Multiple Customer Classes Research Articles

Optimal Control of an Assembly System with Demand for the End-Product and Intermediate Components

We consider the production and admission control of a two-stage manufacturing system where intermediate components are produced to stock in the first stage and an end-product is assembled from these components through a second stage assembly operation which may allow backorders. The manufacturing firm faces two types of demand. The one directed at the end-product is satisfied immediately if there are available products in inventory, and the firm has the option to accept the order for later delivery or to reject the order if no inventory is available. The second type of demand is for any of the intermediate components and the firm again has the option to accept the order or reject it to keep the components available for assembly purposes. We provide structural results for the demand admission, component production and product assembly decisions. We also extend the model to take into account multiple customer classes based on revenue and a more general revenue collecting scheme where only an upfront partial payment for an item is received if a customer demand is accepted for future delivery with the remaining revenue received upon delivery. Since the optimal policy structure is rather complex and defined by switching surfaces in a multidimensional space, we also propose a simple heuristic policy for which the computational load grows linearly with the number of products and test its performance under a variety of example problems.

A simple heuristic for load balancing in parallel processing networks with highly variable service time distributions

Suppose that customers arrive at a service center (call center, web server, etc.) with two stations in accordance with independent Poisson processes. Service times at either station follow the same general distribution, are independent of each other and are independent of the arrival process. The system is charged station-dependent holding costs at each station per customer per unit time. At any point in time, a decision-maker may decide to move, at a cost, some number of jobs in one queue to the other. The goals of this paper are twofold. First, we are interested in providing insights into this decision-making scenario. We do so, in the important case that the service time distribution is highly variable or simply has a heavy tail. Secondly, we propose that the savvy use of Markov decision processes can lead to easily implementable heuristics when features of the service time distribution can be captured by introducing multiple customer classes. To this end, we consider a two-station proxy for the original system, where the service times are assumed to be exponential, but of one of two classes with different rates. We prove structural results for this proxy and show that these results lead to heuristics that perform well.

Partitioning of Servers in Queueing Systems During Rush Hour

This paper is motivated by two phenomena observed in many queueing systems in practice. The first is the partitioning of server capacity among different customers based on their service time requirements. The second is rush hour demand where a large number of customers arrive over a short period of time followed by few or no arrivals for an extended period thereafter. We study a system with multiple parallel servers and multiple customer classes. The servers can be partitioned into server groups, each dedicated to a single customer class. The system operates under a rush hour regime with a large number of customers arriving at the beginning of the rush hour period. We show that this allows us to reduce the problem to one that is deterministic and for which closed-form solutions can be obtained. We compare the performance of the system with and without server partitioning during rush hour and address three basic questions. (1) Is partitioning beneficial to the system? (2) Is it equally beneficial to all customer classes? (3) If it is implemented, what is an optimal partition? We evaluate the applicability of our results to systems where customers arrive over time using (1) deterministic fluid models and (2) simulation models for systems with stochastic interarrival times.

On a Data-Driven Method for Staffing Large Call Centers

We consider a call center model with multiple customer classes and multiple server pools. Calls arrive randomly over time, and the instantaneous arrival rates are allowed to vary both temporally and stochastically in an arbitrary manner. The objective is to minimize the sum of personnel costs and expected abandonment penalties by selecting an appropriate staffing level for each server pool. We propose a simple and computationally tractable method for solving this problem that requires as input only a few system parameters and historical call arrival data for each customer class; in this sense the method is said to be data-driven. The efficacy of the proposed method is illustrated via numerical examples. An asymptotic analysis establishes that the prescribed staffing levels achieve near-optimal performance and characterizes the magnitude of the optimality gap.

Queue-and-Idleness-Ratio Controls in Many-Server Service Systems

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 here to prove stochastic-process limits and in subsequent papers to solve important staffing and control problems for large-scale service systems.

Scheduling Flexible Servers with Convex Delay Costs in Many-Server Service Systems

In a recent paper we introduced the queue-and-idleness ratio (QIR) family of routing rules for many-server service systems with multiple customer classes and server pools. A newly available server serves the customer from the head of the queue of the class (from among those the server is eligible to serve) whose queue length most exceeds a specified proportion of the total queue length. Under fairly general conditions, QIR produces an important state-space collapse as the total arrival rate and the numbers of servers increase in a coordinated way. That state-space collapse was previously used to delicately balance service levels for the different customer classes. In this sequel, we show that a special version of QIR stochastically minimizes convex holding costs in a finite-horizon setting when the service rates are restricted to be pool dependent. Under additional regularity conditions, the special version of QIR reduces 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 server selects a customer from the class experiencing the greatest marginal cost at that time.

