Raga GopalakrishnanUniversity of Colorado at Boulder

Adam Wierman (Caltech)Amy R. Ward (USC)

Sherwin Doroudi (CMU)

Routing and Staffing when Servers are Strategic

server

𝝁

𝝁

Routing and Staffing

∈ 𝐚𝐫𝐠𝐦𝐚𝐱⟨𝒄𝒐𝒏𝒔𝒕𝒓𝒂𝒊𝒏𝒕𝒔 𝒐𝒏𝝁 ⟩𝑼𝒕𝒊𝒍𝒊𝒕𝒚 (𝝁 ;𝒑𝒐𝒍𝒊𝒄𝒊𝒆𝒔 )

strategicserver

is fixed

Routing and Staffing

𝝁strategicserver

• Journal reviews• Call centers• Crowd/Out-sourcing• Cloud computing• Enterprise data centers• …

service systems

systemperformance

strategicserver

systemperformance

Classic Queueing: Assumes fixed (arrival and) service rates, fixed control/policies.

[Hassin & Haviv 2003] [Kalai, Kamien, & Rubinovitch 1992] [Gilbert & Weng 1998][Cachon & Harker 1999] [Chen & Wan 2002] [Cachon & Zhang 2007]

This talk: Impact of strategic servers on optimal system design

Routing and Staffing

CS-Econ Literature: Servers strategically misreport their service rates.[Nisan & Ronen 1999] [Archer & Tardos 2001][Christodoulou & Koutsoupias 2009]

[Halfin & Whitt 1981] [Borst, Mandelbaum, & Reiman 2004][Armony 2005] [Atar 2008] [Armony & Ward 2010] [Armony & Mandelbaum 2011]

Queueing Games: Strategic arrivals and service/pricing amidst competition between different firms.

(within the same firm)

[Zhan & Ward 2014] Compensation and Staffing for Strategic Employees: How to Incentivize a Speed-Quality Trade-off in a Large

Service System. Working Paper.

Outline• The M/M/1 queue – a simple example

• Model for a strategic server

• The strategic M/M/N queue

• Classic policies in non-strategic setting

• Impact of strategic servers• Asymptotically optimal policy

Routing Staffingwhich idle server gets the next job?

how many servers to

hire?

M/M/1/FCFS

mm

𝔼 [𝑾 ]= 𝝀𝝁 (𝝁−𝝀 )

𝑰 (𝝁 )𝑰 (𝝁 )−𝒄 (𝝁)𝑼 (𝝁 )=𝑰 (𝝁 )−𝒄 (𝝁)

strategic serveridleness cost

utility function

𝝁∗∈𝐚𝐫𝐠𝐦𝐚𝐱𝝁>𝝀

𝑼 (𝝁 )

𝔼 [𝑾 ]= 𝝀

𝝁∗ (𝝁∗−𝝀)

𝑼 (𝝁 )=𝟏− 𝝀𝝁−𝒄 (𝝁)

𝝀𝝁∗𝟐=𝒄′ (𝝁∗)

l0

1 / l

m*

LHS

RHS

𝔼 [𝑾 ]= 𝝀𝝁 (𝝁−𝝀 )

l

Outline• The M/M/1 queue – a simple example

• Model for a strategic server

• The strategic M/M/N queue

• Classic policies in non-strategic setting

• Impact of strategic servers• Asymptotically optimal policy

Routing Staffingwhich idle server gets the next job?

how many servers to

hire?

M/M/N/FCFS

strategic servers

routing

𝑼 𝒊 (𝝁𝒊 , �⃗�−𝒊 ;𝚷)=𝑰 𝒊 (𝝁 𝒊 , �⃗�− 𝒊;𝚷 )−𝒄 (𝝁𝒊)𝑼 𝒊 (𝝁𝒊 , �⃗�−𝒊 )=𝑰 𝒊 (𝝁 𝒊 , �⃗�− 𝒊 )−𝒄 (𝝁𝒊)

𝚷

𝔼 [𝑾 ]=𝓒(𝑵 ,

𝝀𝝁∗ )

𝑵𝝁∗−𝝀

𝝁∗∈𝐚𝐫𝐠𝐦𝐚𝐱𝝁𝒊>

𝝀𝑵

𝑼 𝒊 (𝝁𝒊 , �⃗�−𝒊∗ ;𝚷)𝝁𝒊

∗∈𝐚𝐫𝐠𝐦𝐚𝐱𝝁𝒊>

𝝀𝑵

𝑼 𝒊 (𝝁𝒊 , �⃗�−𝒊∗ ;𝚷)

symmetricNash equilibriumNash equilibrium

existence?performance?

