Planning Service Capacity When Demand Is Sensitive to Delay

This paper presents a model designed to help determine the number of clerks or servers in a retail queuing situation. The model demonstrates a realistic treatment of the effect of waiting time on customer demand and revenues by treating the average waiting time as a form of price. Thus, the demand rate and service level are interrelated. The number of servers is obtained by balancing the cost of additional servers against the revenues and profits arising from greater demand. An example is shown, using data obtained from a supermarket, to demonstrate the insight gained from applying the model. Some extensions and research possibilities are considered, including a treatment of ancillary activities. An appendix examines a continuous version of the model which provides additional insight into the character of the model.     

AN IMPROVED HEURISTIC FOR STAFFING TELEPHONE CALL CENTERS WITH LIMITED OPERATING HOURS

Many telephone call centers that experience cyclic and random customer demand adjust their staffing over the day in an attempt to provide a consistent target level of customer service. The standard and widely used staffing method, which we call the stationary independent period by period (SIPP) approach, divides the workday into planning periods and uses a series of stationary independent Erlang‐c queuing models—one for each planning period—to estimate minimum staffing needs. Our research evaluates and improves upon this commonly used heuristic for those telephone call centers with limited hours of operation during the workday. We show that the SIPP approach often suggests staffing that is substantially too low to achieve the targeted customer service levels (probability of customer delay) during critical periods. The major reasons for SIPP‘ s shortfall are as follows: (1) SIPP's failure to account for the time lag between the peak in customer demand and when system congestion actually peaks; and (2) SIPP’ s use of the planning period average arrival rate, thereby assuming that the arrival rate is constant during the period. We identify specific domains for which SIPP tends to suggest inadequate staffing. Based on an analysis of the factors that influence the magnitude of the lag in infinite server systems that start empty and idle, we propose and test two simple “lagged” SIPP modifications that, in most situations, consistently achieve the service target with only modest increases in staffing.     

Optimal Workforce Mix in Service Systems with Two Types of Customers

We consider a service system with two types of customers. In such an environment, the servers can either be specialists (or dedicated) who serve a specific customer type, or generalists (or flexible) who serve either type of customers. Cross‐trained workers are more flexible and help reduce system delay, but also contribute to increased service costs and reduced service efficiency. Our objective is to provide insights into the choice of an optimal workforce mix of flexible and dedicated servers. We assume Poisson arrivals and exponential service times, and use matrix‐analytic methods to investigate the impact of various system parameters such as the number of servers, server utilization, and server efficiency on the choice of server mix. We develop guidelines for managers that would help them to decide whether they should be either at one of the extremes, i.e., total flexibility or total specialization, or some combination. If it is the latter, we offer an analytical tool to optimize the server mix.     

DUAL-RESOURCE PARALLEL QUEUES WITH SERVER TRANSFER AND INFORMATION ACCESS DELAYS*

Dual-resource constrained queuing systems contain fewer servers than service facilities. This study uses computer simulation to evaluate several server assignment procedures in a dual-resource system. A field study serves as the basis for developing a model with two service facilities in parallel, a single server, and deterministic information access and transfer delays that can be applied to job shops, computer operating systems, and elevators. Several findings, useful in server assignment decision making, resulted from the study. If first-come, first-served sequencing is used, delaying server assignment at a facility until all jobs are completed reduces both the mean and the variance of job flow time. If shortest-process-time-first sequencing is used, an assignment rule is tested that delays a server at a facility until a sufficiently short job is estimated to have arrived elsewhere. This rule performs best overall in terms of both the mean and variance of flow time. Methods to implement this decision rule easily are discussed.     

Capacity Planning and Allocation for Web‐Based Applications

Motivated by the technology division of a financial services firm, we study the problem of capacity planning and allocation for Web‐based applications. The steady growth in Web traffic has affected the quality of service (QoS) as measured by response time (RT), for numerous e‐businesses. In addition, the lack of understanding of system interactions and availability of proper planning tools has impeded effective capacity management. Managers typically make decisions to add server capacity on an ad hoc basis when systems reach critical response levels. Very often this turns out to be too late and results in extremely long response times and the system crashes. We present an analytical model to understand system interactions with the goal of making better server capacity decisions based on the results. The model studies the relationships and important interactions between the various components of a Web‐based application using a continuous time Markov chain embedded in a queuing network as the basic framework. We use several structured aggregation schemes to appropriately represent a complex system, and demonstrate how the model can be used to quickly predict system performance, which facilitates effective capacity allocation decision making. Using simulation as a benchmark, we show that our model produces results within 5% accuracy at a fraction of the time of simulation, even at high traffic intensities. This knowledge helps managers quickly analyze the performance of the system and better plan server capacity to maintain desirable levels of QoS. We also demonstrate how to utilize a combination of dedicated and shared resources to achieve QoS using fewer servers.     

