(untitled) (Contemporary Mathematics, 2023)
This paper aims to analyze the steady-state behavior of bulk input general service queue with a second optional service (SOS), balking, and feedback facility. In this study, the server provides two kinds of services such as first essential service (FES) and SOS. The FES is provided to all arriving customers to the system while SOS is only to those customers who demand additional service. When the customer completes FES and is not satisfied with the service, he may choose to rejoin the queue (feedback) or opt for SOS or depart from the system with a certain probability. We have computed the probability-generating function of the queue-length distribution after converting the non-Markovian to Markovian process by using a supplementary variable technique. This technique is used to solve the non-Markov queue model by taking the elapsed service times as the supplementary variable so that the process becomes Markovian. This study contributes to filling the gap in the analysis of batch arrival general service queues with balking, feedback, and SOS. Furthermore, we have presented the numerical results and cost optimization. The results reveal that the higher service rate in both FES and SOS helps the system manager to run the system effectively. Similarly, in cost optimization, the system manager should make emphasize choosing optimal service rates to have a cost-benefit and less congestion in the queueing system.