Post on 30-Aug-2020
Introduction Theoretical Results Time-Varying Queues
Uncertainty in the Demand for Service:The Case of Call Centers and Emergency
Departments
Shimrit Maman
Technion - Israel Institute of Technology
January 28, 2009
Advisors: Prof. Avishai Mandelbaum and Dr. Sergey Zeltyn
Introduction Theoretical Results Time-Varying Queues
Outline
1 IntroductionMotivationModel DefinitionThe Case of a Financial Call CenterThe Case of an Emergency DepartmentPractical Guidelines
2 Theoretical ResultsQueues Performances in a Random EnvironmentQED-c RegimeQED RegimeED Regime
3 Time-Varying Queues
Introduction Theoretical Results Time-Varying Queues
Motivation
A standard assumption in the modeling of a service systempostulates that the arrival process is a Poisson process with knownparameters. For example, the prevalent approach in call centers isto assume known arrival rates for each basic interval (say,half-hour).
However, as a rule, call centers data contradicts this assumptionand shows a larger variability of the arrival process than the oneexpected from the Poisson hypothesis.
We explain this over-dispersion by the natural uncertainty of thearrival rate.
Introduction Theoretical Results Time-Varying Queues
Literature Review
Henderson S.
Input model uncertainty: Why do we care and what should we do aboutit?. 2003.
Whitt W.
Staffing a call center with uncertain arrival rate and absenteeism. 2006.
Halfin S., 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-serverasymptotics of the M/M/n+G queue. 2005.
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.
Introduction Theoretical Results Time-Varying Queues
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 densityexist 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 variesfrom day to day in an i.i.d. fashion:
M = λ + λcX , c ≤ 1,
c ≤ 1/2: Conventional variability.
1/2 < c < 1: Moderate variability.
c = 1: Extreme variability.
Introduction Theoretical Results Time-Varying Queues
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 densityexist 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 variesfrom day to day in an i.i.d. fashion:
M = λ + λcX , c ≤ 1,
c ≤ 1/2: Conventional variability.
1/2 < c < 1: Moderate variability.
c = 1: Extreme variability.
Introduction Theoretical Results Time-Varying Queues
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 densityexist 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 variesfrom day to day in an i.i.d. fashion:
M = λ + λcX , c ≤ 1,
c ≤ 1/2: Conventional variability.
1/2 < c < 1: Moderate variability.
c = 1: Extreme variability.
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Our study focuses on the arrival counts to the Retail queue.
We consider 263 regular weekdays ranging between April 2007and April 2008.
Holidays which exhibit different daily patterns are excludedfrom our analysis.
Each day is divided into 48 half-hour intervals.
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Coefficient of Variationsampled CV- solid line, Poisson CV - dashed lineCoefficient of Variation Per 30 Minutes, seperated weekdays
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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
Coe
ffici
ent o
f Var
iatio
n
Sundays Mondays Tuesdays Wednesdays Thursdays
Sampled CV’s � CV’s of Poisson variables with fixed arrival rates
⇒Over-Dispersion
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Coefficient of Variationsampled CV- solid line, Poisson CV - dashed lineCoefficient of Variation Per 30 Minutes, seperated weekdays
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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
Coe
ffici
ent o
f Var
iatio
n
Sundays Mondays Tuesdays Wednesdays Thursdays
Sampled CV’s � CV’s of Poisson variables with fixed arrival rates
⇒Over-Dispersion
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Coefficient of Variationsampled CV- solid line, Poisson CV - dashed lineCoefficient of Variation Per 30 Minutes, seperated weekdays
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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
Coe
ffici
ent o
f Var
iatio
n
Sundays Mondays Tuesdays Wednesdays Thursdays
Sampled CV’s � CV’s of Poisson variables with fixed arrival rates
⇒Over-Dispersion
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Average Number of ArrivalsAverage Arrival Per 30 Minutes, seperated weekdays
0
200
400
600
800
1000
1200
1400
1600
1800
2000
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
Ave
rage
Arr
ival
Sundays Mondays Tuesdays Wednesdays Thursdays
(1) Sundays; (3) Tuesdays and Wednesdays;(2) Mondays; (4) Thursdays;
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Average Number of ArrivalsAverage Arrival Per 30 Minutes, seperated weekdays
0
200
400
600
800
1000
1200
1400
1600
1800
2000
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
Ave
rage
Arr
ival
Sundays Mondays Tuesdays Wednesdays Thursdays
(1) Sundays; (3) Tuesdays and Wednesdays;(2) Mondays; (4) Thursdays;
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Basic definitions and notation:
λid - The expectation of arrival volume in the i th interval forday type d , i = 1, 2, . . . , 48 and d = 1, 2, 3, 4.
σid - The std of arrival volume for the i th interval for day typed .
cd - The uncertainty coefficient for day type d .
λid - The average call volume in the i th interval over all daysof type d .
σid - The sampled standard deviation of call volumes in thei th interval over all days of type d .
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
1. Relation between λid and σid:
Consider a Poisson mixture variable Y with random rateM = λ + λc · X , where E [X ] = 0, finite σ(X ) > 0 and1/2 < c ≤ 1. Then,
Var(Y ) = λ2c · Var(X ) + λ + λc · E (X ).
Given λ →∞σ(Y ) ∼ λc · σ(X ), 1
andln(σ(Y )) ∼ c · ln(λ) + ln(σ(X )).
1f (λ) ∼ g(λ) denotes that limλ→∞ f (λ)/g(λ) = 1.
