Analysis and Application of a Batch Arrival Queueing Model with the Second Optional Service and Randomized Vacation Policy

This paper aims to investigate M[X]/(G1, G2)/1/VAC(J) queuing system with a random(p) vacation policy and optional second service, where X is the batch arrival number of customers. When no customers are in system, server immediately goes on vacation. And when returns from finds that at least one customer waiting will provide First Essential Service (FES). After complete first essential some continue receive Second Optional service (SOS). completes FES, choose accept additional equipment adjustment or maintenance (the probability θ). In addition, for be idle p enter but there (1-p) pattern until vacations reaches J times. Suppose after Jth system; always service. consider servers unreliable can repaired immediately, establish supplementary variables as well use construct Kolmogorov forward equation governs then variable techniques derive expected customers, time other important characteristics proposed queueing system. The relevant results used performance evaluation decision-making tools require secondary services regular practical applications models.