Introduction
Case Studies
Theoretical Results
Future Research
Uncertainty in the Demand for Service:
The Case of Call Centers and
Emergency Departments
Shimrit Maman, Technion
Avishai Mandelbaum, Technion
Sergey Zeltyn, IBM Research Lab, Haifa
ORSIS Annual Meeting, Herzliya, May 10, 2009
Introduction
Case Studies
Contents of Talk
1
Introduction
Motivation
Research Outline
Related Work
Model Definition
2
Case Studies
Financial Call Center
Emergency Department
3
Theoretical Results
QED-c Regime
Outline of Additional Results
4
Future Research
Theoretical Results
Future Research
Introduction
Case Studies
Theoretical Results
Future Research
Motivation
Standard assumption in service system modeling: arrival
process is Poisson with known parameters.
Example of call centers: known arrival rates for each basic
interval (say, half-hour).
Application of standard approach to basic interval (say, next
Tuesday, 9am-9:30am):
Derive Poisson parameters from historical data.
Plug parameters into a queueing model (M|M|n, M|M|n + M,
Skills-Based Routing models, . . .).
Set staffing levels according to this model and service-level
agreement.
Is standard Poisson assumption valid? As a rule it is not, one
observes larger variability of the arrival process than the one
expected from the Poisson hypothesis.
Introduction
Case Studies
Theoretical Results
Research Outline
Design model for overdispersed arrival rate.
Plug arrival model into M|M|n + G queueing model.
Derive asymptotic results relevant for real-life staffing
problems.
Validate our approach via analysis of real data.
Future Research
Introduction
Case Studies
Theoretical Results
Future Research
Related Work
Henderson S.
Input model uncertainty: Why do we care and what should we do about
it?. 2003.
Steckley S., Henderson S. and Mehrotra V.
Forecast errors in service systems. 2007.
Koole G. and Jongbloed G.
Managing uncertainty in call centers using poisson mixtures. 2001.
Halfin S. and Whitt W.
Heavy-traffic limits for queues with many exponential servers. 1981.
Zeltyn S. and Mandelbaum A.
Call centers with impatient customers: Exact analysis and many-server
asymptotics of the M|M|n + G queue. 2005.
Feldman Z., Mandelbaum A., Massey W. A. and Whitt W.
Staffing of time-varying queues to achieve time-stable performance. 2007.
Introduction
Case Studies
Theoretical Results
Future Research
Model Definition
The M? |M|n + G Queue:
λ - Expected arrival rate of a Poisson arrival process.
µ - Exponential service rate.
n service agents.
G - Patience distribution. Assume that the patience density
exists at the origin and its value g0 is strictly positive.
Random Arrival Rate: Let X be a random variable with cdf F ,
E [X ] = 0, and finite σ(X ). Assume that the arrival rate varies
from interval to interval in an i.i.d. fashion:
Λ = λ + λc X,
c ≤ 1,
c ≤ 1/2: Conventional variability ∼ QED staffing regime.
1/2 < c < 1: Moderate variability ∼ QED-c regime (new).
c = 1: Extreme variability ∼ ED regime.
Introduction
Case Studies
Theoretical Results
Future Research
Financial Call Center: Data Description
Israeli Bank.
Arrival counts to the Retail queue are studied.
263 regular weekdays ranging between April 2007 and April
2008.
Holidays with different daily patterns are excluded.
Each day is divided into 48 half-hour intervals.
Introduction
Case Studies
Theoretical Results
Future Research
Financial Call Center: Arrival Rates
Average Arrival Per 30 Minutes, seperated weekdays
Average Number of Arrivals
2000
1800
Average Arrival
1600
1400
1200
1000
800
600
400
200
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
Time
Sundays
(1) Sundays;
(2) Mondays;
Mondays
Tuesdays
Wednesdays
Thursdays
(3) Tuesdays and Wednesdays;
(4) Thursdays.
