This paper studies the queue length process in series Jackson networks with external input to the first station. We show that its Markov transition probabilities can be written as a finite sum of non-crossing probabilities, so that questions on time-dependent queueing behavior are translated to questions on non-crossing probabilities. This makes previous work on non-crossing probabilities relevant to queueing systems and allows new queueing results to be established. To illustrate the latter, we prove that the relaxation time (i.e., the reciprocal of the `spectral gap') of a positive recurrent system equals the relaxation time of an M/M/1 queue with the same arrival and service rates as the network's bottleneck station. This resolves a conjecture of Blanc, which he proved for two queues in series.