A matrix-analytic method for the steady state analysis of a Markovian queueing system with scrap items

This paper studies a mathematical analysis of scrap item-based single server stochastic queueing, which consists of storage space for the scrap items collected from the customers. The primary customers arrive in the finite waiting hall to sell scrap or recyclable products. The server collects the customer's scrap items and stores them in the storage space. If the waiting hall is full whenever the primary customers arrive, they can join an infinite orbit or permanently leave the system under the Bernoulli trial. Whenever the orbital customers find at least one free space in the waiting hall, they can enter it. The server begins vacation once the storage reaches its maximum capacity or no more customers exist in the waiting hall. The server switches to regular mode whenever at least one free space exists in the storage, and one customer is in the waiting hall. Otherwise, he is not working in vacation mode. Customers in the waiting hall may lose their patience after waiting for a long time and may either enter orbit or leave the system permanently. Whenever the storage level (scrap item level) reaches a certain limit, the server places the procurement order for sending the items to the recycling unit or re-manufacturer. The system's steady-state probability vector and stability condition have been derived by truncating the orbit at a point. Various system performances, such as expected total cost, profit cost, and waiting time for customers in the waiting hall, are defined, and a numerical study is carried out to analyze the performance of the proposed model. The numerical findings indicate that disabling working in vacation mode increases waiting time and total cost while working in vacation mode results in minimized waiting time and total cost.