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Mondaysln(sd) vs ln(average) per 30 minutes. Sundays
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8
ln(Average Arrival)
ln(S
tand
ard
Dev
iatio
n)
Tue-Wedln(sd) vs ln(average) per 30 minutes. Sundays
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8
ln(Average Arrival)
ln(S
tand
ard
Dev
iatio
n)
Results:
Two clusters exists. Denote j(i) = 1 for i = 1, 2, . . . , 21, andj(i) = 1 for i = 22, 23, . . . , 48.Very good fit (R2 > 0.97).Significant linear relations:
ln(σid) = cdj(i) · ln(λid) + ln(σ(Xdj(i))) ∀ d , i
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Mondaysln(sd) vs ln(average) per 30 minutes. Sundays
y = 0.8292x - 0.2882R2 = 0.992
y = 0.8356x - 0.6561R2 = 0.9766
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8
ln(Average Arrival)
ln(S
tand
ard
Dev
iatio
n)
00:00-10:30 10:30-00:00
Tue-Wedln(sd) vs ln(average) per 30 minutes. Sundays
y = 0.8027x - 0.1235R2 = 0.9899
y = 0.8752x - 0.8589R2 = 0.9882
0
1
2
3
4
5
6
0 1 2 3 4 5 6 7 8
ln(Average Arrival)
ln(S
tand
ard
Dev
iatio
n)
00:00-10:30 10:30-00:00
Results:
Two clusters exists. Denote j(i) = 1 for i = 1, 2, . . . , 21, andj(i) = 1 for i = 22, 23, . . . , 48.Very good fit (R2 > 0.97).Significant linear relations:
ln(σid) = cdj(i) · ln(λid) + ln(σ(Xdj(i))) ∀ d , i
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
2. Fitting a Gamma Poisson mixture model to the data:(Jongbloed and Koole [’01])
Assume a prior Gamma distribution for the arrival rate
λ + λcXd= Gamma(a, b). Then, the distribution of Y is Negative
Binomial.
1 Maximum likelihood estimators of a and b.
2 Goodness of fit test including FDR control method to correctthe multiple comparisons.
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
H0,id : Midd= Gamma(aid , bid)
Sundays 10:00-10:30
1500 2000 2500
0.0
0.2
0.4
0.6
0.8
1.0
Mondays 10:00−10:30
CD
F
EmpiricalGPM(29.51,58.5),l=1726.43p.value= 0.674
Monday 14:30-15:00
700 800 900 1000 1100 1200
0.0
0.2
0.4
0.6
0.8
1.0
Mondays 14:30−15:00
CD
F
EmpiricalGPM(47.78,19.01),l=908.22p.value= 0.982
Results:
Very good fit.
Only 13 hypotheses are rejected (out of 192).
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
H0,id : Midd= Gamma(aid , bid)
Sundays 10:00-10:30
1500 2000 2500
0.0
0.2
0.4
0.6
0.8
1.0
Mondays 10:00−10:30
CD
F
EmpiricalGPM(29.51,58.5),l=1726.43p.value= 0.674
Monday 14:30-15:00
700 800 900 1000 1100 1200
0.0
0.2
0.4
0.6
0.8
1.0
Mondays 14:30−15:00
CD
F
EmpiricalGPM(47.78,19.01),l=908.22p.value= 0.982
Results:
Very good fit.
Only 13 hypotheses are rejected (out of 192).
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
3. Relation between our main model and Gamma Poissonmixture model:
Let M = λ + λcXd= Gamma(a, b). Then,
E [M] = ab = λ; Var(M) = λb and Var(X ) = λ1−2c · b.
We derive the following relations
b = σ2(X ) · λ2c−1,
a = σ−2(X ) · λ2−2c .
and conclude that
ln(b) = (2c − 1) · ln(λ) + ln(σ2(X )).
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Mondaysln(sd) vs ln(average) per 30 minutes. mondays
0
1
2
3
4
5
0 1 2 3 4 5 6 7 8
ln(Average Arrival)
ln(b
)
Tue-Wedln(sd) vs ln(average) per 30 minutes. Tuesdays-wednesdays
0
1
2
3
4
5
0 1 2 3 4 5 6 7 8
ln(Average Arrival)
ln(b
)
Results:
Two clusters.
Very good fit.
Significant linear relations:
ln(bid) = cdj(i) · ln(λid) + ln(σ2(Xdj(i))) ∀ d , i
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
Mondaysln(sd) vs ln(average) per 30 minutes. mondays
y = 0.6975x - 0.894R2 = 0.9453
y = 0.7699x - 2.0427R2 = 0.9076
0
1
2
3
4
5
0 1 2 3 4 5 6 7 8
ln(Average Arrival)
ln(b
)
00:00-10:30 10:30-00:00
Tue-Wedln(sd) vs ln(average) per 30 minutes. Tuesdays-wednesdays
y = 0.6146x - 0.5349R2 = 0.9257
y = 0.7825x - 2.0107R2 = 0.9457
0
1
2
3
4
5
0 1 2 3 4 5 6 7 8
ln(Average Arrival)
ln(b
)
00:00-10:30 10:30-00:00
Results:
Two clusters.
Very good fit.
Significant linear relations:
ln(bid) = cdj(i) · ln(λid) + ln(σ2(Xdj(i))) ∀ d , i
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
4. Asymptotic distribution of X:
X =M − λ
λc=
M − ab
(ab)c=
M − E [M]
σ(M)/σ(X )
Let Wad=
Gamma(a, b)− ab
b√
a=
Gamma(a, 1)− a√a
.
Then lima→∞
MGFa(t) = et2/2, t ∈ R, and this limit is the moment
generating function of the standard normal distribution Norm(0, 1).
Conclusion: As λ →∞ (equivalent to a →∞), the randomvariable X/σ(X ) converges in distribution to a standard normaldistributed variable.
X
σ(X )D→ Norm(0, 1).
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
4. Asymptotic distribution of X:
X =M − λ
λc=
M − ab
(ab)c=
M − E [M]
σ(M)/σ(X )
Let Wad=
Gamma(a, b)− ab
b√
a=
Gamma(a, 1)− a√a
.
Then lima→∞
MGFa(t) = et2/2, t ∈ R, and this limit is the moment
generating function of the standard normal distribution Norm(0, 1).
Conclusion: As λ →∞ (equivalent to a →∞), the randomvariable X/σ(X ) converges in distribution to a standard normaldistributed variable.
X
σ(X )D→ Norm(0, 1).
Introduction Theoretical Results Time-Varying Queues
The Case of a Financial Call Center
4. Asymptotic distribution of X:
X =M − λ
λc=
M − ab
(ab)c=
M − E [M]
σ(M)/σ(X )
Let Wad=
Gamma(a, b)− ab
b√
a=
Gamma(a, 1)− a√a
.
Then lima→∞
MGFa(t) = et2/2, t ∈ R, and this limit is the moment
generating function of the standard normal distribution Norm(0, 1).
Conclusion: As λ →∞ (equivalent to a →∞), the randomvariable X/σ(X ) converges in distribution to a standard normaldistributed variable.
X
σ(X )D→ Norm(0, 1).
Introduction Theoretical Results Time-Varying Queues
The Case of an Emergency Department
Consider 194 weeks between from January 2004 till October2007 (five war weeks are excluded from data).
