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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