Forecast Errors and Random Arrival Rates

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Forecast Errors andRandom Arrival Rates

• Samuel G. Steckley
• Shane G. Henderson
• Vijay Mehrotra

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Tuesdays 8-9am

• Average over past weeks 100 calls
• Std deviation 40
• Not Poisson?
• Maybe it is Poisson if take into account forecast

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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

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Outline

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

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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

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Forecast errors are real. What is their impact?
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Performance Measures

• Pr(Abandon)
• Let W Time in queue if dont abandon
• Pr(W 0)
• P(W lt 20 seconds)

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A Model of Forecast Errors

• Xi Poisson(Bi ?i)
• Bis 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?

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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

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Results

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

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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?

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No abandonment, ? 12, P(W0)
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? ? 12, P(W0)
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? ? 12, P(Ab)
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So What Should We Do?

• Ignore it?
• Add agents?
• Agents on call at home?
• Outsourcing excess calls

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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?

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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