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 (FES) in batches, with batch sizes constrained between a lower limit a and an upper limit b. On completion of FES, the same server may offer a Second Optional Service (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 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 FES batch. Service times for both FES and 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 FES and 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.