Empirical Analysis of a Telephone Call Center

1. 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, 29.5.01

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.

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.

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.

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 customer’s 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).

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).

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.

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

9. Description of Data • About 450,000 calls. • All calls during 1999. • 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.

10. Sample Data

11. Calls per Month

12. Types of Service

13. % of Internet Calls per Month

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.

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 ?

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.

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).

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.

19. Waiting Times Survival Curves Survival Time

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.

21. Time Willing to Wait Survival function Survival Time

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).

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.

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.

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.

26. Time Willing To Wait Hazard rate Hazard Time

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.

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.

29. Service Time

30. Short Service Times Jan-Oct Nov-Dec

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

32. Service Time (cont’) Hazard rate Survival Time

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.

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.

35. Arrival Process (cont’) Yearly Monthly Hourly Daily

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.

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.

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.

39. Obsessive Callers