Song-Hee Kim

Are Call Center and Hospital Arrivals Well Modeled by Nonhomogeneous Poisson Processes

Service systems such as call centers and hospitals typically have strongly time-varying arrivals. A natural model for such an arrival process is a nonhomogeneous Poisson process (NHPP), but that should be tested by applying appropriate statistical tests to arrival data. Assuming that the NHPP has a rate that can be regarded as approximately piecewise-constant, a Kolmogorov–Smirnov (KS) statistical test of a Poisson process (PP) can be applied to test for a NHPP by combining data from separate subintervals, exploiting the classical conditional-uniform property. In this paper, we apply KS tests to banking call center and hospital emergency department arrival data and show that they are consistent with the NHPP property, but only if that data is analyzed carefully. Initial testing rejected the NHPP null hypothesis because it failed to account for three common features of arrival data: (i) data rounding, e.g., to seconds; (ii) choosing subintervals over which the rate varies too much; and (iii) overdispersion caused by combining data from fixed hours on a fixed day of the week over multiple weeks that do not have the same arrival rate. In this paper, we investigate how to address each of these three problems.

Statistical Analysis with Little's Law

The theory supporting Littles Law (L = λW) is now well developed, applying to both limits of averages and expected values of stationary distributions, but applications of Littles Law with actual system data involve measurements over a finite-time interval, which are neither of these. We advocate taking a statistical approach with such measurements. We investigate how estimates of L and λ can be used to estimate W when the waiting times are not observed. We advocate estimating confidence intervals. Given a single sample-path segment, we suggest estimating confidence intervals using the method of batch means, as is often done in stochastic simulation output analysis. We show how to estimate and remove bias due to interval edge effects when the system does not begin and end empty. We illustrate the methods with data from a call center and simulation experiments.

Poisson and non-Poisson properties in appointment-generated arrival processes

Previous statistical tests showed that call center arrival data were consistent with a non-homogeneous Poisson process (NHPP) within each day, but exhibit over-dispersion over multiple days. These tests are applied to arrival data from an endocrinology clinic, where arrivals are by appointment. The clinic data are also consistent with an NHPP within each day, but exhibit under-dispersion over multiple days. This analysis supports a new Gaussian-uniform arrival process model, with Gaussian daily totals and uniformly distributed arrivals given the totals.

Association Among Icu Congestion, Icu Admission Decision, and Patient Outcomes.

Objectives:To employ automated bed data to examine whether ICU occupancy influences ICU admission decisions and patient outcomes. Design:Retrospective study using an instrumental variable to remove biases from unobserved differences in illness severity for patients admitted to ICU. Setting:Fifteen hospitals in an integrated healthcare delivery system in California. Patients:Seventy thousand one hundred thirty-three episodes involving patients admitted via emergency departments to a medical service over a 1-year period between 2008 and 2009. Interventions:None. Measurements and Main Results:A third of patients admitted via emergency department to a medical service were admitted under high ICU congestion (more than 90% of beds occupied). High ICU congestion was associated with a 9% lower likelihood of ICU admission for patients defined as eligible for ICU admission. We further found strong associations between ICU admission and patient outcomes, with a 32% lower likelihood of hospital readmission if the first inpatient unit was an ICU. Similarly, hospital length of stay decreased by 33% and likelihood of transfer to ICU from other units—including ICU readmission if the first unit was an ICU—decreased by 73%. Conclusions:High ICU congestion is associated with a lower likelihood of ICU admission, which has important operational implications and can affect patient outcomes. By taking advantage of our ability to identify a subset of patients whose ICU admission decisions are affected by congestion, we found that, if congestion were not a barrier and more eligible patients were admitted to ICU, this hospital system could save approximately 7.5 hospital readmissions and 253.8 hospital days per year. These findings could help inform future capacity planning and staffing decisions.

ESTIMATING WAITING TIMES WITH THE TIME-VARYING LITTLE'S LAW

When waiting times cannot be observed directly, Little’s law can be applied to estimate the average waiting time by the average number in system divided by the average arrival rate, but that simple indirect estimator tends to be biased significantly when the arrival rates are time-varying and the service times are relatively long. Here it is shown that the bias in that indirect estimator can be estimated and reduced by applying the timevarying Little’s law (TVLL). If there is appropriate time-varying staffing, then the waiting time distribution may not be time-varying even though the arrival rate is time varying. Given a fixed waiting time distribution with unknown mean, there is a unique mean consistent with the TVLL for each time t. Thus, under that condition, the TVLL provides an estimator for the unknown mean wait, given estimates of the average number in system over a subinterval and the arrival rate function. Useful variants of the TVLL estimator are obtained by fitting a linear or quadratic function to arrival data. When the arrival rate function is approximately linear (quadratic), the mean waiting time satisfies a quadratic (cubic) equation. The new estimator based on the TVLL is a positive real root of that equation. The new methods are shown to be effective in estimating the bias in the indirect estimator and reducing it, using simulations of multi-server queues and data from a call center.

