Abstract not found

Motivated by telephone call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. For the purpose of delicately balancing service levels of the different customer classes, we propose a family of routing controls called Fixed-Queue-Ratio (FQR) rules. A newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified propor-tion of the total queue length. We show that the proportions can be set to achieve desired service-level targets for all classes; these targets are achieved asymptotically as the total arrival rate increases. The FQR rule is a special case of the Queue-and-Idleness-Ratio (QIR) family of controls which in a pre-vious paper where shown to produce an important state-space collapse (SSC) as the total arrival rate increases. This SSC facilitates establishing asymptotic results. In simplified settings, SSC allows us to solve a combined design-staffing-and-routing problem in a nearly optimal way. Our analysis also establishes a diminishing-returns property of flexibility: Under FQR, very moderate cross-training is sufficient to make the call center as efficient as a single-pool system, again in the limit as the total arrival rate increases.

Motivated by call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. We propose a family of routing rules called Queue-and-Idleness-Ratio (QIR) rules. A newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified state-dependent proportion of the total queue length. An arriving customer is routed to the agent pool whose idleness most exceeds a specified state-dependent proportion of the total idleness. We identify regularity conditions on the network structure and system parameters under which QIR produces an important state-space collapse (SSC) result in the Quality-and-Efficiency-Driven (QED) many-server heavy-traffic limiting regime. The SSC result is applied in two subsequent papers to solve important staffing and control problems for large-scale service systems.

In a call center, there is a natural trade-off between minimizing customer wait time and fairly dividing the workload amongst 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 two heterogeneous agent pools, one that handles calls at a faster speed than the other, and a single customer class. The objective is to minimize steady-state expected customer wait time subject to a “fairness” constraint on the workload division. The optimization problem we formulate is difficult to solve exactly. Therefore, we solve the diffusion control problem that arises in the many-server heavy-traffic QED limiting regime. The resulting routing policy is a threshold policy that prioritizes faster agents when the number of customers in the system exceeds some threshold level and otherwise prioritizes slower agents. We prove our proposed threshold routing policy is near-optimal as the number of agents increases, and the system’s load approaches its maximum processing capacity. We further show simulation results that evidence 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) in terms of the steady-state expected customer waiting time for identical desired workload divisions.

In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stopped arrival processes as in the proof of Theorem 6.3.

We consider a parallel server system that consists of several customer classes and server pools in parallel. We propose a simple robust control policy to minimize the total linear holding and reneging costs. We show that this policy is asymptotically optimal under the many-server heavy traffic regime for parallel server systems when the service times are only server pool dependent and exponentially distributed.

A general model with multiple input flows (classes) and several flexible multi-server pools is considered. We propose a robust, generic scheme for routing new arrivals, which optimally balances server pools’ loads, without the knowledge of the flow input rates and without solving any optimization problem. The scheme is based on Shadow routing in a virtual queueing system. We study the behavior of our scheme in the Halfin–Whitt (or, QED) asymptotic regime, when server pool sizes and the input rates are scaled up simultaneously by a factor r growing to infinity, while keeping the system load within O(√r)of its capacity. The main results are as follows. (i) We show that, in general, a system in a stationary regime has at least O ( √ r) average queue lengths, even if the so called null-controllability (Atar et al., Ann. Appl. Probab. 16, 1764–1804, 2006) on a finite time interval is possible; strategies achieving this O(√r) growth rate we call order-optimal. (ii) We show that some natural algorithms, such as MaxWeight, that guarantee stability, are not order-optimal. (iii) Under the complete resource pooling condition, we prove the diffusion limit of the arrival processes into server pools, under the Shadow routing. (We conjecture that result (iii) leads to order-optimality of the Shadow routing algorithm; a formal proof of this fact is an important subject of future work.) Simulation results demonstrate good performance and robustness of our scheme.

In a call center, arriving customers must be routed to available servers, and servers that have just become available must be scheduled to help waiting customers. These dynamic routing and scheduling decisions are very difficult, because customers have different needs and servers have different skill levels. A further complication is that it is preferable that these decisions are made blindly; that is, they depend only on the system state and not on system parameter information such as call arrival rates and service speeds. This is because this information is generally not known with certainty. Ideally, a dynamic control policy for making routing and scheduling decisions balances customer and server needs, by keeping customer delays low, but still fairly dividing the workload amongst the various servers. In this paper, we propose two blind dynamic control policies for parallel server systems with multiple customer classes and server pools, one that is based on the number of customers waiting and the number of agents idling, and one that is based on customer delay times and server idling times. We show that, in the Halfin-Whitt many-server heavy traffic limiting regime, our proposed blind policies perform extremely well when the objective is to minimize customer holding or delay costs subject to “server fairness”, as defined by how the system idleness is divided among servers. To do this, we formulate an approximating diffusion control problem (DCP), and compare the performance of the non-blind DCP solution to a feasible policy for the DCP that is blind. We establish that the increase in the DCP objective function value is small over a wide range of parameter values. We then use simulation to validate that a small increase in the DCP objective function value is indicative of our proposed blind policies performing very well.

We establish many-server heavy-traffic limits for G/M/n + M queueing models, allowing cus-tomer abandonment (the +M), subject to exogenous regenerative service interruptions. With unscaled service interruption times, we obtain a FWLLN for the queue-length process where the limit is an ordinary differential equation in a two-state random environment. With asymptoti-cally negligible service interruptions, we obtain a FCLT for the queue-length process, where the limit is characterized as the pathwise unique solution to a stochastic integral equation with jumps. When the arrivals are renewal and the interruption cycle time is exponential, the limit is a Markov process, being a jump-diffusion process in the QED regime and an O-U process driven by a Levy process in the ED regime (and for infinite-server queues). A stochastic-decompostion property of the steady-state distribution of the limit process in the ED regime (and for infinite-server queues) is obtained.

Abstract not found