DYNAMIC SCHEDULING OF A PARALLEL
SERVER SYSTEM IN HEAVY TRAFFIC UNDER A
COMPLETE RESOURCE POOLING CONDITION

I
I
1

J
1
K
Ruth J. Williams
University of California, San Diego
http://math.ucsd.edu/~williams
Collaborators: Steven Bell, Maury Bramson
HEAVY TRAFFIC APPROACH TO
DYNAMIC SCHEDULING
1. Formulate stochastic network model
2. Define heavy traffic (flexible servers)
3. Formal diffusion approximation: BCP
(Brownian control problem)
4. Reduce to EWF (equivalent workload
formulation)
5. Solve the BCP (or EWF)
6. Interpret the solution of the BCP
7. Analyze the performance of this policy
• OVERALL APPROACH: Harrison (‘88), Laws (‘92), Kelly-Laws
(‘93), Harrison-Van Mieghem (‘97), Harrison (‘00)
• SOME RELATED WORK ON OPTIMAL CONTROL OF PARALLEL
SERVER MODELS: Harrison (‘98), Harrison-Lopez (‘99), W (‘00),
Squillante, Xia, Yao, Zhang (‘00), Bell-W (‘01).
– Convex holding cost: Stolyar (‘04), Mandelbaum-Stolyar (‘04).
– Kushner-Chen (‘00), Ata-Kumar (‘04).
PARALLEL SERVER SYSTEM

I
I
1

J
1
K
Q(t )  E (t )  CS (T (t ))
I (t )  1 t  AT (t )
J
CS (T (t ))i   Cij S j (T j (t ))
j 1
T j (t )  total time allocated to activity j up to t
Model Primitives
Stochastic Processes
• E I -diml renewal arrival
process, rate   0
• S J-diml renewal potential
service process, rate 
B  diag (  )
Assume certain independence
and finite second moments of
interarrival and service times
Structure
Cij  1 iff activity j processes buffer i
Akj  1 iff server k performs activity j
Assume jobs in buffers queued
in FIFO order (HL service)
Heavy traffic
(Harrison, ’00)
Linear Program (LP)
min 
CBxx   , Ax  1, x  0.
*
*
*
Heavy traffic (HT) There is a unique solution
( x ,  ) of the LP and
*
*
Ax  1,   1.
*
*
Proposition (Bramson-W, ‘03) The heavy traffic
condition holds iff there is a unique vector
x*  ( x1* , , xJ* ) (of allocation rates) such that
*
Ax
1
(i ) x  0,
*
(ii ) C Bxx   x(balanced ) x
*
& in addition,
Ax  1
*
(full server utilization)
Two-server example: Harrison ‘98
1.3
0.4
1
0.5
1
1
x 1
2
*
1
1
2
x  0.6
*
2
x3*  0.4
Holding cost per unit of queuelength per
unit time: h1  3, h2  1
Greedy Scheduling Policy
• Simulation of this policy goes
here
Diffusion approximation
• Assume HT henceforth
• For a suitable cost function, T (t )  x t
is the only reasonable “fluid” allocation
• How should one achieve T (t )  x*t using a
policy in the original system?
*
• Diffusion scaling:
Qˆ r (t )  Q r ( r 2t ) / r
Iˆr (t )  I r ( r 2t ) / r
Yˆ r (t )  ( x*r 2t  T r ( r 2t )) / r
x*j  0, j  1,
,B
• Basic activities:
Non-basic activities: x*j  0, j  B  1, ,J
• Cost function:

 t
r

ˆ
J (T )  E  e h Q (t )dt 
 0

r
r
(Formal) Brownian Control Problem
(Harrison, ‘00)

 t

min E  e h Q(t )dt 
 0

Q(t )  X (t )  RY (t )  0
I (t )  AY (t ), I , I (0)  0, YN 
Q, Y do not anticipate the future of X
where
X ( )  E ( )  CS ( x )
*
is a Brownian motion and
R  CB
“Canonical” non-negative workload
(Harrison-Van Mieghem ‘97, Harrison ’00)
Dualyprogram
R  zA, z(DP)
1  1, z  0
max y
1 1
L
L
Extreme point solutions: ( y , z ), ,( y , z )
Max. lin. indept. subset of { y i } : y1 , , y L
Theorem:
span{ y1 , , y L }  {Rx : Ax  0, xN  0}
and y1 , , y L are all non-negative. If we let
1
L
have
rows
given
by
y , ,y,

