the case of $ s = 1 $

    $ n $ $ \rho = 0.8 $ $ \rho = 1.0 $ $ \rho = 1.2 $
    $ 0 $ 0.386257 0.288225 0.209605
    $ 1 $ 0.309006 0.288225 0.251526
    $ 2 $ 0.174800 0.203806 0.213427
    $ 3 $ 0.080737 0.117668 0.147867
    $ 4 $ 0.032295 0.058834 0.088720
    $ 5 $ 0.011554 0.026311 0.047612
    $ 6 $ 0.003774 0.010742 0.023325
    $ 7 $ 0.001141 0.004060 0.010579
    $ 8 $ 0.000323 0.001435 0.004488
    $ 9 $ 0.000086 0.000478 0.001795
    $ 10 $ 0.000022 0.000151 0.000681
     | Show Table
    DownLoad: CSV

    Table 3.  The arrival rate $ \lambda_{n} $ against various $ r $ in the case of $ s = 3 $

    $ n $ $ r = 0.00 $ $ r = 0.25 $ $ r = 0.50 $ $ r=0.75 $ $ r = 1.00 $ $ r = 1.25 $
    $ 0 $ 20.00 20.00 20.00 20.00 20.00 20.00
    $ 1 $ 20.00 20.00 20.00 20.00 20.00 20.00
    $ 2 $ 20.00 20.00 20.00 20.00 20.00 20.00
    $ 3 $ 20.00 16.82 14.14 11.89 10.00 8.41
    $ 4 $ 20.00 15.20 11.55 8.77 6.67 5.07
    $ 5 $ 20.00 14.14 10.00 7.07 5.00 3.54
    $ 6 $ 20.00 13.37 8.94 5.98 4.00 2.67
    $ 7 $ 20.00 12.78 8.16 5.22 3.33 2.13
    $ 8 $ 20.00 12.30 7.56 4.65 2.86 1.76
    $ 9 $ 20.00 11.89 7.07 4.20 2.50 1.49
    $ 10 $ 20.00 11.55 6.67 3.85 2.22 1.28
     | Show Table
    DownLoad: CSV

    Table 4.  The steady-state probability $ P_{n} $ under some $ \rho $ in the case of $ s = 3 $

    $ n $ $ \rho = 0.8 $ $ \rho = 1.0 $ $ \rho = 1.2 $
    $ 0 $ 0.092113 0.050987 0.028157
    $ 1 $ 0.221072 0.152961 0.101365
    $ 2 $ 0.265287 0.229442 0.182457
    $ 3 $ 0.212229 0.229442 0.218948
    $ 4 $ 0.120055 0.162240 0.185784
    $ 5 $ 0.055451 0.093669 0.128715
    $ 6 $ 0.022180 0.046835 0.077229
    $ 7 $ 0.007936 0.020945 0.041445
    $ 8 $ 0.002592 0.008551 0.020304
    $ 9 $ 0.000784 0.003232 0.009209
    $ 10 $ 0.000222 0.001143 0.003907
     | Show Table
    DownLoad: CSV

    Table 5.  An example of the estimation as $ r^{\ast} = 0.76 $

    $ n $ $ P_{n}^† $ $ P_{n}^{‡} $ $ P_{n}^{\ast} $
    $ 0 $ 0.044992 0.053333 0.045067
    $ 1 $ 0.143973 0.156667 0.144213
    $ 2 $ 0.230357 0.210000 0.230741
    $ 3 $ 0.245715 0.246667 0.246123
    $ 4 $ 0.196572 0.200000 0.196899
    $ 5 $ 0.093506 0.086667 0.093014
    $ 6 $ 0.032816 0.036667 0.032287
    $ 7 $ 0.009282 0.006667 0.009006
    $ 8 $ or more 0.002787 0.003333 0.002650
     | Show Table
    DownLoad: CSV

