We study two classes of stochastic systems, the limited processor sharing system and the multi-server system. They share the common feature that multiple jobs/customers are being processed simultaneously, which makes the study of them intrinsically difficult. In the limited processor sharing system, a limited number of jobs can equally share a single server, and the excess ones wait in a first-in-first-out buffer. The model is mainly motivated by computer related applications, such as database servers and packet transmission over the Internet. This model is studied in the first part of the thesis. The multi-server queue is mainly motivated by call centers, where each customer is handled by an agent. The number of customers being served at any time is limited by number of agents employed. Customers who can not be served upon arrival wait in a first-in-first-out buffer. This model is studied in the second part of the thesis.

Customer call centers with hundreds of agents working in parallel are ubiquitous in many industries. These systems have a large amount of daily traffic that is stochastic in nature. It becomes more and more difficult to manage a call center because of its increasingly large scale and the stochastic variability in arrival and service processes. In call center operations, customer abandonment is a key factor and may significantly impact the system performance. It must be modeled explicitly in order for an operational model to be relevant for decision making. In this thesis, a large-scale call center is modeled as a queue with many parallel servers. To model the customer abandonment, each customer is assigned a patience time. When his waiting time for service exceeds his patience time, a customer abandons the system without service. We develop analytical and numerical tools for analyzing such a queue. We first study a sequence of G/G/n+GI queues, where the customer patience times are independent and identically distributed (iid) following a general distribution. The focus is the abandonment and the queue length processes. We prove that under certain conditions, a deterministic relationship holds asymptotically in diffusion scaling between these two stochastic processes, as the number of servers goes to infinity. Next, we restrict the service time distribution to be a phase-type distribution with d phases. Using the aforementioned asymptotic relationship, we prove limit theorems for G/Ph/n+GI queues in the quality- and efficiency-driven (QED) regime. In particular, the limit process for the customer number in each phase is a d-dimensional piecewise Ornstein-Uhlenbeck (OU) process. Motivated by the diffusion limit process, we propose two approximate models for a GI/Ph/n+GI queue. In each model, a d-dimensional diffusion process is used to approximate the dynamics of the queue. These two models differ in how the patience time distribution is built into them. The first diffusion model uses the patience time density at zero and the second one uses the entire patience time distribution. We also develop a numerical algorithm to analyze these diffusion models. The algorithm solves the stationary distribution of each model. The computed stationary distribution is used to estimate the queue's performance. A crucial part of this algorithm is to choose an appropriate reference density that controls the convergence of the algorithm. We develop a systematic approach to constructing a reference density. With the proposed reference density, the algorithm is shown to converge quickly in numerical experiments. These experiments also show that the diffusion models are good approximations of queues with a moderate to large number of servers.