• Blue for strategic service rates• Yellow for control/policy

parameters

𝝁𝒊∗∈𝐚𝐫𝐠𝐦𝐚𝐱

𝝁𝒊>𝝀𝑵

𝑼 𝒊 (𝝁𝒊 ,�⃗�− 𝒊 ;𝚷 )

m1

m2

mN

l

𝓒 (𝑵 ,𝝆 )=𝑬𝒓𝒍𝒂𝒏𝒈−𝑪𝑭𝒐𝒓𝒎𝒖𝒍𝒂

=

𝝆𝑵

𝑵 !𝑵

𝑵− 𝝆

∑𝒋=𝟎

𝑵−𝟏 𝝆 𝒋

𝒋 !+ 𝝆

𝑵

𝑵 !𝑵

𝑵 −𝝆

Outline• The M/M/1 queue – a simple example

• Model for a strategic server

• The strategic M/M/N queue

• Classic policies in non-strategic setting

• Impact of strategic servers• Asymptotically optimal policy

Routing Staffingwhich idle server gets the next job?

how many servers to

hire?

l

m1

m2

mN

Classical Results: (nonstrategic setting)

[Lin and Kumar 1984] [de Véricourt & Zhou 2005] [Armony 2005]

[Atar 2008]

(1) Fastest Server First (FSF) is “asymptotically optimal” for minimizing the mean response time

(2) Longest Idle Server First (LISF) is “asymptotically fair” in distributing idle time proportionately among the servers

routing

𝚷

M/M/N/FCFS

Rate-basedpolicies

Idle-time-basedpolicies

FSFSSF

LISFSISF

Random

Goal: minimize the mean response time at symmetric Nash equilibrium

l

Our Results:

𝝁∗

𝝁∗

𝝁∗

routing

𝚷

M/M/N/FCFS

Rate-basedpolicies

FSFSSF

Random&

Idle-time-based policies

First order condition:

same uniquesymmetricequilibrium

Goal: minimize the mean response time at symmetric Nash equilibrium

l

Our Results:

𝝁∗

𝝁∗

𝝁∗

routing

𝚷

M/M/N/FCFS

[Haji & Ross 2013]

𝓒 (𝑵 ,𝝆 )=𝑬𝒓𝒍𝒂𝒏𝒈−𝑪𝑭𝒐𝒓𝒎𝒖𝒍𝒂

=

𝝆𝑵

𝑵 !𝑵

𝑵− 𝝆

∑𝒋=𝟎

𝑵−𝟏 𝝆 𝒋

𝒋 !+ 𝝆

𝑵

𝑵 !𝑵

𝑵 −𝝆

same uniquesymmetricequilibrium

Rate-basedpolicies

FSFSSF

Random&

Idle-time-based policies

Can we do better than Random?

Yes, but…

Goal: minimize the mean response time at symmetric Nash equilibrium

l

Our Results:

𝝁∗

𝝁∗

𝝁∗

routing

𝚷

M/M/N/FCFS

Outline• The M/M/1 queue – a simple example

• Model for a strategic server

• The strategic M/M/N queue

• Classic policies in non-strategic setting

• Impact of strategic servers• Asymptotically optimal policy

Routing Staffingwhich idle server gets the next job?

how many servers to

hire?

Goal: minimize the total system cost

m

m

m

Random

per-unit staffing

cost

per-unit waiting

cost

mean waiting

time

Square-root staffing:

“asymptotically optimal”

[Borst, Mandelbaum, & Reiman 2004]

l

𝑵 𝝀

Classical Result: (nonstrategic setting)

M/M/N/FCFS

Randoml

𝝁∗

𝝁∗

𝝁∗

𝑵 𝝀

Goal: minimize the total system cost at Nash equilibrium

Our Result:

Let . Then, the policy with and

is asymptotically optimal in the sense that:

as has 1 solution

M/M/N/FCFS

Suppose for some function .Then, feasibility is satisfied only if .

STEP 1: Discard infeasible policies

Randoml

𝝁∗

𝝁∗

𝝁∗

𝑵 𝝀

Proof Outline:

Feasibility: We are interested in policies for which:

• overstaffing: servers get too lazy• understaffing: servers “work to death”

Recall the FOC:

M/M/N/FCFS

STEP 2: Analyze the limiting cost and the limiting FOC

Randoml

𝝁∗

𝝁∗

𝝁∗

𝑵 𝝀

Proof Outline:

Let . Then, as ,

Limiting FOC:𝟏𝒂−𝟏𝝁∗=

𝝁∗𝟐

𝒂𝟐 𝒄′ (𝝁∗ )

𝟏/𝒂

𝟎

Limiting Cost:𝟏𝒂𝒄𝑺

Pick to optimize limiting costsubject to the limiting FOC

having at least one solution.

Observation:

𝒂∗<𝝁∗

⟹𝑵∗ ,𝝀>𝑵𝑩𝑴𝑹 ,𝝀=𝝀𝝁∗+𝒐(𝝀)

𝑵∗ ,𝝀

M/M/N/FCFS

Concluding remarks

• We need to rethink optimal system design when servers are strategic!

• Joint routing-staffing optimization?• Empirical studies / Experimental evaluation?• Asymmetric models / equilibria?• Interaction between strategic arrivals and

strategic servers?

l𝝁∗

Random𝝁∗

𝝁∗

𝑵∗ ,𝝀𝑵𝑩𝑴𝑹 ,𝝀

loss of efficiency

?

$$$$$

$$

? ?

M/M/N/FCFS

Ragavendran GopalakrishnanUniversity of Colorado at Boulder

Adam Wierman (Caltech)Amy R. Ward (USC)

Sherwin Doroudi (CMU)

Routing and Staffing when Servers are Strategic

[Zhan & Ward 2014] Compensation and Staffing for Strategic Employees: How to Incentivize a Speed-Quality Trade-off in a Large

Service System. Working Paper.

Companion Talk

MSOM: Saturday@11:15am