Publication Archive

Abstract: Motivated by applications of Poisson processes for modelling periodic time-varying phenomena, we study a semi-parametric estimator of the period of cyclic intensity function of a non-homogeneous Poisson process. There are no parametric assumptions on the intensity function which is treated as an infinite dimensional nuisance parameter. We propose a new family of estimators for the period of the intensity function, address the identifiability and consistency issues and present simulations which demonstrate good performance of the proposed estimation procedure in practice. We compare our method to competing methods on synthetic data and apply it to a real data set from a call center.

Abstract: The interface between an emergency department and internal wards is often a hospital's bottleneck. Motivated by this interaction in an anonymous hospital, we analyze queueing systems with heterogeneous server pools, where the pools represent the wards, and the servers are beds. Our queueing system, with a single centralized queue and several server pools, forms an inverted-V model. We introduce the randomized most-idle (RMI) routing policy and analyze it in the quality-and efficiency-driven regime, which is natural in our setting. The RMI policy results in the same server fairness (measured by idleness ratios) as the longest-idle-server-first (LISF) policy, which is commonly used in call centers and considered fair. However, the RMI policy utilizes only the information on the number of idle servers in different pools, whereas the LISF policy requires information that is unavailable in hospitals on a real-time basis.

Abstract: A new innovations state space modeling framework, incorporating Box-Cox transformations, Fourier series with time varying coefficients and ARMA error correction, is introduced for forecasting complex seasonal time series that cannot be handled using existing forecasting models. Such complex time series include time series with multiple seasonal periods, high frequency seasonality, non-integer seasonality and dual-calendar effects. Our new modelling framework provides an alternative to existing exponential smoothing models, and is shown to have many advantages. The methods for initialization and estimation, including likelihood evaluation, are presented, and analytical expressions for point forecasts and interval predictions under the assumption of Gaussian errors are derived, leading to a simple, comprehensible approach to forecasting complex seasonal time series. Our trigonometric formulation is also presented as a means of decomposing complex seasonal time series, which cannot be decomposed using any of the existing decomposition methods. The approach is useful in a broad range of applications, and we illustrate its versatility in three empirical studies where it demonstrates excellent forecasting performance over a range of prediction horizons. In addition, we show that our trigonometric decomposition leads to the identification and extraction of seasonal components, which are otherwise not apparent in the time series plot itself.

Abstract: 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.

Abstract: Analysts are frequently confronted with time-series data. A simple form is magnitude (or count) and time frame, whether the data is number of e-mails sent, number of cell phones called, purchases made by volume or cost, or a variety of other time-derived data. Studying the temporal dimension of data allows analysts more opportunities to find relational ties and trends in data, classify or group like activity, and even help narrow the search space of massively complex and large datasets. This thesis presents a new approach called the Rule Based Intuition (RBI) system that can evaluate time-series data by finding the best fitting rule, from a repository of known rules, to quickly infer information about the data. This approach is most applicable for analysts viewing large sets of data who wish to classify or correlate data from users’ temporal activity.

Abstract: In this work we provide a new class of test statistics for hypothesis testing of the equality of the baseline hazard functions for correlated survival data under frailty models. The asymptotic distribution of the test statistics is investigated theoretically under the null hypothesis and certain local alternatives. We also provide a simple variance estimator. The properties of the test statistics, under finite sample size, is being studied by an extensive simulation study and we verify the control of Type I error and our proposed sample size formula. To the best of our knowledge, this is the first work for comparing the baseline hazard functions of correlated survival outcomes with covariates and frailty models. The utility of our proposed estimating technique is illustrated by the analysis of the call center data of an Israeli commercial company that processes up to 100,000 calls per day and the analysis of the breast cancer data of the Washington Ashkenazi Kin-Cohort family study.

