American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The joint location-transportation model based on grey bi-level programming for early post-earthquake relief
  • JIMO Home
  • This Issue
  • Next Article
    An efficient genetic algorithm for decentralized multi-project scheduling with resource transfers
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 & 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.  Google Scholar

[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.  Google Scholar

[3] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, England, UK, 1987.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10] C. Preston, Gibbs States on Countable Sets, Cambridge University Press, London, England, UK, 1974.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[3] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, England, UK, 1987.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10] C. Preston, Gibbs States on Countable Sets, Cambridge University Press, London, England, UK, 1974.   Google Scholar
[11]

J. Sztrik, Basic Queueing Theory, Faculty of Informatics, University of Debrecen, Hungary, 2012. Google Scholar

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

$ 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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

$ 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
[1]

Ahmed M. K. Tarabia. Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs. Journal of Industrial & Management Optimization, 2011, 7 (4) : 811-823. doi: 10.3934/jimo.2011.7.811
[2]

Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the $ MAP/M/s+G $ queueing model with generally distributed patience times. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021078
[3]

Dequan Yue, Wuyi Yue. Block-partitioning matrix solution of M/M/R/N queueing system with balking, reneging and server breakdowns. Journal of Industrial & Management Optimization, 2009, 5 (3) : 417-430. doi: 10.3934/jimo.2009.5.417
[4]

Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715
[5]

Dequan Yue, Wuyi Yue, Zsolt Saffer, Xiaohong Chen. Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy. Journal of Industrial & Management Optimization, 2014, 10 (1) : 89-112. doi: 10.3934/jimo.2014.10.89
[6]

Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691
[7]

Dequan Yue, Wuyi Yue, Guoxi Zhao. Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states. Journal of Industrial & Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653
[8]

Xiuting Li. The energy conservation for weak solutions to the relativistic Nordström-Vlasov system. Evolution Equations & Control Theory, 2016, 5 (1) : 135-145. doi: 10.3934/eect.2016.5.135
[9]

Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135
[10]

Shihe Xu, Fangwei Zhang, Meng Bai. Stability of positive steady-state solutions to a time-delayed system with some applications. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021286
[11]

Philipp Reiter. Regularity theory for the Möbius energy. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1463-1471. doi: 10.3934/cpaa.2010.9.1463
[12]

Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial & Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1
[13]

Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks & Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011
[14]

Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147
[15]

Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895
[16]

Hideaki Takagi. Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026
[17]

Tuan Phung-Duc, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. State-dependent M/M/c/c + r retrial queues with Bernoulli abandonment. Journal of Industrial & Management Optimization, 2010, 6 (3) : 517-540. doi: 10.3934/jimo.2010.6.517
[18]

Yutaka Sakuma, Atsushi Inoie, Ken’ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time. Journal of Industrial & Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934/jimo.2011.7.593
[19]

Biao Xu, Xiuli Xu, Zhong Yao. Equilibrium and optimal balking strategies for low-priority customers in the M/G/1 queue with two classes of customers and preemptive priority. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1599-1615. doi: 10.3934/jimo.2018113
[20]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945

2020 Impact Factor: 1.801

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy