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This article explores the performance of interconnected queueing systems and queueing-inventory systems (IQSQIS) in stochastic modeling. IQSQIS features two types of service facilities with dual service stations: one providing non-commodity service with a multi-server, and another handling commodity sales with a single server. As directed, a customer type i,i=1,2 can approach the station i,i=1,2 whose arrival pattern follows the marked Markovian arrival process (MMAP) since we assume that both service stations have an equal arrival phase. The IQSQIS gives an offer to the type-2 customer to choose type-1 service at the end of their service completion. Numerical results suggest that this option reduces customer wait times, orbit size, and overall system costs. The comparative analysis of heterogeneous and homogeneous servers in station-1 and queue-dependent and non-queue-dependent service facilities in station-2 is presented and investigated using the numerical outputs.

This paper considers a stochastic queueing-inventory system with dual service stations, two groups of heterogeneous multi-servers, two finite waiting halls, and two classes of customers. Station-1 and station-2 provide an inventory sales service and feedback service, respectively. The feedback service option can be given to customers at the service completion epoch. If a customer requires feedback service, moves to an orbit; otherwise, leaves the system permanently. The classical retrial policy is applied to get feedback service. For the replenishment process, the system follows an (S−1,S) base stock ordering policy. This study analyzes the model under four classifications: 1) orbit size is finite and servers have homogeneous service rate; 2) orbit size is finite and servers have heterogeneous service rate; 3) orbit size is infinite and servers have homogeneous service rate; and 4) orbit size is infinite and servers have heterogeneous service rate. The steady-state probability vector for an infinite-size orbit case is computed using the Neuts and Rao truncation method. An expected total cost function is derived with sufficient system indicators for the four classifications. An optimized expected total cost is gained for classifications 2 and 4. The probability that a server is busy, the customer's waiting time, and the customer's loss rate are minimized for classifications 2 and 4.