# American Institute of Mathematical Sciences

• Previous Article
Incentive contract design for supplier switching with considering learning effect
• JIMO Home
• This Issue
• Next Article
Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy
doi: 10.3934/jimo.2020141

## 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

* Corresponding author: Ikuo Arizono

Received  January 2020 Revised  June 2020 Published  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.

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, 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
The relationships between the number of customers $n$ and the arrival rate $\lambda_{n}$ in the case of $s = 1$
The steady-state probability distributions in the case of $s = 1$ and $r = 0.5$
The relationships between the number of customers $n$ and the arrival rate $\lambda_{n}$ in the case of $s = 3$
The steady-state probability distributions in the case of $s = 3$ and $r = 0.5$
The relationship between the cost $c$ and the profit $T$
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
 $n$ $r = 0.00$ $r = 0.25$ $r = 0.50$ $r=0.75$ $r = 1.00$ $r = 1.25$ $0$ 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
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
 $n$ $\rho = 0.8$ $\rho = 1.0$ $\rho = 1.2$ $0$ 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
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
 $n$ $r = 0.00$ $r = 0.25$ $r = 0.50$ $r=0.75$ $r = 1.00$ $r = 1.25$ $0$ 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
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
 $n$ $\rho = 0.8$ $\rho = 1.0$ $\rho = 1.2$ $0$ 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
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
 $n$ $P_{n}^†$ $P_{n}^{‡}$ $P_{n}^{\ast}$ $0$ 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
The estimated results of $r^{\ast}$
 0.76 0.6 0.7 0.76 0.97 0.94 0.77 0.79 0.8 0.64 0.76 0.73 0.65 0.64 0.66 0.66 0.76 0.66 0.76 0.61 0.8 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 0.7 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 0.8 0.74 0.7 0.81 0.93 0.84 0.77 0.64 0.97 0.83 0.65 0.7 0.72 0.92 0.76 0.73 0.84 0.8 0.86 0.66 0.73 0.75 0.69 0.74 0.64 0.78 0.88 0.89 0.7 0.69 0.52 0.71 0.75 0.86 0.83 0.59 0.62 0.74 0.65
 0.76 0.6 0.7 0.76 0.97 0.94 0.77 0.79 0.8 0.64 0.76 0.73 0.65 0.64 0.66 0.66 0.76 0.66 0.76 0.61 0.8 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 0.7 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 0.8 0.74 0.7 0.81 0.93 0.84 0.77 0.64 0.97 0.83 0.65 0.7 0.72 0.92 0.76 0.73 0.84 0.8 0.86 0.66 0.73 0.75 0.69 0.74 0.64 0.78 0.88 0.89 0.7 0.69 0.52 0.71 0.75 0.86 0.83 0.59 0.62 0.74 0.65
The basic statistics for $r^{\ast}$ in Table \ref{Table 2}
 average 0.7527 standard deviation 0.1021
 average 0.7527 standard deviation 0.1021
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
 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 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
 $s$ $T$ $\bar{\lambda}$ $4$ 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] Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $(n, m)$-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117 [2] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [3] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [4] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [5] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [6] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [7] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [8] Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268 [9] Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346 [10] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [11] Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446 [12] Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158 [13] Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219 [14] Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026 [15] Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321 [16] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [17] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273 [18] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [19] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [20] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

2019 Impact Factor: 1.366

## Tools

Article outline

Figures and Tables