finite buffer fluid networks with overflows yoni nazarathy, swinburne university of technology,...

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.