The analysis presented here considers a queueing system of Markovian nature that contains several important and complicated features like heterogeneous servers, impatience of customers, working vacation, retrial queues, server failures with repairs and a backup server. Such
features are important in practice as many services use queueing systems of this type, for instance in medical and IT systems. Infinite size of the queue is assumed with random service rates. Necessary stability conditions are developed, steady-state probabilities are found using
the matrix geometric approach and a cost function is introduced to obtain efficient tradeoff between efficiency and service performance. Data-driven predictions and conclusions are produced using the Adaptive Neuro Fuzzy Inference System. Results indicate that the increase in arrival rate λ from 0.5 to 1.1 (assuming that β = 0.6) increases the mean queue length from 1.0 to 1.45 customers, while increasing the service rate in working vacation from 0.5 to 2.0 decreases it from 4.0 to 2.1 customers, obtaining accuracies of 99% and 99.8%, correspondingly. The impact of other important parameters is investigated numerically. The originality of this approach is in inclusion of a number of real life features together with data driven predictions which makes the approach more realistic compared to other techniques available in the literature.