Statistics Seminar Forecasting Loads for a Telephone Call Center Sivan Aldor - Noiman Advisors:

1. Statistics Seminar Forecasting Loads for a Telephone Call Center Sivan Aldor - Noiman Advisors: Prof. Paul Feigin Prof. Avishai Mandelbaum

2. Outline • Call Centers Overview • The Cellular Phone Company and Data Description • The Mixed Model • Evaluation Methodology • Goodness of Fit tests • Predicting Average Service Times • Load Forecasts and Staffing Formulae

3. Call Centers - Overview • Call Centers are the primary contact point between service providers and their customers in modern businesses. • Call Centers comprise human agents who provide services, as well as an automated interactive facility which often handles the initial phase of the customer interaction, and which may provide complete service to a significant proportion of the incoming and outgoing calls.

4. Call Centers - Overview • There are approximately 2.68 million operator positions in over 50,000 call centers in the US with some locations employing over 1000 agents. • A typical call center spends between 60% and 70% of its annual budget on staff salary • Call center investment in workforce optimization technologies will exceed $1 billion by 2006.

5. Project Data MOCCA • Goal of Data MOCCA (MOdels for Call Center Analysis) Designing and Implementing a (universal) database and interface for storing, retrieving, analyzing and displaying call- by-call information from Call Centers • Enable studying the behavior of: • Customers • Service Providers / Agents • Managers / System • Wait Time, Abandonment, Retrials • Service Time, Activity Profile • Queues Lengths, Loads, Trends

6. Project Data MOCCA • Israeli Bank • Israeli Cellular Phone Company • Large US Bank • The Wharton School Team , University of Pennsylvania • Technion Team including the Statistical Laboratory group

7. The Cellular Phone Company • The Company’s Call Center • Working hours are 7AM-11PM Sunday-Friday • 750 agents are employed on a regular weekday • Handles 50,000-60,000 calls on regular weekdays • Provides services to 21 different types of queues. The three largest are: • Private queue (30% of incoming calls) • Business queue (15% of incoming calls) • Technical Support queue ( 10% of incoming calls)

8. The Cellular Company • Current Forecasting Procedure: • An automatic “black box” system predicts weekly forecasts. • Each Thursday the following procedure is carried out by the planning group: • 1. Use 6 weeks of past data as the learning data • 2. Predict the week which begins ten days later • 3. Use subjective experience to correct the predictions • 4. Derive the demand and required staffing for that week

9. Sunday Monday Tuesday Wednesday Thursday Friday Saturday 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Learning Period Prediction Lead Time Forecast Period The Cellular Company

10. DataDescription • Arrival counts to the Private queue • Each day divided into 33 half-hour intervals between 7AM-11:30PM • The learning data includes working weekdays ranging between mid-February, 2004 and December 31, 2004 • Irregular days and Holidays were removed • All together, there are 254 days

11. DataDescription

12. DataDescription • The call center has four different billing cycles • Each of its customers belongs to one of the cycles. • Each billing cycle is characterized by two periods: • Delivery Period (1-2 days) • Billing Period (usually one day), about a week later

13. DataDescription August 2004 B=Billing Period D=Delivery Period Cycle 1 Cycle 2 Cycle 3 Cycle 4 We model each cycle using two indicators one for each period

14. Daily Arrivals Weekday effect Billing effect Delivery effect Billing cycles Selection Do we really need all 8 indicators? We employ the following Poisson regression on the daily arrivals to determine the important indicators: We tried different options…

15. Characterizing Billing Cycles • Using the LR statistic, comparing to the full model, we arrived at the following two options: • Weekdays, Four delivery period indicators and Cycle 1 billing period indicator • Weekdays, One global deliveryperiod indicator and Cycle 1 billing period indicator

16. The Mixed Model • Model assumptions: • The arrival process follows an inhomogeneous Poisson process. • During half-hour intervals the rate remains constant We first use the following variance-stabilizing transformation to “change” the Poisson data into approximately normal data : For large enough since the transformed data follows

