考虑等待提示策略的呼叫中心流体近似方法

向顾客公布需等待的排队时间用以缓解系统拥挤是目前呼叫中心运营管理的重要手段之一。为了有效刻画等待提示策略下顾客行为变化对呼叫系统性能的影响,采用流体近似方法建立了呼叫排队系统模型。首先,通过排队分析构造等待提示影响下排队行为框架,包含带有心理行为变化特征的多种行为要素概率函数;其次,利用流体方法构建了考虑顾客重拨影响的连续排队模型,并求解了稳态条件下的系统性能指标;最后,通过与仿真模型的对比,验证了该流体近似方法的有效性与精确性。研究结果对于带有等待时间提示的呼叫中心运营具有重要指导作用。     

Call center service process analysis: Bayesian parametric and semi-parametric mixture modeling

The main focus of the call center research has been on models that assume all input distributions are known in queuing theory which gives birth to staffing and the estimation of operating characteristics. Studies investigating uncertainty of the input distributions and its implications on call center management are scarce. This study attempts to fill this gap by analyzing the call center service distribution behavior by using Bayesian parametric and semi-parametric mixture models that are capable of exhibiting non-standard behavior such as multi-modality, skewness and excess kurtosis motivated by real call center data. The study is motivated by the observation that different customer profiles might require different agent skill sets which can create additional sources of uncertainty in the behavior of service distributions. In estimating model parameters, Markov chain Monte Carlo methods such as the Gibbs sampler and the reversible jump algorithms are presented and the implications of using such models on system performance and staffing are discussed.     

Performance analysis of a call center with interactive voice response units

A Call center may be defined as a service unit where a group of agents handle a large volume of incoming telephone calls for the purpose of sales, service, or other specialized transactions. Typically a call center consists of telephone trunk lines, a switching machine known as the automatic call distributor (ACD) together with a voice response unit (VRU), and telephone sales agents. Customers usually dial a special number provided by the call center; if a trunk line is free, the customer seizes it, otherwise the call is lost. Once the trunk line is seized, the caller is instructed to choose among several options provided by the call center via VRU. After completing the instructions at the VRU, the call is routed to an available agent. If all agents are busy, the call is queued at the ACD until one is free. One of the challenging issues in the design of a call center is the determination of the number of trunk lines and agents required for a given call load and a given service level. Call center industries use the Erlang-C and the Erlang-B formulae in isolation to determine the number of agents and the number of trunk lines needed respectively. In this paper we propose and analyze a flow controlled network model to capture the role of the VRU as well as the agents. Initially, we assume Poisson arrivals, exponential processing time at the VRU and exponential talk time. This model provides a way to determine the number of trunk lines and agents required simultaneously. An alternative simplified model (that ignores the role of the VRU) will be to use anM|M|S|N queueing model (whereS is the number of agents andN is the number of trunk lines) to determine the optimalS andN subject to service level constraints. We will compare the effectiveness of this simplified model and other approximate methods with our model. We will also point out the drawbacks of using Erlang-C and Erlang-B formulae in isolation. Contributed paper for the First Madrid Conference on Queueing Theory, held in Complutense University, Madrid, Spain, July 2–5, 2002     

Robust and data-driven approaches to call centers

We propose both robust and data-driven approaches to a fluid model of call centers that incorporates random arrival rates with abandonment to determine staff levels and dynamic routing policies. We test the resulting models with real data obtained from the call center of a US bank. Computational results show that the robust fluid model is significantly more tractable as compared to the data-driven one and produces overall better solutions to call centers in most experiments.     

Rare event RESTART simulation of two-stage networks

REpetitive Simulation Trials After Reaching Thresholds (RESTART) is a widely applied accelerated simulation technique that allows the evaluation of extremely low probabilities. In this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher. Formulas for evaluating the optimal number of regions and retrials as well as guidelines for obtaining suitable importance functions were provided in previous papers. Nevertheless, further investigations are required to apply these guidelines to practical cases.     

