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Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking
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 |
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.
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. Google Scholar |
[6] |
W. Greiner, L. Neise and H. St{ö}cker, Thermodynamics and Statistical Mechanics, Springer-Verlag, New York, 1995. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
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. Google Scholar |
[6] |
W. Greiner, L. Neise and H. St{ö}cker, Thermodynamics and Statistical Mechanics, Springer-Verlag, New York, 1995. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |


20.00 | 20.00 | 20.00 | 20.00 | 20.00 | 20.00 | |
20.00 | 16.82 | 14.14 | 11.89 | 10.00 | 8.41 | |
20.00 | 15.20 | 11.55 | 8.77 | 6.67 | 5.07 | |
20.00 | 14.14 | 10.00 | 7.07 | 5.00 | 3.54 | |
20.00 | 13.37 | 8.94 | 5.98 | 4.00 | 2.67 | |
20.00 | 12.78 | 8.16 | 5.22 | 3.33 | 2.13 | |
20.00 | 12.30 | 7.56 | 4.65 | 2.86 | 1.76 | |
20.00 | 11.89 | 7.07 | 4.20 | 2.50 | 1.49 | |
20.00 | 11.55 | 6.67 | 3.85 | 2.22 | 1.28 | |
20.00 | 11.25 | 6.32 | 3.56 | 2.00 | 1.12 | |
20.00 | 10.98 | 6.03 | 3.31 | 1.82 | 1.00 |
20.00 | 20.00 | 20.00 | 20.00 | 20.00 | 20.00 | |
20.00 | 16.82 | 14.14 | 11.89 | 10.00 | 8.41 | |
20.00 | 15.20 | 11.55 | 8.77 | 6.67 | 5.07 | |
20.00 | 14.14 | 10.00 | 7.07 | 5.00 | 3.54 | |
20.00 | 13.37 | 8.94 | 5.98 | 4.00 | 2.67 | |
20.00 | 12.78 | 8.16 | 5.22 | 3.33 | 2.13 | |
20.00 | 12.30 | 7.56 | 4.65 | 2.86 | 1.76 | |
20.00 | 11.89 | 7.07 | 4.20 | 2.50 | 1.49 | |
20.00 | 11.55 | 6.67 | 3.85 | 2.22 | 1.28 | |
20.00 | 11.25 | 6.32 | 3.56 | 2.00 | 1.12 | |
20.00 | 10.98 | 6.03 | 3.31 | 1.82 | 1.00 |
0.386257 | 0.288225 | 0.209605 | |
0.309006 | 0.288225 | 0.251526 | |
0.174800 | 0.203806 | 0.213427 | |
0.080737 | 0.117668 | 0.147867 | |
0.032295 | 0.058834 | 0.088720 | |
0.011554 | 0.026311 | 0.047612 | |
0.003774 | 0.010742 | 0.023325 | |
0.001141 | 0.004060 | 0.010579 | |
0.000323 | 0.001435 | 0.004488 | |
0.000086 | 0.000478 | 0.001795 | |
0.000022 | 0.000151 | 0.000681 |
0.386257 | 0.288225 | 0.209605 | |
0.309006 | 0.288225 | 0.251526 | |
0.174800 | 0.203806 | 0.213427 | |
0.080737 | 0.117668 | 0.147867 | |
0.032295 | 0.058834 | 0.088720 | |
0.011554 | 0.026311 | 0.047612 | |
0.003774 | 0.010742 | 0.023325 | |
0.001141 | 0.004060 | 0.010579 | |
0.000323 | 0.001435 | 0.004488 | |
0.000086 | 0.000478 | 0.001795 | |
0.000022 | 0.000151 | 0.000681 |
20.00 | 20.00 | 20.00 | 20.00 | 20.00 | 20.00 | |
20.00 | 20.00 | 20.00 | 20.00 | 20.00 | 20.00 | |
20.00 | 20.00 | 20.00 | 20.00 | 20.00 | 20.00 | |
20.00 | 16.82 | 14.14 | 11.89 | 10.00 | 8.41 | |
20.00 | 15.20 | 11.55 | 8.77 | 6.67 | 5.07 | |
20.00 | 14.14 | 10.00 | 7.07 | 5.00 | 3.54 | |
