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In this thesis, we study the queueing systems with heterogeneous servers andservice rate uncertainty under the Halfin-Whitt heavy traffic regime. First, weanalyse many server queues with abandonments when service rates are i.i.d.random variables. We derive a diffusion approximation using a novel method. Thediffusion has a random drift, and hence depending on the realisations ofservice 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 QEDregime, the abandonments are negligible in the fluid limit, but when it isunder ED regime, the probability of abandonment will converge to a non-zerovalue. We then analyse the optimal staffing levels to balance holding costswith staffing costs combining these three regimes. We also analyse how thevariance of service rates influence abandonment rate.  Next, we focus on the state space collapse (SSC) phenomenon. We prove thatunder some assumptions, the system process will collapse to a lower dimensionalprocess without losing essential information. We first formulate a generalmethod to prove SSC results inside pools for heavy traffic systems using thehydrodynamic limit idea. Then we work on the SSC in multi-class queueingnetworks under the Halfin-Whitt heavy traffic when service rates are i.i.d.random variables within pools. For such systems, exact analysis provideslimited insight on the general properties. Alternatively, asymptotic analysisby diffusion approximation proves to be effective. Further, limit theorems,which state the diffusively scaled system process weakly converges to adiffusion process, are usually the central part in such asymptotic analysis.The SSC result is key to proving such a limit. We conclude by giving exampleson how SSC is applied to the analysis of systems.

Many Server Queueing Models with Heterogeneous Servers and Parameter Uncertainty with Customer Contact Centre Applications