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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.

Calculations are presented for a series of interrelated problems in the theory of disordered solids. The simple mean field theory of tricriticality in the layered Ising metamagnet is modified by inclusion of the Bethe-Peierls equation of state for the planar interactions. The approximation allows for a model of dilution with a finite percolation concentration for the layers. The calculated behaviour as a function of dilution for anisotropic coupling strengths allows comparison with experimental results on dilute ferrous chloride. Estimation is made of the effects of treating fluctuations by theories with mean field singularities. A discussion is given of first order phase transitions in disordered systems. A mean field theory of the implications of introducing quenched disorder to a system undergoing a transition of first order is effected by reformulating the problem in terms of a translationally invariant one via the replica method. The conclusions are examined in terms of a simple scaling theory and criteria derived for smearing of the singularities. Dynamic and thermodynamic properties of diluted magnetic insulators near the percolation concentration are considered in terms of simple geometric models of the percolating cluster as introduced by de Gennes. New scaling relations for the spin wave stiffness and conductivity exponents are derived and differences from previous relations interpreted geometrically. The scaling models of the percolating cluster so found, in which correlations propagate locally via effectively one-dimensional paths, are applied to the determination of the mobility edge for spin wave excitations in a dilute Heisenberg magnet near the percolation threshold. A prediction for the functional form of the mobility edge is made by means of results known for the Anderson model and a localisation length derived from an exact solution of a random problem by Dyson.

This research deals with classical, Bayesian, and generalized estimation of stress-strength reliability parameter, Rs;k = Pr(at least s of (X1;X2; :::;Xk) exceed Y) = Pr(Xks+1:k >Y) of an s-out-of-k : G multicomponent system, based on progressively type-II right-censored samples with random removals when stress and strength are two independent Chen random variables. Under squared-error and LINEX loss functions, Bayes estimates are developed by using Lindley’s approximation and Markov Chain Monte Carlo method. Generalized estimates are developed using generalized variable method while classical estimates - the maximum likelihood estimators, their asymptotic distributions, asymptotic confidence intervals, bootstrap-based confidence intervals - are also developed. A simulation study and a real-world data analysis are provided to illustrate the proposed procedures. The size of the test, adjusted and unadjusted power of the test, coverage probability and expected lengths of the confidence intervals, and biases of the estimators are also computed, compared and contrasted.

This thesis examines random variables of one dimension.