Itai Gurvich

Service-Level Differentiation in Many-Server Service Systems via Queue-Ratio Routing

Motivated by telephone call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. To minimize staffing costs subject to service-level constraints, where we delicately balance the service levels (SLs) of the different classes, we propose a family of routing rules called fixed-queue-ratio (FQR) rules. With FQR, 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 proportion of the total queue length. The proportions can be set to achieve desired SL targets. The FQR rule achieves an important state-space collapse (SSC) as the total arrival rate increases, in which the individual queue lengths evolve as fixed proportions of the total queue length. In the current paper we consider a variety of service-level types and exploit SSC to construct asymptotically optimal solutions for the staffing-and-routing problem. The key assumption in the current paper is that the service rates depend only on the agent pool.

Staffing Call Centers with Uncertain Demand Forecasts: A Chance-Constrained Optimization Approach

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.

On the dynamic control of matching queues

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.

When Promotions Meet Operations: Cross-Selling and Its Effect on Call Center Performance

We study cross-selling operations in call centers. The following questions are 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 center while maintaining a prespecified service level target? We tackle these questions by characterizing control and staffing schemes that are asymptotically optimal in the limit, as the system load grows large. Our main finding is that a threshold priority 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 threshold priority reduces the staffing problem to a solution of a simple deterministic problem in one regime and to a simple search procedure in another. We show that our joint staffing and control scheme is nearly optimal for large systems. Furthermore, it performs extremely well, even for relatively small systems.

Excursion-Based Universal Approximations for the Erlang-A Queue in Steady-State

We revisit many-server approximations for the well-studied Erlang-A queue. This is a system with a single pool of i.i.d. servers that serve one class of impatient i.i.d. customers. Arrivals follow a Poisson process and service times are exponentially distributed as are the customers’ patience times. We propose a diffusion approximation that applies simultaneously to all existing many-server heavy-traffic regimes: quality and efficiency driven, efficiency driven, quality driven, and nondegenerate slowdown. We prove that the approximation provides accurate estimates for a broad family of steady-state metrics. Our approach is “metric-free” in that we do not use the specific formulas for the steady-state distribution of the Erlang-A queue. Rather, we study excursions of the underlying birth-and-death process and couple these to properly defined excursions of the corresponding diffusion process. Regenerative process and martingale arguments, together with derivative bounds for solutions to certain ordinary differential equations, allow us to control the accuracy of the approximation. We demonstrate the appeal of universal approximation by studying two staffing optimization problems of practical interest.

Operations in the On-Demand Economy: Staffing Services with Self-Scheduling Capacity

We study capacity management in a service system in which workers determine their own schedules. Our study is motivated by recent innovations in service delivery such as work-from-home call centers and ride-sharing services. Providers in these settings promise their customers a given level of service but do not directly assign agents to time intervals. Agents choose when to work based on the compensation offered and their individual availability thresholds in the relevant time interval. A provider must recruit enough agents and set the period-by-period compensation so as to generate enough participation to meet the desired service level. We examine how a service provider should manage self-scheduling agents in two settings. The first is inspired by work-from-home call centers and assumes a service firm must staff to achieve contracted service level targets. The second is a newsvendor setting in which the service level is endogenous. In both settings, the firm prefers having a large pool of agents. However, this creates a tension between the firm’s profit and agents’ earnings. If the firm must guarantee agents a minimal level of compensation, the firm recruits a smaller pool of agents and limits the number of agents that can work in some time periods. Finally, we show that allowing self scheduling leads to non-conventional cost structures and can be costly. Specifically, the average cost to serve a customer may increase with the volume of transactions. In other words, the service system does not exhibit the economies of scale one would typically expect. Further, we show that when the service level is endogenous, self scheduling leads to less staffing and hence worse service than a conventional benchmark.

