The research on call centres has attracted many researchers from different disciplines recently. In this thesis, we focus on call centre modelling, analysis and design. In terms of modelling, traditionally call centres have been modelled as single-node queueing systems. Based on the Semiopen Queueing Network (SOQN) model proposed by Srinivasan et al. [42], we propose and study SOQN models with balking and abandonment (both exponential and general patience time distributions). In addition, we study the corresponding single-node queueing systems and obtain new results. For each model, we study the queue length distribution, waiting time distribution and the related performance measures. To facilitate the computation, we express the performance measures in terms of special functions. In terms of call centre design, we develop a design algorithm to determine the minimal number of CSRs (S) and trunk lines (N) to satisfy a given set of service level constraints.<p> The explicit expressions for performance measures obtained allow for theoretical analysis of the performance measures. For example we prove monotonicity and convexity properties of performance measures for the M/M/S/N and M/M/S/N + M models. We also study the comparison of different patience time distributions for the M/M/S/N+G model.<p> We provide numerical examples for each model and discuss numerical results such as monotonicity properties of performance measures. In particular, we illustrate the efficacy of our design algorithm for various models including patient, balking and abandonment models. The impact of model parameters on the design of call centres is also discussed based on the numerical examples. The results are computed using Matlab, where special functions are available.

The central thesis of this dissertation is that insight from queueing analysis can effectively guide standard (combinatorial) scheduling algorithms in dynamic environments. Scheduling is generally concerned with complex combinatorial decisions for static problems, whereas queueing theory simplifies the combinatorics and focuses on dynamic systems. We examine a queueing network with flexible servers under queueing and scheduling techniques. Based on the strengths of queueing analysis and scheduling, we develop a hybrid model that guides scheduling with results from the queueing model. In order to include setup times, we create a logic-based Benders decomposition model for a static representation of the queueing network. Our model is able to find optimal schedules up to 5 orders of magnitude faster than the only other model in the literature. A hybrid model is then developed for the dynamic problem and shown to achieve the best mean flow time while also guaranteeing maximal capacity.

The central thesis of this dissertation is that insight from queueing analysis can effectively guide standard (combinatorial) scheduling algorithms in dynamic environments. Scheduling is generally concerned with complex combinatorial decisions for static problems, whereas queueing theory simplifies the combinatorics and focuses on dynamic systems. We examine a queueing network with flexible servers under queueing and scheduling techniques. Based on the strengths of queueing analysis and scheduling, we develop a hybrid model that guides scheduling with results from the queueing model. In order to include setup times, we create a logic-based Benders decomposition model for a static representation of the queueing network. Our model is able to find optimal schedules up to 5 orders of magnitude faster than the only other model in the literature. A hybrid model is then developed for the dynamic problem and shown to achieve the best mean flow time while also guaranteeing maximal capacity.

We consider three problems on inventory, transportation and location in a supply chain. In Chapter 2, we study Multilevel Rationing (MR) and Strict Priority (SP) stock allocation policies for a centralized single product multi-class M/G/1 make-to-stock queueing systems. To obtain the total cost of the system under these policies, we introduce a new method called “customer composition”. Using this method, we focus on the proportion of customers of each class out of the total number of customers in the queue since the number of customers in M/G/1 queues is invariant for any non-idling and non-anticipating policy. We consider a series of two-priority M/G/1 queues with an exceptional service time in each busy period to characterize the customer composition. We derive closed form expressions for the costs of SP and MR policies using these results. In Chapter 3, we consider a two-echelon inventory system with a congested centralized production facility and several Distribution Centers (DCs). We assume that the production and transportation times are stochastic that are generally distributed, and customers arrive to each DC according to an independent Poisson process. Inventory at DCs is managed using the one-for-one replenishment policy. We use the customer composition approach to characterize the total inventory carrying and backlog costs of the system under the FCFS, SP and MR allocation policies at the warehouse. For the special case of exponentially distributed production and transportation times, we use the unit-flow method and derive closed form expressions for the optimal cost and base-stock level of the DCs. We numerically demonstrate that prioritization using either the SP or the MR policy could be very beneficial in comparison with the FCFS policy. In Chapter 4, we study a two-echelon supply chain with a set of suppliers, a set of retailers and a set of capacitated cross-docks which are to be established. The demand of the retailers could be satisfied from the suppliers through the cross-docks. The objective is to determine the number and location of cross-docks, the assignment of retailers to suppliers so that the total cost of pipeline and retailers’ inventory, transportation, and facility location is minimized. We formulate the problem as a non-linear mixed integer programming and derive several structural results for special cases of the problem. To solve the general problem, we show that it can be written as a cutting stock problem and develop a column generation algorithm to solve it. We investigate the efficiency of the proposed algorithm numerically.

