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.

Queueing theory (msc 60K25), Queues and service (msc 90B22)
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.