On joint distributions of general bulk service Poisson queue with infinite buffer and unique bulk optional service

This paper investigates a batch-service queueing model where a single-server operates in two service stages. Clients arrive at the system according to the Poisson process. The server first provides the First Essential Service (\textit{FES}) in batches, with batch sizes constrained between a lower limit $a$ and an upper limit $b$. On completion of \textit{FES}, the same server may offer a Second Optional Service (\textit{SOS}), where some or all clients from the previous batch may participate, again in batches limited between 1 and $b$, based on a certain probability. If, after completing the \textit{SOS}, the number of clients in the queue is less than $a$, the server remains idle until the queue length reaches $a$; otherwise, it proceeds to the next \textit{FES} batch. Service times for both \textit{FES} and \textit{SOS} follow exponential distributions. The system assumes an infinite-buffer, allowing unrestricted waiting space for incoming clients. Such a queueing structure can model real-world systems like public transportation or communication networks. Using the probability generating function (PGF) approach, we derive steady-state joint distributions of the number of clients in the queue and those being served during both \textit{FES} and \textit{SOS}. Key performance metrics and numerical results are provided to support further research. The paper concludes with a healthcare-associated cost-minimization problem through service rates optimization using metaheuristic PSO technique.