This paper studies many-server limits for multi-server queues that have a phase-type service time distribution and allow for customer abandonment. The first set of limit theorems is for critically loaded G/Ph/n + GI queues, where the patience times are independent and identically distributed following a general distribution. The next limit theorem is for overloaded G/Ph/n + M queues, where the patience time distribution is restricted to be exponential. We prove that a pair of diffusion-scaled total-customer-count and serverallocation processes, properly centered, converges in distribution to a continuous Markov process as the number of servers n goes to infinity. In the overloaded case, the limit is a multi-dimensional diffusion process, and in the critically loaded case, the limit is a simple transformation of a diffusion process. When the queues are critically loaded, our diffusion limit generalizes the result by Puhalskii and Reiman (2000) for GI /Ph/n queues without customer abandonment. When the queues are overloaded, the diffusion limit provides a refinement to a fluid limit and it generalizes a result by Whitt (2004) for M/M/n/ + M queues with an exponential service time distribution. The proof techniques employed in this paper are innovative. First, a perturbed system is shown to be equivalent to the original system. Next, two maps are employed in both fluid and diffusion scalings. These maps allow one to prove the limit theorems by applying the standard continuous-mapping theorem and the standard random-time-change theorem.

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We study G/G/n + GI queues in which customer patience times are independent, identically distributed following a general distribution. When a customer's waiting time in queue exceeds his patience time, the customer abandons the system without service. For the performance of such a system, we focus on the abandonment process and the queue length process. We prove that under some conditions, a deterministic relationship between the two stochastic processes holds asymptotically under the diffusion scaling when the number of servers n goes to infinity. These conditions include a minor assumption on the arrival processes that can be time-nonhomogeneous and a key assumption that the sequence of diffusion-scaled queue length processes, indexed by n, is stochastically bounded. We also establish a comparison result that allows one to verify the stochastic boundedness by studying a corresponding sequence of systems without customer abandonment.

We consider a continuous-review inventory system in which the setup cost of each order is a general function of the order quantity and the demand process is modeled as a Brownian motion with a positive drift. Assuming the holding and shortage cost to be a convex function of the inventory level, we obtain the optimal ordering policy that minimizes the long-run average cost by a lower bound approach. To tackle some technical issues in the lower bound approach under the quantity-dependent setup cost assumption, we establish a comparison theorem that enables one to prove the global optimality of a policy by examining a tractable subset of admissible policies. Since the smooth pasting technique does not apply to our Brownian inventory model, we also propose a selection procedure for computing the optimal policy parameters when the setup cost is a step function.

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