Mathematics and Statistics 11(4): 654-660, 2023 DOI: 10.13189/ms.2023.110406 http://www.hrpub.org A New Type of Single Server Queue Operating in A Multi-level Environment with Customer Impatience Akshaya Ramesh, S. Udayabaskaran * Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, India Received January 23, 2022; Revised May 26, 2023; Accepted June 9, 2023 Cite This Paper in the following Citation Styles (a): [1] Akshaya Ramesh, S. Udayabaskaran, ”A New Type of Single Server Queue Operating in A Multi-level Environment with Customer Impatience,” Mathematics and Statistics, Vol.11, No.3, pp. 654-660, 2023. DOI: 10.13189/ms.2023.110406 (b): Akshaya Ramesh, S. Udayabaskaran, (2023). A New Type of Single Server Queue Operating in A Multi-level Environment with Customer Impatience. Mathematics and Statistics, 11(3), 654-660. DOI: 10.13189/ms.2023.110406 Copyright ©2023 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract A new type of single server queue is considered. In this type, the server asks for an assignment in a multi-level environment and the customer develops impatience during the assignment process. The environment has N levels and the server is assigned to operate in one of these levels with level dependent arrival and service rates. Customers arrive at the system all the time and there is an infinite buffer with the system. The assignment is done by a random switch which can initiate an assignment process only if at least one customer is in the system. The server working in any level of the environment reports to the random switch after serving the last customer in that level. Customers are not flushed out at any time. The random switch initiates an assignment process immediately at the epoch of arrival of a customer to the system. Assignment time is random and during the assignment period, customers are permitted to join the system. Once the assignment process starts, each customer waiting in the buffer clicks on a random impatience timer with him/her and leaves the system in case his/her timer ends before the assignment to the server is made. For this model, steady-state probabilities are found and a performance analysis is also made. Keywords Single Server Queue, Multi-level Environment, Assignment, Random Switch, Customer Impatience 1 Introduction Customer impatience is an inevitable phenomenon which arises in studies of queueing systems with buffer. It arises in the mind of an arriving customer who can not tolerate prolonged waiting in the queue. A huge amount of research on stochastic models of queues with customer impatience is already avail- able in the literature. In his research on congestion in telephone traffic, Palm [1] developed the study of queueing systems sub- ject to customer impatience. He used an M/M/c/∞ queueing model where each one of the customers in the buffer switches on an exponentially distributed impatience timer immediately upon arrival and abandons the queueing system once for all at the epoch when the timer expires before getting service. Palm’s paper triggered a huge variety of problems on queues with customer impatience. Customer impatience in an M/M/1/N queueing model was studied by Ancker and Gafarian [2, 3] by assuming exponentially distributed impatience timer. Daley [4] studied a GI/G/1 queue with general customer impatience. Takacs [5] analysed an M/G/1 queue where customers in the buffer adopt fixed deadlines for waiting in the queue. Baccelli et al. [6] considered GI/G/1 queues with independent and identically distributed impatience times. Boxma and deWall [7] studied multi-server queues with customer impatience. Customer impatience in queueing systems may also arise when the server is unavailable due to several reasons such as vacation or failure. In many situations, customers are al- lowed to join the buffer even though the server is on vaca- tion or undergoing repair due to failure. These customers de- velop impatience as before which results in customer loss. Alt- man and Yechiali [8] analysed customer impatience in queues with server going on vacation. Yechiali [9] studied queues with system disasters and customer getting impatient when the server is undergoing repair. Perel and Yechiali [10] analysed an M/M/c queueing system in a two-phase (fast and slow) Markovian environment where customers in the buffer become impatient in slow phase. Sudhesh [11] provided a transient analysis of an M/M/1 queue with system disasters and cus- tomer impatience. Ammar [12] provided a transient analy- sis of a two-heterogeneous servers queue with customer impa- tience. Ammar [13] studied the transient analysis of impatient