Routing and Staffing to Incentivize Servers
in Many Server Systems
Amy Ward (USC)
Raga Gopalakrishnan (Caltech/CU-Boulder/USC)
Adam Wierman (Caltech)
Sherwin Doroudi (CMU)
Service systems are staffed by humans.
m
strategic servers
system performance
Service systems are staffed by humans.
m
Routing and Staffing to Incentivize
strategic Servers
servers
system performance
Queueing
games:Assumes fixed
Classic
Queueing:
[Hassin and Haviv 2003]
• Strategic
(arrival
and) arrivals
service rates.
This •talk:
Impact of strategic
server on system design
Service/price
competition
• 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
Routing
which idle server gets
the next job?
Staffing
how many
servers to hire?
• Classic policies in non-strategic setting
• Impact of strategic servers
M/M/1/FCFS
λ
m
strategic server
Values idleness
Cost of effort
I( m )
c( m )
U ( m ) = (I1( ml) -/ cm()m-)c ( m )
utility function
l
W=
?
m(m - l)
m
What is the service rate?
{
}
m = arg max U ( m )
*
m >l
( )
l m
*
2
( )
= c' m *
l
W= * *
m m -l
(
)
Outline
• The M/M/1 queue – a simple example
• Model for a strategic server
• The strategic M/M/N queue
Scheduling
Staffing
• Classic policies in non-strategic setting
• Impact of strategic servers
M/M/N/FCFS
l
m
m2
scheduling
𝚷
mN
U i ( m;p ) = I i ( m;p ) - c (Umi () m ) = I ( m ) - c ( m )
i
i
i
strategic servers
{ (
Nash equilibrium
mi* Îarg max U i mi , m-i* ;p
symmetric
m = m for all i
*
i
mi > l / N
*
)}
Why symmetric? This is fair. (Server payment is fixed.)
existence?
performance?
W =?
Outline
• The M/M/1 queue – a simple example
• Model for a strategic server
• The strategic M/M/N queue
Scheduling
Staffing
• Classic policies in non-strategic setting
• Impact of strategic servers
M/M/N/FCFS
l
m
scheduling
m2
mN
When servers are not strategic…
• Fastest-Server-First (FSF) is asymptotically optimal
for .
[Lin and Kumar 1984] [Armony 2005]
• Longest-Idle-Server-First (LISF) is asymptotically
optimal subject to fairness (idleness distribution).
[Atar 2008] [Armony and Ward 2010]
M/M/N/FCFS
l
m
m2
scheduling
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.
Also, (Haji and Ross, 2013).
Q: Can we do better than Random?
Answer: Yes, but …
M/M/N/FCFS
l
m
m2
Random
mN
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.
What is the symmetric equilibrium service rate?
U ( m1 , m ) = I ( m1 , m ) - c ( m1 )
First order
condition:
¶I ( m¶U
,m
1 ()m1 , m )
= c' ( m=)0
¶ m1 ¶ m
m1 = m
1
m1 = m
Proposition: Under Random routing,
æ
æ
öö
ç
÷÷
æ
r öç
ræ
m ö çç
C(N , r )
÷÷
I( m1 , m ) = ç 1- ÷ 1- ç 1- ÷ 1+
æ
Nøç
N è m1 ø ç
è
m1 ö ÷ ÷
N
r
+
1ç
ç
÷÷
ç
÷
m
è
ø
è
øø
è
-1
ö
æ m1 ö m1
¶I
I2 l æ
C(N , r )
C( N , r )
ç 1+
÷
= 2
+ ç 1- ÷
2
¶ m1 m1 N - r çè
N - ( r + 1- m1 / m ) è
m ø m ( N - ( r + 1- m / m ) ) ÷ø
1
where r = l / m and C(N , r ) isthe Erlang - C formula
rN
C(N , r ) =
N
N! N - r
å
N -1
j=0
r
j
j!
