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TwoParameter HeavyTraffic Limits for InfiniteServer Queues
"... Abstract: In order to obtain Markov heavytraffic approximations for infiniteserver queues with general nonexponential servicetime distributions and general arrival processes, possibly with timevarying arrival rates, we establish heavytraffic limits for twoparameter stochastic processes. We ..."
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Cited by 26 (13 self)
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Abstract: In order to obtain Markov heavytraffic approximations for infiniteserver queues with general nonexponential servicetime distributions and general arrival processes, possibly with timevarying arrival rates, we establish heavytraffic limits for twoparameter stochastic processes. We
Are Call Center and Hospital Arrivals Well Modeled by Nonhomogeneous Poisson Processes?
, 2013
"... manuscript (Please, provide the mansucript number!) Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. ..."
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Cited by 16 (8 self)
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manuscript (Please, provide the mansucript number!) Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication.
Choosing arrival process models for service systems: tests of a nonhomogeneous Poisson process.
 Nav. Res. Logist.
, 2014
"... Abstract: Service systems such as call centers and hospital emergency rooms typically have strongly timevarying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to per ..."
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Cited by 12 (10 self)
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Abstract: Service systems such as call centers and hospital emergency rooms typically have strongly timevarying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to perform statistical tests with service system data to confirm that an NHPP is actually appropriate, as emphasized by Brown et al.
Appendix to: Choosing Arrival Process Models for Service Systems: Tests of a Nonhomogeneous Poisson Process
, 2013
"... Service systems such as call centers and hospital emergency rooms typically have strongly timevarying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to perform statist ..."
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Cited by 3 (3 self)
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Service systems such as call centers and hospital emergency rooms typically have strongly timevarying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to perform statistical tests with service system data to confirm that an NHPP is actually appropriate, as emphasized by Brown et al. (2005). They suggested a specific statistical test based on the KolmogorovSmirnov statistic after exploiting the conditionaluniform property to transform the NHPP into a sequence of i.i.d. random variables uniformly distributed on [0, 1] and then performing a logarithmic transformation of the data. We conduct extensive simulation experiments to study the power of that statistical test and various alternatives. We conclude that the general approach of Brown et al. (2005) is excellent, but that an alternative KolmogorovSmirnov test proposed by Lewis (1965), exploiting a different transformation due to Durbin (1961), consistently has greater power. This appendix provides additional details for the
The Power of Alternative KolmogorovSmirnov Tests Based on Transformations of the Data
, 2013
"... The KolmogorovSmirnov (KS) statistical test is commonly used to determine if data can be regarded as a sample from a sequence of i.i.d. random variables with specified continuous cdf F, but with small samples it can have insufficient power, i.e., its probability of rejecting natural alternatives ca ..."
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Cited by 3 (3 self)
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The KolmogorovSmirnov (KS) statistical test is commonly used to determine if data can be regarded as a sample from a sequence of i.i.d. random variables with specified continuous cdf F, but with small samples it can have insufficient power, i.e., its probability of rejecting natural alternatives can be too low. However, Durbin [1961] showed that the power of the KS test often can be increased, for given significance level, by a wellchosen transformation of the data. Simulation experiments reported here show that the power can often be more consistently and substantially increased by modifying the original Durbin transformation by first transforming the given sequence to a sequence of mean1 exponential random variables, which is equivalent to a rate1 Poisson process, and then applying the classical conditionaluniform transformation to convert the arrival times into the order statistics of i.i.d. uniform random variables. The new KS test often has much more power, because it focuses on the cumulative sums rather than the random variables themselves.
Infiniteserver queues with batch arrivals and dependent service times
, 2011
"... Motivated by largescale service systems, we consider an infiniteserver queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be timevarying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d ..."
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Cited by 2 (2 self)
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Motivated by largescale service systems, we consider an infiniteserver queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be timevarying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavytraffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavytraffic approximations for the stationary model and the model with a timevarying arrival rate. In the simulation experiments, we use the MarshallOlkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of MarshallOlkin multivariate hyperexponential distributions to model dependent hyperexponential service times within a batch.
