*Wissenschaftliches Programm* *Liste der Vortragenden*

**Sektion 9**

Donnerstag, 21.09.2000,
16.30–16.50 Uhr, WIL C 133

#### Manfred Brandt, Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

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.