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Scheduling and staffing strategic servers. Raga Gopalakrishnan Caltech CU-Boulder / USC Adam Wierman (Caltech) Amy R. Ward (USC) Sherwin Doroudi (CMU). Journal reviews Call centers Crowdsourcing Cloud computing Enterprise data centers …. service systems. m. strategic servers. - PowerPoint PPT Presentation
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Raga GopalakrishnanCaltech CU-Boulder / USC
Adam Wierman (Caltech)Amy R. Ward (USC)
Sherwin Doroudi (CMU)
Scheduling and staffing strategic servers
strategic servers
system performance
• Journal reviews• Call centers• Crowdsourcing• Cloud computing• Enterprise data centers• …
service systems
m
strategic servers
system performance
• Journal reviews• Call centers• Crowdsourcing• Cloud computing• Enterprise data centers• …
service systems
This talk: Impact of strategic servers
on optimal system design
m
Classic Queueing: Assumes fixed (arrival and) service rates, fixed control/policies.
Queueing games:• Strategic arrivals• Service/price
competition
[Hassin and Haviv 2003]
Scheduling and staffing strategic servers
Outline• The M/M/1 Queue – a simple example
• Model for a strategic server
• The M/M/N Queue
• Classic policies in non-strategic setting
• Impact of strategic servers
Scheduling Staffing
which idle server gets the next job?
how many servers to
hire?
lM/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 )
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
Scheduling Staffing
lM/M/N/FCFS
m1
strategic servers
scheduling
m2
mN
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
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
Scheduling Staffing
lM/M/N/FCFS
scheduling
m1
m2
mN
When servers are not strategic…• Fastest-Server-First (FSF) is
asymptotically optimal for .
• Longest-Idle-Server-First (LISF) is asymptotically fair (idleness distribution).
• Random is naïve, and easily implementable.
[Lin et al. 1984] [Véricourt et al. 2005] [Armony 2005]
[Atar 2008] [Armony et al. 2010]
lM/M/N/FCFS
m1
scheduling
m2
mN
Q: Which policy does better – FSF or its counterpart, SSF?
Theorem: No symmetric equilibrium exists
under either FSF or SSF.Q: How about Longest-Idle-Server-First (LISF)?
Theorem: All idle-time-order-based policies result in the same symmetric
equilibrium as Random.Q: Can we do better than Random?
Answer: Yes!
(ask me later!)
lM/M/N/FCFS
m1
Randomm2
mN
Theorem: For every ,, under mild conditions on c, there exists a unique symmetric equilibrium
under Random.
𝑼 𝒊 (𝝁𝒊 , �⃗�−𝒊 )=𝑰 𝒊 (𝝁 𝒊 , �⃗�− 𝒊 )−𝒄 (𝝁𝒊)𝑼 (𝝁𝒊 , �⃗�−𝒊 )=𝑰 (𝝁 𝒊 , �⃗�− 𝒊 )−𝒄 (𝝁𝒊)
Q: What does look like?
First order
condition:
𝑼 𝒊 (𝝁𝒊 , �⃗�−𝒊 )=𝑰 (𝝁 𝒊 , �⃗�− 𝒊 )−𝒄 (𝝁𝒊)
𝝏 𝑰 (𝝁𝟏 , �⃗�−𝟏∗ )
𝝏𝝁𝟏|𝝁𝟏=𝝁∗
=𝒄′ (𝝁∗ )First order
condition:Theorem: Under Random scheduling, suppose a
tagged server works at rate , and the other servers work at rate . Then,
where , and is the Erlang-C formula,
Problem: This is a mess!!!
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
Scheduling Staffing
lM/M/N/FCFS
m
m
m
When servers are not strategic…
Randomstaffin
g 𝑵 𝝀
Q: What staffing policy should the system manager adopt?
Objective: minimize total system cost:
𝑵𝒐𝒑𝒕 ,𝝀=𝐚𝐫𝐠𝐦𝐢𝐧𝑵 𝝀
𝑪 (𝝀 ,𝑵 𝝀 )
Answer: Square-root staffing:
asymptotically optimal
[Borst et al. 2004]
lM/M/N/FCFS
m
Randomm
m
staffing 𝑵 𝝀
When servers are strategic…
Q: What staffing policy should the system manager adopt?
𝑵𝒐𝒑𝒕 ,𝝀=𝐚𝐫𝐠𝐦𝐢𝐧𝑵 𝝀
𝑪 (𝝀 ,𝑵 𝝀 )
Objective: minimize total system cost:
Hope: Perhaps feasible to solve when is large.
Problem: Explicit expression unknown!
approximate by taking the limit as
lM/M/N/FCFS
m
Randomm
m
staffing 𝑵 𝝀
When servers are strategic…
1. Rate-independent staffing for some function
Need to staff in a very narrow way in order to
ensure unique equilibrium2. Rate-dependent staffing
for some function
(ask me later!)
for some function
Theorem: Feasibility is satisfied if and only if as . Furthermore, , where:
Feasibility: We are interested in solutions for which:
𝑵𝒐𝒑𝒕 ,𝝀=𝐚𝐫𝐠𝐦𝐢𝐧𝑵 𝝀
𝑪 (𝝀 ,𝑵 𝝀 )
• Eliminates square-root staffing ()• Need to staff more servers!
for some function
Feasibility: We are interested in solutions for which:
𝑵𝒐𝒑𝒕 ,𝝀=𝐚𝐫𝐠𝐦𝐢𝐧𝑵 𝝀
𝑪 (𝝀 ,𝑵 𝝀 )
Theorem (asymptotic optimality): Suppose as , where . Then,
as
Concluding remarks
• We need to rethink optimal system design when servers are strategic!
lM/M/N/FCFS
m
Randomm
m
𝑵 𝝀𝑵𝑩𝑴𝑹 ,𝝀
loss of efficiency
?
$$$$$
$$
? ?
Ragavendran GopalakrishnanCaltech CU-Boulder / USC
Adam Wierman (Caltech)Amy R. Ward (USC)
Sherwin Doroudi (CMU)
Scheduling and staffing strategic servers
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