In this paper, we present a diffusion limit for the timedependent distribution of the number of customers in the orbit for a tandem queueing system with one orbit, Poisson arrival process of incoming calls and two sequentially connected servers using a characteristic function approach. Under the condition that the mean time of a customer in the orbit tends to infinity, the number of customers in the orbit explodes. Using a proper scaling, we prove that the scaled version of the number of customers in the orbit asymptotically follows a diffusion process. Using the steady-state solution of the diffusion process, we build an approximation for the steady-state distribution of the number of customers in the orbit. We compare this new approximation with the traditional approximation based on the central limit theorem and with simulation. Numerical results show that the new approximation has higher accuracy than that based on the central limit theorem.
Performance Engineering and Stochastic Modeling : 17th European Workshop, EPEW 2021 and 26th International Conference, ASMTA 2021 Virtual Event, December 9–10 and December 13–14, 2021 : proceedings. Cham, 2021. P. 441-456