+
r
N
N
N! N - r
.
Gumbel (1960) for the
fully heterogeneous case.
Problem: This is a mess!!! There is no hope to
use this to decide on a staffing level.
Outline
• The M/M/1 queue – a simple example
• Model for a strategic server
• The strategic M/M/N queue
Scheduling
Staffing
• Classic policies in non-strategic setting
• Impact of strategic servers
M/M/N/FCFS
l
m
m
Random
When servers are not strategic…
staffing
N
l
m
Q: How many servers to staff?
Objective: Minimize total system cost
C l (N ) = cS N + wlWml
N opt ,l = arg min C l (N )
N >l / m
Answer: Square root staffing is asymptotically optimal.
Halfin and Whitt (1981) and Borst, Mandelbaum and Reiman (2004)
N
BMR,l
l
l
*
= + b (cS , w)
m
m
M/M/N/FCFS
l
m
m
Random
staffing
N
m
When servers are strategic…
Q: How many servers to staff?
Objective: Minimize total system cost
C (N ) = cS N + wlW
l
N
opt ,l
l
m*
l
= arg min C (N )
N >l / m
Problem: Explicit expression is unknown.
Fortunately, there is hope if we let λbecome large.
l
M/M/N/FCFS
l
m
Random
m
N
m
When servers are strategic…
1. Rate-independent staffing
N = f (l ) + o ( f (l ))
l
2. Rate-dependent staffing
æ
æ l ö
N = f ç l ,* ÷ + o ç
èm ø
è
l
æ l öö
f ç l ,* ÷ ÷
è m øø
staffing
l
M/M/N/FCFS
l
m
m
Random
æ
æ l ö
N = f ç l ,* ÷ + o ç
èm ø
è
l
staffing
N
æ l öö
f ç l ,* ÷ ÷
è m øø
In order that there exists μ*,λ with
l
m
Such a solution
is not desirable.
0 < lim inf m *,l < limsup m *,l < ¥
l ®¥
Eliminates square-root staffing.
Must staff order λmore.
The cost function
blows up at rate λ.
we must staff
l ®¥
ææ l öö
æ l ö
N = a ç l ,* ÷ + o ç ç l ,* ÷ ÷ for a > 1.
èm ø
èè m øø
l
M/M/N/FCFS
l
m
m
Random
staffing
N
ææ l öö
æ l ö
N = a ç l ,* ÷ + o ç ç l ,* ÷ ÷ for a > 1.
èm ø
èè m øø
What is a?
Fluid scale cost.
l
m
l
ì a a -1
ü
Set a = a* = arg min íc
: 2 = mc'( m ) ý
S
a>1
î m a
þ
Since servers are strategic.
Theorem: The staffing Nλ is asymptotically optimal
in the sense that
Cl N l
lim
l ®¥
( )
C
l
(N )
opt ,l
= 1.
M/M/N/FCFS
l
m
m
Random
staffing
N
ææ l öö
æ l ö
N = a ç l ,* ÷ + o ç ç l ,* ÷ ÷ for a > 1.
èm ø
èè m øø
Example:
p
Suppose c m = m for some p ³ 1.
l
l
m
( )
ìï 1.5 as p ¯ 1
1+ 2 / p
®í
.
Then a =
1+ 1/ p
ïî 1.0 as ­ ¥
l
1
l
Note that r = l *,l ® < 1 as l ® ¥. Convexity helps.
a
N m
2
If p = 1, then a = 1.5 and m * = 0.222, and r l ®
Efficiency is decreased.
3
as l ® ¥.
Concluding remarks
• We need to rethink optimal system design to
account for how servers respond to incentives
(i.e., when servers are strategic)!
$$$$
l
M/M/N/FCFS
m
FSF,SSF
LISF =
Random
?
m
m
We solved for an asymptotically optimal staffing
There is a loss of efficiency.
l
N >
. N
BMR,l
.