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Forecast Errors and Random Arrival Rates

Explore the extent of forecast errors in call volume data and their impact on performance measures. Identify minimal agent levels for good service and consider options such as outsourcing or having agents on call at home. Address the challenges and issues associated with ignoring or adding agents to the system.

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Forecast Errors and Random Arrival Rates

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  1. Forecast Errors andRandom Arrival Rates Samuel G. Steckley Shane G. Henderson Vijay Mehrotra

  2. volume week Tuesdays 8-9am • Average over past weeks = 100 calls • Std deviation = 40 • Not Poisson? • Maybe it is Poisson if take into account forecast

  3. Big Picture • Want to identify minimal agent levels to achieve good service • Not easy - call volume forecast errors • Contribution: • Measure extent of these errors in dataset, explicitly using forecasts as baseline • Identify good performance measures • Demonstrate impact of forecast errors on performance. Do they matter? • Reinforces JK01, CH01, ADL04, BGMSSZZ

  4. Outline • Are forecast errors real? • Performance measures • Model of forecast errors • Estimation of busyness factor • Impact on performance measures • So what do we do?

  5. Are Forecast Errors Real? • Have n instances of a period • i = call volume forecast for ith instance • Xi = actual call volume in ith instance • Xi» Poisson(i) • Set Zi = (Xi – i) / i • Zs should have mean 0, std deviation 1

  6. Forecast errors are real. What is their impact?

  7. Performance Measures • Pr(Abandon) Let W = Time in queue if don’t abandon • Pr(W = 0) • P(W <= 20 seconds)

  8. A Model of Forecast Errors • Xi» Poisson(Bi i) • Bi‘s are i.i.d., gamma(, 1/) • For instance i of the period: • Generate Bi as gamma • Generate Xi as Poisson(Bii) • How do we fit  from data?

  9. MLE and MoM Estimators • Maximum likelihood is straightforward, involves 1-D numerical optimization • Also Method of Moments: Xi» Poisson(Bii), E Bi = 1 Zi = (Xi – i) / ihas E Zi = 0, var Zi = 1 + i /  Proposition: n! a.s. if i is bounded

  10. Results •  varies from around 4 and up • Values around 10 are not unusual • Some values are over 100

  11. Weighted Performance • Let f(.) be performance as fn of arr rate • Expected long-run performance is • Choose # servers assuming  = 1 • How good/bad do we do?

  12. No abandonment,  = 12, P(W=0)

  13.  = = 12, P(W=0)

  14.  = = 12, P(Ab)

  15. So What Should We Do? • Ignore it? • Add agents? • Agents on call at home? • Outsourcing excess calls

  16. Issues: Ignore / Add agents • Performance on any day is random • Short-run performance measures i.e., What might happen tomorrow? • Choose # agents to ensure service tomorrow is good with high probability?

  17. Agents on call/outsourcing • Characterize the overflow process • When should you ask for help? • How to structure contracts so that no incentive to game the system • What is a fair payment structure and amount? • Recourse problem

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