Malaya Journal of Matematik, Vol. 7, No. 4, 795-807, 2019 https://doi.org/10.26637/MJM0704/0027 Analysis of an M/G/1 retrial queue with second optional service and customer feedback, under Bernoulli vacation schedule S. Pavai Madheswari 1 and S.A. Josephine 2 * Abstract A single server retrial queueing system with second optional service under Bernoulli vacation schedule is investigated. The customer is permitted to balk if his service is not immediate upon arrival and allowed to join the orbit for repeating his service. Instead, if the server is free the customer’s service is started immediately. Every customer is provided with a first phase of essential service followed by a second phase of optional service. After a service completion if the system is found to be empty then the server begins a vacation period. On the other hand if the system is not empty, the server chooses to either continue serving the customer with probability (1 − a) or goes on vacation with probability a(0 ≤ a ≤ 1). After a service completion, a customer opts to either exit the system or chooses to join the orbit for repeating service. The joint generating functions of orbit size and server status are derived using supplementary variable technique. Some important performance measures have been derived and the effect of various parameters on the system performance has been analysed numerically. Stochastic decomposition law has been established in the absence of balking. Keywords Retrial queue, Balking, Second optional service, Bernoulli vacation, Feedback. AMS Subject Classification 60K25, 90B22, 68M20. 1 Department of Mathematics, R.M.K. Engineering College, Kavaraipettai-601206, Tamil Nadu, India. 2 Department of Mathematics, Loyola-ICAM College of Engineering and Technology, Chennai-600034, Tamil Nadu, India. *Corresponding author: 1 pavai.arunachalam@gmail.com; 2 josemat4142@gmail.com Article History: Received 24 March 2019; Accepted 09 October 2019 c 2019 MJM. Contents 1 Introduction ....................................... 795 2 Model description and stability condition ........ 796 3 Steady state distribution .......................... 798 4 Performance measures ........................... 802 5 Stochastic decomposition ........................ 804 6 Numerical illustrations ............................ 804 7 Conclusion ........................................ 806 References ........................................ 806 1. Introduction Retrial queueing system is characterised by the phenomenon that an arriving customer who finds the server busy upon ar- rival may join the virtual group of blocked customers, called orbit and retry for service after a random amount of time. Study of retrial queues has gained more importance due to the potential real time applications in telephone services, com- puter and communication networks. In the so called classical retrial policy the interval between successive repeated cus- tomers are exponentially distributed with rate nθ when the number of customers in the retrial group i.e., orbit size is n, studied by Falin [5]. However, there is a second kind of discipline, in which intervals separating successive repeated attempts are independent of the orbit size. This policy is known as the constant retrial policy. The latter discipline was introduced by Fayolle [7], who investigated an M/M/1 retrial queue in which repeated customers form a queue and only the customers at the head of the orbit can request a service after an exponentially distributed retrial time. Farah- mand [6] calls this discipline a retrial queue with FCFS or- bit. Artalejo and Gomez-Corral [1] introduced the linear retrial policy by incorporating both the possibilities assuming