Analysis of junior servers approaching a senior server in the multi-server queueing-inventory system.

This paper studies the multi-server queueing-inventory system, which consists of n junior servers, a senior server, and two distinct waiting halls dedicated for customers and junior servers. The junior servers sometimes face difficulties while providing the service to the customers. To tackle the issues, junior servers need guidance from the senior server. When the senior server is already assisting a junior server, the other junior servers who are approaching must wait in a waiting area of size n until they consult with the senior server. We analyze the performance of such a system by modeling it into a quasi-birth-and-death process, and its steady-state probability vector has been computed. The convexity in the expected total cost for systems employing both homogeneous and state-dependent service rates has been established. Our findings demonstrate that systems employing state-dependent service rate exhibit superior operational efficiency compared to homogeneous service rates. Additionally, we conducted a comparative analysis of the expected waiting times on both homogeneous and state-dependent service rates. The expected number of junior servers in both server and consultation modes is analyzed. Furthermore, we examined the influence of changing probabilities of junior servers seeking the consultation with the senior server and compared our model with the traditional queueing-inventory model with multiple servers. In the comparison, our model performs more efficiently which shows that implementing a senior server in the multi-server system would improve the efficiency of the system in many ways to boost up the business profit.