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January  2022, 18(1): 25-44. doi: 10.3934/jimo.2020141

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

Ikuo Arizono 1,, and  Yasuhiko Takemoto 2,

1. 

Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530, Japan
2. 

Kindai University, 3-4-1 Kowakae, Higashi-Osaka, Osaka, 577-8502, Japan

* Corresponding author: Ikuo Arizono

Received  January 2020 Revised  June 2020 Published  January 2022 Early access  September 2020

Fund Project: 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"

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.

Keywords: M/M/s queueing system, balking, statistical mechanics, entropy, potential energy, steady-state analysis.
Mathematics Subject Classification: Primary: 60K25; Secondary: 60K20, 60K35.
Citation: Ikuo Arizono, Yasuhiko Takemoto. Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking. Journal of Industrial and Management Optimization, 2022, 18 (1) : 25-44. doi: 10.3934/jimo.2020141
References:
[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. Arizono, Y. 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. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, England, UK, 1987. 
[4]

C. Chen, Z. 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. Jain, R. 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-Haghighi, J. 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. Preston, Gibbs 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.

show all references

References:
[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. Arizono, Y. 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. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, England, UK, 1987. 
[4]

C. Chen, Z. 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. Jain, R. 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-Haghighi, J. 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. Preston, Gibbs 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.

Figure 1.  The relationships between the number of customers $ n $ and the arrival rate $ \lambda_{n} $ in the case of $ s = 1 $
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Figure 2.  The steady-state probability distributions in the case of $ s = 1 $ and $ r = 0.5 $
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Figure 3.  The relationships between the number of customers $ n $ and the arrival rate $ \lambda_{n} $ in the case of $ s = 3 $
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Figure 4.  The steady-state probability distributions in the case of $ s = 3 $ and $ r = 0.5 $
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Figure 5.  The relationship between the cost $ c $ and the profit $ T $
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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
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$ 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
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
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$ 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
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
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$ 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
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
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$ 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
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
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$ 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
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
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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
Table 7.  The basic statistics for $ r^{\ast} $ in Table \ref{Table 2}
average 0.7527
standard deviation 0.1021
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average 0.7527
standard deviation 0.1021
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
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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
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
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$ 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
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