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

• * Corresponding author: Ikuo Arizono

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.

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

 Citation:

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

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

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

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

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

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 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 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 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 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 6.  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

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

 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 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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