International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Volume 5 Issue 2, February 2016 www.ijsr.net Licensed Under Creative Commons Attribution CC BY M/D/1 Feedback Queueing Models with Retention of Reneged Customers Dr. S. K. Tiwari 1 , Dr. V. K. Gupta 2 , Tabi Nandan Joshi 3 1 School of Studies in Mathematics, Vikram University, Ujjain, India 2 Govt. Madhav Scinnce Colleges, Vikram University, Ujjain, India 3 School of Studies in Mathematics, Vikram University, Ujjain, India Abstract: Every organization is facing the problem of customer impatience. Customer retention is the key issue in this context. Organizations are applying strategies to sustain their businesses. An impatient customer (due to reneging) may be convinced to stay in service system for his service by utilizing certain convincing mechanisms. Such customers are termed as retained customers. Queueing with feedback represents customer dissatisfaction because of unsuitable quality of service. In case of feedback, after getting partial or incomplete service, customer retries for service. Customer Retention is incorporated in a single-server Markovian feedback queueing model. The steady-state solution of the models is derived. Some useful measures of performance are derived. A particular case of the model is discussed. Keywords: Customer retention, Feedback, Reneging; Steady-State Solution. 1. Introduction A queue, or a waiting line, involves arriving items that wait to be served at the facility which provides the service they seek. Queuing theory is concerned with the statistical description of the behavior of the queues with result, e.g., the probability distribution of the number in the queue from which the mean and variance of queue length and the probability distribution of waiting time for a customer, or the distribution of a server’s busy period can be found. In this paper we have discussed about a steady state solution of the ordered queuing problem with reneging. A customer may enter the queue, but after a time lose patience and decide to leave. In this case the customer is said to have reneged. A unit reneges (i.e., becomes impatient and leaves without having been served) after joining the queue if it is decided that the wait will be longer than can be tolerated. Here the waiting line is of Poisson balking probability which depend not only on the number of customers in the system, but also on the rate of service in the system. A queuing situation with the following characteristics has been considered. A customer receives the service immediately, when the system is empty. But a joining customer that has to wait for service due to impatience may leave i.e., the customer may renege. Service is performed on the customer at the head of the line. One service has commenced on a customer, it remains until the completion of service. Barrer [7] has studied the problem of a unit leaving a queue after having waited longer than an acceptable time. O. Brien [9] has also found the solution of some queuing problem. Miller [20] and Konigsberg [8] have also studied about balk queue and queuing with special service. In this paper we have attempted to find out a steady state solution of queue, when the Poisson probabilities depend not only on the number of the customers in the system. Customers are the backbone of any business, because without customers there will be no reason for a business to operate. Customer impatience leads to loss of potential customers. It has become a highly challenging problem in the current era of cut-throat competition. Queueing with customer impatience has special significance for the business world as it has a very negative effect on the revenue generation of a firm. Therefore, the concept of customer retention assumes a tremendous importance for the business management. Customer retention is the key issue in the organizations facing the problem of customer impatience. Firms are employing a number of customer retention strategies to maintain their businesses. An impatient customer (due to reneging) may be convinced to stay in service system for his service by utilizing certain persuasive mechanisms. Such customers are termed as retained customers. When a customer gets impatient, he may leave the queue with some probability, say  2 and may remain in the queue for service with some complementary probability  2 (=1− 2 ). Recently, Kumar and Sharma [14] study the retention of reneged customers in an M/M/1/N queuing model and perform sensitivity analysis of the model. Kumar and Sharma [15] study M/M/1/N queuing system with retention of reneged customers and balking. They extend the work of Kumar and Sharma [14] by taking balking aspects in their model to study the effect of probability of retaining the reneged customers on expected system size. They perform the sensitivity analysis of the model. We assume that after the completion of service, each customer may rejoin the system as a feedback customer for receiving another regular service with probability p 1 and may not join with complementary probability 1- p 1 . Sharma and Kumar [21] study single-server finite capacity, Markovian queue with feedback and retention of reneged customers. They perform steady-state analysis of the model. Paper ID: SUB157725 405