Age of information (AoI) is an effective performance metric measuring the freshness of information and is popular for applications involving status update. Most of the existing works have adopted average AoI as the metric, which cannot provide strict performance guarantees. In this work, the outage probability of the peak AoI exceeding a given threshold is analyzed in a multi-source system under round robin scheduling. Two queueing disciplines are considered, namely the first-come-first-serve (FCFS) queue and the single packet queue. For FCFS, upper and lower bounds on the outage probability are derived which coincides asymptotically, characterizing its true scaling. For the single packet queue, an upper bound is derived whose effectiveness is validated by the simulation results. The analysis concretizes the common belief that single packet queueing has a better AoI performance than FCFS. Moreover, it also reveals that the two disciplines would have similar asymptotic performance when the inter-arrival time is much larger than the total transmission time.