mathematics Article The Stability Analysis of a Double-X Queuing Network Occurring in the Banking Sector Hong Zhang 1 , Saviour Worlanyo Akuamoah 2 , Wilson Osafo Apeanti 3 , Prince Harvim 4 , David Yaro 5 and Paul Georgescu 6, *   Citation: Zhang, H.; Akuamoah, S.W.; Apeanti, W.O.; Harvim, P.; Yaro,D.; Georgescu, P. The Stability Analysis of a Double-X Queuing Network Occurring in the Banking Sector. Mathematics 2021, 9, 1957. https://doi.org/10.3390/ math9161957 Academic Editor: Mark Kelbert Received: 7 July 2021 Accepted: 11 August 2021 Published: 16 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 School of Economics and Management, Changzhou Institute of Technology, Changzhou 213032, China; zhanghong2018@czu.cn 2 Department of Mathematics and Statistics, Ho Technical University, Ho Volta Region P.O. Box HP 217, Ghana; wakuamoah@htu.edu.gh 3 Faculty of Science, University of Education, Winneba P.O. Box 25, Ghana; woapeanti@uew.edu.gh 4 Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada; princeharvim@yahoo.com 5 Department of Mathematics and Statistics, School of Applied Sciences and Technology, Cape Coast Technical University, Cape Coast P.O. Box DL 50, Ghana; ortaega36@yahoo.com 6 Department of Mathematics, Technical University of Ia¸ si, Bd. Copou 11A, 700506 Ia¸ si, Romania * Correspondence: vpgeo@tuiasi.ro Abstract: We model a common teller–customer interaction occurring in the Ghanaian banking sector via a Double-X queuing network consisting of three single servers with infinite-capacity buffers. The servers are assumed to face independent general renewal of customers and independent identically distributed general service times, the inter-arrival and service time distributions being different for each server. Servers, when free, help serve customers waiting in the queues of other servers. By using the fluid limit approach, we find a sufficient stability condition for the system, which involves the arrival and service rates in the form of a set of inequalities. Finally, the model is validated using an illustrative example from a Ghanaian bank. Keywords: Double-X cascade network; Lyapunov function; fluid limit approach; interacting server 1. Introduction The competition resulting from a decade of deregulation in the Ghanaian banking industry is now becoming more intense, due to the regulatory imperatives of worldwide banking and to the customers becoming increasingly aware of their rights [1,2]. On the one hand, bank customers have become increasingly demanding, requiring high quality, low priced and immediate service delivery [3], a key development being the entry of private banks into the market and the expansion of branches of existing banks. On the other hand, banks have found new methods to differentiate their products and services to attract new customers [4]. Queuing theory aims at developing mathematical and numerical models to investigate the formation and congestion of waiting lines when a service is requested, its basic ideas being proposed by A. K. Erlang. In this regard, there is a high probability for one to queue at least once during everyday activities before a service is rendered [5,6]. Common experience suggests that the act of queuing is associated with waiting, which is an inevitable part of modern life. This leads to the idea that queuing theory has very wide applications [7]. For example, customers waiting to be served at grocery stores, banks and post offices, people having to wait for an operator during a telephone call, people waiting for a taxi or a bus on the way to the workplace can be thought as subjects of queuing theory [8,9]. Even queues that are generally non-problematic may become a nuisance if not man- aged well. For the banking sector, this requires a swift management of the server–customer relationship since queuing cannot be excluded from the daily operation of a bank [10,11]. Mathematics 2021, 9, 1957. https://doi.org/10.3390/math9161957 https://www.mdpi.com/journal/mathematics