Critically loaded queueing models that are throughput suboptimal

This paper introduces and analyzes the notion of throughput suboptimality for many-server queueing systems in heavy traffic. The queueing model under consideration has multiple customer classes, indexed by a finite set $\mathcal{I}$, and heterogenous, exponential servers. Servers are dynamically chosen to serve customers, and buffers are available for customers waiting to be served. The arrival rates and the number of servers are scaled up in such a way that the processes representing the number of class-i customers in the system, $i\in\mathcal{I}$, fluctuate about a static fluid model, that is assumed to be critically loaded in a standard sense. At the same time, the fluid model is assumed to be throughput suboptimal. Roughly, this means that the servers can be allocated so as to achieve a total processing rate that is greater than the total arrival rate. We show that there exists a dynamic control policy for the queueing model that is efficient in the following strong sense: Under this policy, for every finite T, the measure of the set of times prior to T, at which at least one customer is in the buffer, converges to zero in probability as the arrival rates and number of servers go to infinity. On the way to prove our main result, we provide a characterization of throughput suboptimality in terms of properties of the buffer-station graph.

Competition in Service Industries with Segmented Markets

We develop a model for the competitive interactions in service industries where firms cater to multiple customer classes or market segments with the help of shared service facilities or processes so as to exploit pooling benefits. Different customer classes typically have distinct sensitivities to the price of service as well as the delays encountered. In such settings firms need to determine (i) the prices charged to all customer classes; (ii) the waiting time standards, i.e., expected steady state waiting time promised to all classes; (iii) the capacity level; and (iv) a priority discipline enabling the firm to meet the promised waiting time standards under the chosen capacity level, all in an integrated planning model that accounts for the impact of the strategic choices of all competing firms. We distinguish between three types of competition: depending on whether firms compete on the basis of their prices only, waiting time standards only, or on the basis of prices and waiting time standards. We establish in each of the three competition models that a Nash equilibrium exists under minor conditions regarding the demand volumes. We systematically compare the equilibria with those achieved when the firms service each market segment with a dedicated service process.

Dynamic Control of a Make-to-Order, Parallel-Server System with Cancellations

Motivated by make-to-order production systems, we consider a dynamic control problem for a multiclass, parallel-server queueing system. The production system serves multiple classes of customers who require rigid due-date lead times and may cancel their order subject to a cancellation penalty. To meet the due-date constraints, a system manager may outsource orders when the backlog of work is judged excessive, thereby incurring outsourcing costs. The system manager strives to minimize long-run average costs by dynamically making outsourcing and resource allocation decisions. Under heavy-traffic conditions, the scheduling problem is approximated by a Brownian control problem. Interpreting the solution of the Brownian control problem in the context of the original queueing system, a nongreedy outsourcing and resource allocation policy is proposed. A simulation experiment is performed to demonstrate the effectiveness of this policy.

Technical Note—A Make-to-Stock System with Multiple Customer Classes and Batch Ordering

This paper examines the impact of customer order sizes on a make-to-stock system with multiple demand classes. We first characterize the manufacturer's optimal production and rationing policies when the demand is nonunitary and lost if unsatisfied. We also investigate the optimal policies of a backorder system with two demand classes and fixed order sizes. Through a numerical study, we show the effects of batch orders on the manufacturer's inventory cost as well as on the benefit of optimal stock rationing. It is shown that batch ordering may reduce the manufacturer's overall cost if carefully introduced in a first-come-first-served (FCFS) system. With the same effective demand rates, the customers' order sizes also have a strong impact on the benefit of optimal stock rationing.

