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Empirical Analysis of a Telephone Call Center Anat Sakov Joint work with Avishai Mandelbaum, Sergey Zeltyn, Larry Brown, Linda Zhao and Heipeng Shen Statistics Seminar, Tel Aviv University,

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2 Introduction A call center is a service network in which agents provide telephone-based services. Consists of: callers (customers), servers (agents) and queues. Growing rapidly. Sources of data: ACD, CTI, surveys. Availability of data.

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3 Israeli Bank Call Center Types of service Regular services on checking/saving accounts. Regular services in English. Internet technical support. Information for prospective customers. Stock trading. Outgoing calls.

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4 Israeli Bank Call Center (cont) Agents 8 regular agents. 5 Internet-support agents. 1 shift supervisor. Working hours Weekdays: 7a.m.-24a.m. Friday: 7a.m.-14p.m. Saturday: 8p.m.-24p.m.

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5 Research Goals Standard assumptions: Arrival rate is a Poisson process. Service time is Exponential. Little empirical evidence to support assumptions. Checking assumptions, and developing (with OR researchers) realistic models. Estimation of customers patience while waiting, has not been studied. When better understood, can be used by call center managers (e.g. for better staffing). Requires data on the individual call basis. As a first of its kind, can serve as a prototype for future work (e.g. larger centers).

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6 Specific Questions Checking standard queueing theory assumptions. Interrelation between components of queueing model. What can be said on behavior of customers. Are they patience ? Do customers with different service type, behave differently ? Analyzing individual customers (e.g. to set priorities). Analysis of an individual agent (e.g. learning curve of new agents; skill-based routing).

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7 Statistical Challenges Massive amounts of data (our center is very small compared to centers in US, and even Israel). Issues in Survival Analysis High % of censoring (we have 80%, can reach 95% or even higher). Smooth estimation of hazard function and confidence bands for it. Dependence between censoring and failure times. More … Interdisciplinary research.

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8 Incoming Call – Event History Abandon ~15% Abandon ~5% End of Service ~80% End of Service

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9 Description of Data About 450,000 calls. All calls during Information on calls Customer ID. Priority (high/low). Type of service. Date. Time call enters VRU. Time spent in VRU. Time in queue (could be 0). Service time. Outcome: HANG / AGENT. Server name.

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10 Sample Data

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11 Calls per Month

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12 Types of Service

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13 % of Internet Calls per Month

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14 Queueing Process Three components Arrival to the system. Queue while waiting to an agent. Service. Interrelations between the three components. Waiting time is a function of arrival rate and of service time.

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15 Queueing Time 60% of calls waited in queue (positive wait) Average wait – 98 seconds. SD – 105. Median – 62. Exponential looking. Can we say that the customers are patience ? How can we define patience ?

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16 Queueing Time (cont) Average waiting time and SD for Customers reaching an agent: 105 (111). Customers abandoning: 79 (104). Numbers vary by type and priorities. Can 105 and 79 be estimates for the mean waiting time until reaching an agent and until abandoning ? No. To estimate mean time until abandoning, customers reaching an agent are censored by abandoning customers, and vice versa.

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17 Queueing Time (Cont) Denote time willing to wait, by R. Denote time needed to wait, by V. Observe W=min(R,V ) and δ=1(W=V ). To estimate the distribution of R, about 75%- 80% censoring. If assumes R and V are both Exponential and independent E(V ) = 131 (compared to 105). E(R ) = 393 (compared to 78).

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18 Queueing Time (Cont) To avoid parametric assumptions, use Kaplan-Meier, to estimate survival function. E(V ) = 141 (compared to 105). E(R ) = 741 (compared to 78). Depending on which observation is last, either E(R ) or E(V ) is downward biased.

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19 Waiting Times Survival Curves Time Survival

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20 Stochastic Ordering The stochastic ordering says that customers are willing to wait, more than what they need to wait. This suggests that customers are patience. We obtained the same picture for different types of service and different months.

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21 Time Willing to Wait Time Survival Survival function

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22 Hazard Estimation Shows local behavior. Raw hazard are building blocks for Kaplan- Meier. Noisy and unstable at tails. Would like to estimate hazard function smoothly (later, construct confidence bands).

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23 HEFT /HARE Let Model (Kooperberg, Stone and Truong (1995 JASA) ), The are splines basis functions. Plug into joint likelihood, and estimate coefficients using maximum-likelihood.

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24 HEFT/HARE (cont) HARE – HAzard REgression Use linear splines in time and covariates, and their interactions. Cox proportional model is a special case. Additivity in time and covariates indicates that proportionality assumption holds. HEFT – Hazard Estimation with Flexible Tails. No covariates. Cubic splines in time. Include additional two log terms. Fit Weibull and Pareto very well. Can use bootstrap to construct confidence bands.

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25 HEFT/HARE (software) Implementation in Splus. Pick model in an adaptive manner. Using stepwise addition/deletion. Add/drop terms to maintain hierarchy. Use BIC criteria. Fits the tails well.

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26 Time Willing To Wait Hazard rate Time Hazard

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27 Validity of Analysis A basic assumption in Survival Analysis is independence of time to failure and censoring time. A message which informs customers about their location in queue, might affect their patience. Nevertheless, the picture is informative. We ignore this and other types of dependence, as well.

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28 Other Approaches Apply nonparametric regression to obtain smooth estimates of hazard (regress raw hazard on time). Super-smoother; Kernel; LOWESS. Not as good at the tails. Local polynomial (LOCFIT). Has a module to estimate hazard. Gave qualitatively same picture.

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29 Service Time

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30 Short Service Times Jan-OctNov-Dec

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31 Service Time (cont) Survival curve, by types Time Survival

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32 Service Time (cont) Hazard rate Time Survival

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33 Service Time (cont) Standard assumption is that service time distribution is Exponential (for mathematical convenience). Density, survival function and hazard, do not support this assumption. We found that log-normal is a very good fit to service time. Holds for different types of service. Holds for different times of days. We are in the process of examining how service time vary by time of day. Can use regression.

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34 Arrival process Four levels of presentation. Differ by their time scale. Top three levels are required to support staffing. Yearly – supports strategic decisions; how many agents are needed (affects hiring and training). Monthly – supports tactical decisions; given total number of agents needed, how many permanent. Daily – supports operational decisions; staffing is made to fit rush hours, weekdays, weekends. All the above exhibit predictable variability.

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35 Arrival Process (cont) YearlyMonthly Daily Hourly

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36 Arrival process (cont) Hourly picture – depict stochastic pattern. Arrivals are typically random. Usual assumptions: Many potential, statistically identical callers. Very small probability for each to call, at any given minute. Decisions to call are independent of each other. Under the above assumptions, Poisson process. Further decomposition by types, shows that behavior vary by type of service.

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37 Checking Poisson Assumption Arrival rate being a Poisson process is a standard assumption in queueing processes. The daily picture suggests that the rate is not constant over the day, hence, inhomogeneous Poisson. We are in the process of checking this. Consider the difference in times between successive calls. Expect to behave like an exponential sample. Our checks indicates that indeed inhomogeneous Poisson.

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38 Analyzing Individual Customer Analysis of an individual customer can be used for example to update his priority. Most customers calls more than once during the year. AverageSDMedian Overall16645 Regular service14514 Stock trading831885

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39 Obsessive Callers Calls/yearCalls/day (ave) ServiceAverage Service Stock Trading %9.5 min. Regular Service %2 min.

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