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.