We consider a s-server system with two FCFS queues, where the arrival rates at the queues and the service rate may depend on the number n of customers being in service or in the first queue, but the service rate is assumed to be constant for n ? s. The customers in the first queue are impatient. If the offered waiting time exceeds a random maximal waiting time I , then the customer leaves the first queue after time I . If I is less than a given deterministic time then he leaves the system else he transits to the end of the second queue. The customers in the first queue have priority. The service of a customer from the second queue will be started if the first queue is empty and more than a given number of servers become idle. For the model being a generalization of the M(n)=M(n)=s+GI system balance conditions for the density of the stationary state process are derived yielding the stability conditions and the probabilities that precisely n customers are in service or in the first queue...

This paper gives a pathwise construction of Jackson-type queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMP’s. The paper also provides new results on the Jackson-type networks with i.i.d. driving sequences which were studied in the past.

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