Abstract: Today’s call center managers face multiple operational decision-making tasks. One of the most common is determining the weekly staffing levels to ensure customer satisfaction and meeting their needs while minimizing service costs. An initial step for producing the weekly schedule is forecasting the future system loads which involves predicting both arrival counts and average service times. We introduce an arrival count model which is based on a mixed Poisson process approach. The model is applied to data from an Israeli Telecom company call center. In our model, we also consider the effect of events such as billing on the arrival process and we demonstrate how to incorporate them as exogenous variables in the model. After obtaining the forecasted system load, in large call centers, a manager can choose to apply the QED (Quality-Efficiency Driven) regime’s “squareroot staffing” rule in order to balance the offered-load per server with the quality of service. Implementing this staffing rule requires that the forecasted values of the arrival counts and average service times maintain certain levels of precision. We develop different goodness of fit criteria that help determine our model’s practical performance under the QED regime. These show that during most hours of the day the model can reach desired precision levels.

Abstract: Bayesian inference for the transient behavior and duration of a busy period in a single server queueing system with general, unknown distributions for the interarrival and service times is investigated. Both the interarrival and service time distributions are approximated using the dense family of Coxian distributions. A suitable reparameterization allows the definition of a non-informative prior and Bayesian inference is then undertaken using reversible jump Markov chain Monte Carlo methods. An advantage of the proposed procedure is that heavy tailed interarrival and service time distributions such as the Pareto can be well approximated. The proposed procedure for estimating the system measures is based on recent theoretical results for the Coxian/Coxian/1 system. A numerical technique is developed for every MCMC iteration so that the transient queue length and waiting time distributions and the duration of a busy period can be estimated. The approach is illustrated with both simulated and real data.

Abstract: Accurate forecasting of call arrivals is critical for staffing and scheduling of a telephone call center. We develop methods for interday and dynamic intraday forecasting of incoming call volumes. Our approach is to treat the intraday call volume profiles as a high-dimensional vector time series. We propose first to reduce the dimensionality by singular value decomposition of the matrix of historical intraday profiles and then to apply time series and regression techniques. Our approach takes into account both interday (or day-to-day) dynamics and intraday (or within-day) patterns of call arrivals. Distributional forecasts are also developed. The proposed methods are data driven, appear to be robust against model assumptions in our simulation studies, and are shown to be very competitive in out-of-sample forecast comparisons using two real data sets. Our methods are computationally fast; it is therefore feasible to use them for real-time dynamic forecasting.

Abstract: Motivated by applications in call center management, we propose a framework based on empirical process techniques for inference about waiting time and patience distributions in multiserver queues with abandonment. The framework rigorises heuristics based on survival analysis of independent and identically distributed observations by allowing correlated waiting times. Assuming a regenerative structure of offered waiting times, we establish asymptotic properties of estimators of limiting distribution functions and derived functionals. We discuss construction of bootstrap confidence intervals and statistical tests, including a simple bootstrap two-sample test for comparing patience distributions. A small simulation study and a real data example are presented.

Abstract: Call centers becoming increasingly important in our modern commerce. We are interested in modeling the time-varying pattern of average customer service times at a bank call center. Understating such a pattern is essential for efficient operation of a call center. The call service times are show to be lognormally distributed. Motivated by this observation and the important application, we propose a new method for inference about nonparametric regression curves when the errors are lognormally distributed. Estimates and pointwise confidence bands are developed. The method builds upon the special relationship between the lognormal distribution and the normal distribution, and improves upon a naïve estimation procedure that ignores this distributional structure. Our approach includes local nonparametric estimation for both the mean function and the heteroscedastic variance function of the logged data, and uses local polynomial regression as a fitting tool. A simulation study is performed to illustrate the method. We then apply the method to model the time-varying patterns of mean service times for different types of customer calls. Several operationally interesting finding are obtained and discussed.

