Diffusion approximations for open Jackson networks with reneging

Abstract

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 (Queueing Syst 50:371---400, 2005) and the patience time with hazard rate scaling and unbounded support adopted by Reed and Ward (Math Oper Res 33:606---644, 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 (Math Oper Res 35:347---362, 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.