A queueing model for time-dependent rental systems with phase-type distributed rentals and substitutions of items

Society currently experiences a growing interest in renting as an alternative to exclusive ownership (Fota et al., 2019; Yeganeh, 2021). This paper supports rental companies in this transition by providing a method to evaluate rental system occupancy. The method considers time-dependent systems in which customers rent substitutes during stockouts. Specifically, the paper provides a queueing model based on a continuous-time Markov chain for systems where rentals occur according to Poisson processes and rental times follow phase-type distributions. Additionally, the paper provides a heuristic optimization algorithm to minimize inventory capacity while maintaining an upper stockout probability limit. The paper evaluates these methods through numerical experiments, including a company case. The results indicate that the queueing model adequately reflects the occupancy distributions of the investigated items, and that the optimization algorithm minimizes the inventory capacity to near-optimality. In the company case, these methods reduce the inventory capacity while simultaneously improving the maximum stockout probability.