M\(^{\mathrm{X}}\)/G/1 queuing model with state dependent arrival and second optional vacation (Q1753741)

Summary: This investigation deals with single server state dependent queuing systems, wherein the arrivals of units are in batches and follow the Poisson process with state dependent arrival rates. After availing of the First Regular Vacation (FRV) in a case when there is no customer in the system, the server may also take a Second Optional Vacation (SOV). By using supplementary variable techniques, the probability generating function of the queue length distribution is established to study various performance measures. The maximum entropy approach is also used to find queue length distribution for evaluation of steady state probabilities in all different states. Numerical illustrations are provided to verify the tractability of performance measures obtained analytically.