Online Appendix to: The Power of Alternative KolmogorovSmirnov Tests Based on Transformations of the Data
"... In this appendix, we present supporting materials complementing the main paper, Kim and Whitt [2013d]. In §B, we present detailed results for our main experimental setting in Section 4.1 of the main paper; §B.1 provides additional plots that supplement Section 4.2 of the main paper. We test for Erla ..."
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Cited by 1 (1 self)
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In this appendix, we present supporting materials complementing the main paper, Kim and Whitt [2013d]. In §B, we present detailed results for our main experimental setting in Section 4.1 of the main paper; §B.1 provides additional plots that supplement Section 4.2 of the main paper. We test for Erlang, Hyperexponential, and Lognormal alternatives with different parameters in §C (supplementing Section 4.3 of the paper), and §C.1 and C.2 provide supporting average empirical distribution plots for the case of E2 and H2 with c 2 = 2. In §C.3, we take a closer look at the results of the test for LN(1, 1), since it is often the specific model suggested for the service times (e.g., see Brown et al. [2005]). In §D that complement Section 4.4 of the main paper, we see how the power increases as the sample size increases for E2, H2 with c 2 = 2, and LN(1, 4) null hypotheses. §E provides supplementary materials for Section 5 of the main paper, which is on the second normal experiment.
Submitted to Manufacturing & Service Operations Management manuscript (Please, provide the mansucript number!) Are Call Center and Hospital Arrivals Well Modeled by Nonhomogeneous Poisson Processes?
, 2014
"... Service systems such as call centers and hospitals typically have strongly timevarying 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 ..."
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Service systems such as call centers and hospitals typically have strongly timevarying 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 piecewiseconstant, a KolmogorovSmirnov (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 conditionaluniform 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 take account of 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. Key words: arrival processes, nonhomogeneous Poisson process, KolmogorovSmirnov statistical test, data rounding, overdispersion
B. BASE CASE: TEST FOR EXP
"... In this appendix, we present supporting materials complementing the main paper. In §B, we present detailed results for our main experimental setting (described in Section 5.1 of the main paper); §B.1 provides additional plots that supplement Section 5.2 of the main paper. We test for Erlang, Hyperex ..."
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In this appendix, we present supporting materials complementing the main paper. In §B, we present detailed results for our main experimental setting (described in Section 5.1 of the main paper); §B.1 provides additional plots that supplement Section 5.2 of the main paper. We test for Erlang, Hyperexponential, and Lognormal alternatives with different parameters in §C (supplementing Section 5.3 of the paper), and §C.1 and C.2 provide supporting average empirical distribution plots for the case of E2 and H2 with c2 = 2. In §C.3, we take a closer look at the results of the test for LN(1, 1), since it is often the specific model suggested for the service times (e.g., see Brown et al. [2005]). In §D—which complements Section 5.4 of the main paper—, we see how the power increases as the sample size increases for E2, H2 with c2 = 2, and LN(1, 4) null hypotheses. §E provides supplementary materials for Section 6 of the main paper on the second normal experiment and §F provides supplementary materials for Section 7 of the main paper on estimating parameters.
Fluid Approximation of a Call Center Model with Redials and Reconnects
, 2013
"... In many call centers, callers may call multiple times. Some of the calls are reattempts after abandonments (redials), and some are reattempts after connected calls (reconnects). The combination of redials and reconnects has not been considered when making staffing decisions, while ignoring them wi ..."
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In many call centers, callers may call multiple times. Some of the calls are reattempts after abandonments (redials), and some are reattempts after connected calls (reconnects). The combination of redials and reconnects has not been considered when making staffing decisions, while ignoring them will inevitably lead to under or overestimation of call volumes, which results in improper and hence costly staffing decisions. Motivated by this, in this paper we study call centers where customers can abandon, and abandoned customers may redial, and when a customer finishes his conversation with an agent, he may reconnect. We use a fluid model to derive first order approximations for the number of customers in the redial and reconnect orbits in the heavy traffic. We show that the fluid limit of such a model is the unique solution to a system of three differential equations. Furthermore, we use the fluid limit to calculate the expected total arrival rate, which is then given as an input to the Erlang A model for the purpose of calculating service levels and abandonment rates. The performance of such a procedure is validated in the case of single intervals as well as multiple intervals with changing parameters. 1