Analysis of a Batch Arrival, Batch Service Queuing-Inventory System with Processing of Inventory While on Vacation

A single-server queuing-inventory system in which arrivals are governed by a batch Markovian arrival process and successive arrival batch sizes form a finite first-order Markov chain is considered in this paper. Service is provided in batches according to a batch Markovian service process, with consecutive service batch sizes forming a finite first-order Markov chain. A service starts for the next batch on completion of the current service, provided that inventory is available at that epoch; otherwise, there will be a delay in starting the next service. When the service of a batch is completed, the inventory decreases by 1 unit, irrespective of batch size. A control policy in which the server goes on vacation when a service process is frozen until a quorum can initiate the next batch service is proposed to ensure idle-time utilization. During the vacation, the server produces inventory (items) for future services until it hits a specified level L or until the number of customers in the system reaches a maximum service batch size N, with whichever occurring first. In the former case, a server stays idle once the processed inventory level reaches L until the number of customers reaches (or even exceeds because of batch arrival) a maximum service batch size N. The time required for processing one unit of inventory follows a phase-type distribution. In this paper, the steady-state probability vector of this infinite system is computed. The distributions of inventory processing time in a vacation cycle, idle time in a vacation cycle, and vacation cycle length are found. The effect of correlation in successive inter-arrival times and service times on performance measures for such a queuing system is illustrated with a numerical example. An optimization problem is considered. The proposed system is then compared with a queuing-inventory system without the Markov-dependent assumption on successive arrivals as well as service batch sizes using numerical examples. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

Krishnamoorthy A. 1 , Joshua A.N.2 , Kozyrev D. 3, 4
Номер выпуска
  • 1 Centre for Research in Mathematics, C.M.S. College, Kottayam, 686001, India
  • 2 Department of Mathematics, Union Christian College, Aluva, 683102, India
  • 3 Applied Probability and Informatics Department, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russian Federation
  • 4 V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Street, Moscow, 117997, Russian Federation
Ключевые слова
Batch Markovian arrival process; Batch Markovian service process; Markov-dependent arrival and service batches; N−policy; Queuing-inventory system; Vacation
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