ISSN 0005-1179, Automation and Remote Control, 2019, Vol. 80, No. 1, pp. 53–65. c  Pleiades Publishing, Ltd., 2019. Russian Text c  A.Z. Melikov, M.O. Shahmaliyev, 2019, published in Avtomatika i Telemekhanika, 2019, No. 1, pp. 67–82. STOCHASTIC SYSTEMS Queueing System M/M/1/∞ with Perishable Inventory and Repeated Customers A. Z. Melikov ∗,a and M. O. Shahmaliyev ∗∗,b ∗ Institute of Control Systems, Azerbaijan National Academy of Sciences, Baku, Azerbaijan ∗∗ National Aviation Academy, Baku, Azerbaijan e-mail: a agassi.melikov@gmail.com, b mamed.shahmaliyev@gmail.com Received April 9, 2018 Revised July 23, 2018 Accepted November 8, 2018 Abstract—We propose a model of a queueing system with a single server, perishable inventory and repeated customers that can form an orbit of infinite size. In the absence of inventory in the system, primary customers according to the Bernoulli scheme either enter the queue or go into the orbit. The system uses the (s, S)-policy of replenishing the inventory. We develop a method for calculating system characteristics and solve the problem of minimizing total costs by choosing the critical level of inventory. Keywords : queueing-inventory systems, perishable inventory, repeated customers, computation algorithm, optimization DOI: 10.1134/S0005117919010053 1. INTRODUCTION The first publications devoted to the study of queuing-inventory systems (QIS) with durable inventory in the presence of repeated customers were [1, 2]. In these works, the authors, indepen- dently from each other and using different approaches, studied the Markov model of a QIS with instant servicing and (s, S )-policy of replenishment. In this model, it is assumed that if at the mo- ment of receiving the primary customer (p-customer) there are no inventory in the system, then it enters an orbit of infinite size; repeated customers from the orbit (r-customers) repeat an attempt to be serviced after some random time that has an exponential distribution function (CDF). As a mathematical model of the considered system, these works used a two-dimensional Markov chain (MC), while in [1] the method of generating functions was used to find the system characteristics, and the work [2] used the Laplace transform method. Similar models with a finite orbit size for persistent r-customers and (S − 1,S )-policies were studied in [3, 4]. QIS models with durable inventory and positive servicing time for customers (i.e., customer servicing time greater than zero) in the presence of repeated customers have been studied in [5–9]. In particular, the work [5] studied three classes of Markov QIS models with a single server, r-customers and (s, S )-policy of replenishment. For each type of models, it constructs the corresponding three- dimensional MCs and shows that they have a three-diagonal generating matrix (GM); the matrix- geometric method [10] was applied to find stationary distributions and characteristics of the studied QIS models are found. The work [6] considered a Markov QIS model with the (s, S )-policy, where servers of the system are considered as inventory, and the inventory are replenished instantly. If at the time of a customer’s arrival all servers of the system are busy, then the customer goes into an infinite size orbit. In [6], the corresponding three-dimensional MC was constructed and system characteristics were computed. The work [7] considers a Markov QIS model with a finite queue 53