Present-day queuing inventory systems (QIS) do not utilize two multi-server service channels. We proposed two multi-server service channels referred to as $T1S$ (Type 1 n-identical multi-server) and $T2S$ (Type 2 m-identical multi-server). It includes an optional interconnected service connection between $T1S$ and $T2S$, which has a finite queue of size N. An arriving customer either uses the inventory (basic service or main service) for their demand, whom we call $T1$, or simply uses the service only, whom we call $T2$. Customer $T1$ will utilize the server $T1S$, while customer $T2$ will utilize the server $T2S$, and $T1$ can also get the second optional service after completing their main service. If there is a free server with a positive inventory, there is a chance that $T1$ customers may go to an infinite orbit whenever they find that either all the servers are busy or no sufficient stock. The orbital customer can request for $T1S$ service under the classical retrial policy. $Q(=S-s)$ items are replaced into the inventory whenever it falls into the reorder level s such that the inequality always holds $n<s$. We use the standard $(s,Q)$ ordering policy to replace items into the inventory. By varying S and s, we investigate to find the optimal cost value using stationary probability vector $\varphi$. We used the Neuts Matrix geometric approach to derive the stability condition and steady-state analysis with R-matrix to find $\varphi$. Then, we perform the waiting time analysis for both $T1$ and $T2$ customers using Laplace transform technique. Further, we computed the necessary system characteristics and presented sufficient numerical results.

This paper deals with an integrated and interconnected stochastic queuing-inventory system with a fresh item, a returned item, and a refurbished item. This system provides a multi-type service facility to an arriving multi-class customer through a dedicated channel. It sells fresh and refurbished items, buys used items from customers, refurbishes the used items for resale, and provides a repair service for defective items. The assumption of purchasing a used item from the customer and allowing them to buy a fresh item is a new idea in stochastic queuing-inventory modeling. To do so, this system has four parallel queues to receive four classes of customers and five dedicated servers to provide a multi-type service facility. Customers are classified according to the type of service they require. Each class of arrival follows an independent Poisson process. The service time of each dedicated server is assumed to be exponentially distributed and independent. This system assumes an instantaneous ordering policy for the replenishment of a fresh item. In the long run of this considered system, the joint probability distribution of the seven-dimensional stochastic process, significant system performance measures, and the optimum total cost are to be derived using the Neuts matrix geometric technique. The main objective of the system was to increase the occurrence of all kinds of customers by providing a multi-type service facility in one place. Buying a used item is unavoidable in an emerging society because it helps form a green society. Furthermore, the numerical result shows that the assumption of a system that allows a customer to sell their used item and purchase a new item will increase the number of customers approaching the system.

This article analyses a four-dimensional stochastic queueing-inventory system with multiple server vacations and a state-dependent arrival process. The server can start multiple vacations at a random time only when there is no customer in the waiting hall and the inventory level is zero. The arrival flow of customers in the system is state-dependent. Whenever the arriving customer finds that the waiting hall is full, they enter into the infinite orbit and they retry to enter the waiting hall. If there is at least one space in the waiting hall, the orbital customer enters the waiting hall. When the server is on vacation, the primary (retrial) customer enters the system with a rate of ${\lambda }_1\left({\theta }_1\right)$. If the server is not on vacation, the primary (retrial) arrival occurs with a rate of ${\lambda }_2\left({\theta }_2\right)$. Each arrival rate follows an independent Poisson distribution. The service is provided to customers one by one in a positive time with the rate of $\mu$, which follows exponential distribution. When the inventory level drops to a fixed s, reorder of Q items is triggered immediately under $(s,Q)$ ordering policy. The stability of the system has been analysed, and using the Neuts matrix geometric approach, the stationary probability vectors have been obtained. Moreover, various system performance measures are derived. The expected total cost analysis explores and verifies the characteristics of the assumed parameters of this model. The average waiting time of a customer in the waiting hall and orbit are investigated using all the parameters. The monotonicity of the parameters is verified with its characteristics by the numerical simulation. The discussion about the fraction time server being on vacation suggests that as the server’s vacation duration reduces, its fraction time also reduces. The mean number of customers in the waiting hall and orbit is reduced whenever the average service time per customer and average replenishment time are reduced.

The use of a Markovian inventory system is a critical part of inventory management. The purpose of this study is to examine the demand for two commodities in a Markovian inventory system, one of which is designated as a major item (Commodity-I) and the other as a complimentary item (Commodity-II). Demand arrives according to a Poisson process, and service time is exponential at a queue-dependent rate. We investigate a strategy of $(s,Q)$ type control for commodity-I with a random lead time but instantaneous replenishment for commodity-II. If the waiting hall reaches its maximum capacity of N, any arriving primary client may enter an infinite capacity orbit with a specified ratio. For orbiting consumers, the classical retrial policy is used. In a steady-state setting, the joint probability distributions for commodities and the number of demands in the queue and the orbit, are derived. From this, we derive a waiting time analysis and a variety of system performance metrics in the steady-state. Additionally, the physical properties of various performance measures are evaluated using various numerical assumptions associated with diverse stochastic behaviours.

This study presents and discusses the home delivery services in stochastic queuing-inventory modeling (SQIM). This system consists of two servers: one server manages the inventory sales processes, and the other server provides home delivery services at the doorstep of customers. Based on the Bernoulli schedule, a customer served by the first server may opt for a home delivery service. If any customer chooses the home delivery option, he hands over the purchased item for home delivery and leaves the system immediately. Otherwise, he carries the purchased item and leaves the system. When the delivery server returns to the system after the last home delivery service and finds that there are no items available for delivery, he goes on vacation. Such a vacation of a delivery server is to be interrupted compulsorily or voluntarily, according to the prefixed threshold level. The replenishment process is executed due to the ($s,Q$) reordering policy. The unique solution of the stationary probability vector to the finite generator matrix is found using recursive substitution and the normalizing condition. The necessary and sufficient system performance measures and the expected total cost of the system are computed. The optimal expected total cost is obtained numerically for all the parameters and shown graphically. The influence of parameters on the expected number of items that need to be delivered, the probability that the delivery server is busy, and the expected rate at which the delivery server’s self and compulsory vacation interruptions are also discussed.