Introduction
Case Studies
Theoretical Results
Future Research
Financial Call Center: Over-Dispersion Phenomenon
Coefficient of Variation
Coefficient of Variation Per 30 Minutes, seperated weekdays
sampled CV- solid line, Poisson CV - dashed line
0.7
Coefficient of Variation
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
Time
Sundays
Mondays
Tuesdays
Wednesdays
Thursdays
√
Poisson CV = 1/ mean arrival rate.
Sampled CV’s Poisson CV’s ⇒ Over-Dispersion
Introduction
Case Studies
Theoretical Results
Financial Call Center:
Relation between Mean and Standard Deviation
Consider a Poisson mixture variable Y with random rate
Λ = λ + λc · X , where E [X ] = 0, finite σ(X ) > 0 and
1/2 < c ≤ 1. Then,
Var (Y ) = λ2c · Var (X ) + λ + λc · E [X ]
and
lim (ln(σ(Y )) − c ln(λ)) = ln(σ(X )).
λ→∞
Therefore, for large λ,
ln(σ(Y)) ≈ c · ln(λ) + ln(σ(X)).
Future Research
Introduction
Case Studies
Theoretical Results
Future Research
Financial Call Center: Fitting Regression Model
Tue-Wed, 30 min resolution
Tue-Wed, 5 min resolution
ln(sd) vs ln(average) per 30 minutes. Sundays
ln(Standard Deviation) vs ln(Average) per 5 minutes, thu-wed
5
ln(Standard Deviation)
ln(Standard Deviation)
6
5
4
3
2
1
4
3
2
1
0
0
0
1
2
3
4
5
ln(Average Arrival)
6
7
8
0
1
2
3
ln(Average)
4
5
6
Introduction
Case Studies
Theoretical Results
Future Research
Financial Call Center: Fitting Regression Model
Tue-Wed, 30 min resolution
Tue-Wed, 5 min resolution
ln(sd) vs ln(average) per 30 minutes. Sundays
ln(Standard Deviation) vs ln(Average) per 5 minutes, thu-wed
5
5
y = 0.8027x - 0.1235
R2 = 0.9899
4
y = 0.8752x - 0.8589
R2 = 0.9882
3
2
1
ln(Standard Deviation)
ln(Standard Deviation)
6
4
y = 0.7228x - 0.0025
R2 = 0.9937
3
2
y = 0.7933x - 0.5727
R2 = 0.9783
1
0
0
0
1
2
3
4
5
6
ln(Average Arrival)
00:00-10:30
10:30-00:00
7
8
0
1
2
3
4
5
ln(Average)
00:00-10:30
10:30-00:00
Results:
Two clusters exist: midnight-10:30am and 10:30am-midnight.
Very good fit (R 2 > 0.97).
Significant linear relations for different weekdays and
time-resolution (5-30 min):
ln(σ(Y)) = c · ln(λ) + ln(σ(X)).
6
Introduction
Case Studies
Theoretical Results
Future Research
Financial Call Center: Outline of Additional Results
Fitting a Gamma Poisson mixture model to the data:
(Jongbloed and Koole [’01])
Assume a prior Gamma distribution for the arrival rate
d
d
Λ = Gamma(a, b). Then, the distribution of Y = Poiss(Λ)
is Negative Binomial.
Good fit of Gamma Poisson mixture model to data of the
Financial Call Center for most intervals.
Relation between our main model and Gamma Poisson
mixture model is established.
The distribution of X is derived under Gamma assumption:
it is asymptotically normal, given λ → ∞.
Introduction
Case Studies
Theoretical Results
Future Research
Emergency Department: Data Description
Israeli Emergency Department.
194 weeks between from January 2004 till October 2007 (five
war weeks are excluded from data).
The analysis is performed using two resolutions: hourly arrival
rates (168 intervals in a week) and three-hour arrival rates (56
intervals in a week).
Holidays are not excluded (results with excluded holidays are
similar).
Peaks
Introduction
at graphs correspond
to night periods withTheoretical
small arrival
rates.
