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Poisson process and queuing system

In a queuing system, the inter-arrival times are known to be exponentially distributed. My textbook states: "It can be shown that, if the underlying distribution of inter-arrival times { T1, T2, ..., Tn } is exponential, the arrival times are uniformly distributed on the interval (0, T). The arrival times- T1, (T1 + T2), (T1 + T2 + T3), ... , (T1 + ... + Tn) are obtained by adding inter-arrival times." But isn't the sum of exponential random variables distributed as Erlang? Implying that arrival times should be Erlang distributed? How are arrival times distributed? Is the distribution of arrival times Uniform or Erlang? Why so? I'd like some clarity on the topic.