Modeling and Managing the Percentage of Satisfied Customers in Hidden and Revealed Waiting Line Systems

We perform an analysis of various queueing systems with an emphasis on estimating a single performance metric. This metric is defined to be the percentage of customers whose actual waiting time was less than their individual waiting time threshold. We label this metric the Percentage of Satisfied Customers (PSC.) This threshold is a reflection of the customers' expectation of a reasonable waiting time in the system given its current state. Cases in which no system state information is available to the customer are referred to as “hidden queues.” For such systems, the waiting time threshold is independent of the length of the waiting line, and it is randomly drawn from a distribution of threshold values for the customer population. The literature generally assumes that such thresholds are exponentially distributed. For these cases, we derive closed form expressions for our performance metric for a variety of possible service time distributions. We also relax this assumption for cases where service times are exponential and derive closed form results for a large class of threshold distributions. We analyze such queues for both single and multi‐server systems. We refer to cases in which customers may observe the length of the line as “revealed” queues.“ We perform a parallel analysis for both single and multi‐server revealed queues. The chief distinction is that for these cases, customers may develop threshold values that are dependent upon the number of customers in the system upon their arrival. The new perspective this paper brings to the modeling of the performance of waiting line systems allows us to rethink and suggest ways to enhance the effectiveness of various managerial options for improving the service quality and customer satisfaction of waiting line systems. We conclude with many useful insights on ways to improve customer satisfaction in waiting line situations that follow directly from our analysis.     

Coping with Time‐Varying Demand When Setting Staffing Requirements for a Service System

We review queueing‐theory methods for setting staffing requirements in service systems where customer demand varies in a predictable pattern over the day. Analyzing these systems is not straightforward, because standard queueing theory focuses on the long‐run steady‐state behavior of stationary models. We show how to adapt stationary queueing models for use in nonstationary environments so that time‐dependent performance is captured and staffing requirements can be set. Relatively little modification of straightforward stationary analysis applies in systems where service times are short and the targeted quality of service is high. When service times are moderate and the targeted quality of service is still high, time‐lag refinements can improve traditional stationary independent period‐by‐period and peak‐hour approximations. Time‐varying infinite‐server models help develop refinements, because closed‐form expressions exist for their time‐dependent behavior. More difficult cases with very long service times and other complicated features, such as end‐of‐day effects, can often be treated by a modified‐offered‐load approximation, which is based on an associated infinite‐server model. Numerical algorithms and deterministic fluid models are useful when the system is overloaded for an extensive period of time. Our discussion focuses on telephone call centers, but applications to police patrol, banking, and hospital emergency rooms are also mentioned.     

Supporting Quick Response Through Scheduling of Make-to-Stock Production/Inventory Systems*

Scheduling of traditional job shops in make-to-order systems has seen extensive research over the past three decades. In such systems, performance is often related to various job completion metrics such as average flow time, average lateness, etc. This paper examines a scheduling problem in a make-to-stock environment where individual job completion measures are irrelevant. In this case, customer orders are satisfied through on-hand inventory where customer service is more closely related to the manufacturer's ability to quickly satisfy demand. We consider the role of scheduling in reducing inventories and improving customer service in the context of a manufacturer who assembles several different products on a single assembly line. We develop scheduling rules for such a system and experimentally compare their performance to those typically used in such environments. Our results indicate that rules which consider the inventory position and demand forecast outperform traditional fixed cycle rules.     

INSIGHTS ON SERVICE SYSTEM DESIGN FROM A NORMAL APPROXIMATION TO ERLANG'S DELAY FORMULA

We show how a simple normal approximation to Erlang's delay formula can be used to analyze capacity and staffing problems in service systems that can be modeled as M/M/s queues. The numbers of servers, s, needed in an M/M/s queueing system to assure a probability of delay of, at most, p can be well approximated by s ≅ p + z_***I-p+, where z_1-p, is the (1 - p)th percentile of the standard normal distribution and ρ, the presented load on the system, is the ratio of Λ, the customer arrival rate, to μ, the service rate. We examine the accuracy of this approximation over a set of parameters typical of service operations ranging from police patrol, through telemarketing to automatic teller machines, and we demonstrate that it tends to slightly underestimate the number of servers actually needed to hit the delay probability target—adding one server to the number suggested by the above formula typically gives the exact result. More importantly, the structure of the approximation promotes operational insight by explicitly linking the number of servers with server utilization and the customer service level. Using a scenario based on an actual teleservicing operation, we show how operations managers and designers can quickly obtain insights about the trade-offs between system size, system utilization and customer service. We argue that this little used approach deserves a prominent role in the operations analyst's and operations manager's toolbags.     

