Routing and Staffing to Incentivize Servers in Many Server Systems
Feb 24, 2016
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
m
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 Staffingwhich idle server gets the next job?
how many servers to
hire?
λ
M/M/1/FCFS
mm=1/m
strategic server
Values idlenessCost 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…
Random staffing
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.
m
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
m
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
m
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
m
FSF,SSFLISF
m
m
There is a loss of efficiency.
$$$$
?
We solved for an asymptotically optimal staffing
=Random