Online reservation queueing–inventory system with two distinct services and client types

This study examines a queueing–inventory system offering premium and non-premium services for a single commodity, where both services have individual waiting areas with finite capacities. To enhance premium service, the system includes an online reservation facility with a limited level. An online client initially pre-books a spot in the premium waiting area (PWA) and, after a random duration, either joins the PWA or cancels the reservation. An offline client directly visits the system and chooses any one of the services based on their needs. The arrivals of both client types follow independent Markovian arrival processes (MAPs). Further, the waiting times of a client in the premium and non-premium waiting areas are derived using the Laplace–Stieltjes transform. The steady-state probabilities are computed, and the system's essential performance metrics are calculated. Subsequently, the optimal total expected cost is determined through numerical analysis and visually represented in a graph.