Integrated management and optimisation of vaccination and treatment processes in a batching system with observation and limited space

This study examines a single-server system that incorporates multiple vacations. Customer arrivals adhere to a Markovian arrival process (MAP), and two types of service are applied, both following a phase-type distribution. Upon arrival, customers are grouped into batches of size L. Following service, customers enter an observation room, waiting for a predetermined duration. Subsequently, customers either leave the system if comfortable, or they join queue 2 for further service if not. Queue 2 follows a non-preemptive policy guided by the server. In the absence of customers, the server experiences multiple vacations following an exponential distribution. The server remains idle if customers are in the observation hall or if the number of customers in queue 1 is below L. This intricate system is represented as a quasi-birth-and-death (QBD) process, which we analyse using matrix-based methods. Also analysing the busy period, invariant probability, and performance metrics through graphical and numerical approaches.