A Finite Capacity Finite Source Single Server Markovian Queueing System with Discouraged Arrivals Retention of Reneged Customers and Controllable Arrival Rates with Feedback

Objectives: The purpose of this study is to (i) introduce a finite capacity finite source of interdependent queueing model and retention of reneged customers with feedback controllable arrival rates, (ii) compute the mean quantity of clients within the system and determine the anticipated waiting time of those clients, (iii) deal with the model descriptions, steady-state equations, and characteristics, which are expressed in terms and (iv) analyze the probabilities of the queueing system and its characteristics with numerical verification of the obtained results. (v) Observe graphical representation for Methods : The Poisson process is used to manage the arrival rates through quicker and slower arrival rates while delivering the input. The service additionally offers an exponential distribution. The service is supplied by the server using FCFS. In this article, two types of models are used: one that represents the system’s conditions by the number of units present in the queue, and another that distributes all probabilities based on the speed of arrival using this concept. Then, the steady-state probabilities are computed using a recursive approach. Findings: This paper explores the average number of clients using the system and the expected number of clients in the system, based on the probability derived from the steady-state calculation. Little’s formula is used to derive the expected waiting period of the clients in the system. Novelty: There are articles connected to the finite capacity of failed service in functioning and malfunctioning, but this takes the initiative to provide a link in connection with the rates of the controllable arrivals and interdependency in the arrival and service processes. This is the new model of M/M/1/K/N controllable arrival rates derived. Mathematics Subject allocation: 60K25, 68M20, 90B22. Keywords: Finite Capacity, Finite Source, Interdependent Controllable, Arrival and Service rates, Reverse balking, Single Server, Bivariate Poisson process and feedback