dispatching to incentivize fast service in multi-server queues raga gopalakrishnan and adam wierman...

Dispatching to Incentivize Fast Service in Multi-Server Queues

Raga Gopalakrishnan and Adam WiermanCalifornia Institute of Technology

Sherwin DoroudiCarnegie Mellon University

6/8/2011 MAMA 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

Dispatching to Incentivize Fast Service in Multi-Server Queues

Raga Gopalakrishnan and Adam WiermanCalifornia Institute of Technology

Sherwin DoroudiCarnegie Mellon University

6/8/2011 MAMA 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