Routing and Staffing to Incentivize Servers i n Many Server Systems
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Transcript of Routing and Staffing to Incentivize Servers i n Many Server Systems
Routing and Staffing to Incentivize Serversin Many Server Systems
strategic servers
system performance
Service systems are staffed by humans.
m
strategic servers
system performance
This talk: Impact of strategic server on system design
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Classic Queueing: Assumes fixed (arrival and) service rates.Queueing games:• Strategic arrivals• Service/price
competition
[Hassin and Haviv 2003]
Routing and Staffing to Incentivize Servers
Service systems are staffed by humans.
• Blue for strategic service rates• Yellow for routing/staffing policy
parameters• Pink is to highlight.
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
Routing Staffing
which idle server gets the next job?
how many servers to
hire?
λ
M/M/1/FCFS
m=1/mμ
strategic server
Values idleness
Cost of effort
utility function
?
What is the service rate?
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
M/M/N/FCFS
m1
strategic servers
scheduling
m2
mN
𝚷
symmetric
Nash equilibrium
existence? performance?
Why symmetric? This is fair. (Server payment is fixed.)
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
M/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 optimal subject to fairness (idleness distribution).
[Lin and Kumar 1984] [Armony 2005]
[Atar 2008] [Armony and Ward 2010]
M/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, but …
Also, (Haji and Ross, 2013).
M/M/N/FCFS
m1
Randomm2
mN
First order
condition:
What is the symmetric equilibrium service rate?
Theorem: For every λand N, under mild conditions on c,there exists a unique symmetric equilibrium service rate μ*
under Random. Furthermore, U(μ*)>0.
Problem: This is a mess!!! There is no hope to use this to decide on a staffing
level.
Proposition: Under Random routing,
Gumbel (1960) for the fully heterogeneous case.
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
M/M/N/FCFS
m
m
mWhen servers are not strategic…
Random
Q: How many servers to staff?
Objective: Minimize total system cost
Answer: Square root staffing is asymptotically optimal.Halfin and Whitt (1981) and Borst, Mandelbaum and Reiman (2004)
staffing
M/M/N/FCFS
When servers are strategic…
Randomstaffin
g
Q: How many servers to staff?
Objective: Minimize total system cost
Problem: Explicit expression is unknown.
Fortunately, there is hope if we let λbecome large.
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m
m
M/M/N/FCFS
m
Randomm
m
When servers are strategic…
1. Rate-independent staffing
2. Rate-dependent staffing
staffing
M/M/N/FCFS
m
Randomm
m
staffing
In order that there exists μ*,λ with
Such a solutionis not desirable.
The cost functionblows up at rate λ.
Eliminates square-root staffing.Must staff order λmore.
we must staff
M/M/N/FCFS
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Randomm
m
staffing
Set
Theorem: The staffing Nλ is asymptotically optimalin the sense that
Fluid scale cost.
Since servers are strategic.
What is a?
M/M/N/FCFS
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Randomm
m
staffing
Example:Suppose
Then
Convexity helps.
Efficiency is decreased.
Concluding remarks
• We need to rethink optimal system design to account for how servers respond to incentives (i.e., when servers are strategic)!
M/M/N/FCFS
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FSF,SSFLISF
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There is a loss of efficiency.
$$$$
?
We solved for an asymptotically optimal staffing
=Random