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Overflow Queueing Networks: Limiting Traffic Equations, Trajectories and Sojourn Times
Stijn Fleuren, Yoni Nazarathy, Erjen Lefeber
Open Problem SessionEURANDOM
October 28, 2010
* Supported by NWO-VIDI Grant 639.072.072
Overview• Overflow queueing networks• Large buffer fluid scaling• Limiting: traffic equations, trajectories, sojourn times
(conjectures 1, 2, 3)• Some items for discussion (problem session):
– Related work? Where to take this?– Approaches for the limit proofs?– Generalizing DPH distributions?– “Almost Discrete” Sojourn Time Phenomenon
Disclaimer: Conjectures 1,2,3 are rough…
Open Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991
1
1
'
( ')
M
i i j j ij
p
P
I P
, ,M M M MP
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
Modification: Finite Buffers and Overflows Wolff, 1988, Chapter 8 & references there in & after
ii
Exact Traffic Equations:
i jp
M
1
1M
i jij
p p
Problem Data:
, , , ,M M M M M M MP 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)
Large Buffer Fluid Scaling
N
N
N
N
N K
1,2,...N
And maybe scale space and initial conditions when needed
Limiting Traffic Equations
1 1
M M
i i j j ji j j jij j
p q
limiting out rate
limiting overflow rate ( )
' '( )P Q or
1 1( ') ( ( ') ) , ( ') ( ')LCP I Q I P I Q I P
or
Properties of the Limiting Traffic Equations
Proposition: Unique solution exists under certain non-singularity assumptions of P and Q
Proposition: An algorithm in at most iterations (as opposed to )
Conjecture 1: Under general processing time and arrival assumptions
lim N
N
' '( )P Q
2M2M
Limiting TrajectoriesIn similar spirit to the traffic equations, limiting trajectories, , may be calculated…
Conjecture 2: Under general assumptions,
( )lim sup ( ) 0
N
tN
X tx t
N
( )x t
Sojourn Times
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
Construction of Limiting Sojourn Times
time through i F i
i
K
{1,..., }
{ 1,..., }
F s
F s M
i i
i i
for i S
for i S
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
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
“Almost Discrete” Sojourn Time PhenomenonTaken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82):