## 5 Hybrid Modeling of Heterogeneous Servers with General Service Time Distributions  
### Uncertainty of the Offered Load  
The stationary solution of the M/M/c/c + k system of identical servers has been studied analytically and is included in manuals of Markovian queues. The analytical multiclass, multichannel, or heterogeneous servers in model E can be used to study the performance of M/Mi/3/3 + k, but when using the Monte Carlo method (MCM), it can approach M/Gi/3/3 + k. The combination of MCM and model E can offer the resulting distribution of the stationary solution for each system setup of mean arrival rate and service rates Gi coming from general (non-exponential) distributions. Regardless of the distribution from which the service rates come, model E offers one solution (stationary), and thus, the variability effect of service rates under general distribution on performance can be assessed by generating the resulting distribution of performance results. We proposed the use of the precise stationary solution from the transition matrix in combination with MCM to evaluate the effect of the uncertainty in the offered load or the service rates, which approached the solution of the system for other service distributions. The transition matrix was developed with the strong assumption of events being independent or memoryless, and thus, the independent state probabilities were evaluated in a Markovian process. The MCM offers the distribution of solutions from independent trials of shots to the non-exponential distributions of arrival and/or service rates. The MCM is a standard method for uncertainty evaluation in other fields, like metrology, and its use together with the analytical Markovian model can allow for evaluating the effect of service time variability or arrival rate uncertainty without the time-consuming discrete-event simulation to obtain the stationary solution under general distributions. For instance, it might be of interest in facilitating system control or decision making in logistic or manufacturing activities in cyberphysical systems. While the transition matrix gives the stationary solution of the temporal series, the output of the MCM is a distribution around the average stationary solutions that show the expected spread around the mean value under the ergodic assumption, which is due to the variation in the service times under different distributions. The hybrid method is considered very appropriate because the variability effect in the response to the uncertainty of the variables is also based on its variance, like the arrival or service rates variability. In addition, it is easy to implement for the proposed analytical model to produce a quick resolution, with no need for solution initialization or other previous algorithm adjustments, and thus, the method directly propagates the spread of stochastic variables through the analytical model into the solution. Hybrid modeling of accurate Markovian chain models in combination with the MCM can facilitate approximate studies by alleviating the limitation of the distribution with memory that non-exponential distributions impose because each shot in the MCM simulation will be calculated independently with the accurate stationary result (expected asymptotic mean in time of an ergodic process), and the resulting distribution from the MCM will be the convolution of stationary results generated from shots to the distribution of the arrival rate or the service rate. The alternative of direct discrete-event simulation will require looking for the average solution when a simultaneous change occurs in the mean values and their variance; therefore, the effect of the mean and variance are combined in every solution of the simulation run. In addition, real systems are better represented for decision making by the mean value of service time or demand in short-term periods with some uncertainty, but with a drift in longer temporal periods.