Optimizing the Performance of Multi-server Heterogeneous Queueing Systems with Dynamic Customer Behaviour

This article addresses the pressing issue of customer impatience and dissatisfaction with queueing systems, a critical aspect of modern businesses. It delves into specific aspects of customer behavior, viz., reverse balking, reneging, and retention of reneged customers, which significantly influence queue dynamics. A novel finite-buffer two-server heterogeneous queueing model is introduced, taking into account reverse balking, and position-dependent reneging (reneging till the beginning of service i.e., \(R\_BOS\) $R\_BOS$ and reneging till the end of service i.e., \(R\_EOS\) $R\_EOS$ ) along with the retention of reneged customers. The probability distribution of the number of customers in the system is ferret out and the key performance measures are derived along with sensitivity analysis. Extending the analysis, this study tackles a cost optimization problem using an algorithm based on the Quasi-Newton Method (QNM) to achieve optimal solutions for both the server that optimize the cost function. A numerical example is provided to illustrate the practical implications and benefits of the proposed optimization approach for the queueing model.