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 Networks
Jackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991
Problem Data:
, , P
1
Assume: open, no “dead” nodes
Traffic Equations (Stable Case):
M
i
i
pi j
M
pi  1   pi j
j 1
i   i    j p j i
j 1
    P '
  ( I  P ') 1
Traffic Equations (General Case):
i   i     j   j  p j i
M
M
j 1
    P '   
LCP   ( I  P ')  , ( I  P ')

Open Jackson Networks
Jackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991
Problem Data:
, , P
1
Assume: open, no “dead” nodes
Traffic Equations (Stable Case):
M
i
i
pi j
M
pi  1   pi j
j 1
i   i    j p j i
j 1
    P '
  ( I  P ') 1
Product Form “Miracle”:
M
 j
lim P  X 1 (t )  k1 ,..., X M (t )  k M    1 
 
t 
j 1 
j
M
  j

 
 j



kj
Modification: Finite Buffers and Overflows
Problem Data:
K1
 ,  , P, K , Q
1
Assume: open, no “dead” nodes,
no “jam” (open overflows)
qi j
i
Ki
i
pi j
Explicit Solutions:
M
pi  1   pi j
j 1
M
qi  1   qi j
j 1
K M M
Generally No
Exact Traffic Equations:
Generally No
A Practical (Important) Model:
Yes
Our Contribution
(in progress)
Limiting Traffic Equations:
K1
1
qi j
i
Ki
    P '       Q '(   )
i
Efficient Algorithm for Unique Solution:
pi j
M
pi  1   pi j
j 1
M
qi  1   qi j
j 1
K M M
Limiting Deterministic Trajectories
(N )
 X (t )

lim supt 
 x(t )   0
N 
 N

Limiting Sojourn Time Distribution
S (N )  S
P( S  k )  1    T k 1
Interlude: How I got to this problem
Control of queueing networks:
Server 1
Server 2
PUSH

PULL
1
1
PULL
2
PUSH
2

Output process, D(t), asymptotic variance:
E  D (t ) 
lim
t 
t
vs.
Var  D (t ) 
lim
t 
t
2
3
BRAVO
effect for
M/M/1/K
load
When K is Big, Things are “Simpler”
for K big,
out rate    
overflow rate        (   ) 
Scaling Yields a Fluid System
A sequence of systems: N  1, 2,...
 (N )   
 (N )  N 
 (N )  N K
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
 X ( N ) (t )

lim supt 
 x(t )   0
N 
 N

Traffic Equations (at equib. point)
out rate    
overflow rate  (   )
i  i     j   j  p ji     j   j  q ji
M
M
j 1
j 1

or

    P '       Q '(   )
or
LCP ( I  Q ') (  ( I  P ')  ) , ( I  Q ') ( I  P ')
1
1

LCP (Linear Complementarity Problem)
a
M
,G 
M M
LCP (a, G ) :Find z , w 
M
such that,
w  Gz  a,
w  0, z  0,
w ' z  0.
The last (complemenatrity) condition reads:
wi  0  zi  0 and zi  0  wi  0.
Min-Linear Equations as LCP
Find  :     B (   )
    B
0 
0  
(   ) '(    )  0
w  ( I  B) z    ( I  B) 
 
w    , z   
z  0, w  0
w' z  0
LCP(  ( I  B)  , I  B)
Existence, Uniqueness and Solution
Definition: A matrix, G 
M M
is a "P"-matrix if the
determinants of all (2n  1) principal submatrices are positive.
Theorem (1958): LCP(a, G) has a unique solution
for all a 
M
if and only if G is a "P"-matrix.
"P"-matrix means that the complementary cones "parition"
Immediate naive algorithm
with 2M steps
1 0  w1   g11
0 1   w    g

  2   21
g12   z1   a1 

g22   z2   a2 
 0
 
1
We essentially assume that our
1
matrix ( G  ( I  Q ') ( I  P ') )
is a “P”-Matrix
We have an algorithm
(for our type of G)
taking M2 steps
n
C{1,2}
C{2}
1 
 
0
 g 
 11 
  g 21 
C
 g 
 12 
 g22 
a 
 1
a2 
C{1}
Sojourn Time  Time in system of customer arriving
to steady state FCFS system
S ( N )  Sojourn time of customer in N'th scaled system
We want to find the limiting distribution of S
(N )
Sojourn Times Scale to a Discrete Distribution!!!
P  S (N )  x
x
“Molecule” Sojourn Times
F  {1,..., s}
i  i for i  F
F  {s  1,..., M }
i  i for i  F
Observe,
time through i  F 
For job at entrance of buffer
iF
w. p. 
:
Ki
i
time through i  F  0
i
enters buffer i
i
 
w. p.  1  i
 i

 qij routed to entrace of buffer j

 
w. p.  1  i
 i

 q i leaves the system

A job at entrance of buffer
iF:
A “fast” chain and “slow” chain…
routed almost immediately according to
P
The “Fast” Chain and “Slow” Chain
Example: M
 4
,
4

 1,  i 1,
1
2
i 1
K1
K2
F  {1, 2}, F  {3, 4}
4
 a
4
p
start
j 1
4
 a
j 1
j
1
1
ji
:
1’
1
1j
a j1
 j 1 1 j j 0
1   q1
 4 1 
p1   p1 j a j 0
j 1
4
p
“Fast” chain
on {0, 1, 2, 1’, 2’, 3’, 4’}:
ai j  Absorbtion probability
in j {0,1, 2} starting in i'
j 1
2’
 1 
1   q1i
 1 
1j
a j2
0
2
3’
“Slow” chain on {0, 1, 2}
DPH distribution (hitting time of 0)
transitions based on “Fast” chain
p4 i
p4
4’
E.g: Moshe Haviv (soon) book: Queues, Section on “Shortcutting states”
The DPH Parameters (Details)
F  {1,..., s}, F  {s  1,..., M }
“Fast” chain
BM s
 1
0 
 1







s 
 0
s 


0M s s


CM  M
 1
1  
1







0 M  s s



0



0 M  M  s   Q   0 M  s s M  s   P

I M s 


1 s

s


AM s  ( I  C ) 1  B
“Slow” chain
Tss
  I s
0s  M  s   P  A
 s1 
1
M

j 1
S ~ DPH (Tss ,1s )
T  A
j
P( S  k )  1    T k 1s1
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.