The analysis is performed using two resolutions: hourly arrivalrates (168 intervals in a week) and three-hour arrival rates (56intervals in a week).
Meanwhile, we do not clean our data (Jewish holidays are notexcluded as they should be).
Introduction Theoretical Results Time-Varying Queues
The Case of an Emergency Department
Coefficient of Variation
1 Arrival-Rate Variability: The Case of Emergency
Department
Consider 194 weeks between from January 2004 till October 2007 (five war weeks are excludedfrom data). The analysis is performed using two resolutions: hourly arrival rates (168intervals in a week) and three-hour arrival rates (56 intervals in a week). Meanwhile, we donot clean our data (Jewish holidays are not excluded as they should be).
First, we redraw Figure 1.2 from Section 1.6 on Financial call center, comparing betweencoefficient of variation and inverted square root of mean arrival rates. Figure 1 shows that,in contrast to the call center study, inverted square root of mean is relatively close to CV’s.Peaks at graphs correspond to night periods with small arrival rates.
Therefore, variability of arrival rates at this resolution is somewhat larger (but not muchlarger) the the variability of iid Poisson random variables.
Figure 1: Coefficient of variation
One-hour resolution Three-hour resolution
20 40 60 80 100 120 140 1600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
interval
coefficient of variationinverted sq. root of mean
5 10 15 20 25 30 35 40 45 50 550
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
interval
coefficient of variationinverted sq. root of mean
If we fit regression model of ln(σ) versus ln(λ), Figure 2 demonstrates a linear patternwith the slope that is very close to 0.5.
Note. Derivation of asymptotic linear relation (1.35) in Section 1.6 is based on c > 1/2.It is unclear how it works for c that is close to 1/2 and relatively small λ.
Finally, Figure 3 shows that if we increase time resolution, variability of arrival rates issignificantly larger than the variability of iid Poisson random variables.
1
In contrast to the call center study, inverted square root ofmean is relatively close to CV’s.
Peaks at graphs correspond to night periods with small arrivalrates.
Introduction Theoretical Results Time-Varying Queues
The Case of an Emergency Department
Coefficient of Variation
1 Arrival-Rate Variability: The Case of Emergency
Department
Consider 194 weeks between from January 2004 till October 2007 (five war weeks are excludedfrom data). The analysis is performed using two resolutions: hourly arrival rates (168intervals in a week) and three-hour arrival rates (56 intervals in a week). Meanwhile, we donot clean our data (Jewish holidays are not excluded as they should be).
First, we redraw Figure 1.2 from Section 1.6 on Financial call center, comparing betweencoefficient of variation and inverted square root of mean arrival rates. Figure 1 shows that,in contrast to the call center study, inverted square root of mean is relatively close to CV’s.Peaks at graphs correspond to night periods with small arrival rates.
Therefore, variability of arrival rates at this resolution is somewhat larger (but not muchlarger) the the variability of iid Poisson random variables.
Figure 1: Coefficient of variation
One-hour resolution Three-hour resolution
20 40 60 80 100 120 140 1600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
interval
coefficient of variationinverted sq. root of mean
5 10 15 20 25 30 35 40 45 50 550
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
interval
coefficient of variationinverted sq. root of mean
If we fit regression model of ln(σ) versus ln(λ), Figure 2 demonstrates a linear patternwith the slope that is very close to 0.5.
Note. Derivation of asymptotic linear relation (1.35) in Section 1.6 is based on c > 1/2.It is unclear how it works for c that is close to 1/2 and relatively small λ.
Finally, Figure 3 shows that if we increase time resolution, variability of arrival rates issignificantly larger than the variability of iid Poisson random variables.
1
In contrast to the call center study, inverted square root ofmean is relatively close to CV’s.
Peaks at graphs correspond to night periods with small arrivalrates.
Introduction Theoretical Results Time-Varying Queues
The Case of an Emergency Department
ln(σ) versus ln(λ) plotsFigure 2: ln(σ) versus ln(λ) plots
One-hour resolution Three-hour resolutiony = 0.497x+ 0.102, R2 = 0.968 y = 0.527x+ 0.087, R2 = 0.947
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
ln(Mean Arrival Rate)
ln(S
tand
ard
Dev
iatio
n)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.5
1
1.5
2
2.5
ln(Mean Arrival Rate)ln
(Sta
ndar
d D
evia
tion)
Figure 3: Coefficient of variation
Eight-hour resolution One-day resolution
2
A linear pattern with the slope that is very close to 0.5.
Derivation of the asymptotic linear relation is based onc > 1/2. It is unclear how it works for c that is close to 1/2and relatively small λ.
Introduction Theoretical Results Time-Varying Queues
The Case of an Emergency Department
ln(σ) versus ln(λ) plotsFigure 2: ln(σ) versus ln(λ) plots
One-hour resolution Three-hour resolutiony = 0.497x+ 0.102, R2 = 0.968 y = 0.527x+ 0.087, R2 = 0.947
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
ln(Mean Arrival Rate)
ln(S
tand
ard
Dev
iatio
n)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.5
1
1.5
2
2.5
ln(Mean Arrival Rate)ln
(Sta
ndar
d D
evia
tion)
Figure 3: Coefficient of variation
Eight-hour resolution One-day resolution
2
A linear pattern with the slope that is very close to 0.5.
Derivation of the asymptotic linear relation is based onc > 1/2. It is unclear how it works for c that is close to 1/2and relatively small λ.
Introduction Theoretical Results Time-Varying Queues
Practical Guidelines
Determine ”uncertainty coefficient c” via regression analysis.
Check if Gamma model is reasonable.
Calculate X distribution (asymptotic analysis).
Apply our QED-c results in order to determine appropriatestaffing.
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
Assume a set of D days. At each day an arrival rate, m, isindependently generated. Let Y be a system performance measure,and denote by YM the corresponding random performance measurein the random environment.
Denote:
arrj - Number of arrivals on day j
yj - The value of the performance measure YM , on day j .
If the performance measure is related to system performance, e.g.offered load and queue length, by SLLN
Y =1
D
D∑j=1
yjD↑∞−→ E [YM ] .
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
Assume a set of D days. At each day an arrival rate, m, isindependently generated. Let Y be a system performance measure,and denote by YM the corresponding random performance measurein the random environment.
Denote:
arrj - Number of arrivals on day j
yj - The value of the performance measure YM , on day j .
