On a two-queue priority system with impatience and its application to a call center

We consider an s-server priority system with a protected and an unprotected queue. The arrival rates at the queues and the service rate may depend on the number n of customers being in service or in the protected queue, but the service rate is assumed to be constant for n > s. As soon as any server is idle, a customer from the protected queue will be served according to the first-come-first-served discipline. However, the customers in the protected queue are impatient. If the offered waiting time exceeds a random maximal waiting time I, then the customer leaves the protected queue after time I. If I is less than a given deterministic time, then he leaves the system, else he will be transferred by the system to the unprotected queue. The service of a customer from the unprotected queue will be started if the protected queue is empty and more than a given number of servers become idle. The model is a generalization of the many-server queue with impatient customers. The global balance conditions seem to have no explicit solution. However, the balance conditions for the density of the stationary state process for the subsystem of customers being in service or in the protected queue can be solved. This yields the stability conditions and the probabilities that precisely n customers are in service or in the protected queue. For obtaining performance measures for the unprotected queue, a system approximation based on fitting impatience intensities is constructed. The results are applied to the performance analysis of a call center with an integrated voice-mail-server.