The M(n)/M(n)/s + GI system is a s-server queueing system with a (potential) unlimited waiting room with FCFS queueing discipline where the calls waiting in the queue for service are impatient. The arrival and service processes are allowed to be state dependent with respect to the number n of calls in the system, but the cumulative service rate is assumed to be constant for n > s. Each call arriving at the system has a maximal waiting time I. If the offered waiting time exceeds I, then the call departs from the system after having waited time I. The maximal waiting times are assumed to be i.i.d. with a general distribution C(u).
Various performance measures for the general M(n)/M(n)/s + GI system are known, in particular the density of the detailed state process and the occupancy distribution. For the departure intensities of calls due to impatience new monotonicity and asymptotic results are derived.
Using these results a simple Markovian approximation for the M(n)/M(n)/s + GI system is constructed. The idea is to replace the individual maximal waiting times of the calls by waiting place dependent impatience rates, i.e., with each waiting place, which are numbered by i = 1, 2, . . . , there is associated an impatience rate i. The relevant performance measures for the new system can easily be obtained since the process of the number of calls in the system is a simple birth-death process.
The impatience rates i can be chosen such that the occupancy distribution of this system is fitted to those of a given M(n)/M(n)/s + GI system. The fitting of the occupancy distribution implies the fitting of other performance measures. From the asymptotic results for the departure intensities of calls due to impatience it follows that the i converge to the intensity corresponding to an exponential fitting if the third moment of the distribution C(u) is finite. However, numerical examples show that the i may significantly differ from their limit for smaller i.
The proposed approximation is useful if in a given queueing network one wants to approximate a ./M(n)/s + GI node by a simple birth-death node. The use of the approximation lies in reducing the numerical complexity for computing performance measures in queueing networks.