If the performance measure is related to system performance, e.g.offered load and queue length, by SLLN
Y =1
D
D∑j=1
yjD↑∞−→ E [YM ] .
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
We classify performance measures that relate to the customers,e.g. delay probability and waiting time, into two classes:
Short-Term Performance Measure: What will happentomorrow?
Y =1
D
D∑j=1
yjD↑∞−→ E [YM ] .
Long-Term Performance Measure: What would be theperformances in the long-run?
Y =
∑Dj=1 arrj · yj∑D
j=1 arrj
D↑∞−→ E [M · YM ]
E [M]> Y .
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
We classify performance measures that relate to the customers,e.g. delay probability and waiting time, into two classes:
Short-Term Performance Measure: What will happentomorrow?
Y =1
D
D∑j=1
yjD↑∞−→ E [YM ] .
Long-Term Performance Measure: What would be theperformances in the long-run?
Y =
∑Dj=1 arrj · yj∑D
j=1 arrj
D↑∞−→ E [M · YM ]
E [M]> Y .
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
We classify performance measures that relate to the customers,e.g. delay probability and waiting time, into two classes:
Short-Term Performance Measure: What will happentomorrow?
Y =1
D
D∑j=1
yjD↑∞−→ E [YM ] .
Long-Term Performance Measure: What would be theperformances in the long-run?
Y =
∑Dj=1 arrj · yj∑D
j=1 arrj
D↑∞−→ E [M · YM ]
E [M]> Y .
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
We classify performance measures that relate to the customers,e.g. delay probability and waiting time, into two classes:
Short-Term Performance Measure: What will happentomorrow?
Y =1
D
D∑j=1
yjD↑∞−→ E [YM ] .
Long-Term Performance Measure: What would be theperformances in the long-run?
Y =
∑Dj=1 arrj · yj∑D
j=1 arrj
D↑∞−→ E [M · YM ]
E [M]> Y .
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
We classify performance measures that relate to the customers,e.g. delay probability and waiting time, into two classes:
Short-Term Performance Measure: What will happentomorrow?
Y =1
D
D∑j=1
yjD↑∞−→ E [YM ] .
Long-Term Performance Measure: What would be theperformances in the long-run?
Y =
∑Dj=1 arrj · yj∑D
j=1 arrj
D↑∞−→ E [M · YM ]
E [M]> Y .
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
Lemma
Assume that the system is in steady state. The long-term valueand the short-term value of YM are asymptotically equivalent for allc < 1.
limλ→∞
E [M · YM ]
E [M]= lim
λ→∞E [YM ] .
Remark: Note that the above does not hold for c = 1. In thiswork we focus on long-term performance measures. Thetechniques of solutions, when considering the short-termperformance measures, are similar.
Introduction Theoretical Results Time-Varying Queues
Queues Performances in a Random Environment
Lemma
Assume that the system is in steady state. The long-term valueand the short-term value of YM are asymptotically equivalent for allc < 1.
limλ→∞
E [M · YM ]
E [M]= lim
λ→∞E [YM ] .
Remark: Note that the above does not hold for c = 1. In thiswork we focus on long-term performance measures. Thetechniques of solutions, when considering the short-termperformance measures, are similar.
Introduction Theoretical Results Time-Varying Queues
QED-c Regime
QED-c staffing rule:
n =λ
µ+ β
(λ
µ
)c
+ 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 provided asymptotically equivalent expressions(or bounds) for P{Wq > 0}, P{Ab|V > 0} and E [V |V > 0].Calculations, based on building blocks from Zeltyn[’05], are carriedout via the Laplace Method.
Introduction Theoretical Results Time-Varying Queues
QED-c Regime
QED-c staffing rule:
n =λ
µ+ β
(λ
µ
)c
+ 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 provided asymptotically equivalent expressions(or bounds) for P{Wq > 0}, P{Ab|V > 0} and E [V |V > 0].Calculations, based on building blocks from Zeltyn[’05], are carriedout via the Laplace Method.
Introduction Theoretical Results Time-Varying Queues
QED-c Regime
QED-c staffing rule:
n =λ
µ+ β
(λ
µ
)c
+ 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 provided asymptotically equivalent expressions(or bounds) for P{Wq > 0}, P{Ab|V > 0} and E [V |V > 0].Calculations, based on building blocks from Zeltyn[’05], are carriedout via the Laplace Method.
Introduction Theoretical Results Time-Varying Queues
Square-Root Staffing versus QED-c staffing
QED-c StaffingR = λ/µ β SRS2
c = 0.6 c = 0.75 c = 0.9
1000.5 105 108 (+3%) 116 (+10%) 132 (+25%)1 110 116 (+5%) 132 (+20%) 163 (+48%)
1.5 115 124 (+8%) 147 (+28%) 195 (+69%)
5000.5 511 521 (+2%) 553 (+8%) 634 (+24%)1 522 542 (+4%) 606 (+16%) 769 (+47%)
1.5 534 562 (+5%) 659 (+23%) 903 (+69%)
10000.5 1016 1032 (+2%) 1089 (+7%) 1251 (+23%)1 1032 1063 (+3%) 1178 (+14%) 1501 (+46%)
1.5 1047 1095 (+5%) 1267 (+21%) 1752 (+67%)
20000.5 2022 2048 (+1%) 2150 (+6%) 2468 (+22%)1 2045 2096 (+2%) 2300 (+12%) 2936 (+43%)
1.5 2067 2143 (+4%) 2449 (+18%) 3403 (+65%)
2Square-Root-Staffing
Introduction Theoretical Results Time-Varying Queues
QED-c Regime
Theorem
Assume random arrival rate M = λ + λcµ1−cX , c ∈ (1/2, 1),E [X ] = 0, finite σ(X ) > 0, and staffing according to the QED-cstaffing rule with the corresponding c . Then, as λ →∞,
a. Delay probability: PM,n{Wq > 0} ∼ 1− F (β).
b. Abandonment probability: PM,n{Ab} ∼ E [X − β]+n1−c
.
c. Average waiting time: EM,n[Wq] ∼E [X − β]+n1−c · g0
Remark: For simplifying the formulae we takeM = λ + µ1−c · λcX instead of M = λ + λcX .
Introduction Theoretical Results Time-Varying Queues
QED-c Regime
Examples: Consider two distributions of X
Uniform distribution on [-1,1],
Standard Normal distribution.
