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Stability analysis of a two-station cascade queueing network

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Abstract

We consider a two-station cascade network, where the first station has Poisson input and the second station has renewal input, with i.i.d. service times at both stations. The following partial interaction exists between stations: whenever the second station becomes empty while customers are awaiting service at the first one, one customer can jump to the second station to be served there immediately. However, the first station cannot assist the second one in the opposite case. For this system, we establish necessary and sufficient stability conditions of the basic workload process, using a regenerative method. An extension of the basic model, including a multiserver first station, a different service time distribution for customers jumping from station 1 to station 2, and an arbitrary threshold d 1≥1 on the queue-size at station 1 allowing jumps to station 2, are also treated.

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Acknowledgements

The authors thank the anonymous referees for very useful comments which have led to considerable improvement of the paper. Special thanks also to Rosario Delgado for a valuable discussion of the fluid approach.

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Authors and Affiliations

  1. Institute of Applied Mathematical Research, Karelian Research Centre RAS, Pushkinskaya 11, Petrozavodsk, 185610, Russia

    E. Morozov

  2. SMACS Research Group, Department of Telecommunications and Information Processing, Ghent University, Sint-Pietersnieuwstraat 41, 9000, Ghent, Belgium

    B. Steyaert

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  1. E. Morozov
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  2. B. Steyaert
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Correspondence to E. Morozov.

Additional information

The work of E. Morozov is supported by the Russian Foundation for Basic Research under grant 10-07-00017.

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Morozov, E., Steyaert, B. Stability analysis of a two-station cascade queueing network. Ann Oper Res 202, 135–160 (2013). https://doi.org/10.1007/s10479-011-1034-9

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