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

• * Corresponding author: Jian Zhang
• 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.

Mathematics Subject Classification: Primary: 60K20, 60K25; Secondary: 68M20.

 Citation: • • Figure 1.  The continuous time Markov chain of the server process

Figure 2.  The continuous-time Markov chain of the queueing model

Figure 3.  The fluctuation of service rate $\mu$ over the time with parameter $\frac{\mu_c}{\mu_s} = 10$, $\lambda_s = 10$

Figure 4.  The first and second moments of service time increase with the variance of the service rate

Figure 5.  Service rate becomes a constant when system reaches equilibrium

Figure 6.  The two-state extreme scenario

Figure 7.  The transition diagram of the two service states

Figure 8.  Mean queue length $L$ is bounded by $L_1$ and $L_2$

Figure 10.  Overload probability $a$ vs. parameter $\alpha$

Figure 11.  Mean queue length in overload region $L_{overload}$ and overload probability $a$

Figure 12.  Simulation and approximation results of mean queue lengths

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