Case Studies
Results
Future Research
Therefore, variability of arrival rates at this resolution is somewhat larger (but not much
larger) the the variability of iid Poisson random variables.
Emergency Department: Over-Dispersion Phenomenon
Figure 1: Coefficient of variation
Coefficient of Variation
One-hour resolution
Three-hour resolution
0.8
0.5
coefficient of variation
inverted sq. root of mean
0.45
0.7
0.4
0.6
0.35
0.5
0.3
0.4
0.25
0.2
0.3
0.15
0.2
0
0.1
coefficient of variation
inverted sq. root of mean
0.1
20
40
60
80
100
interval
120
140
0.05
160
0
5
10
15
20
25
30
interval
35
40
45
50
55
Moderate over-dispersion.
is observed
at ln(λ),
daily Figure
level 2⇒
scaling problem
IfOverdispersion
we fit regression model
of ln(σ) versus
demonstrates
a linear pattern
withshould
the slopebe
thatstudied.
is very close(Dependence
to 0.5.
of c on the interval length.)
Note. Derivation of asymptotic linear relation (1.35) in Section 1.6 is based on c > 1/2.
It is cunclear
howseems
it works to
for be
c that
is close to 1/2assumption
and relatively small
λ.
= 1/2
reasonable
for hourly
resolution.
Introduction
Case Studies
Theoretical Results
Future Research
Emergency Department: Fitting Regression Model
ln(σ(Y)) = c · ln(λ) + ln(σ(X))
Linear Regression:
Figure 2: ln(σ) versus ln(λ) plots
One-hour resolution
y = 0.497x + 0.102, R2 = 0.968
Three-hour resolution
y = 0.527x + 0.087, R2 = 0.947
1.8
2.5
1.6
2
ln(Standard Deviation)
ln(Standard Deviation)
1.4
1.2
1
0.8
0.6
0.4
1.5
1
0.5
0.2
0
0
0.5
1
1.5
2
ln(Mean Arrival Rate)
2.5
3
3.5
0
0
0.5
1
1.5
2
2.5
3
ln(Mean Arrival Rate)
3.5
4
4.5
A linear pattern with the slope that is very close to 0.5 while
derivation of the asymptotic relation is based on c > 1/2.
Figure 3: Coefficient of variation
Introduction
Case Studies
Theoretical Results
Future Research
Emergency Department: Fitting Regression Model
Non-Linear Regression:
Linear Regression
Non-Linear Regression
ln(σ(Y)) = 0.5 · ln(λ2c σ 2 (X) + λ)
One-hour
ĉ
0.497
0.481
resolution
σ̂(X)
1.108
0.476
Three-hour
ĉ
0.527
0.595
resolution
σ̂(X)
1.087
0.466
Estimates for c are close but the estimates for σ(X ) are
significantly higher in the case of linear regression.
The two regression curves are almost identical.
Introduction
Case Studies
Theoretical Results
Future Research
QED-c Regime: Fixed Arrival Rate
QED-c staffing rule:
n =
λ
µ
+β
c
λ
µ
√
+ o( λ),
β ∈ R, c ∈ (1/2, 1).
Assume an M|M|n + G queue with fixed arrival rate λ.
Take λ to ∞:
β > 0: Over-staffing.
β < 0: Under-staffing.
For both cases we provide asymptotically equivalent expressions (or
bounds) for P{Wq > 0}, P{Ab} and E [V ], where
Wq - waiting time, V - offered wait (wait given infinite patience).
Proofs: based on M|M|n + G building blocks from Zeltyn and
Mandelbaum [’05], carried out via the Laplace Method for
asymptotic calculation of integrals.