1
L
have
rows
given
by
z , ,z ,

then there is a matrix   0 :
W (t )  Q(t )
 X (t )  I (t )  YN (t )
COMPLETE RESOURCE POOLING

I
I
1

J
1
K
Theorem (Harrison-Lopez ‘99)
The following are equivalent.
(i) the workload is one-dimensional,
(ii) B=I+K-1,
(iii) there is a unique solution of (DP),
(iv) all servers communicate via basic activities.
In fact, under any of these conditions, the
server-buffer graph with basic activities as
edges is a tree.
(W, ’00; Squillante-Xia-Yao-Zhang, ’00)
Solution of the BCP under complete
resource pooling (Harrison-Lopez ’99)
• Unique soln of (DP): ( y*, z*)
• One-dimensional workload process:
W (t )  y * Q(t )  y * X (t )  z * I (t )  YN (t )
• Holding cost:
h Q(t )  c *W (t )
c*  min{hi / yi* : i  1, , I }
• Minimum workload:
*
*
W (t )  y * X (t )  V (t )
V * (t )  max{ y * X ( s) : 0  s  t}
• Optimal queuelength and idletime:
i*  arg min{hi / yi* : i  1, , I },
k * serves i * via basic activity
Qi** (t )  W * (t ) / yi** , Qi* (t )  0 for i  i *
I k** (t )  V * (t ) / zk** ,
I k* (t )  0 for k  k *, YN*  0
How can one interpret the solution of
the BCP?
• Seek a policy that
(a) keeps the bulk of the work in a buffer i* with
smallest ratio of holding cost to workload
contribution,
(b) incurs idleness only when the system is
nearly empty,
(c) incurs the bulk of the idleness at a server k*
that serves i* via a basic activity.
Two-server example
1.3
0.4
1
0.5
1
1
h1  3, h2  1
2
i  2, k  2
*
1
*
2
Threshold policy (Bell-W, ‘01)
Q2r
Priority to buffer 2
Priority to buffer 1
0.4
0.4
1.3
1.0
1.5
1.3
1.0
Q1r
r
L
Asymptotically optimal if L  c log r
(and finite exponential moments)
r
Parallel server example
Threshold policy: W ‘00
1
2
3
1
2
3
Threshold policy:
4
Suppose i*= 4
Then k*=3
3
T
Server 3 is the root.
3
2
Buffer 4 has lowest priority.
Place thresholds on transition
activities. Servers give priority to
2
transition activities below them in
the tree, except suspend such an activity
T
when the associated buffer is below
1
threshold. Next priority goes to
non-transition activities below server.
Lowest priority to activities above server.
1
4
Asymptotic optimality of tree based
threshold policy
• Theorem (Bell-W, ’04)
Assume finite exponential moments.
• If T denotes the threshold control in
the rth system, then for any other
r
sequence of control policies {T }, we
have
r ,*
liminf r  J (T )  J  limr  J (T )
r
r
*
r
r ,*

 t
*

 E  e h Q (t )dt 
 0

where

r
r
J (T )  E   e t h Qˆ r (t )dt 
 0

Complete Resource Pooling (CRP)
CRP: workload is one-dimensional and non-negative.
Solution of the Brownian control problem “keep all of
the work in the buffer i* with the smallest ratio of
holding cost hi* to workload contribution yi*, and
only allow server idling when the whole system is
empty.
How should one interpret this solution?
Parallel server system: P=0
• Harrison-Lopez ‘99:
– CRP iff all servers communicate via basic activities
– proposed discrete review policy (proof of asymptotic
optimality – special two server case: Harrison ’98)
• W. ‘00 (see also Squillante, Xia, Yao, Zhang ‘00):
– CRP iff server-buffer graph with edges given by basic
activities is a tree
– proposed threshold policy (continuous review)
• Bell-W. ’01, ‘04
– Proof that threshold policy of W. ‘00 is asymptotically
optimal (two server (AAP) & multiserver (in prep.))
General network with CRP: 0  P
– S. Kumar ‘99: proposed discrete review policy (example)
– Ata-Kumar ’04: discrete review policy and proof of asymptotic
optimality
– Kang-W. (in prep.): proposed threshold policy (continuous
review)