A Data-Driven Model of an Appointment-Generated Arrival Process at an Outpatient Clinic

We develop a high-fidelity simulation model of the patient arrival process to an endocrinology clinic by carefully examining appointment and arrival data from that clinic. The data include the time that the appointment was originally made as well as the time that the patient actually arrived, as well as if the patient did not arrive at all, in addition to the scheduled appointment time. We take a data-based approach, specifying the schedule for each day by its value at the end of the previous day. This data-based approach shows that the schedule for a given day evolves randomly over time. Indeed, in addition to three recognized sources of variability—(i) no-shows, (ii) extra unscheduled arrivals, and (iii) deviations in the actual arrival times from the scheduled times—we find that the primary source of variability in the arrival process is variability in the daily schedule itself. Even though service systems with arrivals by appointment can differ in many ways, we think that our data-based approach to mo...

The Power of Alternative Kolmogorov-Smirnov Tests Based on Transformations of the Data

The Kolmogorov-Smirnov (KS) statistical test is commonly used to determine if data can be regarded as a sample from a sequence of independent and identically distributed (i.i.d.) random variables with specified continuous cumulative distribution function (cdf) F, but with small samples it can have insufficient power, that is, its probability of rejecting natural alternatives can be too low. However, in 1961, Durbin showed that the power of the KS test often can be increased, for a given significance level, by a well-chosen transformation of the data. Simulation experiments reported here show that the power can often be more consistently and substantially increased by a different transformation. We first transform the given sequence to a sequence of mean-1 exponential random variables, which is equivalent to a rate-1 Poisson process. We then apply the classical conditional-uniform transformation to convert the arrival times into i.i.d. random variables uniformly distributed on [0, 1]. And then, after those two preliminary steps, we apply the original Durbin transformation. Since these KS tests assume a fully specified cdf, we also investigate the consequence of having to estimate parameters of the cdf.

Maximizing Intervention Effectiveness

Frequently, policymakers seek to roll out an intervention previously proven effective in a research study, perhaps subject to resource constraints. However, since different subpopulations may respond differently to the same treatment, there is no a priori guarantee that the intervention will be as effective in the targeted population as it was in the study. How then should policymakers target individuals to maximize intervention effectiveness? We propose a novel robust optimization approach that leverages evidence typically available in a published study. Our approach is tractable -- real-world instances are easily optimized in minutes with off-the-shelf software -- and flexible enough to accommodate a variety of resource and fairness constraints. We compare our approach with current practice by proving tight, performance guarantees for both approaches which emphasize their structural differences. We also prove an intuitive interpretation of our model in terms of regularization, penalizing differences in the demographic distribution between targeted individuals and the study population. Although the precise penalty depends on the choice of uncertainty set, we show for special cases that we can recover classical penalties from the covariate matching literature on causal inference. Finally, using real data from a large teaching hospital, we compare our approach to current practice in the particular context of reducing emergency department utilization by Medicaid patients through case management. We find that our approach can offer significant benefits over current practice, particularly when the heterogeneity in patient response to the treatment is large.

Using simulation to study statistical tests for arrival process and service time models for service systems

When fitting queueing models to service system data, it can be helpful to perform statistical tests to confirm that the candidate model is appropriate. The Kolmogorov-Smirnov (KS) test can be used to test whether a sample of interarrival times or service times can be regarded as a sequence of i.i.d. random variables with a continuous cdf, and also to test a nonhomogeneous Poisson Process (NHPP). Using extensive simulation experiments, we study the power of various alternative KS tests based on data transformations. Among available alternative tests, we find the one with the greatest power in testing a NHPP. Furthermore, we devise a new method to test a sequence of i.i.d. random variables with a specified continuous cdf; it first transforms a given sequence to a rate-1 Poisson process (PP) and then applies the existing KS test of a PP. We show that it has greater power than direct KS tests.