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- Title
Diffusion approximations for open Jackson networks with reneging.
- Authors
Huang, Junfei; Zhang, Hanqin
- 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, ) and the patience time with hazard rate scaling and unbounded support adopted by Reed and Ward (Math Oper Res 33:606-644, ). 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, ), 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.
- Subjects
DIFFUSION; APPROXIMATION theory; GENERALIZATION; CONSUMERS; QUEUING theory; MATHEMATICAL mappings
- Publication
Queueing Systems, 2013, Vol 74, Issue 4, p445
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-012-9335-5