Staffing and Control of Many-server Service Systems

Staffing and Control of Many-Server Service Systems Itai Gurvich This dissertation considers large-scale service systems with multiple customer classes and agent types. Customers are classified according to their processing requirements, service-level guarantees, or both. The customers are served by agents of different types. These are classified according to the subset of customer classes that they can serve. We consider the problem of optimally choosing the capacity allocations of the different agent types and the real-time routing of costumers to agents. As a first step towards solving two such optimization problems, we introduce and analyze the Queue-and-Idleness-Ratio (QIR) family of routing rules. The QIR rules are defined as follows: (i) an arriving customer is routed to the agent pool (among those that are eligible to serve him) whose idleness most exceeds a specified state-dependent proportion of the total number of idle agents, summed over all types; (ii) a newly-available agent serves the customer from the head of the queue (from among those he is eligible to serve) whose length most exceeds a specified state-dependent proportion of total queue length, summed over all classes. We identify regularity conditions on the network structure and on the system parameters under which QIR produces an important State-Space Collapse (SSC) in the Quality-and-Efficiency-Driven (QED) many-server heavy-traffic regime. We also provide convergence results for various performance measures. The QIR family of rules is central to the solution of two optimization problems: Staffing subject to service-level targets: We consider the problem of minimizing labor costs subject to service-level constraints that are defined through probabilistic bounds on the waiting time. Agents of different types can have different salary costs. We show that a special case of QIR in which the queue ratios are fixed, i.e, a Fixed-Queue-Ratio (FQR) routing rule, can be used to construct solutions for this practical problem. The proportions can be set to achieve desired service-level constraints for all classes; these targets are achieved asymptotically as the total arrival rate increases. The SSC obtained under QIR facilitates establishing asymptotic results. In simplified settings, SSC allows us to solve a combined design-staffing-and-routing problem in a nearly optimal way. In other cases it provides a simple yet feasible solution at reasonable costs, namely, a solution that guarantees the achievement of the service-levels for the different classes while keeping the staffing costs reasonably close to optimality. Holding cost minimization: We consider also the problem of minimizing convex holdingcosts. In the case where service rates depend on the agent type but not on the customer class (pool-dependent service rates), QIR with appropriately-chosen ratio functions is shown to be asymptotically optimal in the QED many-server heavy-traffic regime. In special cases, the QIR solution is reduced to a simple policy: linear costs produce a priority-type rule, in which the least-cost customers are given low priority. Strictly convex costs (plus other regularity conditions) produce a many-server analogue of the generalized-cμ (Gcμ) rule, under which a newly-available agent selects a customer from the class experiencing the greatest marginal cost at that time.