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January  2021, 17(1): 185-204. doi: 10.3934/jimo.2019106

## An approximate mean queue length formula for queueing systems with varying service rate

 1 Shanghai Jiao Tong University, 800 Dongchuan Rd, Shanghai, China 2 The Chinese University of Hong Kong(Shenzhen), Shanghai Jiao Tong University, Zhejiang University of Technology

* Corresponding author: Jian Zhang

Received  October 2018 Revised  May 2019 Published  September 2019

In this paper, we analyze the delay performance of queueing systems in which the service rate varies with time and the number of service states may be infinite. Except in some simple special cases, in general, the queueing model with varying service rate is mathematically intractable. Motivated by the P-K formula for M/G/1 queue, we developed a limiting analysis approach based on the connection between the fluctuation of service rate and the mean queue length. Considering the two extreme service rates, we provide a lower bound and upper bound of mean queue length. Furthermore, an approximate mean queue length formula is derived from the convex combination of these two bounds. The accuracy of our approximation has been confirmed by extensive simulation studies with different system parameters. We also verified that all limiting cases of the system behavior are consistent with the predictions made by our formula.

Citation: Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106
##### References:

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##### References:
The continuous time Markov chain of the server process
The continuous-time Markov chain of the queueing model
The fluctuation of service rate $\mu$ over the time with parameter $\frac{\mu_c}{\mu_s} = 10$, $\lambda_s = 10$
The first and second moments of service time increase with the variance of the service rate
Service rate becomes a constant when system reaches equilibrium
The two-state extreme scenario
The transition diagram of the two service states
Mean queue length $L$ is bounded by $L_1$ and $L_2$
Overload probability $a$ vs. parameter $\alpha$
Mean queue length in overload region $L_{overload}$ and overload probability $a$
Simulation and approximation results of mean queue lengths
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