Optimal State Dependent Dual Sourcing Policies in Queueing Inventory Systems with General Lead Times and Cost Optimization

Abstract

his paper develops a rigorous analytical framework for a queueing inventory system governed by a state dependent dual sourcing replenishment policy under general lead time distributions. Customers arrive according to a Markovian arrival process, while service times follow a phase type distribution, allowing a flexible representation of stochastic variability. The inventory system operates under an $(s,S)$ structure, where replenishment decisions are dynamically influenced by the queue length through a threshold parameter that determines the activation of either a regular or an expedited supply mode. Unlike existing studies that rely on exponential lead times and purely numerical optimization, the present work incorporates general lead time structures and formulates an explicit expected total cost functional that integrates holding, shortage, waiting, and emergency procurement costs. The system is modeled as a multi dimensional continuous time Markov chain with a quasi birth death structure, and its steady state distribution is obtained using matrix analytic techniques. Beyond computational analysis, the paper establishes structural properties of the optimal policy, including monotonic behavior of cost with respect to control parameters and the existence of an optimal threshold pair. Numerical investigations reveal intricate interactions between congestion, replenishment speed, and cost trade offs, providing clear operational insights into when expedited sourcing becomes economically justified. The results offer both theoretical advancement and practical guidance for managing inventory systems under demand uncertainty and service delays.