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Diffusion approximations for open Jackson networks with reneging

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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.

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Acknowledgments

The authors would like to express their deep appreciation to Professor Jim Dai at Georgia Institute of Technology, Professor Avi Mandelbaum at Technion, and Professor David Yao at Columbia University for stimulating discussion and guidance. The authors would like to thank the anonymous associate editor and two anonymous referees for commenting on the earlier version of the manuscript and offering very helpful suggestions that have led to its much improved present version.

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  1. NUS Business School, National University of Singapore, Singapore, Singapore

    Junfei Huang & Hanqin Zhang

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  1. Junfei Huang
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  2. Hanqin Zhang
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Correspondence to Junfei Huang.

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Huang, J., Zhang, H. Diffusion approximations for open Jackson networks with reneging. Queueing Syst 74, 445–476 (2013). https://doi.org/10.1007/s11134-012-9335-5

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  • DOI: https://doi.org/10.1007/s11134-012-9335-5

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