Social Optimal Location of Facilities with Fixed Servers,Stochastic Demand,and Congestion

We consider two capacity choice scenarios for the optimal location of facilities with fixed servers, stochastic demand, and congestion. Motivating applications include virtual call centers, consisting of geographically dispersed centers, walk‐in health clinics, motor vehicle inspection stations, automobile emissions testing stations, and internal service systems. The choice of locations for such facilities influences both the travel cost and waiting times of users. In contrast to most previous research, we explicitly embed both customer travel/connection and delay costs in the objective function and solve the location–allocation problem and choose facility capacities simultaneously. The choice of capacity for a facility that is viewed as a queueing system with Poisson arrivals and exponential service times could mean choosing a service rate for the servers (Scenario 1) or choosing the number of servers (Scenario 2). We express the optimal service rate in closed form in Scenario 1 and the (asymptotically) optimal number of servers in closed form in Scenario 2. This allows us to eliminate both the number of servers and the service rates from the optimization problems, leading to tractable mixed‐integer nonlinear programs. Our computational results show that both problems can be solved efficiently using a Lagrangian relaxation optimization procedure.     

Real‐Time Delay Estimation Based on Delay History in Many‐Server Service Systems with Time‐Varying Arrivals

Motivated by interest in making delay announcements in service systems, we study real‐time delay estimators in many‐server service systems, both with and without customer abandonment. Our main contribution here is to consider the realistic feature of time‐varying arrival rates. We focus especially on delay estimators exploiting recent customer delay history. We show that time‐varying arrival rates can introduce significant estimation bias in delay‐history‐based delay estimators when the system experiences alternating periods of overload and underload. We then introduce refined delay‐history estimators that effectively cope with time‐varying arrival rates together with non‐exponential service‐time and abandonment‐time distributions, which are often observed in practice. We use computer simulation to verify that our proposed estimators outperform several natural alternatives.     

An appointment scheduling policy for healthcare systems with parallel servers and pre-determined quality of service

The appointment scheduling problem is well-known in the literature. The use of appointment systems has been adopted widely in many different fields, including service industries and especially healthcare.This research focuses on healthcare systems where patients arrive according to pre-assigned appointments. We consider healthcare systems with several parallel servers, where a given sequence of patients, with randomly distributed service durations and a possibility of no-shows, is to be scheduled. The aim is to minimize the end of day and increase resource utilization while a minimal probability of each appointment starting on time (quality of service) is required.We formulated the problem using mathematical programing and developed a multi-server numerical-based (MSN) algorithm to solve it. We conducted some experimental runs and checked the impact of the problem parameters on the end of day, customers’ average waiting time and the percentage of customers that waited for service. We also show how server pooling improves the above system measures. Finally, once the appointments are set, we develop a methodology to determine the shift length so as to balance overtime costs (costs of overtime hours) against undertime costs (costs of regular, unused hours).     

Revenue Management with End‐of‐Period Discounts in the Presence of Customer Learning

Consider a firm that sells identical products over a series of selling periods (e.g., weekly all‐inclusive vacations at the same resort). To stimulate demand and enhance revenue, in some periods, the firm may choose to offer a part of its available inventory at a discount. As customers learn to expect such discounts, a fraction may wait rather than purchase at a regular price. A problem the firm faces is how to incorporate this waiting and learning into its revenue management decisions. To address this problem we summarize two types of learning behaviors and propose a general model that allows for both stochastic consumer demand and stochastic waiting. For the case with two customer classes, we develop a novel solution approach to the resulting dynamic program. We then examine two simplified models, where either the demand or the waiting behavior are deterministic, and present the solution in a closed form. We extend the model to incorporate three customer classes and discuss the effects of overselling the capacity and bumping customers. Through numerical simulations we study the value of offering end‐of‐period deals optimally and analyze how this value changes under different consumer behavior and demand scenarios.     

Design Principles for Flexible Systems

A fundamental aspect of designing systems with dedicated servers is identifying and improving the system bottlenecks. We extend the concept of a bottleneck to networks with heterogeneous, flexible servers. In contrast with a network with dedicated servers, the bottlenecks are not a priori obvious, but can be determined by solving a number of linear programming problems. Unlike the dedicated server case, we find that a bottleneck may span several nodes in the network. We then identify some characteristics of desirable flexibility structures. In particular, the chosen flexibility structure should not only achieve the maximal possible capacity (corresponding to full server flexibility), but should also have the feature that the entire network is the (unique) system bottleneck. The reason is that it is then possible to shift capacity between arbitrary nodes in the network, allowing the network to cope with demand fluctuations. Finally, we specify when certain flexibility structures (in particular chaining, targeted flexibility, and the “N” and “W” structures from the call center literature) possess these desirable characteristics.     