Introduction
Case Studies
Theoretical Results
Future Research
QED-c Regime: Random Arrival Rate
Theorem
Assume random arrival rate Λ = λ + λc µ1−c X , c ∈ (1/2, 1),
E [X ] = 0, finite σ(X ) > 0, and staffing according to the QED-c
staffing rule with the corresponding c. Then, as λ → ∞,
a. Delay probability:
PΛ,n {Wq > 0} ∼ 1 − F (β).a
b. Abandonment probability:
c. Average offered waiting time:
a
PΛ,n {Ab} ∼
E [X − β]+
.
n1−c
EΛ,n [V ] ∼
E [X − β]+
.
n1−c · g0
f (λ) ∼ g (λ) denotes that limλ→∞ f (λ)/g (λ) = 1.
Proofs: based on conditioning on values of X and results for
QED-c staffing rule.
Introduction
Case Studies
Theoretical Results
Future Research
QED-c Regime: Numerical Experiments
Examples: Consider two distributions of X
Uniform distribution on [-1,1],
Standard Normal distribution.
β = −0.5, c = 0.7
P{W>0}
c=0.7, β=-0.5
β=-0.5
P{W>0} vs. λ, c=0.7,
P{Abandon}
c=0.7,
β=-0.5
P{Abandon}
vs.vs.
λ, λ,
c=0.7,
β=-0.5
Abandonment
Probability
Delay Probability
0.9
0.9
short-term
short-termU(-1,1)
U(-1,1)
long-term
long-termU(-1,1)
U(-1,1)
approx
approxU(-1,1)
U(-1,1)
short-term
short-termN(0,1)
N(0,1)
long-term
long-termN(0,1)
N(0,1)
approx
approxN(0,1)
N(0,1)
0.8
0.8
0.25
0.25
0.75
0.75
0.2
0.2
0.15
0.15
0.7
0.7
0.65
0.65
0.6
0.6 0
0
short-term
U(-1,1)
short-term
U(-1,1)
long-term
U(-1,1)
long-term
U(-1,1)
approx
U(-1,1)
approx
U(-1,1)
short-term
N(0,1)
short-term
N(0,1)
long-term
N(0,1)
long-term
N(0,1)
approx
N(0,1)
approx
N(0,1)
0.3
0.3
P{Abandon}
P{Abandon}
0.85
0.85
P{W>0}
P{W>0}
0.35
0.35
0.1
0.1
0.05
0.05
1000
1000
2000
2000
3000
3000
4000
4000
λ
λ
P{W>0} vs. λ, c=0.7, β=-0.5
5000
5000
0
00
0
1000
2000
3000
4000
5000
1000
2000
3000
4000
5000
λ
λ
P{Abandon} vs. λ, c=0.7, β=-0.5
Introduction
Case Studies
Theoretical Results
Future Research
QED-c Regime: Practical Guidelines
Determine ”uncertainty coefficient c” via regression analysis.
Check if Gamma Poisson mixture model is reasonable.
Assume that X is asymptotically normal, calculate standard
deviation from regression model.
Apply our QED-c asymptotic results in order to determine
appropriate staffing.
Introduction
Case Studies
Theoretical Results
Future Research
Outline of Additional Results
Queueing Theory. Asymptotic performance measures derived
and constraint satisfaction problems solved for:
QED regime (c = 1/2).
ED regime (c = 1), discrete and continuous distribution of X .
Numerical Experiments. Very good fit between asymptotic
results and the exact ones.
Iterative Staffing Algorithm (ISA), a simulation code
developed by Feldman et al. [’07] with the features of random
arrival rate in the time-varying M|M|n + G queue.
Goal: determine time-dependent staffing levels aiming to
achieve a time-stable delay probability.
Introduction
Case Studies
Theoretical Results
Future Research
Future Research Challenges
Incorporating forecasting errors into our model (in the spirit
of Steckley et al. [’07]).
Scaling problem: dependence of c on the basic interval
duration.
ISA: achieving time-stable performance measures (probability
to abandon, average wait).
Validation of M ? |M|n + M (or M ? |M|n + G ) model in call
center environment (and probably other service systems).
Introduction
Case Studies
Theoretical Results
Thank You
Future Research