Diffusion approximations for open Jackson networks with reneging

We consider generalized Jackson networks with reneging in which the customer patience times follow a general distribution that unifies the patience time without scaling adopted by Ward and Glynn (2005) and the patience time with hazard rate scaling and unbounded support adopted by Reed and Ward (2008). The diffusion approximations for both the queue length process and the abandonment‐count process are established under the conventional heavy traffic limit regime. In light of the recent work by Dai and He (2010), the diffusion approximations are obtained by the following four steps: first, establishing the stochastic boundedness for the queue length process and the virtual waiting time process; second, obtaining the $C$ ‐tightness and fluid limits for the queue length process and the abandonment‐count process; then third, building an asymptotic relationship between the abandonment‐count process and the queue length process in terms of the customer patience time. Finally, the fourth step is to get the diffusion approximations by invoking the continuous mapping theorem.