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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
π
π
Routing and Staffing
β ππ«π π¦ππ±β¨πππππππππππ πππ β©πΌππππππ (π ;ππππππππ )
strategicserver
is fixed
Routing and Staffing
πstrategicserver
β’ Journal reviewsβ’ Call centersβ’ Crowd/Out-sourcingβ’ Cloud computingβ’ Enterprise data centersβ’ β¦
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.
Outlineβ’ The M/M/1 queue β a simple example
β’ Model for a strategic server
β’ The strategic M/M/N queue
β’ Classic policies in non-strategic setting
β’ Impact of strategic serversβ’ Asymptotically optimal policy
Routing Staffingwhich idle server gets the next job?
how many servers to
hire?
M/M/1/FCFS
mm
πΌ [πΎ ]= ππ (πβπ )
π° (π )π° (π )βπ (π)πΌ (π )=π° (π )βπ (π)
strategic serveridleness cost
utility function
πββππ«π π¦ππ±π>π
πΌ (π )
πΌ [πΎ ]= π
πβ (πββπ)
πΌ (π )=πβ ππβπ (π)
ππβπ=πβ² (πβ)
l0
1 / l
m*
LHS
RHS
πΌ [πΎ ]= ππ (πβπ )
l
Outlineβ’ The M/M/1 queue β a simple example
β’ Model for a strategic server
β’ The strategic M/M/N queue
β’ Classic policies in non-strategic setting
β’ Impact of strategic serversβ’ Asymptotically optimal policy
Routing Staffingwhich idle server gets the next job?
how many servers to
hire?
M/M/N/FCFS
strategic servers
routing
πΌ π (ππ , οΏ½βοΏ½βπ ;π·)=π° π (π π , οΏ½βοΏ½β π;π· )βπ (ππ)πΌ π (ππ , οΏ½βοΏ½βπ )=π° π (π π , οΏ½βοΏ½β π )βπ (ππ)
π·
πΌ [πΎ ]=π(π΅ ,
ππβ )
π΅πββπ
πββππ«π π¦ππ±ππ>
ππ΅
πΌ π (ππ , οΏ½βοΏ½βπβ ;π·)ππ
ββππ«π π¦ππ±ππ>
ππ΅
πΌ π (ππ , οΏ½βοΏ½βπβ ;π·)
symmetricNash equilibriumNash equilibrium
existence?performance?
β’ Blue for strategic service ratesβ’ Yellow for control/policy
parameters
ππββππ«π π¦ππ±
ππ>ππ΅
πΌ π (ππ ,οΏ½βοΏ½β π ;π· )
m1
m2
mN
l
π (π΅ ,π )=π¬πππππβπͺπππππππ
=
ππ΅
π΅ !π΅
π΅β π
βπ=π
π΅βπ π π
π !+ π
π΅
π΅ !π΅
π΅ βπ
Outlineβ’ The M/M/1 queue β a simple example
β’ Model for a strategic server
β’ The strategic M/M/N queue
β’ Classic policies in non-strategic setting
β’ Impact of strategic serversβ’ 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 VΓ©ricourt & Zhou 2005] [Armony 2005]
[Atar 2008]
(1) Fastest Server First (FSF) is βasymptotically optimalβ for minimizing the mean response time
(2) Longest Idle Server First (LISF) is βasymptotically fairβ 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:
πβ
πβ
πβ
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:
πβ
πβ
πβ
routing
π·
M/M/N/FCFS
[Haji & Ross 2013]
π (π΅ ,π )=π¬πππππβπͺπππππππ
=
ππ΅
π΅ !π΅
π΅β π
βπ=π
π΅βπ π π
π !+ π
π΅
π΅ !π΅
π΅ βπ
same uniquesymmetricequilibrium
Rate-basedpolicies
FSFSSF
Random&
Idle-time-based policies
Can we do better than Random?
Yes, butβ¦
Goal: minimize the mean response time at symmetric Nash equilibrium
l
Our Results:
πβ
πβ
πβ
routing
π·
M/M/N/FCFS
Outlineβ’ The M/M/1 queue β a simple example
β’ Model for a strategic server
β’ The strategic M/M/N queue
β’ Classic policies in non-strategic setting
β’ Impact of strategic serversβ’ 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:
βasymptotically optimalβ
[Borst, Mandelbaum, & Reiman 2004]
l
π΅ π
Classical Result: (nonstrategic setting)
M/M/N/FCFS
Randoml
πβ
πβ
πβ
π΅ π
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
πβ
πβ
πβ
π΅ π
Proof Outline:
Feasibility: We are interested in policies for which:
β’ overstaffing: servers get too lazyβ’ understaffing: servers βwork to deathβ
Recall the FOC:
M/M/N/FCFS
STEP 2: Analyze the limiting cost and the limiting FOC
Randoml
πβ
πβ
πβ
π΅ π
Proof Outline:
Let . Then, as ,
Limiting FOC:ππβππβ=
πβπ
ππ πβ² (πβ )
π/π
π
Limiting Cost:ππππΊ
Pick to optimize limiting costsubject to the limiting FOC
having at least one solution.
Observation:
πβ<πβ
βΉπ΅β ,π>π΅π©π΄πΉ ,π=ππβ+π(π)
π΅β ,π
M/M/N/FCFS
Concluding remarks
β’ We need to rethink optimal system design when servers are strategic!
β’ Joint routing-staffing optimization?β’ Empirical studies / Experimental evaluation?β’ Asymmetric models / equilibria?β’ Interaction between strategic arrivals and
strategic servers?
lπβ
Randomπβ
πβ
π΅β ,ππ΅π©π΄πΉ ,π
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