We consider three problems on inventory, transportation and location in a supply chain. In Chapter 2, we study Multilevel Rationing (MR) and Strict Priority (SP) stock allocation policies for a centralized single product multi-class M/G/1 make-to-stock queueing systems. To obtain the total cost of the system under these policies, we introduce a new method called “customer composition”. Using this method, we focus on the proportion of customers of each class out of the total number of customers in the queue since the number of customers in M/G/1 queues is invariant for any non-idling and non-anticipating policy. We consider a series of two-priority M/G/1 queues with an exceptional service time in each busy period to characterize the customer composition. We derive closed form expressions for the costs of SP and MR policies using these results. In Chapter 3, we consider a two-echelon inventory system with a congested centralized production facility and several Distribution Centers (DCs). We assume that the production and transportation times are stochastic that are generally distributed, and customers arrive to each DC according to an independent Poisson process. Inventory at DCs is managed using the one-for-one replenishment policy. We use the customer composition approach to characterize the total inventory carrying and backlog costs of the system under the FCFS, SP and MR allocation policies at the warehouse. For the special case of exponentially distributed production and transportation times, we use the unit-flow method and derive closed form expressions for the optimal cost and base-stock level of the DCs. We numerically demonstrate that prioritization using either the SP or the MR policy could be very beneficial in comparison with the FCFS policy. In Chapter 4, we study a two-echelon supply chain with a set of suppliers, a set of retailers and a set of capacitated cross-docks which are to be established. The demand of the retailers could be satisfied from the suppliers through the cross-docks. The objective is to determine the number and location of cross-docks, the assignment of retailers to suppliers so that the total cost of pipeline and retailers’ inventory, transportation, and facility location is minimized. We formulate the problem as a non-linear mixed integer programming and derive several structural results for special cases of the problem. To solve the general problem, we show that it can be written as a cutting stock problem and develop a column generation algorithm to solve it. We investigate the efficiency of the proposed algorithm numerically.

The research on call centres has attracted many researchers from different disciplines recently. In this thesis, we focus on call centre modelling, analysis and design. In terms of modelling, traditionally call centres have been modelled as single-node queueing systems. Based on the Semiopen Queueing Network (SOQN) model proposed by Srinivasan et al. [42], we propose and study SOQN models with balking and abandonment (both exponential and general patience time distributions). In addition, we study the corresponding single-node queueing systems and obtain new results. For each model, we study the queue length distribution, waiting time distribution and the related performance measures. To facilitate the computation, we express the performance measures in terms of special functions. In terms of call centre design, we develop a design algorithm to determine the minimal number of CSRs (S) and trunk lines (N) to satisfy a given set of service level constraints.<p> The explicit expressions for performance measures obtained allow for theoretical analysis of the performance measures. For example we prove monotonicity and convexity properties of performance measures for the M/M/S/N and M/M/S/N + M models. We also study the comparison of different patience time distributions for the M/M/S/N+G model.<p> We provide numerical examples for each model and discuss numerical results such as monotonicity properties of performance measures. In particular, we illustrate the efficacy of our design algorithm for various models including patient, balking and abandonment models. The impact of model parameters on the design of call centres is also discussed based on the numerical examples. The results are computed using Matlab, where special functions are available.

We consider the optimal pricing problem in a service facility in order to maximize its long-run average profit per unit time. We model the facility as a queueing process that may have finite or infinite capacity. Customers are admitted into the system if it is not full and if they are willing to pay the price posted by the service provider. Moreover, the congestion level in the facility incurs penalties that greatly influence profit. We model congestion penalties in three different manners: holding costs, balking customers and impatient customers. First, we assume that congestion-dependent holding costs are incurred per unit of time. Second, we consider that each customer might be deterred by the system congestion level and might balk upon arrival. Third, customers are impatient and can leave the system with a full refund before being serviced. We are interested in both static and dynamic pricing for all three types of congestion penalties. In the static case, we demonstrate that there is a unique optimal price that maximizes the long-run average profit per unit time. We also investigate how optimal prices vary as system parameters change. In the dynamic case, we show the existence of an optimal stationary policy in a continuous and unbounded action space that maximizes the long-run average profit per unit time. We provide explicit expressions for this policy under certain conditions. We also analyze the structure of this policy and investigate its relationship with our optimal static price.

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In the overlay Cognitive Radio (CR) networks, the low priority Secondary Users (SUs) must constantly monitor the occupied spectrum to detect the possible appearances of the high priority Primary Users (PUs) within the same spectrum portion. On detection, the SUs must vacate the occupied spectrum portion without interfering with the PUs beyond a certain threshold duration and must opportunistically access another idle spectrum portion to guarantee their seamless communication. This mechanism is known as the spectrum handoff process. In this thesis, we first introduce a novel approach to model the CR channel which is capable of capturing a more realistic behavior of the spectrum occupancy by both user types and that is more suitable for modeling the spectrum handoff process as opposed to the existing approaches. Then using that as a base we focus on building analytical models to capture the various aspects of the spectrum handoff process in a realistic manner.

In the overlay Cognitive Radio (CR) networks, the low priority Secondary Users (SUs) must constantly monitor the occupied spectrum to detect the possible appearances of the high priority Primary Users (PUs) within the same spectrum portion. On detection, the SUs must vacate the occupied spectrum portion without interfering with the PUs beyond a certain threshold duration and must opportunistically access another idle spectrum portion to guarantee their seamless communication. This mechanism is known as the spectrum handoff process. In this thesis, we first introduce a novel approach to model the CR channel which is capable of capturing a more realistic behavior of the spectrum occupancy by both user types and that is more suitable for modeling the spectrum handoff process as opposed to the existing approaches. Then using that as a base we focus on building analytical models to capture the various aspects of the spectrum handoff process in a realistic manner.