Fluid Models for Overloaded Multiclass Many-Server Queueing Systems with First-Come, First-Served Routing

Motivated by models of tenant assignment in public housing, we study approximating deterministic fluid models for overloaded queueing systems having multiple customer classes (classes of tenants) and multiple service pools (housing authorities), each with many servers (housing units). Customer abandonment acts to keep the system stable, yielding a proper steady-state description. Motivated by fairness considerations, we assume that customers are selected for service by newly available servers on a first-come, first-served (FCFS) basis from all classes the corresponding service pools are allowed to serve. In this context, it is challenging to determine stationary routing flow rates between customer classes and service pools. Given those routing flow rates, each single fluid queue can be analyzed separately using previously established methods. Our ability to determine the routing flow rates depends on the structure of the network routing graph. We obtain the desired routing flow rates in three cases: when the routing graph is (i) a tree (sparsely connected), (ii) complete bipartite (fully connected), and (iii) an appropriate combination of the previous two cases. Other cases remain unsolved. In the last two solved cases, the routing flow rates are actually not uniquely determined by the fluid model, but become so once we make stochastic assumptions about the queueing models that the fluid model approximates.

Profit-oriented shift scheduling of inbound contact centers with skills-based routing, impatient customers, and retrials

This paper presents a profit-oriented shift scheduling approach for inbound contact centers. The focus is on systems in which multiple agent classes with different qualifications serve multiple customer classes with different needs. We assume that customers are impatient, abandon if they have to wait, and that they may retry. A discrete-time modeling approach is used to capture the dynamics of the system due to time-dependent arrival rates. Staffing levels and shift schedules are simultaneously optimized over a set of different approximate realizations of the underlying stochastic processes to consider the randomness of the system. The numerical results indicate that the presented approach works best for medium-sized and large contact centers with skills-based routing of customers for which stochastic queueing models are rarely applicable.

A note on optimal pricing for finite capacity queueing systems with multiple customer classes

This article investigates optimal static prices for a finite capacity queueing system serving customers from different classes. We first show that the original multi-class formulation in which the price for each class is a decision variable can be reformulated as a single dimensional problem with the total load as the decision variable. Using this alternative formulation, we prove an upper bound for the optimal arrival rates for a fairly large class of queueing systems and provide sufficient conditions that ensure the existence of a unique optimal arrival rate vector. We show that these conditions hold for M/M/1/m and M/G/s/s systems and prove structural results on the relationships between the optimal arrival rates and system capacity. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008

Pricing and Operational Recourse in Coproduction Systems

Coproduction systems, in which multiple products are produced simultaneously in a single production run, are prevalent in many industries. Such systems typically produce a random quantity of vertically differentiated products. This product hierarchy enables the firm to fill demand for a lower-quality product by converting a higher-quality product. In addition to the challenges presented by random yields and multiple products, coproduction systems often serve multiple customer classes that differ in their product valuations. Furthermore, the sizes of these classes are uncertain. Employing a utility-maximizing customer model, we investigate the production, pricing, downconversion, and allocation decisions in a two-class, stochastic-demand, stochastic-yield coproduction system. For the single-class case, we establish that downconversion will not occur if prices are set optimally. In contrast, we show that downconversion can be optimal in the two-class case, even if prices are set optimally. We consider the benefit of postponing certain operational decisions, e.g., the pricing or allocation-rule decisions, until after uncertainties are resolved. We use the term recourse to denote actions taken after uncertainties have been resolved. We find that recourse pricing benefits the firm much more than either downconversion or recourse allocation do, implying that recourse demand management is more valuable than recourse supply management. Special cases of our model include the single-class and two-class random-yield newsvendor models.

Service-Level Differentiation in Call Centers with Fully Flexible Servers

We study large-scale service systems with multiple customer classes and many statistically identical servers. The following question is addressed: How many servers are required (staffing) and how does one match them with customers (control) to minimize staffing cost, subject to class-level quality-of-service constraints? We tackle this question by characterizing scheduling and staffing schemes that are asymptotically optimal in the limit, as system load grows to infinity. The asymptotic regimes considered are consistent with the efficiency-driven (ED), quality-driven (QD), and quality-and-efficiency-driven (QED) regimes, first introduced in the context of a single-class service system. Our main findings are as follows: (a) Decoupling of staffing and control, namely, (i) staffing disregards the multiclass nature of the system and is analogous to the staffing of a single-class system with the same aggregate demand and a single global quality-of-service constraint, and (ii) class-level service differentiation is obtained by using a simple idle-server-based threshold-priority (ITP) control (with state-independent thresholds); and (b) robustness of the staffing and control rules: our proposed single-class staffing (SCS) rule and ITP control are approximately optimal under various problem formulations and model assumptions. Particularly, although our solution is shown to be asymptotically optimal for large systems, we numerically demonstrate that it performs well also for relatively small systems.