17. Fixed Effects Random Effects - No. of days between day i and day j The Mixed Model

18. The Benchmark Model

19. Evaluation Methodology • To test the prediction models use the out-of-sample performance measures. Predict each week (6 days) between April 12 and December 24, 2004. There are 203 days. • For each week use the same data and lead time as the cellular company. • At the end of the process we have for each period k during each day d:

20. Evaluation Methodology • Evaluate the model using the following measures: • For each period k during day d compute: 2. For each day summaries:

21. The Mixed Model – Billing cycles RMSE APE

22. The Mixed Model – Billing Cycles Coverage Probability Width

23. The Mixed Model – Weekdays • Weekdays patterns: • Fridays are different • Sundays are different • All other weekdays are • not significantly different • during 76% of the periods. Is it important to have 6 different patterns?

24. The Mixed Model – Weekdays RMSE APE

25. The Mixed Model – Weekdays Coverage Probability Width

26. The Mixed Model – Within Periods Comparing different within-periods covariance structures 1. AR(1) 2. ARMA(1,1) 3. Toeplitz

27. Coverage Probability RMSE AR(1) ARMA(1,1) AR(1) ARMA(1,1) Lower Quartile 0.88 0.88 Lower Quartile 32.65 32.39 Median 0.97 0.97 Median 38.35 38.47 Mean 0.93 0.93 Mean 43.40 43.23 Upper Quartile 1 1 Upper Quartile 50.94 50.74 APE Width AR(1) ARMA(1,1) AR(1) ARMA(1,1) Lower Quartile 7.83 7.83 Lower Quartile 141.53 138.98 Median 160.76 158.76 Median 9.68 9.54 Mean 162.84 161.48 Mean 10.8 10.73 Upper Quartile 185.4 161.7 Upper Quartile 12.86 12.83 The Mixed Model – Within Periods • Is ARMA really worth it? • AR(1) Procedure takes 1 hour • ARMA(1,1) takes 6-7 hours

28. V d The Mixed Model – Daily Effect • How important is the random daily effect if we are • predicting 10-days-ahead? • We might expect it to have a very small influence on the • predictions… • We test two models: • The Three-Pattern Model with the random daily effect • The Three-Pattern Model without the random daily effect

29. Coverage Probability RMSE Daily No Daily Daily No Daily Lower Quartile 0.88 0.85 Lower Quartile 32.65 32.03 Median 0.97 0.94 Median 38.35 38.63 Mean 0.93 0.91 Mean 43.40 44.53 Upper Quartile 1 1 Upper Quartile 50.94 54.12 APE Width Daily No Daily Daily No Daily Lower Quartile 7.83 7.86 Lower Quartile 141.53 136.12 Median 160.76 151.61 Median 9.68 10.09 Mean 162.84 154.63 Mean 10.8 11.03 Upper Quartile 185.4 174.95 Upper Quartile 12.86 13.2 The Mixed Model – Daily Effect When we carried out one-day-ahead predictions there were bigger differences

30. Testing for Goodness of Fit The Mixed Model Normal Assumption First we check the QQ-plot of the prediction residuals to see if it is normally distributed.

31. Is the estimated value of this variance close to 0.25? Testing for Goodness of Fit The Mixed Model We can use ARMA(1,1) to our advantage and check this out

32. Testing for Goodness of Fit The Mixed Model

33. Operational Regimes • Efficiency Driven • Quality Driven • QED (Quality-Efficiency Driven)

34. QED Regime • To predict the load (R) • We need to predict the arrival rate • We need to predict the average service

35. Average Service Time Patterns We fitted a quadratic regression based on the weekday and period The final model has the following components: Period Period*Period Weekday Weekday*Period Date

36. Average Service Time Patterns The Predicted Average Service patterns for a typical week:

37. predicted no. of required agents actual no. of required agents Predicting Staffing Levels Given the user supplied we estimate as follows:

38. Predicting Staffing Levels Averaging the difference separately for each period over the days we get the following graph:

39. Predicting Staffing Levels

40. Work in Progress • Comparison with other models • Comparison with industry models • Trying to answer: “What is the effect of different time resolutions on prediction performance?” • Tying up some loose ends…

41. NOT

42. The Prediction Lead time What is the influence of the lead time on the APE and RMSE? Lead time (Days)