Steady-State Convergence of the Continuous-Time Routing System with General Distributions in Heavy Traffic

This paper examines a continuous-time routing system with general interarrival and service time distributions, operating under either the join-the-shortest-queue policy or the power-of-two-choices policy. Under a weaker set of assumptions than those commonly established in the literature, we prove that the scaled steady-state queue length at each station converges weakly to a common exponential random variable in heavy traffic. Specifically, our results hold under the assumption of the (2 + ε)th moment for the interarrival and service distributions with some ε > 0. The proof leverages the Palm version of the basic adjoint relationship (BAR) as a key technique.