Validity of Heavy-Traffic Steady-State Approximations in Multiclass Queueing Networks: The Case of Queue-Ratio Disciplines

A class of stochastic processes known as semi-martingale reflecting Brownian motions SRBMs is often used to approximate the dynamics of heavily loaded queueing networks. In two influential papers, Bramson [Bramson M 1998 State space collapse with applications to heavy-traffic limits for multiclass queueing networks. Queueing Systems 30:89--148] and Williams [Williams RJ 1998b Diffusion approximations for open multiclass queueuing networks: Sufficient conditions involving state space collapse. Queueing Systems 30:27--88] laid out a general and structured approach for proving the validity of such heavy-traffic approximations, in which an SRBM is obtained as a diffusion limit from a sequence of suitably normalized workload processes. However, for multiclass networks it is still not known in general whether the steady-state distribution of the SRBM provides a valid approximation for the steady-state distribution of the original network. In this paper we study the case of queue-ratio disciplines and provide a set of sufficient conditions under which the above question can be answered in the affirmative. In addition to standard assumptions made in the literature towards the stability of the pre-and post-limit processes and the existence of diffusion limits, we add a requirement that solutions to the fluid model are attracted to the invariant manifold at a linear rate. For the special case of static-priority networks such linear attraction is known to hold under certain conditions on the network primitives. The analysis elucidates interesting connections between stability of the pre-and post-limit processes, their respective fluid models and state-space collapse, and identifies the respective roles played by all of the above in establishing validity of heavy-traffic steady-state approximations.

Overflow Networks: Approximations and Implications to Call Center Outsourcing

Motivated by call center cosourcing problems, we consider a service network operated under an overflow mechanism. Calls are first routed to an in-house (or dedicated) service station that has a finite waiting room. If the waiting room is full, the call is overflowed to an outside provider (an overflow station) that might also be serving overflows from other stations. We establish approximations for overflow networks with many servers under a resource-pooling assumption that stipulates, in our context, that the fraction of overflowed calls is nonnegligible. Our two main results are (i) an approximation for the overflow processes via limit theorems and (ii) asymptotic independence between each of the in-house stations and the overflow station. In particular, we show that, as the system becomes large, the dependency between each in-house station and the overflow station becomes negligible. Independence between stations in overflow networks is assumed in the literature on call centers, and we provide a rigorous support for those useful heuristics.

Collaboration and Multitasking in Networks: Architectures, Bottlenecks and Capacity

Motivated by the trend towards more collaboration in work flows, we study stochastic processing networks where some activities require the simultaneous collaboration of multiple human resources. Collaboration introduces resource synchronization requirements that are not captured in the standard procedure (formalized through a static planning problem) to identify bottlenecks and theoretical capacity. We introduce the notions of collaboration architecture and unavoidable idleness. In general, collaboration architectures may feature unavoidable idleness so that the theoretical capacity exceeds the maximal achievable throughput or actual capacity. This fundamental tradeoff between collaboration and throughput does not disappear in multi-server networks and has important ramifications to service-system staffing. We identify a special class of collaboration architectures that have no unavoidable idleness and present a condition on this architecture that guarantees, regardless of the processing times of the various activities, that the standard bottleneck procedure in fact identifies the actual capacity of the network. In multi-server cases this class of networks guarantees that the theoretical capacity is achievable provided one has the right number of floaters. Finally, we study the subtleties that collaboration introduces to questions of flexibility investment. Unavoidable idleness may limit the ability to materialize the benefits of flexibility. We study the interplay of flexibility and unavoidable idleness and offer remedies derived from collaboration architecture.

Scheduling parallel servers in the nondegenerate slowdown diffusion regime: Asymptotic optimality results

We consider the problem of minimizing queue-length costs in a system with heterogenous parallel servers, operating in a many-server heavy-traffic regime with nondegenerate slowdown. This regime is distinct from the well-studied heavy traffic diffusion regimes, namely the (single server) conventional regime and the (many-server) Halfin-Whitt regime. It has the distinguishing property that waiting times and service times are of comparable magnitudes. We establish an asymptotic lower bound on the cost and devise a sequence of policies that asymptotically attain this bound. As in the conventional regime, the asymptotics can be described by means of a Brownian control problem, the solution of which exhibits a state space collapse.