Markov Models of Systems with Demands of Two Types and Different Restocking Policies
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SYSTEMS ANALYSIS MARKOV MODELS OF SYSTEMS WITH DEMANDS OF TWO TYPES AND DIFFERENT RESTOCKING POLICIES A. Z. Melikov,1 L. A. Ponomarenko,2 and I. A. Aliyev3
UDC 519.21:658.7
Abstract. Markov models of queuing-inventory systems with demands of two types are proposed. In these systems, two restocking policies are used: policies with fixed and variable demand size. High-priority demands have no access constraints while low-priority demands are accepted if the total number of demands in the system is less than a given threshold. Methods are developed to calculate performance measures of the systems under study and problems of their optimization are solved. Results of the numerical experiments are shown. Keywords: queuing-inventory system, restocking policy, demands of different types, calculation method, optimization. INTRODUCTION Inventory systems with a server, where inventory demands arrive at random instants of time are called queuing-inventory systems (QIS). For the first time, term QIS was used in [1] though the fundamentals of studying such systems can be found in earlier studies [2–4]. A review of publications devoted to different aspects of the analysis of such systems is presented in [5]. Over the last years, these systems are being intensively analyzed. In the majority of studies, the authors consider that demands are identical in all the parameters: size (i.e., volume of stock they demand), service time, importance, etc. However, in practice, suppliers distinguish their clients. For example, clients may be of different size, be permanent or occassional, some of them may pay for the same goods more than other clients do, etc. In such cases, to encourage favourable clients, suppliers use different schemes for their priority service. Though QIS with demands of different types are often used, they are poorly studied. For example, [6–11] analyze models of QIS with instant service and two demand classes when ( S -1, S ) restocking policy (RP) is used. Similar models where ( s, S ) -policy is used are studied in [12, 13]. In [14], a model of QIS with two types of customers and ( s, S ) -policy is analyzed, where demands of low priority go to an orbit of infinite size if system’s inventory level at the moments of their arrival is less than s; high-priority demands are accepted if the inventory level in the system is greater than zero. (For publications in this field, see references in the above-mentioned studies). A model of QIS with demands of different types and positive service time is analyzed in [15]. In the system, K ³ 2 types of Poisson flows of demands are serviced, and a demand of ith type with probability q i will require inventory bi , 1 £ bi £ S , where S is the maximum size of system’s warehouse, and q1 + q 2 + K + q K = 1 . In [15], a randomized RP is considered and the problem of finding optimal demand 1
Institute of Control Systems, National Academy of Sciences of Azerbaijan, Baku, Azerbaijan, [email protected]. 2International Scientific and Training Center of Information Technologies and
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