Asymptotic analysis for waiting time in a retrial queue with multiple input streams

Under the classical retrial policy, we consider a single-server M/G/ 1 queue with multiple input streams and orbits. Different types of customers have corresponding arrival rates, general distributions of service time and retrial rates. Assume that the retrial rates for different types of customers linearly converge to zero. We firstly derive the first-order asymptotics of the orbit queue lengths. Subsequently, we find that the joint asymptotic distribution of the number of retrials follows a multidimensional geometric distribution. Finally, we obtain the joint asymptotic distribution of waiting times, which follows a multidimensional exponential distribution. This result indicates that the waiting times for different types of customers are independent of each other.