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Forecast Errors andRandom Arrival Rates

Samuel G. SteckleyShane 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

week

volume

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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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Histogram of Std Devs

0

50

100

150

200

250

300

3500

.5

1.5

2.6

3.7

4.7

5.8

6.9

7.9

9.0

10

.1

11

.1

12

.2

13

.3

14

.3

Mo

re

std(Z)

Fre

qu

en

cy

Forecast errors are real.

What is their impact?

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Performance Measures

1. Pr(Abandon)

Let W = Time in queue if don’t abandon

2. Pr(W = 0)3. P(W <= 20 seconds)

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9

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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?

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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 Z

i = 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

Gamma densities

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

00.

25 0.5

0.75 1

1.25 1.

51.

75 22.

25 2.5

2.75 3

alpha = 4

alpha=10

alpha=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(W=0)

100 500 1000

c 12 49 94

1 0.82 0.81 0.82

10 0.62 0.5 0.48

4 0.51 0.42 0.41

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= 12, P(W=0) 100 500 1000

c 12 48 92

1 0.82 0.81 0.82

10 0.62 0.5 0.48

4 0.51 0.42 0.41

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= 12, P(Ab) 100 500 1000

c 12 48 92

1 0.02 0.01 0.01

10 0.05 0.08 0.09

4 0.10 0.15 0.16

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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