  In this thesis, we study the queueing systems with heterogeneous servers and
service rate uncertainty under the Halfin-Whitt heavy traffic regime. First, we
analyse many server queues with abandonments when service rates are i.i.d.
random variables. We derive a diffusion approximation using a novel method. The
diffusion has a random drift, and hence depending on the realisations of
service rates, the system can be in Quality Driven (QD), Efficiency Driven (ED)
or Quality-Efficiency-Driven (QED) regime. When the system is under QD or QED
regime, the abandonments are negligible in the fluid limit, but when it is
under ED regime, the probability of abandonment will converge to a non-zero
value. We then analyse the optimal staffing levels to balance holding costs
with staffing costs combining these three regimes. We also analyse how the
variance of service rates influence abandonment rate.
  Next, we focus on the state space collapse (SSC) phenomenon. We prove that
under some assumptions, the system process will collapse to a lower dimensional
process without losing essential information. We first formulate a general
method to prove SSC results inside pools for heavy traffic systems using the
hydrodynamic limit idea. Then we work on the SSC in multi-class queueing
networks under the Halfin-Whitt heavy traffic when service rates are i.i.d.
random variables within pools. For such systems, exact analysis provides
limited insight on the general properties. Alternatively, asymptotic analysis
by diffusion approximation proves to be effective. Further, limit theorems,
which state the diffusively scaled system process weakly converges to a
diffusion process, are usually the central part in such asymptotic analysis.
The SSC result is key to proving such a limit. We conclude by giving examples
on how SSC is applied to the analysis of systems.
