  We analyze the latency or sojourn time L(m,n) for the last customer in a
batch of n customers to exit from the m-th queue in a tandem of m queues in the
setting where the queues are in equilibrium before the batch of customers
arrives at the first queue. We first characterize the distribution of L(m,n)
exactly for every m and n, under the assumption that the queues have unlimited
buffers and that each server has customer independent, exponentially
distributed service times with an arbitrary, known rate. We then evaluate the
first two leading order terms of the distributions in the large m and n limit
and bring into sharp focus the existence of phase transitions in the system
behavior. The phase transition occurs due to the presence of either slow
bottleneck servers or a high external arrival rate. We determine the critical
thresholds for the service rate and the arrival rate, respectively, about which
this phase transition occurs; it turns out that they are the same. This
critical threshold depends, in a manner we make explicit, on the individual
service rates, the number of customers and the number of queues but not on the
external arrival rate.
