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

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 9.  Overload and underload regions

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