The Impact of Server Incentives on Scheduling
Raga Gopalakrishnan and Adam WiermanCalifornia Institute of Technology
Sherwin DoroudiCarnegie Mellon University
7/7/2011 INFORMS APS 2011
Scheduling in Multi-Server Queues
How should the dispatcher be designed?
l
FCFS
dispatcher
m1
m2
mm
Commonly Studied Dispatch Policies
• Fastest Server First (FSF) [Lin et al. 1984] [Véricourt et al. 2005] [Armony 2005]
• RANDOM
Dispatch Policy (P)
l
FCFS
dispatcher
m1
m2
mm
P
What if servers are people?
• Fair distribution of idle time is an important measure of employee satisfaction. [Cohen-Charash et al. 2001] [Colquitt et al. 2001] [Whitt 2006]
• FSF is not a “fair” policy. [Armony 2005]
Example: Call Centers
l
FCFS
dispatcher
m1
m2
mm
P
What if servers are people?
• Longest Idle Server First (LISF) [Atar 2008] [Armony et al. 2010]
• LISF has good “fairness” properties. [Atar 2008]
Example: Call Centers
l
FCFS
dispatcher
m1
m2
mm
P
What if people can react?
This Talk:How should the dispatcher be designed
if servers are strategic?
l
FCFS
dispatcher
m1
m2
mm
P
M/M/m/FCFS
Model
servers choose mi є [1/m,∞) to maximize:Ui(m1,m2,…,mm; P) = Ii(m1,m2,…,mm; P) –
c(mi)utility idle time cost
Note: We assume a fixed payment model.
(increasing, convex)
dispatcher
m1
m2
mm
P
= l1
M/M/2/FCFS
Model
servers choose mi є [1/2,∞) to maximize:Ui(m1,m2; P) = Ii(m1,m2; P) – c(mi)
utility idle time cost(increasing, convex)
dispatcher
m1
m2
P
= l1
Note: We assume a fixed payment model.
Goal
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
Design a dispatch policy that:
• leads to a symmetric Nash equilibrium in the service rates: (m*,m*)
• minimizes the mean response time, E[T], at (m*,m*)
Design a dispatch policy that:
• leads to a symmetric Nash equilibrium in the service rates: (m*,m*)
• minimizes the mean response time, E[T], at (m*,m*)
Design a dispatch policy that:
• leads to a symmetric Nash equilibrium in the service rates: (m*,m*)
• minimizes the mean response time, E[T], at (m*,m*)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
(m1,m2) is a Nash equilibrium if, for each server, Ui (m1,m2 ; P) = maxm’i ≥ ½ Ui (m’i,m3-i ;
P)
What about well-known policies?
• Fastest Server First (FSF)• Wrong incentive• No symmetric equilibrium
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
What about well-known policies?
• Slowest Server First (SSF)• Right incentive• No symmetric equilibrium
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
What about well-known policies?
• RANDOM• Unique symmetric equilibrium under mild assumptions that
guarantee voluntary participation: c’(½) < 5/6, c”’(m) > 0.
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
Can we do better than RANDOM?
• Longest Idle Server First (LISF)• Equivalent to RANDOM.
Can we do better than RANDOM?
• Suppose there are |I(t)| idle servers in the system (1 ≤ |I(t)| ≤ 2).• These servers are ranked in the order in which they last became idle.• The next job in the queue is then routed according to a probability
distribution on this ranking.
What about idle-time-based policies in general?
All idle-time-based policies are equivalent and result in the same unique symmetric equilibrium as RANDOM.
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
Can we do better than RANDOM?
• The probability that an idle server i gets the next job is proportional to mir,
where r e R is a policy parameter.
What about rate-based policies in general?
∞0∞–
SSF FSFRANDOM
Policy parameter (r)
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
Can we do better than RANDOM?
Any rate-based policy with r є {-2,-1,0,1} admits a unique symmetric Nash equilibrium.
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
What about rate-based policies in general?
∞0∞–
SSF FSFRANDOM
Policy parameter (r)
Can we do better than RANDOM?
There exists a bounded interval for r outside of which, no rate-based policy admits a symmetric Nash equilibrium.
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
What about rate-based policies in general?
∞0∞–
SSF FSFRANDOM
Policy parameter (r)
Can we do better than RANDOM?
Any rate-based policy that admits a symmetric Nash equilibrium, admits a unique symmetric Nash equilibrium. Further, among all
such policies, E[T] at symmetric equilibrium is increasing in r.
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
What about rate-based policies in general?
Simulation
1
2
3
–20 20 40 60
–1 Policy parameter (r)
Log [Mean response time]
–10
Summary
∞0∞–
SSF FSF
Random, Idle-time-
basedRandom
Policy parameter (r)
Policy parameter (r)
Mea
n re
spon
se ti
me
∞0∞–
∞
Ui(m1,m2;P) =Ii(m1,m2;P) –
c(mi)
M/M/2/FCFS
dispatcher
m1
m2
P
= l1
Design a dispatch policy that:
• leads to a symmetric Nash equilibrium in the service rates: (m*,m*)
• minimizes the mean response time, E[T], at (m*,m*)
M/M/2/FCFS
Model
servers choose mi є [1/2,∞) to maximize:Ui(m1,m2; P) = Ii(m1,m2; P) – c(mi)
utility idle time cost(increasing, convex)
dispatcher
m1
m2
P
= l1
Note: We assume a fixed payment model.
M/M/2/FCFS
Future Work
servers choose mi є [1/2,∞) to maximize:Ui(m1,m2; P) = Ii(m1,m2; P) – c(mi)
utility idle time cost(increasing, convex)
dispatcher
m1
m2
P
= l1
Note: We assume a fixed payment model.
• More than 2 servers• More general queueing models
• Other payment models
• Other utility functions
The Impact of Server Incentives on Scheduling
Raga Gopalakrishnan and Adam WiermanCalifornia Institute of Technology
Sherwin DoroudiCarnegie Mellon University
7/7/2011 INFORMS APS 2011
• [Lin et al. 1984] Optimal control of a queueing system with two heterogeneous servers.
• [Cohen-Charash et al. 2001] The role of justice in organizations: A meta-analysis.
• [Colquitt et al. 2001] Justice at the millennium: A meta-analytic review of 25 years of organizational justice research.
• [Véricourt et al. 2005] Managing response time in a call-routing problem with service failure.
• [Armony 2005] Dynamic routing in large-scale service systems with heterogeneous servers.
• [Whitt 2006] The impact of increased employee retention on performance in a customer contact center.
• [Atar 2008] Central limit theorem for a many-server queue with random service rates.
• [Armony et al. 2010] Fair dynamic routing in large-scale heterogeneous-server systems.
• [Armony et al. 2010] Blind fair routing in large-scale service systems.
References