  We are interested in a large queue in a $GI/G/k$ queue with heterogeneous
servers. For this, we consider tail asymptotics and weak limit approximations
for the stationary distribution of its queue length process in continuous time
under a stability condition. Here, two weak limit approximations are
considered. One is when the variances of the inter-arrival and/or service times
are bounded, and the other is when they get large. Both require a heavy traffic
condition. Tail asymptotics and heavy traffic approximations have been
separately studied in the literature. We develop a unified approach based on a
martingale produced by a good test function for a Markov process to answer both
problems.
