Consider a single-server queue with a renewal arrival process and generally distributed processing times in which each customer independently reneges if service has not begun within a generally distributed amount of time. We establish that both the workload and queue-length processes in this system can be approximated by a regulated Ornstein-Uhlenbeck (ROU) process when the arrival rate is close to the processing rate and reneging times are large. We further show that a ROU process also approximates the queue-length process, under the same parameter assumptions, in a balking model. Our balking model assumes the queue-length is observable to arriving customers, and that each customer balks if his or her conditional expected waiting time is too large.

In a call center, there is a natural trade-off between minimizing customer wait time and fairly dividing the workload among agents of different skill levels. The relevant control is the routing policy, that is, the decision concerning which agent should handle an arriving call when more than one agent is available. We formulate an optimization problem for a call center with heterogeneous agent pools, in which each pool is distinguished by the speed at which agents in that pool handle calls. The objective is to minimize steady-state expected customer wait time subject to a “fairness” constraint on the workload division. We first solve the optimization problem by formulating it as a Markov decision process (MDP), and solving a related linear program. We note that this approach does not in general lead to an optimal policy that has a simple structure. Fortunately, the optimal policy does appear to have a simple structure as the system size grows large, in the Halfin-Whitt many-server heavy-traffic limit regime. Therefore, we solve the diffusion control problem that arises in this regime and interpret its solution as a policy for the original system. The resulting routing policy is a threshold policy that determines server pool priorities based on the total number of customers in the system. We prove that a continuous modification of our proposed threshold routing policy is asymptotically optimal in the Halfin-Whitt limit regime. We furthermore present simulation results to illustrate that our proposed threshold routing policy outperforms a common routing policy used in call centers (that routes to the agent that has been idle the longest).

We study a single-server queue, operating under the first-in-first-out (FIFO) service discipline, in which each customer independently abandons the queue if his service has not begun within a generally distributed amount of time. Under some mild conditions on the abandonment distribution, we identify a limiting heavy-traffic regime in which the resulting diffusion approximation for both the offered waiting time process (the process that tracks the amount of time an infinitely patient arriving customer would wait for service) and the queue-length process contain the entire abandonment distribution. To use a continuous mapping approach to establish our weak convergence results, we additionally develop existence, uniqueness, and continuity results for nonlinear generalized regulator mappings that are of independent interest. We further perform a simulation study to evaluate the quality of the proposed approximations for the steady-state mean queue length and the steady-state probability of abandonment suggested by the limiting diffusion process.

We consider the optimal control of matching queues with random arrivals. In this model, items arrive to dedicated queues, and wait to be matched with items from other (possibly multiple) queues. A match type corresponds to the set of item classes required for a match. Once a decision has been made to perform a match, the matching itself is instantaneous and the matched items depart from the system. We consider the problem of minimizing finite-horizon cumulative holding costs. The controller must decide which matchings to execute given multiple options. In principle, the controller may choose to wait until some "inventory" of items builds up to facilitate more profitable matches in the future.We introduce a multi-dimensional imbalance process, that at each time t, is given by a linear function of the cumulative arrivals to each of the item classes. A non-zero value of the imbalance at time t means that no control could have matched all the items that arrived by time t. A lower bound based on the imbalance process can be specified, at each time point, by a solution to an optimization problem with linear constraints. While not achievable in general, this lower bound can be asymptotically approached under a dedicated item condition (an analogue of the local traffic condition in bandwidth sharing networks). We devise a myopic discrete-review matching control that asymptotically-as the arrival rates become large-achieves the imbalance-based lower bound.