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Finite Buffer Fluid Networks with Overflows
Yoni Nazarathy,Swinburne University of Technology, Melbourne.
Stijn Fleuren and Erjen Lefeber,Eindhoven University of Technology, the Netherlands.
Talk Outline
• Background: Open Jackson networks
• Introducing finite buffers and overflows
–Interlude: How I got to this problem
• Fluid networks as limiting approximations
• Traffic equations and their solution
• Almost discrete sojourn times
Open Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991
1
1
'
( ')
M
i i j j ij
p
P
I P
, ,P
1
'
( ') , ( ')
M
i i j j j ij
p
P
LCP I P I P
ii
Traffic Equations (Stable Case):
Traffic Equations (General Case):
i jp
1
M
1
1M
i jij
p p
Problem Data:
Assume: open, no “dead” nodes
Open Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991
1
1
'
( ')
M
i i j j ij
p
P
I P
, ,P
ii
Traffic Equations (Stable Case):
i jp
1
M
1
1M
i jij
p p
Problem Data:
Assume: open, no “dead” nodes
1 11
lim ( ) ,..., ( ) 1jk
Mj j
M Mt
j j j
P X t k X t k
Product Form “Miracle”:
Modification: Finite Buffers and Overflows
ii
Exact Traffic Equations:
i jp
M
1
1M
i jij
p p
Problem Data:
, , , ,P K Q
Explicit Solutions:
Generally NoiK
MK1
1M
i jij
q q
i jq
11K
Generally No
Assume: open, no “dead” nodes, no “jam” (open overflows)
A Practical (Important) Model:
Yes
Our Contribution (in progress)
ii
Efficient Algorithm for Unique Solution:
i jp
M
1
1M
i jij
p p
Limiting Traffic Equations:
iK
MK1
1M
i jij
q q
i jq
11K
Limiting Sojourn Time Distribution
' '( )P Q
( ) ( )lim sup ( ) 0
N
tN
X tx t
N
Limiting Deterministic Trajectories
P( ) 1 1kS k T
( )NS S
Interlude: How I got to this problem
Output process, D(t), asymptotic variance:
Control of queueing networks:
BRAVO effect for M/M/1/K
( )lim
.
( )lim
t
t
E D t
tvs
Var D t
t
load
23
1 1
22
Server 2Server 1
PUSH
PULL
PULL
PUSH
When K is Big, Things are “Simpler”
out rate overflow rate ( )
for big,K
Scaling Yields a Fluid System( )
( )
( )
N
N
N
N
N K
1,2,...N A sequence of systems:
Make the jobs fast and the buffers big by taking N
The proposed limiting model is a deterministic fluid system:
Fluid Trajectories as an Approximation
( ) ( )lim sup ( ) 0
N
tN
X tx t
N
Traffic Equations (at equib. point)
1 1
M M
i i j j ji j j jij j
p q
out rate
overflow rate ( )
' '( )P Q or
1 1( ') ( ( ') ) , ( ') ( ')LCP I Q I P I Q I P
or
LCP,
( , ) :Find , such that,
,
0, 0,
' 0.
M M M
M
a G
LCP a G z w
w Gz a
w z
w z
The last (complemenatrity) condition reads:
0 0 and 0 0.i i i iw z z w
(Linear Complementarity Problem)
Min-Linear Equations as LCP( )B
0
0
( ) '( ) 0
B
,w z ( ) ( )
0, 0
' 0
w I B z I B
z w
w z
( ( ) , )LCP I B I B
Find :
Existence, Uniqueness and SolutionDefinition: A matrix, is a "P"-matrix if the
determinants of all (2 1) principal submatrices are positive.
M M
n
G
Theorem (1958): ( , ) has a unique solution
for all a if and only if is a "P"-matrix.M
LCP a G
G
{1,2}C
1
0
0
1
12
22
g
g
11
21
g
g
1
2
a
a
{1}C
{2}C
C
"P"-matrix means that the complementary cones "parition" n
Immediate naive algorithm with 2M steps
We essentially assume that our matrix ( ) is a “P”-Matrix
We have an algorithm(for our type of G)taking M2 steps
1( ') ( ')G I Q I P
1 11 12 1 1
2 21 22 2 2
1 0
0 1
w g g z a
w g g z a
Sojourn Time Time in system of customer arriving
to steady state FCFS system
( ) Sojourn time of customer in 'th scaled systemNS N
( )We want to find the limiting distribution of NS
Sojourn Times Scale to a Discrete Distribution!!!
( )NP S x
x
“Molecule” Sojourn Times
time through i F i
i
K
{1,..., }
{ 1,..., }
F s
F s M
i i
i i
for i F
for i F
Observe,
time through i F 0 For job at entrance of buffer :
. . enters buffer i
. . 1 routed to entrace of buffer j
. . 1 leaves the system
i
i
iij
i
ii
i
w p
w p q
w p q
i F
A “fast” chain and “slow” chain…
A job at entrance of buffer : routed almost immediately according toi F P
The “Fast” Chain and “Slow” Chain
1’
2’
3’
4’
1
2
0
4
41 21, 1,
11 2
{1, 2}, {3, 4}
Example: ,
:
M
K K
ii
F F
11
1
1 iq
4p
4
1 011
j jj
p p a
4
1 11
j jj
p a
Absorbtion probability
in {0,1,2} starting in i'
i ja
j
“Fast” chain on {0, 1, 2, 1’, 2’, 3’, 4’}:
“Slow” chain on {0, 1, 2}
start
4
1 21
j jj
p a
1
1
11
1
1 q
4 ip
4
1j ji
j
a
4
01
j jj
a
DPH distribution (hitting time of 0)transitions based on “Fast” chain
E.g: Moshe Haviv (soon) book: Queues, Section on “Shortcutting states”
The DPH Parameters (Details)
1~ ( , )s s sS DPH T
{1,..., }, { 1,..., }F s F s M
1P( ) 1 1ksS k T
1
1
1
00 0
1
0
s M sM M M M s M s
s M s
s
M s s
C Q PI
1
10
0
0
M ss
s
M s s
B
1( )M sA I C B
0s s s s M sT I P A 1
1
1 Ts M
jj
A
“Fast” chain
“Slow” chain
Sojourn Times Scale to a Discrete Distribution!!!
“Almost Discrete” Sojourn Time Phenomena
Taken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82).
Summary
– Trend in queueing networks in past 20 years: “When don’t have product-form…. don’t give up: try asymptotics”
– Limiting traffic equations and trajectories
– Molecule sojourn times (asymptotic) – Discrete!!!
– Future work on the limits.
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