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In this paper, the transient and steady state of a single server batch service queueing system with second optional service is analyzed. All incoming units receive essential service, once the essential service is completed, the units may opt the optional service. The study applies the probability generating function, Rouche’s theorem, and Laplace transform techniques to obtain the transient state probabilities. The stationary probabilities are obtained by using the Tauberian property in the Laplace transform expressions. This study contributes to filling the gap on investigation the batch queue with essential and optional services with interesting practical application in health care systems. Furthermore, we have presented numerical results and cost optimization. The results reveal that a higher service rate in the essential service helps the system manager to run the system effectively and emphasize on optimal service rates in order to have a cost benefit and less congestion in the queue.

Transient and steady state analysis of M/M^([b])/1 queue with second optional service