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