20.00 | 13.37 | 8.94 | 5.98 | 4.00 | 2.67 | |
20.00 | 12.78 | 8.16 | 5.22 | 3.33 | 2.13 | |
20.00 | 12.30 | 7.56 | 4.65 | 2.86 | 1.76 | |
20.00 | 11.89 | 7.07 | 4.20 | 2.50 | 1.49 | |
20.00 | 11.55 | 6.67 | 3.85 | 2.22 | 1.28 |
20.00 | 20.00 | 20.00 | 20.00 | 20.00 | 20.00 | |
20.00 | 20.00 | 20.00 | 20.00 | 20.00 | 20.00 | |
20.00 | 20.00 | 20.00 | 20.00 | 20.00 | 20.00 | |
20.00 | 16.82 | 14.14 | 11.89 | 10.00 | 8.41 | |
20.00 | 15.20 | 11.55 | 8.77 | 6.67 | 5.07 | |
20.00 | 14.14 | 10.00 | 7.07 | 5.00 | 3.54 | |
20.00 | 13.37 | 8.94 | 5.98 | 4.00 | 2.67 | |
20.00 | 12.78 | 8.16 | 5.22 | 3.33 | 2.13 | |
20.00 | 12.30 | 7.56 | 4.65 | 2.86 | 1.76 | |
20.00 | 11.89 | 7.07 | 4.20 | 2.50 | 1.49 | |
20.00 | 11.55 | 6.67 | 3.85 | 2.22 | 1.28 |
0.092113 | 0.050987 | 0.028157 | |
0.221072 | 0.152961 | 0.101365 | |
0.265287 | 0.229442 | 0.182457 | |
0.212229 | 0.229442 | 0.218948 | |
0.120055 | 0.162240 | 0.185784 | |
0.055451 | 0.093669 | 0.128715 | |
0.022180 | 0.046835 | 0.077229 | |
0.007936 | 0.020945 | 0.041445 | |
0.002592 | 0.008551 | 0.020304 | |
0.000784 | 0.003232 | 0.009209 | |
0.000222 | 0.001143 | 0.003907 |
0.092113 | 0.050987 | 0.028157 | |
0.221072 | 0.152961 | 0.101365 | |
0.265287 | 0.229442 | 0.182457 | |
0.212229 | 0.229442 | 0.218948 | |
0.120055 | 0.162240 | 0.185784 | |
0.055451 | 0.093669 | 0.128715 | |
0.022180 | 0.046835 | 0.077229 | |
0.007936 | 0.020945 | 0.041445 | |
0.002592 | 0.008551 | 0.020304 | |
0.000784 | 0.003232 | 0.009209 | |
0.000222 | 0.001143 | 0.003907 |
0.044992 | 0.053333 | 0.045067 | |
0.143973 | 0.156667 | 0.144213 | |
0.230357 | 0.210000 | 0.230741 | |
0.245715 | 0.246667 | 0.246123 | |
0.196572 | 0.200000 | 0.196899 | |
0.093506 | 0.086667 | 0.093014 | |
0.032816 | 0.036667 | 0.032287 | |
0.009282 | 0.006667 | 0.009006 | |
0.002787 | 0.003333 | 0.002650 |
0.044992 | 0.053333 | 0.045067 | |
0.143973 | 0.156667 | 0.144213 | |
0.230357 | 0.210000 | 0.230741 | |
0.245715 | 0.246667 | 0.246123 | |
0.196572 | 0.200000 | 0.196899 | |
0.093506 | 0.086667 | 0.093014 | |
0.032816 | 0.036667 | 0.032287 | |
0.009282 | 0.006667 | 0.009006 | |
0.002787 | 0.003333 | 0.002650 |
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 |
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 |
average | 0.7527 |
standard deviation | 0.1021 |
average | 0.7527 |
standard deviation | 0.1021 |
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 |
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 |
1611.263 | 67.04209 | |
1696.563 | 73.21876 | |
1713.902 | 77.13008 | |
1659.655 | 78.65517 | |
1584.739 | 79.49130 | |
1494.792 | 79.82639 |
1611.263 | 67.04209 | |
1696.563 | 73.21876 | |
1713.902 | 77.13008 | |
1659.655 | 78.65517 | |
1584.739 | 79.49130 | |
1494.792 | 79.82639 |
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