Analysis of Stochastic M/M/c/N Inventory System with Queue-Dependent Server Activation, Multi-Threshold Stages and Optional Retrial Facility

The purpose of this article is to examine the server activation policy (SAP) in a multi-server queuing-inventory system (MQIS). The queue has a total of c number of multi-threshold stages as well as c-homogeneous servers. The activation of each server begins one by one if there is an adequate queue length and inventory in the system; otherwise, they remain idle. The server deactivation process continues until the queue length exceeds the manageable level (predetermined stages) or there is insufficient stock. In addition, when we assume the length of the two successive threshold levels is one, the server activation policy model becomes a regular multi-server model. The Neuts matrix geometric approach is used to discuss the stability condition, stationary probability vector. The Laplace–Stieltjes transform (LST) is used to analyse the waiting time distributions of the queue and orbital customers. Additionally, significant system performance metrics and sensitivity analysis are used to investigate the effects of various parameters and cost values. In the comparative result between the server activation model (SAM) and without the server activation model (WSAM) on the expected total cost, we obtain the minimised cost in the SAM. Moreover, the results are obtained by assuming that the length of the intervals between the two successive threshold levels is to be taken into account as the non-uniform length. The expected inventory level, reorder rate, and waiting time of a customer in the waiting hall and orbit were explored numerically by the parameter analysis.