Queueing Model with Second Phase Optional Service and Bernoulli Vacation Schedule Using PGF Approach

Abstract Present paper describes with the bulk arrival retrial queueing M / G1, G2, / 1 model with two phase service and Bernoulli vacation schedule wherein first phase service is essential and the next second phase service is optional. If the second phase service is not demanded by arriving customer then the single server takes a vacation period according to Bernoulli vacation schedule in order to utilize it to complete some supplementary work and such type of vacation is assumed working vacation. In the queueing model taken into present consideration, the concepts of Bernoulli vacation schedule and next optional service have been incorporated along with realistic provision that the server has an option to avail a vacation with probability p (q) or may continue to serve the next customer, if any with complementary probability ( ) p q just after the completion of first phase essential service i.e. before the commencement of second phase optional service. In the present paper, our central goal is to investigate the steady state behavior of the bulk arrival retrial queueing M / G1, G2, / 1 model with two phase service and Bernoulli vacation schedule. By introducing supplementary variables, Chapman Kolmogorov equations are established and then the probability generating functions (PGFs) for first phase essential service (FPES), next phase optional service (NPOS) and working vacation and for the number of the customers in the orbit at an arbitrary epoch are investigated successfully. Besides investigating PGF for different states of the queueing system taken into consideration, performance measures such as the long run probabilities for which the server is in idle state, FPES state, NPOS state and in working vacation state are also explored in order to focus the application aspect of the investigated PGFs. By the end of the present paper, some numerical illustrations of the investigated results for M / Ek / 1 model as special case of M / G1, G2, / 1 have been presented in Table 1 and Table 2 for varying parameters.