    Table 6.  The estimated results of $ r^{\ast} $

    0.76 0.60 0.70 0.76 0.97
    0.94 0.77 0.79 0.80 0.64
    0.76 0.73 0.65 0.64 0.66
    0.66 0.76 0.66 0.76 0.61
    0.80 0.84 0.86 0.75 0.77
    0.94 0.65 0.72 0.91 0.76
    0.79 0.82 0.76 0.93 0.88
    0.70 0.83 0.76 0.75 0.74
    0.57 0.75 0.86 0.58 0.61
    0.73 0.75 0.67 0.68 0.68
    0.73 0.93 0.61 0.84 0.79
    0.63 0.72 0.77 1.05 0.59
    0.72 0.80 0.74 0.70 0.81
    0.93 0.84 0.77 0.64 0.97
    0.83 0.65 0.70 0.72 0.92
    0.76 0.73 0.84 0.80 0.86
    0.66 0.73 0.75 0.69 0.74
    0.64 0.78 0.88 0.89 0.70
    0.69 0.52 0.71 0.75 0.86
    0.83 0.59 0.62 0.74 0.65
     | Show Table
    DownLoad: CSV

    Table 7.  The basic statistics for $ r^{\ast} $ in Table \ref{Table 2}

    average 0.7527
    standard deviation 0.1021
     | Show Table
    DownLoad: CSV

    Table 8.  The averages and standard deviations by the similar experiments

    the number of observation 300 500 1000 10000
    average 0.7527 0.7446 0.7477 0.7516
    standard deviation 0.1021 0.0909 0.0466 0.0188
     | Show Table
    DownLoad: CSV

    Table 9.  The profit $ T $ and average arrival rate when increasing the number of servers

    $ s $ $ T $ $ \bar{\lambda} $
    $ 4 $ 1611.263 67.04209
    $ 5 $ 1696.563 73.21876
    $ 6 $ 1713.902 77.13008
    $ 7 $ 1659.655 78.65517
    $ 8 $ 1584.739 79.49130
    $ 9 $ 1494.792 79.82639
     | Show Table
    DownLoad: CSV
  • [1] M. O. Abou-El-Ata and A. M. A. Hariri, The M/M/c/N queue with balking and reneging, Computers & Operations Research, 19 (1992), 713-716.  doi: 10.1016/0305-0548(92)90010-3.
    [2] I. ArizonoY. Cui and H. Ohta, An analysis of M/M/$ s $ queueing systems based on the maximum entropy principle, Journal of the Operational Research Society, 42 (1991), 69-73.  doi: 10.1057/jors.1991.8.
    [3] D. ChandlerIntroduction to Modern Statistical Mechanics, Oxford University Press, Oxford, England, UK, 1987. 
    [4] C. ChenZ. Jia and P. Varaiya, Causes and cures of highway congestion, IEEE Control Systems Magazine, 21 (2001), 26-32.  doi: 10.1109/37.969132.
    [5] A. A. El-Sherbiny, The truncated heterogeneous two-server queue: M/M/2/N with reneging and general balk function, International Journal of Mathematical Archive, 3 (2012), 2745-2754. 
    [6] W. Greiner, L. Neise and H. St{ö}cker, Thermodynamics and Statistical Mechanics, Springer-Verlag, New York, 1995.
    [7] N. K. JainR. Kumar and B. Kumar Som, An M/M/1/N queuing system with reverse balking, American Journal of Operational Research, 4 (2014), 17-20. 
    [8] A. Montazer-HaghighiJ. Medhi and S. G. Mohanty, On a multiserver Markovian queueing system with balking and reneging, Computers & Operations Research, 13 (1986), 421-425.  doi: 10.1016/0305-0548(86)90029-8.
    [9] B. Natvig, On the transient state probabilities for a queueing model where potential customers are discouraged by queue length, Journal of Applied Probability, 11 (1974), 345-354.  doi: 10.2307/3212755.
    [10] C. PrestonGibbs States on Countable Sets, Cambridge University Press, London, England, UK, 1974. 
    [11] J. Sztrik, Basic Queueing Theory, Faculty of Informatics, University of Debrecen, Hungary, 2012.
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