(1) β = −0.5, c = 0.7
Delay Probability
0 1000 2000 3000 4000 50000.6
0.65
0.7
0.75
0.8
0.85
0.9
P{W
>0}
λ
P{W>0} vs. λ, c=0.7, β=-0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P{A
band
on}
λ
P{Abandon} vs. λ, c=0.7, β=-0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000.6
0.65
0.7
0.75
0.8
0.85
0.9
P{W
>0}
λ
P{W>0} vs. λ, c=0.7, β=-0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P{A
band
on}
λ
P{Abandon} vs. λ, c=0.7, β=-0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
Abandonment Probability
0 1000 2000 3000 4000 50000.6
0.65
0.7
0.75
0.8
0.85
0.9
P{W
>0}
λ
P{W>0} vs. λ, c=0.7, β=-0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P{A
band
on}
λ
P{Abandon} vs. λ, c=0.7, β=-0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000.6
0.65
0.7
0.75
0.8
0.85
0.9
P{W
>0}
λ
P{W>0} vs. λ, c=0.7, β=-0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P{A
band
on}
λ
P{Abandon} vs. λ, c=0.7, β=-0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
Introduction Theoretical Results Time-Varying Queues
QED-c Regime
(2) β = 0.5, c = 0.6
Delay Probability
0 1000 2000 3000 4000 50000.2
0.25
0.3
0.35
0.4
0.45
0.5
P{W
>0}
λ
P{W>0} vs. λ, c=0.6, β=0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000
0.02
0.04
0.06
0.08
0.1
P{A
band
on}
λ
P{Abandon} vs. λ, c=0.6, β=0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000.2
0.25
0.3
0.35
0.4
0.45
0.5
P{W
>0}
λ
P{W>0} vs. λ, c=0.6, β=0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000
0.02
0.04
0.06
0.08
0.1
P{A
band
on}
λ
P{Abandon} vs. λ, c=0.6, β=0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
Abandonment Probability
0 1000 2000 3000 4000 50000.2
0.25
0.3
0.35
0.4
0.45
P{W
>0}
λ
P{W>0} vs. λ, c=0.6, β=0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000
0.02
0.04
0.06
0.08
0.1
P{A
band
on}
λ
P{Abandon} vs. λ, c=0.6, β=0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000.2
0.25
0.3
0.35
0.4
0.45
P{W
>0}
λ
P{W>0} vs. λ, c=0.6, β=0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
0 1000 2000 3000 4000 50000
0.02
0.04
0.06
0.08
0.1
P{A
band
on}
λ
P{Abandon} vs. λ, c=0.6, β=0.5
short-term U(-1,1)long-term U(-1,1)approx U(-1,1)short-term N(0,1)long-term N(0,1)approx N(0,1)
Introduction Theoretical Results Time-Varying Queues
Constraint Satisfaction
Formulation of the Problem:
Define the optimal staffing level by
n∗λ = argminn
{PM,n{Wq > 0} ≤ α
}.
The staffing level nλ is called asymptotically feasible if
lim supλ→∞
PM,nλ{Wq > 0} ≤ α.
In addition, nλ is asymptotically optimal if
|n∗λ − nλ| = o(f (λ)).
Introduction Theoretical Results Time-Varying Queues
Constraint Satisfaction
Formulation of the Problem:
Define the optimal staffing level by
n∗λ = argminn
{PM,n{Wq > 0} ≤ α
}.
The staffing level nλ is called asymptotically feasible if
lim supλ→∞
PM,nλ{Wq > 0} ≤ α.
In addition, nλ is asymptotically optimal if
|n∗λ − nλ| = o(f (λ)).
Introduction Theoretical Results Time-Varying Queues
Constraint Satisfaction
Formulation of the Problem:
Define the optimal staffing level by
n∗λ = argminn
{PM,n{Wq > 0} ≤ α
}.
The staffing level nλ is called asymptotically feasible if
lim supλ→∞
PM,nλ{Wq > 0} ≤ α.
In addition, nλ is asymptotically optimal if
|n∗λ − nλ| = o(f (λ)).
Introduction Theoretical Results Time-Varying Queues
QED Regime
c = 1/2Assume M = λ +
√µλ · X , E [X ] = 0 and finite σ(X ) > 0. Let
λ →∞.
Theorem (Staffing)
a. The optimal staffing level satisfies
n∗ =λ
µ+ β∗
sλ
µ+ o(
√λ),
where β∗ is the unique solution of the equation
α = E [α(β − X )]
with respect to the unknown β.α(·) is the Garnett function
α(β) =
"1 +
rg0
µ·h(βp
µ/g0)
h(−β)
#−1
.
Introduction Theoretical Results Time-Varying Queues
QED Regime
Theorem (Staffing). Continued.
b. Introduce the staffing level
n∗β =
&λ
µ+ β∗
sλ
µ
’.
Then the staffing level n∗β is both asymptotically feasible and
asymptotically optimal (f (λ) =√
λ)
Proof Outline
Given X=x,
nβ =λ
µ+ β ·
sλ
µ=
λ +√
µλ · xµ
+ b(β, x) ·
sλ +
√µλ · x
µ
⇒ as λ →∞, b(β, x) ∼ β − x .
Introduction Theoretical Results Time-Varying Queues
QED Regime
Theorem (Performance Measures)
Under the square-root staffing level, as λ →∞,
a. PM,n∗β{Ab} ∼ 1√
n· E (γ∗X ). γ∗X
b. ρn∗β= 1− β∗√
n+ o
(1√n
).
c.PM,n∗β
{Ab}EM,n∗β
[Wq]∼ g0.
Introduction Theoretical Results Time-Varying Queues
QED Regime
c < 1/2
Assume Λ = λ + µ1−cλc · X , c < 1/2, E [X ] = 0 and finiteσ(X ) > 0. Let λ →∞.
Lemma
Assume the square-root staffing level
n =λ
µ+ β
sλ
µ+ o(
√λ), −∞ < β < ∞
Then, asymptotically, the random part of the arrival rate does notaffect the system’s performances. Namely, the queue is asymptoti-cally equivalent to the M|M|n + G queue with deterministic arrivalrate λ, for all c < 1/2.The M|M|n+G queue was analyzed comprehensively by Zeltyn[’05].
Introduction Theoretical Results Time-Varying Queues
QED Regime
c < 1/2
Assume Λ = λ + µ1−cλc · X , c < 1/2, E [X ] = 0 and finiteσ(X ) > 0. Let λ →∞.
