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.

This is the technical appendix accompanying the paper, “Cross-Selling in a Call Center with a Heterogeneous Customer Population, ” [3]. The organization of this appendix is as follows: we begin in §B with the completion of the proof of Proposition 1, whose sketch was given in §A of [3]. We continue in §C with some preliminaries required for the performance analysis of (S)-(C).

Given a random variable N with values in N, and N i.i.d. positive random variables {µk}, we consider a queue with renewal arrivals and N exponential servers, where server k serves at rate µk, under two work conserving routing schemes. In the first, the service rates {µk} need not be known to the router, and each customer to arrive at a time when some servers are idle is routed to the server that has been idle for the longest time (or otherwise it is queued). In the second, the service rates are known to the router, and a customer that arrives to find idle servers is routed to the one whose service rate is greatest. In the many-server heavy traffic regime of Halfin and Whitt, the process that represents the number of customers in the system is shown to converge to a one-dimensional diffusion with a random drift coefficient, where the law of the drift depends on the routing scheme. A related result is also provided for nonrandom environments. 1. Introduction. Many-server

doi 10.1287/mnsc.1070.0825

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 consider the problem of staffing call centers with multiple customer classes and agent types operating under quality-of-service (QoS) constraints and demand rate uncertainty. We introduce a formulation of the staffing problem that requires that the QoS constraints are met with high probability with respect to the uncertainty in the demand rate. We contrast this chance-constrained formulation with the average-performance constraints that have been used so far in the literature. We then propose a two-step solution for the staffing problem under chance constraints. In the first step, we introduce a random static planning problem (RSPP) and discuss how it can be solved using two different methods. The RSPP provides us with a first-order (or fluid) approximation for the true optimal staffing levels and a staffing frontier. In the second step, we solve a finite number of staffing problems with known arrival rates—the arrival rates on the optimal staffing frontier. Hence, our formulation and solution approach has the important property that it translates the problem with uncertain demand rates to one with known arrival rates. The output of our procedure is a solution that is feasible with respect to the chance constraint and nearly optimal for large call centers.

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During the last century and the beginning of the present one, the service sector has grown significantly and now accounts for approximately 70 % of the national income in the United States, and similarly in many other Western countries. The service sector covers a wide spectrum of activities, e.g. education, professional services, financial services and

We study cross-selling operations in call centers. The following question is addressed: How many customer service representatives are required (staffing) and when should cross-selling opportunities be exercised (control) in a way that will maximize the expected profit of the firm while maintaining a pre-specified service level target. We tackle these questions by characterizing scheduling and staffing schemes that are asymptotically optimal in the limit, as the system load grows to infinity. Our main finding is that a threshold priority (TP) control, in which cross-selling is exercised only if the number of callers in the system is below a certain threshold, is asymptotically optimal in great generality. The asymptotic optimality of TP reduces the staffing problem to the solution of a simple deterministic problem, in some cases, and to a simple search procedure in others. Our asymptotic approach establishes that our staffing and control scheme is near-optimal for large systems. In addition, we numerically demonstrate that TP performs extremely well even for relatively small systems.