Open Access

Brandt Manfred, Brandt Andreas

• We deal with an infinite-server system where the service speed is governed by a stationary and ergodic process with countably many states. Applying a random time transformation such that the service speed becomes one, the sojourn time of a class of virtual requests with given required service time is equal in distribution to an additive functional defined via a stationary version of the time-changed process. Thus bounds for the expectation of functions of additive functionals yield bounds for the expectation of functions of virtual sojourn times, in particular bounds for fractional moments and the distribution function. Interpreting the $GI(n)/GI(n)/\infty$ system or equivalently the $GI(n)/GI$ system under state-dependent processor sharing as an infinite-server system with random states given by the number $n$ of requests in the system provides results for sojourn times of virtual requests. In case of $M(n)/GI(n)/\infty$, the sojourn times of arriving and added requests are equal in distribution to sojourn times of virtual requests in modified systems, which yields many results for the sojourn times of arriving and added requests. In case of integer moments, the bounds generalize earlier results for $M/GI(n)/\infty$. In particular, the mean sojourn times of arriving and added requests in $M(n)/GI(n)/\infty$ are proportional to the required service time, generalizing Cohen's famous result for $M/GI(n)/\infty$.