Lemma
Assume the square-root staffing level
n =λ
µ+ β
sλ
µ+ o(
√λ), −∞ < β < ∞
Then, asymptotically, the random part of the arrival rate does notaffect the system’s performances. Namely, the queue is asymptoti-cally equivalent to the M|M|n + G queue with deterministic arrivalrate λ, for all c < 1/2.The M|M|n+G queue was analyzed comprehensively by Zeltyn[’05].
Introduction Theoretical Results Time-Varying Queues
ED Regime
c = 1, Discrete Random Arrival Rate
Assume M = λX , where X is a discrete random variable whichtakes values x1 > x2 > . . . > xI > 0, with probabilitiesp1, p2, . . . , pI , respectively. In addition, let E [X ] = 1, σ(X ) < ∞and λ →∞.
Let
κ = argmins
{s∑
i=1
xipi ≥ α
}.
Assume that the inequality is strict.
Define
α∗4=
α−∑κ−1
i=1 xipi
xκpκ.
Introduction Theoretical Results Time-Varying Queues
ED Regime
c = 1, Discrete Random Arrival Rate
Assume M = λX , where X is a discrete random variable whichtakes values x1 > x2 > . . . > xI > 0, with probabilitiesp1, p2, . . . , pI , respectively. In addition, let E [X ] = 1, σ(X ) < ∞and λ →∞.
Let
κ = argmins
{s∑
i=1
xipi ≥ α
}.
Assume that the inequality is strict.
Define
α∗4=
α−∑κ−1
i=1 xipi
xκpκ.
Introduction Theoretical Results Time-Varying Queues
ED Regime
Theorem (Staffing)
a. The optimal staffing level satisfies
n∗ =λxκ
µ+ β∗
sλxκ
µ+ o(
√λ);
here β∗ is the unique solution of the equation
α∗ =
"1 +
rg0
µ·h(βp
µ/g0)
h(−β)
#−1
.
b. Introduce the staffing level
n∗β =
&λxκ
µ+ β∗
sλxκ
µ
’.
Then the staffing level n∗β is both asymptotically feasible, as
well as asymptotically optimal with f (λ) =√
λ.Proof
Introduction Theoretical Results Time-Varying Queues
ED Regime
What if α =κ∑
i=1
xipi ?
n =λxκ+1
µ+ β · h(λ),
1h(λ)√
λ→ constant ⇒ PM,n{Wq > 0} > α. (β ∈ R)
2h(λ)√
λ→ ∞ ⇒ PM,n{Wq > 0} = α. (β > 0)
Introduction Theoretical Results Time-Varying Queues
ED Regime
What if α =κ∑
i=1
xipi ?
n =λxκ+1
µ+ β · h(λ),
1h(λ)√
λ→ constant ⇒ PM,n{Wq > 0} > α. (β ∈ R)
2h(λ)√
λ→ ∞ ⇒ PM,n{Wq > 0} = α. (β > 0)
Introduction Theoretical Results Time-Varying Queues
ED Regime
What if α =κ∑
i=1
xipi ?
n =λxκ+1
µ+ β · h(λ),
1h(λ)√
λ→ constant
⇒ PM,n{Wq > 0} > α. (β ∈ R)
2h(λ)√
λ→ ∞ ⇒ PM,n{Wq > 0} = α. (β > 0)
Introduction Theoretical Results Time-Varying Queues
ED Regime
What if α =κ∑
i=1
xipi ?
n =λxκ+1
µ+ β · h(λ),
1h(λ)√
λ→ constant ⇒ PM,n{Wq > 0} > α. (β ∈ R)
2h(λ)√
λ→ ∞ ⇒ PM,n{Wq > 0} = α. (β > 0)
Introduction Theoretical Results Time-Varying Queues
ED Regime
What if α =κ∑
i=1
xipi ?
n =λxκ+1
µ+ β · h(λ),
1h(λ)√
λ→ constant ⇒ PM,n{Wq > 0} > α. (β ∈ R)
2h(λ)√
λ→ ∞
⇒ PM,n{Wq > 0} = α. (β > 0)
Introduction Theoretical Results Time-Varying Queues
ED Regime
What if α =κ∑
i=1
xipi ?
n =λxκ+1
µ+ β · h(λ),
1h(λ)√
λ→ constant ⇒ PM,n{Wq > 0} > α. (β ∈ R)
2h(λ)√
λ→ ∞ ⇒ PM,n{Wq > 0} = α. (β > 0)
Introduction Theoretical Results Time-Varying Queues
ED Regime
Theorem (Performance Measures)
Under the staffing level n∗β, as λ →∞,
a. The long-term abandonment probability:
ePM,n∗β{Ab} ∼
κ−1Xi=1
pi
`xi − xκ).
b. Mean server’s utilization:
EM,n∗β[U] ∼
κXi=1
pi +IX
i=κ+1
pi ·xi
xκ.
c. Assume that for each i < κ the equation G (y) = 1− xκxi
has
a unique solution y∗i , and g(y∗i ) > 0. Then, the long-termaverage waiting time
eEM,n∗β[Wq] ∼
κ−1Xi=1
"xipi ·
Z y∗i
0
G(u)du
#.
Introduction Theoretical Results Time-Varying Queues
ED Regime
Example: X takes the values 1.5, 1 and 0.5, with probability 1/3for each.
Delay Probability
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
long
-term
P{W
>0}
β
Long-Term P{W>0} vs. β
XK=1.5
XK=1
XK=0.5
Abandonment Probability
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
long
-term
P{W
>0}
β
Long-Term P{W>0} vs. β
0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
long
-term
P{A
band
on}
XK
Long-Term P{Abandon} vs. XK
XK=1.5
XK=1
XK=0.5
Introduction Theoretical Results Time-Varying Queues
ED Regime
β = −1
Delay Probability
0 500 1000 1500 20000.4
0.5
0.6
0.7
0.8
0.9
1
long
-term
P{W
>0}
λ
Long-Term P{W>0} vs. λ, c=1, β=-1
exact XK=1.5
approx XK=1.5
exact XK=1
approx XK=1
exact XK=0.5
approx XK=0.5
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
long
-term
P{A
band
on}
λ
Long-Term P{Abandon} vs. λ, c=1, β=-1
exact XK=1.5
approx XK=1.5
exact XK=1
approx XK=1
exact XK=0.5
approx XK=0.5
Abandonment Probability
0 500 1000 1500 20000.4
0.5
0.6
0.7
0.8
0.9
1
long
-term
P{W
>0}
λ
Long-Term P{W>0} vs. λ, c=1, β=-1
exact XK=1.5
approx XK=1.5
exact XK=1
approx XK=1
exact XK=0.5
approx XK=0.5
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
long
-term
P{A
band
on}
λ
Long-Term P{Abandon} vs. λ, c=1, β=-1
exact XK=1.5
approx XK=1.5
exact XK=1
approx XK=1
exact XK=0.5
approx XK=0.5
Introduction Theoretical Results Time-Varying Queues
ED Regime
c = 1, Continuous Random Arrival Rate
Assume M = λX , where X is a continuous random variable withstrictly continuous cdf F over the distribution support. LetE [X ] = 1 and σ(X ) < ∞. Assume λ →∞.