Towards “zero abandonments” in call center performance

The call center industry is a big business in today's global economy. Staffing costs account for over half of a call center's total operations costs. Some large call centers, in practice, operate at very close to maximum capacity, believing that such an operations policy is efficient. However, by operating at levels close to 100% utilization, a call center is “living dangerously”. If, for example, call volumes even slightly exceed forecasts, customer calls will queue. As queue lengths and durations increase, customers will tend to abandon their calls. We provide some “rule-of-thumb” formulas that evaluate the cost of abandonments. These formulas may be used to justify an investment in additional agents required to improve the quality of service and reduce abandonments. Standard Erlang-C queueing formulas imply that abandonments can be significantly reduced with a small investment in additional agents. Thus, by improving customer service and hiring additional staff, a call center can improve profitability. We illustrate our analysis with realistic data, based on our work with large-scale customer service centers.     

On the distribution of the number of retrials

We consider queuing systems where customers are not allowed to queue, instead of that they make repeated attempts, or retrials, in order to enter service after some time. We obtain the distribution of the number of retrials produced by a tagged customer, until he finds an available server.     

A stochastic programming model for scheduling call centers with global Service Level Agreements

We consider the issue of call center scheduling in an environment where arrivals rates are highly variable, aggregate volumes are uncertain, and the call center is subject to a global service level constraint. This paper is motivated by work with a provider of outsourced technical support services where call volumes exhibit significant variability and uncertainty. The outsourcing contract specifies a Service Level Agreement that must be satisfied over an extended period of a week or month. We formulate the problem as a mixed-integer stochastic program. Our model has two distinctive features. Firstly, we combine the server sizing and staff scheduling steps into a single optimization program. Secondly, we explicitly recognize the uncertainty in period-by-period arrival rates. We show that the stochastic formulation, in general, calculates a higher cost optimal schedule than a model which ignores variability, but that the expected cost of this schedule is lower. We conduct extensive experimentation to compare the solutions of the stochastic program with the deterministic programs, based on mean valued arrivals. We find that, in general, the stochastic model provides a significant reduction in the expected cost of operation. The stochastic model also allows the manager to make informed risk management decisions by evaluating the probability that the Service Level Agreement will be achieved.     

A new model for call centre queue management

A new model for call centre queue management is described. It incorporates important features of call centre queues and is shown to produce results that are very different from those produced by the more usual models. The analytic approach is easy to apply, and is used to offer some interesting insights for call center queue management.     

Designing a call center with an IVR (Interactive Voice Response)

A call center is a service operation that caters to customer needs via the telephone. Call centers typically consist of agents that serve customers, telephone lines, an Interactive Voice Response (IVR) unit, and a switch that routes calls to agents. In this paper we study a Markovian model for a call center with an IVR. We calculate operational performance measures, such as the probability for a busy signal and the average wait time for an agent. Exact calculations of these measures are cumbersome and they lack insight. We thus approximate the measures in an asymptotic regime known as QED (Quality and Efficiency Driven) or the Halfin–Whitt regime, which accommodates moderate to large call centers. The approximations are both insightful and easy to apply (for up to 1000’s of agents). They yield, as special cases, known and novel approximations for the M/M/N/N (Erlang-B), M/M/S (Erlang-C) and M/M/S/N queue.     

Some analytical results for congestion in subscriber line modules

In modern telephone exchanges, subscriber lines are usually connected to the so-called subscriber line modules. These modules serve both incoming and outgoing traffic. An important difference between these two types of calls lies in the fact that in the case of blocking due to all channels busy in the module, outgoing calls can be queued whereas incoming calls get busy signal and must be re-initiated in order to establish the required connection. The corresponding queueing model was discussed recently by Lederman, but only the model with losses has been studied analytically. In the present contribution, we study the model which takes into account subscriber retrials and investigate some of its properties such as existence of stationary regime, derive explicit formulas for the system characteristics, limit theorems for systems under high repetition intensity of blocked calls and limit theorems for systems under heavy traffic.     

A G/M/1-queue with exponential retrial

Summary We consider a G/M/1 retrial model in which the delays between retrials are i.i.d. exponentially distributed random variables. We investigate the steady-state distribution of the embedded Markov chain at completion service epochs, the stationary distribution at anytime and the virtual waiting time.     

A G/M/1 retrial queue with constant retrial rate

In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented.     

