טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentIshay Eva
SubjectFitting Phase-Type Distributions to Data from a Telephone
Call-Center
DepartmentDepartment of Industrial Engineering and Management
Supervisor Professor Emeritus Avishai Mandelbaum


Abstract

We analyze Service Times and Customers’ Patience at a Call Center of one of Israel‘s banks. This is done by modeling and fitting Phase-type (PH) distributions to its data. The motivation is the optimization of call center performance.


Service time is the positive time a customer spends with an agent, until departure from the service/system. Whereas the patience is the time a customer is willing to wait in queue before being served. For customers who abandon the system before being served, the patience is their positive waiting time in queue before abandoning the system. On the other hand, the patience of customers who get the service is larger than their waiting time in queue and hence, the corresponding data constitutes right-censored observations. The parameters of PH-distributions, for both censored and non-censored observations, are estimated via the EM-algorithm using the EMpht-program. The Kaplan-Meier setup for estimating the patience and the Kernel density estimator for estimating the density of service data are implemented, using S-PLUS and Matlab. Simultaneous confidence interval for the empirical cumulative distribution function (CDF) provides heuristic stopping rules for adding phases of the fitted PH-distribution. We implement the Kolmogorov-Smirnov and Anderson-Darling goodness-of-fit tests to evaluate quantitative aspects of the produced fits.


We found that the general structure of order k = 3 already provides a reasonable fit to the service time. Moreover, the Coxian structure of the same order is also appropriate. The PH-model that provides a perfect fit to patience is the general Coxian structure of order k = 30, which captures peaks that take place, around 15 and 60 seconds, while the overall time-interval is over 1000 seconds. We fit PH-distributions to four major service types of customers and priorities. We note a stochastic ordering between types and priorities. For example, it is demonstrated that high priority customers are more patient.


In view of the fact that Service Times of our call center turn out to be Lognormal distributed we compared Phase-Type to Lognormal. Method of moments has been used for comparing between these two distributions. Furthermore, we found the optimal parameters of the PH-distribution numerically, for specific parameters of Lognormal distribution and for a given order of the PH-distribution.