We consider a polling model of two $M/G/1$ queues, served by a single server. The service policy for this polling model is of threshold type. Service at queue 1 is exhaustive. Service at queue 2 is exhaustive unless the size of queue 1 reaches some level $T$ during a service at queue 2; in the latter case the server switches to queue 1 at the end of that service. Both zero- and nonzero switchover times are considered. We derive exact expressions for the joint queue length distribution at customer departure epochs, and for the steady-state queue-length and sojourn time distributions. In addition, we supply a simple and very accurate approximation for the mean queue lengths, which is suitable for optimization purposes.

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CWI
Department of Operations Research, Statistics, and System Theory [BS]
Combinatorial Optimization and Algorithmics

Boxma, O.J, & Down, D.G. (1995). Dynamic server assignment in a two-queue model. Department of Operations Research, Statistics, and System Theory [BS]. CWI.