A stochastic analysis of a network with two levels of service

Queueing Systems - In this paper, a stochastic model of a call center with a two-level architecture is analyzed. A first-level pool of operators answers calls, identifies, and handles non-urgent...     

Discrete-time analysis of the GI/G/1 system with Bernoulli retrials: An algorithmic approach

In this paper, we show that the discrete GI/G/1 system with Bernoulli retrials can be analyzed as a level-dependent QBD process with infinite blocks; these blocks are finite when both the inter-arrival and service times have finite supports. The resulting QBD has a special structure which makes it convenient to analyze by the Matrix-analytic method (MAM). By representing both the inter-arrival and service times using a Markov chain based approach we are able to use the tools for phase type distributions in our model. Secondly, the resulting phase type distributions have additional structures which we exploit in the development of the algorithmic approach. The final working model approximates the level-dependent Markov chain with a level independent Markov chain that has a large set of boundaries. This allows us to use the modified matrix-geometric method to analyze the problem. A key task is selecting the level at which this level independence should begin. A procedure for this selection process is presented and then the distribution of the number of jobs in the orbit is obtained. Numerical examples are presented to demonstrate how this method works.     

A Discrete-Time Geo/G/1 retrial queue with the server subject to starting failures

This paper studies a discrete-time Geo/G/1 retrial queue where the server is subject to starting failures. We analyse the Markov chain underlying the regarded queueing system and present some performance measures of the system in steady-state. Then, we give two stochastic decomposition laws and find a measure of the proximity between the system size distributions of our model and the corresponding model without retrials. We also develop a procedure for calculating the distributions of the orbit and system size as well as the marginal distributions of the orbit size when the server is idle, busy or down. Besides, we prove that the M/G/1 retrial queue with starting failures can be approximated by its discrete-time counterpart. Finally, some numerical examples show the influence of the parameters on several performance characteristics. This work is supported by the DGINV through the project BFM2002-02189.     

Discrete-time GI/G/1 retrial queues with time-controlled vacation policies

A discrete-time GI/G/1 retrial queue with Bernoulli retrials and time-controlled vacation policies is investigated in this paper. By representing the inter-arrival, service and vacation tlmes using a Markov-based approach, we are able to analyze this model as a level-dependent quasi-birth-and-death （LDQBD） process which makes the model algorithmically tractable. Several performance measures such as the stationary probability distribution and the expected number of customers in the orbit have been discussed with two different policies： deterministic time-controlled system and random time-controlled system. To give a comparison with the known vacation policy in the literature, we present the exhaustive vacation policy as a contrast between these policies under the early arrival system （EAS） and the late arrival system with delayed access （LAS-DA）. Significant difference between EAS and LAS-DA is illustrated by some numerical examples.     

Bayesian Analysis of Abandonment in Call Center Operations

In this paper, we consider the modeling and the inference of abandonment behavior in call centers. We present several time to event modeling strategies, develop Bayesian inference for posterior and predictive analyses, and discuss implications on call center staffing. Different family of distributions, piecewise time to abandonment models, and mixture models are introduced, and their posterior analysis with censored abandonment data is carried out using Markov chain Monte Carlo methods. We illustrate the implementation of the proposed models using real call center data, present additional insights that can be obtained from the Bayesian analysis, and discuss implications for different customer profiles. Copyright © 2012 John Wiley & Sons, Ltd.     

Queueing Models of Call Centers: An Introduction

This is a survey of some academic research on telephone call centers. The surveyed research has its origin in, or is related to, queueing theory. Indeed, the queueing-view of call centers is both natural and useful. Accordingly, queueing models have served as prevalent standard support tools for call center management. However, the modern call center is a complex socio-technical system. It thus enjoys central features that challenge existing queueing theory to its limits, and beyond.The present document is an abridged version of a survey that can be downloaded from www.cs.vu.nl/obp/callcenters and ie.technion.ac.il/serveng.     

多目标决策中的力学思想及其交互重心移动法

利用力学中的重心概念 ,提出求多目标规划问题满意解的交互重心移动法 .它不仅简便易行 ,而且可以有效地利用以前的信息 ,防止决策者前后判断不一致情况的出现 .