Optimal service design for heterogeneous queueing system with reverse balking and reneging

This article investigates an M/M/2/K heterogeneous queueing system, specifically focusing on two aspects of customer impatience: reverse balking and reneging till the end of service ( $$R\_EOS$$ R _ E O S ). Reverse balking refers to the phenomenon where the probability of not joining a queueing system decreases as system size increases. Reneging refers to the phenomenon where a customer leaves the system before service completion. There are two types of reneging: reneging till the beginning of service ( $$R\_BOS$$ R _ B O S ) and reneging till the end of service ( $$R\_EOS$$ R _ E O S ). Reneging has been predominantly studied with $$R\_BOS$$ R _ B O S , but $$R\_EOS$$ R _ E O S has received limited attention despite its relevance in practical situations. $$R\_EOS$$ R _ E O S is commonly observed in various domains such as healthcare facilities, theme parks, restaurants, and others. The novelty of this article lies in the investigation of $$R\_EOS$$ R _ E O S in a heterogeneous server queueing model along with reverse balking. The steady-state probability distribution of the number of customers in the system is extracted, and key performance measures are derived. Sensitivity analysis is performed to assess how critical parameters affect the system performance, followed by a cost analysis which includes cost optimization of the system using quadratic fit search method (QFSM). Additionally, a real-life application of the proposed queueing model is presented. By offering practical insights into service design that considers the aspects of customer impatience, this model finds application in various real-world scenarios like healthcare, call centers, retail, and transportation.