Theorem (Staffing)
a. The optimal staffing level satisfies
n∗ =λ
µ· δ∗ + o(λ), δ∗ ∈ supp(f ),
where δ∗ is the unique solution of the equation
α =
Z ∞δ
xdF (x).
b. Introduce the staffing level n∗δ =
‰λδ∗
µ
ı.
Then the staffing level n∗δ is both asymptotically feasible, aswell as asymptotically optimal (f (λ) = λ).
Introduction Theoretical Results Time-Varying Queues
ED Regime
Theorem (Performance Measures)
Under the staffing level n∗δ , as λ →∞,
a. The short-term delay probability:
PM,n∗δ{Wq > 0} ∼ F (δ).
b. The long-term abandonment probability:
ePM,n∗δ{Ab} ∼ (E [X |X > δ]− δ) · F (δ).
c. The short-term abandonment probability:
PM,n∗δ{Ab} ∼ E [1− δ/X |X > δ] · F (δ).
Introduction Theoretical Results Time-Varying Queues
ED Regime
Theorem (Performance Measures). Continued.
d. Mean server’s utilization:
EM,n∗δ[U] ∼ 1
δ· E [X |X < δ] · F (δ) + F (δ).
e. Assume that for each x > δ the equation G (yx) = 1− δx has
a unique solution y∗x , and g(y∗x ) > 0. Then, the long-termaverage waiting time
eEM,n∗δ[Wq] ∼
Z ∞δ
x ·Z y∗x
0
G(u)du
!dF (x).
Introduction Theoretical Results Time-Varying Queues
ED Regime
Examples: Consider three distributions of X
Uniform distribution on [0,2].
Uniform distribution on [0.5,1.5].
Normal distribution with mean=1 and std=0.25.
δ = 0.75
Long-Term P{Wq > 0}
0 500 1000 1500 20000.6
0.7
0.8
0.9
1
long
-term
P{W
>0}
λ
Long-Term P{W>0} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.6
0.7
0.8
0.9
1
shor
t-ter
m P
{W>0
}
λ
Short-Term P{W>0} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.2
0.25
0.3
0.35
0.4
long
-term
P{A
band
on}
λ
Long-Term P{Abandon} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.2
0.25
0.3
0.35
0.4
shor
t-ter
m P
{Aba
ndon
}
λ
Short-Term P{Abandon} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
Short-Term P{Wq > 0}
0 500 1000 1500 20000.6
0.7
0.8
0.9
1
long
-term
P{W
>0}
λ
Long-Term P{W>0} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.6
0.7
0.8
0.9
1
shor
t-ter
m P
{W>0
}
λ
Short-Term P{W>0} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.2
0.25
0.3
0.35
0.4
long
-term
P{A
band
on}
λ
Long-Term P{Abandon} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.2
0.25
0.3
0.35
0.4
shor
t-ter
m P
{Aba
ndon
}
λ
Short-Term P{Abandon} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
Introduction Theoretical Results Time-Varying Queues
ED Regime
δ = 0.75
Long-Term P{Ab}
0 500 1000 1500 20000.6
0.7
0.8
0.9
1
long
-term
P{W
>0}
λ
Long-Term P{W>0} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.6
0.7
0.8
0.9
1
shor
t-ter
m P
{W>0
}
λ
Short-Term P{W>0} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.2
0.25
0.3
0.35
0.4
long
-term
P{A
band
on}
λ
Long-Term P{Abandon} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.2
0.25
0.3
0.35
0.4
shor
t-ter
m P
{Aba
ndon
}
λ
Short-Term P{Abandon} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
Short-Term P{Ab}
0 500 1000 1500 20000.6
0.7
0.8
0.9
1
long
-term
P{W
>0}
λ
Long-Term P{W>0} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.6
0.7
0.8
0.9
1
shor
t-ter
m P
{W>0
}
λ
Short-Term P{W>0} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.2
0.25
0.3
0.35
0.4
long
-term
P{A
band
on}
λ
Long-Term P{Abandon} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
0 500 1000 1500 20000.2
0.25
0.3
0.35
0.4
shor
t-ter
m P
{Aba
ndon
}
λ
Short-Term P{Abandon} vs. λ, c=1, δ=0.75
exact U(0,2)approx U(0,2)exact U(0.5,1.5)approx U(0.5,1.5)exact Norm(1,0.252)approx Norm(1,0.252)
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Based on the ISA (Iterative Staffing Algorithm), a simulationcode developed by Feldman[’04] with the features of randomarrival rate in the time varying M/M/n + G queue.
The goal is to determine time-dependence staffing levelsaiming to achieve a given constant-over-time long- term delaydelay probability, α.
Example 1: c = 1, Discrete Random Arrival Rate1 Mt = λt · X , where
λt = 100− 20 · cos(t), and X =
1.5 w .p. 1/31 w .p. 1/30.5 w .p. 1/3
2 Service time and patience are distributed exponentially withmean 1 (µ = θ = 1).
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Based on the ISA (Iterative Staffing Algorithm), a simulationcode developed by Feldman[’04] with the features of randomarrival rate in the time varying M/M/n + G queue.
The goal is to determine time-dependence staffing levelsaiming to achieve a given constant-over-time long- term delaydelay probability, α.