Dynamic pricing for multiple class deterministic demand fulfillment

We consider how a firm should allocate inventory to multiple customer classes that differ based on the price they pay and their willingness to incur delay in fulfillment of their demand. The problem is set in a deterministic demand, economic-order-quantity-like environment with holding, backorder, lost demand and setup costs. The firm either fulfills demand or offers a price discount to induce the demand to wait for fulfillment from the next reorder. We determine the optimal policy and discuss how changes in various parameters affect profitability, customer service, and operational measures such as order frequency and base stock levels. We compare the results to a policy that only rations inventory without dynamic discounting and to a policy that only provides discounts. Through the comparison, we observe that dynamic pricing can be seen as a combination of a pricing mechanism which determines demand and an allocation mechanism that differentiates between customer classes, serving each ones needs. We show that if lower-value customers are distinguished by accepting reduced service, it is possible that both high and low-value customer classes see better levels of service under the optimal policy than under a discounting only policy. In addition we demonstrate the applicability of the results to a stochastic version of the problem.

Production and Inventory Control of a Single Product Assemble-to-Order System with Multiple Customer Classes

We consider the optimal production and inventory control of an assemble-to-order system with m components, one end-product, and n customer classes. A control policy specifies when to produce each component and, whenever an order is placed, whether or not to satisfy it from on-hand inventory. We formulate the problem as a Markov decision process and characterize the structure of an optimal policy. We show that a base-stock production policy is optimal, but the base-stock level for each component is dynamic and depends on the inventory level of all other components (more specifically, it is nondecreasing). We show that the optimal inventory allocation for each component is a rationing policy with different rationing levels for different demand classes. The rationing levels for each component are dynamic and also nondecreasing in the inventory level of all other components. We compare the performance of the optimal policy to heuristic policies, including the commonly used base-stock policy with fixed base-stock levels, and find them to perform surprisingly well, especially for systems with lost sales.

Optimal management of a finite M/M/R queueing system with multiple customer classes

We analyzed an M/M/R queueing system with finite capacity N where customers have multiple classes with no priorities under steady-state conditions. It is assumed that any one class of arriving customer is serviced by one or more servers, and that each customer class has equal probability of service. Analytic explicit solutions are obtained with the assumption of exponential distributions for arrival times and service times. A cost model is developed to determine the optimal system capacity and the optimal number of servers. The minimum expected cost, the optimal system capacity, the optimal number of servers, and various system performance measures are obtained for three customer classes and four customer classes, based on assumed numerical values given to the designated system parameters.

Performance analysis of an ingress switch in a JumpStart optical burst switching network

We consider an ingress optical burst switching (OBS) node employing the JumpStart signaling protocol. The switch serves a number of users, each connected to the switch with a fiber link that supports multiple wavelengths. Each wavelength is associated with a 3-state Markovian burst arrival process which permits short and long bursts to be modeled. We model the ingress switch as a closed multi-class non-product-form queueing network, which we analyze approximately by decomposition. Specifically, we develop new techniques to analyze the queueing network, first assuming a single class of customers, and subsequently multiple classes of customers. These analytical techniques have applications to general queueing networks beyond the one studied in this paper. We also develop computationally efficient approximate algorithms to analyze an ingress switch in the limiting case where the number of wavelengths is large. The algorithms have a good accuracy, and they provide insight into the effect of various system parameters on the performance of an ingress OBS switch.

Competition in Service Industries with Segmented Markets

We develop a model for the competitive interactions in service industries where firms cater to multiple customer classes or market segments with the help of shared service facilities or processes, so as to exploit pooling benefits. Different customer classes typically have distinct sensitivities to the price of service as well as the delays encountered. In such settings firms need to determine: (i) the prices charged to all customer classes, (ii) the waiting time standards, i.e. expected steady-state waiting time promised to all classes, (iii) the capacity level and (iv) a priority discipline enabling the firm to meet the promised waiting time standards under the chosen capacity level, all in an integrated planning model which accounts for the impact of the strategic choices of all competing firms. We distinguish between three types of competition: depending upon whether firms compete on the basis of their prices only, waiting time standards only, or, on the basis of price and waiting time standard. We establish in each of the three competition models that a Nash equilibrium exists under minor conditions regarding the demand volumes. We systematically compare the equilibria with those achieved when the firms service each market segment with a dedicated service process.