Queuing-Inventory Models with MAP Demands and Random Replenishment Opportunities
12 May 2021-Vol. 9, Iss: 10, pp 1092
TL;DR: This paper looks 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.
Abstract: 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.
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TL;DR: In this article , double-source queuing-inventory systems are studied in the presence of a finite buffer for waiting in the queue of consumer customers, where instant destruction of inventory is possible.
Abstract: Models of double-source queuing-inventory systems are studied in the presence of a finite buffer for waiting in the queue of consumer customers, where instant destruction of inventory is possible. It is assumed that the lead times of orders, as well as the cost of delivery from various sources, differ from each other. Replenishment of stocks from various sources is carried out according to the following scheme: if the inventory level drops to the reorder point s, then a regular order for the supply of inventory to a slow source is generated; if the inventory level falls below a certain threshold value r, where r < s, then the system instantly cancels the regular order and generates an emergency order to the fast source. Models of systems that use (s, S) or (s, Q) replenishment policies are studied. Exact and approximate methods for finding the performance measures of the models under study are proposed. The problems of minimizing the total cost are solved by choosing the appropriate values of the parameters s and r when using different replenishment policies. Numerical examples demonstrated the high accuracy of an approximate method as well as compared performance measures of the system under various replenishment policies.
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Journal Article•
TL;DR: In this article, the authors consider a single-product fluid-inventory model in which the procurement price of the product fluctuates according to a continuous time Markov chain and derive the associated steady-state distributions and cost functionals.
Abstract: We study a single-product fluid-inventory model in which the procurement price of the product fluctuates according to a continuous time Markov chain. We assume that a fixed order price, in addition to state-dependent holding costs are incurred, and that the depletion rate of inventory is determined by the sell price of the product. Hence, at any time the controller has to simultaneously decide on the selling price of the product and whether to order or not, taking into account the current procurement price and the inventory level. In particular, the controller is faced with the question of how to best exploit the random time windows in which the procurement price is low. We consider two policies, derive the associated steady-state distributions and cost functionals, and apply those cost functionals to study the two policies.© 2017 Wiley Periodicals, Inc. Naval Research Logistics, 2017
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TL;DR: In this paper , a continuous review finite capacity production-inventory system with two products in inventory is analyzed, where the inventory is replenished using an independent order-up-to (s, S) policy or a can-order (c, S, C) joint policy in which the endogenously determined lead times drive the parameters of the replenishment policy.
Abstract: Abstract In this paper we analyze a continuous review finite capacity production-inventory system with two products in inventory. With stochastic order quantities and time between orders, the model reflects a supply chain that operates in an environment with high levels of volatility. The inventory is replenished using an independent order-up-to (s, S) policy or a can-order (s, c, S) joint replenishment policy in which the endogenously determined lead times drive the parameters of the replenishment policy. The production facility is modeled as a multi-type MMAP[K]/PH[K]/1 queue in which there are K possible inventory positions when the order is placed and the age process of the busy queue has matrix-exponential distribution. We characterize the system and determine the steady state distribution using matrix analytic methods. Using numerical methods we obtain the inventory parameters that minimize the total ordering and inventory related costs. We present numerical comparisons of independent and joint replenishment policies with varying lead times, order quantities, and cost reductions. We further demonstrate the interplay between the two products in terms of lead times, order quantities and costs.
1 citations
TL;DR: In this paper , the authors study an inventory control problem with two storage facilities: a primary warehouse (PW) of limited capacity M and a subsidiary one (SW) of sufficiently large capacity S, and derive the optimal thresholds that minimize the expected overall cost under the discounted criterion.
Abstract: We study an inventory control problem with two storage facilities: a primary warehouse (PW) of limited capacity M, and a subsidiary one (SW) of sufficiently large capacity. Two types of customers are considered: individual customers arriving at (positive and negative) linear rates governed by a Markov chain, and retailers arriving according to a Markov arrival process and bringing a (positive and negative) random number of items. The PW is managed according to a triple-parameter band policy (M,S,s),0≤s
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TL;DR: In this paper, a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond is presented, which is a book of applicable mathematics, which avoids the approach of listing only the techniques, followed by a few examples.
Abstract: This is a mathematical text suitable for students of engineering and science who are at the third year undergraduate level or beyond. It is a book of applicable mathematics. It avoids the approach of listing only the techniques, followed by a few examples, without explaining why the techniques work. Thus, it provides not only the know-how but also the know-why. Equally, the text has not been written as a book of pure mathematics with a list of theorems followed by their proofs. The authors' aim is to help students develop an understanding of mathematics and its applications. They have refrained from using clichés like “it is obvious” and “it can be shown”, which may be true only to a mature mathematician. On the whole, the authors have been generous in writing down all the steps in solving the example problems.The book comprises ten chapters. Each chapter contains several solved problems clarifying the introduced concepts. Some of the examples are taken from the recent literature and serve to illustrate the applications in various fields of engineering and science. At the end of each chapter, there are assignment problems with two levels of difficulty. A list of references is provided at the end of the book.This book is the product of a close collaboration between two mathematicians and an engineer. The engineer has been helpful in pinpointing the problems which engineering students encounter in books written by mathematicians.
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TL;DR: This chapter discusses quasi-Birth-and-Death Processes, a large number of which are based on the Markovian Point Processes and the Matrix-Geometric Distribution, as well as algorithms for the Rate Matrix.
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01 Jan 1991
TL;DR: This work generalizes results to the single server queue with the batch arrival process and emphasizes the resulting simplifications of the computational complexity of the algorithmic solution of single server queues with a general Markovian arrival process.
Abstract: The versatile Markovian point process was introduced by M. F. Neuts in 1979. This is a rich class of point processes whichcontains many familiar arrival process as very special cases. Recently, the Batch Markovian Arrival Process, a class of point processes which was subsequently shown to be equivalent to Neuts’ point process, has been studied using a more transparent notation. Recent results in the matrix-analytic approach to queueing theory have substantially reduced the computational complexity of the algorithmic solution of single server queues with a general Markovian arrival process. We generalize these results to the single server queue with the batch arrival process and emphasize the resulting simplifications. Algorithms for the special cases of the PH/G/l and MMPP/G/1 queues are highlighted as these models are receiving renewed attention in the literature and the new algorithms proposed here are simpler than existing ones. In particular, the PH/G/1 queue has additional structure which further enh...
1,038 citations
962 citations