Abstract: A call center is a centralized hub where customer and other telephone calls are dealt with by an organization. In today’s economy, they have become the primary point of contact between customers and businesses. Accurate prediction of the call arrival rate is therefore indispensable for call center practitioners to staff their call center efficiently and cost effectively. This article proposes a multiplicative model for modeling and forecasting within-day arrival rates to a US commercial bank’s call center. Markov chain Monte Carlo sampling methods are used to estimate both latent states and model parameters. One-day-ahead density forecasts for the rates and counts are provided. The calibration of these predictive distributions is evaluated through probability integral transforms. Furthermore, we provide one-day-ahead forecasts comparisons with classical statistical models. Our predictions show significant improvements of up to 25% over these standards. A sequential Monte Carlo algorithm is also proposed for sequential estimation and forecasts of the model parameters and rates.

Abstract: A call center is a service network in which agents provide telephone-based services. Customers who seek these services are delayed in tele-queues. This article summarizes an analysis of a unique record of call center operations. The data comprise a complete operational history of a small banking call center, call by call, over a full year. Taking the perspective of queueing theory, we decompose the service process into three fundamental components: arrivals, customer patience, and service durations. Each component involves different basic mathematical structures and requires a different style of statistical analysis. Some of the key empirical results are sketched, along with descriptions of the varied techniques required. Several statistical techniques are developed for analysis of the basic components. One of these techniques is a test that a point process is a Poisson process. Another involves estimation of the mean function in a nonparametric regression with lognormal errors. A new graphical technique is introduced for nonparametric hazard rate estimation with censored data. Models are developed and implemented for forecasting of Poisson arrival rates. Finally, the article surveys how the characteristics deduced from the statistical analyses form the building blocks for theoretically interesting and practically useful mathematical models for call center operations.

Abstract:The subject of the present research is the M/M/n + G queue. This queue is characterized by Poisson arrivals at rate λ, exponential service times at rate μ, n service agents and generally distributed patience times of customers. The model is applied in the call center environment, as it captures the tradeoff between operational efficiency (staffing cost) and service quality (accessibility of agents).
In our research, three asymptotic operational regimes for medium to large call centers are studied. These regimes correspond to the following three staffing rules, as λ and n increase indefinitely and μ held fixed:
Efficiency-Driven (ED): n ≈ (λ/μ)⋅(1−γ),γ>0,
Quality-Driven (QD): n ≈ (λ/μ)⋅(1+γ),γ>0, and
Quality and Efficiency Driven (QED): n ≈ λ/μ+β√(λ/μ),−∞<β<∞.
In the ED regime, the probability to abandon and average wait converge to constants. In the QD regime, we observe a very high service level at the cost of possible overstaffing. Finally, the QED regime carefully balances quality and efficiency: agents are highly utilized, but the probability to abandon and the average wait are small (converge to zero at rate 1/√n).
Numerical experiments demonstrate that, for a wide set of system parameters, the QED formulae provide excellent approximation for exact M/M/n + G performance measures. The much simpler ED approximations are still very useful for overloaded queueing systems.
Finally, empirical findings have demonstrated a robust linear relation between the fraction abandoning and average wait. We validate this relation, asymptotically, in the QED and QD regimes.

Abstract: We consider the problem of testing whether a sample of observations comes from a single Poisson distribution. Of particular interest is the alternative that the observations come from Poisson distributions with different parameters. Such a situation would correspond to the frequently discussed situation of overdispersion. We propose a new test for this problem that is based on Anscombe’s variance stabilizing transformation. There are a number of tests commonly proposed, and we compare the performance of these tests under the null hypothesis with that of our new test. We find that the performance of our test is competitive with the two best of these. The asymptotic distribution of the new test is derived and discussed. Use of these tests is illustrated through two examples of analysis of call-arrival times from a telephone call center. The example facilitates careful discussion of the performance of the tests for small parameter values and moderately large sample sizes.

Abstract: Estimation from sample data and density approximation with phase-type distributions are considered. Maximum likelihood estimation via the EM algorithm is discussed and performed for some data sets. An extended EM algorithm is used to minimize the information divergence (maximize the relative entropy) in the density approximation case. Fits to Weibull, log normal, and Erlang distributions are used as illustrations of the latter.