Example 1: c = 1, Discrete Random Arrival Rate1 Mt = λt · X , where
λt = 100− 20 · cos(t), and X =
1.5 w .p. 1/31 w .p. 1/30.5 w .p. 1/3
2 Service time and patience are distributed exponentially withmean 1 (µ = θ = 1).
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Arrivals, Offered Load andStaffing Level
Target α=0.1Target alpha=0.1: Arrivals, Offered Load and Staffing Level
0
20
40
60
80
100
120
140
160
180
200
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
Arrived Staffing Offered Load
Target α=0.5Target alpha=0.5: Arrivals, Offered Load and Staffing Level
0
20
40
60
80
100
120
140
160
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
Arrived Staffing Offered Load
Target α=0.9Target alpha=0.9: Arrivals, Offered Load and Staffing Level
0
20
40
60
80
100
120
140
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
Arrived Staffing Offered Load
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Theoretical staffing level (homogenous arrival rate) dependsboth on X and β:
n =λxκ
µ+ β
√λxκ
µ.
β for xκ = 1.5Beta for X=1.5
-10
-8
-6
-4
-2
0
2
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
⇒ β is stabilized for targetα = 0.1, 0.2, 0.3 and 0.4.
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
β for xκ = 1Beta for X=1
-6
-4
-2
0
2
4
6
8
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
⇒ β is stabilized for targetα = 0.6, 0.7 and 0.8.
What about target α = 0.5 ?
Extreme values (-2.3, 2.1 &10) and noisy
β for xκ = 0.5Beta for X=0.5
0
2
4
6
8
10
12
14
16
18
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
⇒ β is stabilized for targetα = 0.9
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
β for xκ = 1Beta for X=1
-6
-4
-2
0
2
4
6
8
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
⇒ β is stabilized for targetα = 0.6, 0.7 and 0.8.
What about target α = 0.5 ?
Extreme values (-2.3, 2.1 &10) and noisyβ for xκ = 0.5Beta for X=0.5
0
2
4
6
8
10
12
14
16
18
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
⇒ β is stabilized for targetα = 0.9
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Delay Probability (Target)Delay Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
Theoretical and Empirical Delay
Probability vs. xκ and βTheoretical and Empirical Delay Probability vs. Beta
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-3 -2 -1 0 1 2 3
Beta
Del
ay P
roba
bilit
y
Theoretical Empirical
Xk=0.5
Xk=1
Xk=1.5
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Abandonment ProbabilityAbandonment Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
Abandonment Probability - Low α’sAbandonment Probability
0
0.01
0.02
0.03
0.04
0.05
0.06
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
Target alpha=0.1 Target alpha=0.2
Target alpha=0.3 Target alpha=0.4
Servers’ UtilizationServers' Utilization
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Abandonment ProbabilityAbandonment Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
Abandonment Probability - Low α’sAbandonment Probability
0
0.01
0.02
0.03
0.04
0.05
0.06
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
Target alpha=0.1 Target alpha=0.2
Target alpha=0.3 Target alpha=0.4
Servers’ UtilizationServers' Utilization
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Example 2: c = 1, Continuous Random Arrival Rate
1 Mt = λt · X , where
λt = 100− 20 · cos(t), and X ∼ Uni[0.5, 1.5].
2 Service time and patience are distributed exponentially withmean 1 (µ = θ = 1).
Theoretical staffing level (homogenous arrival rate) dependson the QOS parameter δ:
n =λ
µ· δ
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Example 2: c = 1, Continuous Random Arrival Rate
1 Mt = λt · X , where
λt = 100− 20 · cos(t), and X ∼ Uni[0.5, 1.5].
2 Service time and patience are distributed exponentially withmean 1 (µ = θ = 1).
Theoretical staffing level (homogenous arrival rate) dependson the QOS parameter δ:
n =λ
µ· δ
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Delay Probability (Target)Delay Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
QOS δdelta
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
Delay Probability vs. δTheoretical and Empirical Delay Probability vs. delta
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
delta
Del
ay P
roba
bilit
y
Empirical Theoretical
Introduction Theoretical Results Time-Varying Queues
Time - Varying Queues
Delay Probability (Target)Delay Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
QOS δdelta
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
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
Target alpha=0.1 Target alpha=0.2 Target alpha=0.3
Target alpha=0.4 Target alpha=0.5 Target alpha=0.6
Target alpha=0.7 Target alpha=0.8 Target alpha=0.9
Delay Probability vs. δTheoretical and Empirical Delay Probability vs. delta
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
delta
Del
ay P
roba
bilit
y
Empirical Theoretical
Introduction Theoretical Results Time-Varying Queues
Thank You
Introduction Theoretical Results Time-Varying Queues
Appendix 1
c = 1/2, γ∗X
Denote
β∗x = β∗ − x ,
β∗x = β∗x
rµ
g0= (β∗ − x) ·
rµ
g0.
Then,
γ∗x = α · β∗x ·
"h(β∗x )
β∗x− 1
#.
Back
Introduction Theoretical Results Time-Varying Queues
Appendix 2
Proof Outline (c = 1, Discrete Random Arrival Rate)
1. n =λxκ
µ+ β
sλxκ
µ+ o(
√λ) =
λxi
µ· xκ
xi+ o(λ)
⇒ for i < κ (xi > xκ) we are in the ED regime, for i = κ in the QEDregime, and for i > κ (xi < xκ) in the QD regime.
2. limλ→∞
ePM,n{Wq > 0} = limλ→∞
E [XPM,n{Wq > 0}]
= E limλ→∞
{XPM,n{Wq > 0}} =κ−1Xi=1
xipi + xκpκ · α∗.
3. For a certain value xs of X , the asymptotic long-term delay probability isbounded in the open interval As =
`Ps−1i=1 xipi ,
Psi=1 xipi
´. ⇒ There is a
unique s which for α ∈ As .
Introduction Theoretical Results Time-Varying Queues
Appendix 2
Proof Outline (c = 1, Discrete Random Arrival Rate)
4. For some ε > 0 define
n14=
&λxκ
µ+ (β∗ − ε)
sλxκ
µ
’; n2
4=
$λxκ
µ+ (β∗ + ε)
sλxκ
µ
%.
Garnett function is strictly decreasing in β
⇒ PM,n1{Wq > 0} → α + τ1 ; PM,n2{Wq > 0} → α− τ2.
⇒ For large λ
PM,n1{Wq > 0} > α + τ1/2 ; PM,n2{Wq > 0} < α− τ2/2.
Delay probability is monotonically decreasing in number ofservers
⇒ n1 ≤ n∗ ≤ n2.
ε > 0 is arbitrary ⇒ q.e.d. Back