Stochastic modeling on M/M/1/N inventory system with queue-dependent service rate and retrial facility

Abstract

This paper investigates the queue-dependent service rates(QDSR) in the stochastic queueing-inventory system(SQIS). This SQIS consists a single server service channel, $ S $ number of inventories, and a finite queue. An arriving customer gets the service immediately if the server is free and there exists a positive stock in the SQIS. When the server is busy, they have to wait in the finite queue. Suppose they find that the waiting hall is full, either they leave the system or enter into an infinite orbit under the Bernoulli trial. The service rate of any arrival is dependent on the number of customers in the queue at present. The orbital customer can compete for the service only through joining into the waiting hall based on the classical retrial policy. Whenever the number of inventories in SQIS is reached a certain limit $ s $, the replenishment of $ Q( = S-s) $ items is placed. Due to the structure of rate matrix, the stability analysis, minimal non-negative solutions of the quadratic systems are derived through the Neuts matrix-geometric approximation(MGA). Further, the waiting time distribution(WTD) of arrival and necessary system characteristics are derived. Finally, adequate numerical examples are presented to highlight the proposed SQIS.