Retrial Queuing-Inventory Systems with Delayed Feedback and Instantaneous Damaging of Items

Abstract

This paper studies a Markov model of a queuing-inventory system with primary, retrial, and feedback customers. Primary customers form a Poisson flow, and if an inventory level is positive upon their arrival, they instantly receive the items. If the inventory level is equal to zero upon arrival of a primary customer, then this customer, according to the Bernoulli scheme, either leaves the system or goes into an infinite buffer to repeat their request in the future. The rate of retrial customers is constant, and if the inventory level is zero upon arrival of a retrial customer, then this customer, according to the Bernoulli scheme, either leaves orbit or remains in orbit to repeat its request in the future. According to the Bernoulli scheme, each served primary or retrial customer either leaves the system or feedbacks into orbit to repeat their request. Destructive customers that form a Poisson flow cause damage to items. Unlike primary, retrial, and feedback customers, destructive customers do not require items, since, upon arrival of such customers, the inventory level instantly decreases by one. The system adopted one of two replenishment policies: (s, Q) or (s, S). In both policies, the lead time is a random variable that has an exponential distribution. It is shown that the mathematical model of the system under study was a two-dimensional Markov chain with an infinite state space. Algorithms for calculating the elements of the generating matrices of the constructed chains were developed, and the ergodicity conditions for both policies were found. To calculate the steady-state probabilities, a matrix-geometric method was used. Formulas were found for calculating the main performance measures of the system. The results of the numerical experiments, including the minimization of the total cost, are demonstrated.

Application of Mass Service Theory to Economic Systems Optimization Problems—A Review

“…Various studies of the inventory management system are presented in the works of Melikov. These works cover the modeling of the system with the formation of stock shortages due to production delays [10]; the problems of stock shortage and the search for the optimal threshold level of stocks [43]; a warehouse system with catastrophes (in the form of two-dimensional Markov chains) and the optimization of the reorder point [44]; and systems with flows of primary customers, repeat sales customers, repeat service customers, and destructive customers (customers who do not purchase an item but destroy stock, after which stock is reduced by one) [45], among others. The models proposed by the author represent a Markov chain based on random factors and are aimed at minimizing expected total costs and improving the efficiency of the inventory management system.…”

Single-Server Queuing-Inventory Systems with Negative Customers and Catastrophes in the Warehouse

“…(iii) If ϕ 2 = 0 (pure backorder scheme is used), the ergodicity condition is dependent on all indicated parameters of the system even for case λ − = 0, see Formula (14).…”

Modeling of Junior Servers Approaching a Senior Server in the Retrial Queuing-Inventory System

“…Sugapriya et al [31] examined the stock-dependent demand in a retrial QIS. Melikov et al [32] assumed that in the event of a primary customer arriving when the inventory level is zero, this customer would, according to the Bernoulli scheme, either depart from the system or enter an unbounded buffer to reiterate their request at a later time.…”