Abstract: 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
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. For obtaining performance measures for the second queue
a system approximation basing on fitting impatience intensities is
constructed. The results are applied to the performance analysis of a
call center with an integrated voice-mail-server. For an important special
case a stochastic decomposition is derived illuminating the connection
to the dynamics of the 
system.
Keywords: two queues, many server, server reservation, impatience, occupancy distribution, waiting time distribution, approximate system, M(n)/M(n)/s+GI, stochastic decomposition, call center application
MSC: 60K25, 68M20, 60G10