finite buffer fluid networks with overflows

Finite Buffer Fluid Networks with Overflows

Yoni Nazarathy,Swinburne University of Technology, Melbourne.

Stijn Fleuren and Erjen Lefeber,Eindhoven University of Technology, the Netherlands.

The University of Sydney,Operations Management and Econometrics Seminar,

July 29, 2011.

“Almost Discrete” Sojourn Time Phenomena

Taken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82).

Outline

• Background: Open Jackson networks• Introducing overflows• Fluid networks as limiting approximations• Traffic equations and their solution• 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 ( ) ,..., ( ) 1jkM

j jM 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:

We say Yes

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 ( ) 0N

tN

X t x tN

Not proved in this current work, yet similar statement appears in a different model (and rigorously proved). Come to 14:00 Stats Seminar, Carslaw 173.

Traffic Equations

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 ww Gz aw zw 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

00( ) '( ) 0

B

,w z ( ) ( )0, 0

' 0

u I B v I Bz ww 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

10

01

12

22

gg

11

21

gg

1

2

aa

{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 G) taking M2 steps

1( ') ( ')G I Q I P

1 11 12 1 1

2 21 22 2 2

1 00 1

w g g z aw g g z a

Back To Sojourn Times….

Taken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82).

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!!!

The “Fast” Chain and “Slow” Chain

1’

2’

3’

4’

1

2

0

4

41 2 1, 1,11 2

{1, 2}, {3, 4}

Example: ,

:

M

K Kii

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!!!

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:

• More standard: E.g. convergence of trajectories (2:00 talk)• Hi-tech (I don’t know how to approach): Weak convergence

of sojourn times (we will leave it as a conjecture for now)

“Molecule” 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