Equivalent Alternate Solutions for the Tour Scheduling Problem*

Achieving minimum staffing costs, maximum employee satisfaction with their assigned schedules, and acceptable levels of service are important but potentially conflicting objectives when scheduling service employees. Existing employee scheduling models, such as tour scheduling or general employee scheduling, address at most two of these criteria. This paper describes a heuristic to improve tour scheduling solutions provided by other procedures, and generate a set of equivalent cost feasible alternatives. These alternatives allow managers to identify solutions with attractive secondary characteristics, such as overall employee satisfaction with their assigned tours or consistent employee workloads and customer response times. Tests with both full-time and mixed work force problems reveal the method improves most nonoptimal initial heuristic solutions. Many of the alternatives generated had more even distributions of surplus staff than the initial solutions, yielding more consistent customer response times and employee workloads. The likelihood of satisfying employee scheduling preferences may also be increased since each alternative provides a different deployment of employees among the available schedules.     

Flexible Servers in Understaffed Tandem Lines

We study the dynamic assignment of cross‐trained servers to stations in understaffed lines with finite buffers. Our objective is to maximize the production rate. We identify optimal server assignment policies for systems with three stations, two servers, different flexibility structures, and either deterministic service times and arbitrary buffers or exponential service times and small buffers. We use these policies to develop server assignment heuristics for Markovian systems with larger buffer sizes that appear to yield near‐optimal throughput. In the deterministic setting, we prove that the best possible production rate with full server flexibility and infinite buffers can be attained with partial flexibility and zero buffers, and we identify the critical skills required to achieve this goal. We then present numerical results showing that these critical skills, employed with an effective server assignment policy, also yield near‐optimal throughput in the Markovian setting, even for small buffer sizes. Thus, our results suggest that partial flexibility is sufficient for near‐optimal performance, and that flexibility structures that are effective for deterministic and infinite‐buffered systems are also likely to perform well for finite‐buffered stochastic systems.     

A non-queueing model to predict teller requirements in retail bank branches

Two models are developed in this paper to determine optimum teller scheduling at bank branches. The first is a parametric model consisting of three phases: predicting the aggregate number of transactions for the bank based on the factors influencing customer demand and arrival rates; determining the percent of the aggregate transactions volume performed at each retail branch based on customer profiles and branch locations; formulating a transaction sensitive teller staffing model for each retail branch based upon predetermined customer service time and personnel utilization criteria. The second model is an integer programming model to determine the optimum teller staffing requirements based on the results of the first model and constrained by traditional current banking practices.     

An Analysis of Dual‐Kanban Just‐In‐Time Systems in a Non‐Repetitive Environment

Recent advances in approaches and production technologies for the production of goods and services have made just‐in‐time (JIT) a strong alternative for use in intermittent and small batch production systems, especially when time‐based competition is the norm and a low inventory is a must. However, the conventional JIT system is designed for mass production with a stable master production schedule. This paper suggests supplementing the information provided by production kanbans with information about customer waiting lines to be used by operators to schedule production in each work‐station of intermittent and small batch production systems. This paper uses simulation to analyze the effect of four scheduling policy variables—number of kanbans, length of the withdrawal cycle, information about customer waiting lines, and priority rules on two performance measures—customer wait‐time and inventory. The results show that using information about customer waiting lines reduces customer wait‐time by about 30% while also reducing inventory by about 2%. In addition, the effect of information about customer waiting lines overshadows the effect of priority rules on customer wait‐time and inventory.     

APPROXIMATIONS FOR THE GI/G/m QUEUE

Queueing models can usefully represent production systems experiencing congestion due to irregular flows, but exact analyses of these queueing models can be difficult. Thus it is natural to seek relatively simple approximations that are suitably accurate for engineering purposes. Here approximations for a basic queueing model are developed and evaluated. The model is the GI/G/m queue, which has m identical servers in parallel, unlimited waiting room, and the first-come first-served queue discipline, with service and interarrival times coming from independent sequences of independent and identically distributed random variables with general distributions. The approximations depend on the general interarrival-time and service-time distributions only through their first two moments. The main focus is on the expected waiting time and the probability of having to wait before beginning service, but approximations are also developed for other congestion measures, including the entire distributions of waiting time, queue-length and number in system. These relatively simple approximations are useful supplements to algorithms for computing the exact values that have been developed in recent years. The simple approximations can serve as starting points for developing approximations for more complicated systems for which exact solutions are not yet available. These approximations are especially useful for incorporating GI/G/m models in larger models, such as queueing networks, wherein the approximations can be components of rapid modeling tools.     

Order Promising and the Master Production Schedule*

Previous research on material requirements planning (MRP) systems has rarely considered the impact of the master production scheduling method used to promise customer orders and to allocate production capacity. Based on a simulation study of an MRP environment, we show that the correct selection of a master production schedule (MPS) method depends on the variance of end-item demand. In addition, we find evidence that the effectiveness of a particular MPS method can be enhanced by holding buffer inventory at the same level in the product structure as in the MPS.