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This article studied a queueing-inventory system with batch demands, randomized order policy, and multiple vacations. Customers arrive in this system according to a Poisson process and take away batch items after a service time. The system sends a replenishment order and starts multiple vacations simultaneously once the on-hand inventory level drops to zero. The replenishment is based on a random order size policy. Customers arriving during this period are assumed to be lost. Under the assumption that all the underlying random variables are exponential, the stationary joint probability is obtained in product form. Based on the stationary distribution, the expected sojourn time of a permitted entering customer is derived in recursive form. Several numerical examples are processed to analyze the effect of variant parameters on the performance measures. The numerical results show that the expected total cost function is a convex function with respect to the maximum allowable inventory. The influence of various parameters on the optimal value is also displayed visually. [ABSTRACT FROM AUTHOR]

We consider queueing-inventory models in which customers arrive according to a point process and each customer demands for one or more items but not exceeding a predetermined (finite) value, say, N. The demands of the customers require positive service times. Replenishments are based on (s, S)-type policy and the lead times are assumed to be random. We consider two models. In Model 1, any arriving customer finding the inventory level to be zero will be lost. In Model 2, the loss of customers occur in two ways. First, an arriving customer finding the inventory level to be zero with the server being idle will be lost, and secondly, the customers, if any, present at a service completion with zero inventory will all be lost. We assume that in both the models the demands may be met partially based on the requests and the available inventory levels at that time. The inventory level is reduced by the amount to meet (fully or partially) the requisite demand of the customer at the beginning of the service. Under the assumption that all underlying random variables are exponential, we perform the steady-state analysis of the models using the classical matrix-analytic methods. Illustrative examples comparing the two models are presented. [ABSTRACT FROM AUTHOR]

Combining the study of queuing with inventory is very common and such systems are referred to as queuing-inventory systems in the literature. These systems occur naturally in practice and have been studied extensively in the literature. The inventory systems considered in the literature generally include (s,S)-type. However, in this paper we look at opportunistic-type inventory replenishment in which there is an independent point process that is used to model events that are called opportunistic for replenishing inventory. When an opportunity (to replenish) occurs, a probabilistic rule that depends on the inventory level is used to determine whether to avail it or not. Assuming that the customers arrive according to a Markovian arrival process, the demands for inventory occur in batches of varying size, the demands require random service times that are modeled using a continuous-time phase-type distribution, and the point process for the opportunistic replenishment is a Poisson process, we apply matrix-analytic methods to study two of such models. In one of the models, the customers are lost when at arrivals there is no inventory and in the other model, the customers can enter into the system even if the inventory is zero but the server has to be busy at that moment. However, the customers are lost at arrivals when the server is idle with zero inventory or at service completion epochs that leave the inventory to be zero. Illustrative numerical examples are presented, and some possible future work is highlighted.

This paper studies an M/M/1 queueing-inventory system with batch demands. Customers arrive in the system according to a compound Poisson process, where the size of the batch demands for each arrival is a random variable that follows a geometric distribution. The inventory is replenished according to the standard (s,S) policy. The replenishment time follows an exponential distribution. Two models are considered. In the first model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer takes away all the items in the inventory, and a part of the customer’s batch demands is lost. In the second model, if the on-hand inventory is less than the size of the batch demands of an arrived customer, the customer leaves without taking any item from the inventory, and all of the customer’s batch demands are lost. For these two models, the authors derive the stationary conditions of the system. Then, the authors derive the stationary distributions of the product-form of the joint queue length and the on-hand inventory process. Besides this, the authors obtain some important performance measures and the average cost functions by using these stationary distributions. The results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

The Last Mile Problem refers to the provision of travel service from the nearest public transportation node to a home or office. We study the supply side of this problem in a stochastic setting, with batch demands resulting from the arrival of groups of passengers who request last-mile service at urban rail stations or bus stops. Closed-form approximations are derived for the performance of Last Mile Transportations Systems (LMTS) as a function of the fundamental design parameters of such systems. An initial set of results is obtained for the case wherein a fleet of vehicles of unit capacity provides the Last-Mile service, and each delivery route consists of a simple round trip between the rail station or bus stop and a single passenger's destination. These results are then extended to the general case in which the capacity of a vehicle is a small number (up to 20). It is shown through comparisons with simulation results that the approximations perform consistently well for a broad and realistic range of input values and conditions. These expressions can therefore be used for the preliminary planning and design of an LMTS, especially for determining approximate resource requirements, such as the number of vehicles/servers needed to achieve some prespecified level of service, as measured by the expected waiting time until a passenger is picked up from the station or delivered to her destination. [ABSTRACT FROM AUTHOR]

Combining the study of queuing with inventory is very common and such systems are referred to as queuing-inventory systems in the literature. These systems occur naturally in practice and have been studied extensively in the literature. The inventory systems considered in the literature generally include (s , S) -type. However, in this paper we look at opportunistic-type inventory replenishment in which there is an independent point process that is used to model events that are called opportunistic for replenishing inventory. When an opportunity (to replenish) occurs, a probabilistic rule that depends on the inventory level is used to determine whether to avail it or not. Assuming that the customers arrive according to a Markovian arrival process, the demands for inventory occur in batches of varying size, the demands require random service times that are modeled using a continuous-time phase-type distribution, and the point process for the opportunistic replenishment is a Poisson process, we apply matrix-analytic methods to study two of such models. In one of the models, the customers are lost when at arrivals there is no inventory and in the other model, the customers can enter into the system even if the inventory is zero but the server has to be busy at that moment. However, the customers are lost at arrivals when the server is idle with zero inventory or at service completion epochs that leave the inventory to be zero. Illustrative numerical examples are presented, and some possible future work is highlighted. [ABSTRACT FROM AUTHOR]

• Production-inventory system with batch demands studied by matrix-analytic method. • Losses at arrivals (or also at departures) happen due to inventory exhaustion. • Performance of two types of models compared both analytically and by simulation. • Simulation in multi-server system helps to identify server dependent performance. • Optimization problem in comparing the models. Queueing-inventory models have many practical applications and have been studied extensively in the literature. Most of the studies focus on models in which the demands occur singly. Only a very few papers analyze models wherein the demands occur in batches. In this paper we consider batch demands in the context of two models, both of which assume that the demands occur according to a versatile Markovian point process. The demands need to be serviced with items from the inventory, and the service times are assumed to be of phase type. The replenishment of the inventory is based on the (s, S)-type policy and the lead times are assumed to be random. These two models are such that in the first model an arriving customer finding the inventory level to be zero will be lost; and in the second model a customer can be lost either at the time of an arrival (wherein the server is idle due to zero inventory) or at the time of a service completion (at which time the inventory level becomes zero). In the second model, all waiting customers are removed from the system due to zero inventory. These two models are studied in steady-state using the classical matrix-analytic methods in single server case, and in the case of multi-server systems we resort to simulation using ARENA. Illustrative examples, including an optimization problem, comparing the two models are presented. [ABSTRACT FROM AUTHOR]