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

Ikuo Arizono; Yasuhiko Takemoto

doi:10.3934/jimo.2020141

Article Contents

2022, Volume 18, Issue 1: 25-44. Doi: 10.3934/jimo.2020141

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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
^* Corresponding author: Ikuo Arizono

Received: December 31, 2019

Revised: May 31, 2020

Early access: September 2020

Published: January 2022

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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Abstract

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.

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

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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 $

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

average	0.7527
standard deviation	0.1021

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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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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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