  We consider a heterogeneous queueing system consisting of one large pool of
$O(r)$ identical servers, where $r\to\infty$ is the scaling parameter. The
arriving customers belong to one of several classes which determines the
service times in the distributional sense. The system is heavily loaded in the
Halfin-Whitt sense, namely the nominal utilization is $1-a/\sqrt{r}$ where
$a>0$ is the spare capacity parameter. Our goal is to obtain bounds on the
steady state performance metrics such as the number of customers waiting in the
queue $Q^r(\infty)$. While there is a rich literature on deriving process level
(transient) scaling limits for such systems, the results for steady state are
primarily limited to the single class case.
  This paper is the first one to address the case of heterogeneity in the
steady state regime. Moreover, our results hold for any service policy which
does not admit server idling when there are customers waiting in the queue. We
assume that the interarrival and service times have exponential distribution,
and that customers of each class may abandon while waiting in the queue at a
certain rate (which may be zero). We obtain upper bounds of the form
$O(\sqrt{r})$ on both $Q^r(\infty)$ and the number of idle servers. The bounds
are uniform w.r.t. parameter $r$ and the service policy. In particular, we show
that $\limsup_r E \exp(\theta r^{-1/2}Q^r(\infty))<\infty$. Therefore, the
sequence $r^{-1/2}Q^r(\infty)$ is tight and has a uniform exponential tail
bound. We further consider the system with strictly positive abandonment rates,
and show that in this case every weak limit $\hat{Q}(\infty)$ of
$r^{-1/2}Q^r(\infty)$ has a sub-Gaussian tail. Namely $E[\exp(\theta
(\hat{Q}(\infty))^2)]<\infty$, for some $\theta>0$.
