A two-warehouse queueing-inventory system with orbital search for customers returning items

This paper examines a two-warehouse queueing-inventory model, with one being the own warehouse (OW), which stores fresh items, and the other being the rented warehouse (RW), which stores returned items. Fresh inventory in the OW is controlled by an external replenishment policy with exponentially distributed lead times, whereas the RW inventory has a finite storage capacity. Customer arrivals are modeled by a Markovian arrival process. The OW is assumed to have multiple parallel homogeneous servers with phase-type (PH) distributed service times, while the RW is assumed to have a single server with a different PH-distributed service time. The system dynamics are represented by a continuous-time Markov chain with a block-structured infinitesimal generator. By exploiting this structure, the steady-state distribution is obtained using matrix-analytic methods, and several important performance measures are derived. In addition, a special case with Poisson arrivals and exponential service times is presented to illustrate the analytical framework. Numerical experiments are conducted to examine the influence of key system parameters on the operational performance and to provide managerial insights for effective inventory and service management in dual-warehouse environments.