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