In this work, transient behavior of multi-line QS with a limited number of sources and a smooth change of arrival rate is analyzed. This system adequately describes a computing system consisting of several servers with a limited number of workstations. Investigating transition probabilities of system states in the case of a jump-like arrival rate, as well as a smoothly increasing arrival rate is conducted. The main performance characteristics of the system in a transient mode, such as failure probability and throughput are investigated.