Analysis of Interconnected Arrivals on Queueing-Inventory System with Two Multi-Server Service Channels and One Retrial Facility

Present-day queuing inventory systems (QIS) do not utilize two multi-server service channels. We proposed two multi-server service channels referred to as T1S (Type 1 n-identical multi-server) and T2S (Type 2 m-identical multi-server). It includes an optional interconnected service connection between T1S and T2S, which has a finite queue of size N. An arriving customer either uses the inventory (basic service or main service) for their demand, whom we call T1, or simply uses the service only, whom we call T2. Customer T1 will utilize the server T1S, while customer T2 will utilize the server T2S, and T1 can also get the second optional service after completing their main service. If there is a free server with a positive inventory, there is a chance that T1 customers may go to an infinite orbit whenever they find that either all the servers are busy or no sufficient stock. The orbital customer can request for T1S service under the classical retrial policy. Q(=S−s) items are replaced into the inventory whenever it falls into the reorder level s such that the inequality always holds n<s. We use the standard (s,Q) ordering policy to replace items into the inventory. By varying S and s, we investigate to find the optimal cost value using stationary probability vector ϕ. We used the Neuts Matrix geometric approach to derive the stability condition and steady-state analysis with R-matrix to find ϕ. Then, we perform the waiting time analysis for both T1 and T2 customers using Laplace transform technique. Further, we computed the necessary system characteristics and presented sufficient numerical results.