Malaya Journal of Matematik, Vol. 6, No. 3, 664-677, 2018 https://doi.org/10.26637/MJM0603/0031 Analysis of an M [X ] /G 1 (a, b), G 2 (a, b)/1 unreliable G-queue with optional re-service, Bernoulli vacation, delay time to two phase of repair G. Ayyappan 1 and R. Supraja 2 * Abstract In this paper, we consider the queueing system where the batch of customers arrive at the system according to the compound Poisson process and two types of service, each of which has an optional reservice is provided to the server under Bernoulli vacation. After completion of each type of service, the customer may go for reservice of the same type of service without joining the tail of the queue or they may depart the system. An unpredictable breakdown may occur at any moment during the functioning of the server with any type of service or re-service and at that situation, the service channel will breakoff for a short period of time. A breakdown in a busy server is represented by the arrival of a negative customer which consequently leads to the loss of the customer who is in service. Delay time is referred to as the waiting time of the server for the two phase of repair to start. By considering elapsed service time as the supplementary variable, the PGF of the number of customers in the queue at a random epoch is derived and this PGF is further used to establish explicitly some of the following performance measures namely various states of the system, the mean queue length, and the mean waiting time in the queue. At last, some particular cases are discussed and the numerical illustrations are provided. Keywords Two types of service, Re-service, Bernoulli vacation, G-queue, Delay time to repair, Two phase of repair time. AMS Subject Classification (2010) 60K25, 90B22, 68M20. 1,2 Department of Mathematics, Pondicherry Engineering College, Pillaichavady, Puducherry - 605 014, India. *Corresponding author: 1 ayyappan@pec.edu; 2 suprajaa11@gmail.com Article History: Received 24 July 2018; Accepted 15 September 2018 c 2018 MJM. Contents 1 Introduction ....................................... 664 2 Model Description ................................. 665 3 Equations Governing the Systems ............... 666 4 The steady state results ........................... 671 4.1 Queue size distribution at a random epoch . . 672 5 Stability condition ................................. 673 6 Performance measures ........................... 673 6.1 System state probabilities ............... 674 6.2 Mean queue size ..................... 674 7 Particular cases ................................... 675 8 Numerical results ................................. 676 9 Conclusion and further work ...................... 677 References ........................................ 677 1. Introduction A considerable amount of work has been done on the modelling and analysis for the queueing system using the sup- plementary variable technique where the service is rendered in bulk. Most of the queueing models assume that customers are served singly which is a contradiction to some of the real- life situations where the service is provided in bulk. Bulk service queue was first dealt by Bailey [2]. The “General Bulk Service Rule ” (GBSR) was proposed by Neuts [13] in which service initiates only when a certain number of customers in the queue are available. A detailed survey on bulk queueing models can be seen in the studies of Chaudhury and Temple- ton [3]. Lee et al. [11] discussed the decompositions of the batch service queue with server vacations. Recently, Haridass and Arumuganathan [8] studied a batch arrival general bulk service queueing system by considering the supplementary variable as the remaining service time.