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Finite Buffer Queueing/Fluid Networkswith Overflows

Erjen Lefeber, Yoni Nazarathy.

Swinburne Applied Mathematics Seminar,

April Fools Day, 2011.

* Supported by NWO-VIDI Grant 639.072.072

Background Jackson Networks and LCP

Open Jackson NetworksJackson 1957, Goodman & Massey 1984, Chen & Mandelbaum 1991

1

1

'( ')

M

i i j j ij

p

PI 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

11

M

i jij

p p=

=

Problem Data:

Assume: open, no dead nodes

The Linear Complementarity Problem (LCP)

The last (complemenatrity) condition reads:0 0 and 0 0.i i i iw z z w> = > =

Min-Linear Equations (Using LCP)( )B = +

00( ) '( ) 0

B

= + =

,w z = =

( ( ) , )LCP I B I B

= Find :

Classic Product Form Results Jackson 1957, Goodman & Massey 1984

( )( )

( )

1

'

( ') , ( ')

M

i i j j j ij

p

P

LCP I P I P

=

= +

= +

Assume arrivals are Poisson processes and i.i.d. exponential service durations

Again the Traffic Equations :

Our Contribution:Finite Buffers with Overflows

Modification: Finite Buffers and Overflows Practically important but not as tractable

iiExact Traffic Equations:i jp

M

11

M

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)

Nico van Dijk, 1988. Yes if P=Q.

So scale the system with :

When K is Big, Things are Simpler

out rate overflow rate ( ) + =

N

N

N

NN K

=

=

=

1,2,...N =

For K big:

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

Digression:The Linear Complementarity Problem (LCP)

The Linear Complementarity Problem (LCP)

The last (complemenatrity) condition reads:0 0 and 0 0.i i i iw z z w> = > =

Its all about Choosing a SubsetFor {1,..., } denote by ( ) a matrix withcollumns taken from (identity matrix)and collumns {1,..., } \ taken from .

n BI

n M

is about finding and 0such that

( )In this case:

LCP x

B x q

=

0, .

0i

i ii

ix iw z

x ii

= =

Illustration: n=2

1 0 11 20 1 2

1 12 11 20 22 2

011 11 2121 2

11 12 11 2

21 22 2

{1,2}:

{1}:

{2}:

:

qw w

q

m qw z

m q

m qz w

m q

m m qz z

m m q

+ =

+ =

+ =

+ =

=

=

=

=

1 11 12 1 1

2 21 22 2 2

1 00 1

w m m z qw m m z q

=

{1,2}C

Complementary cones:

10

01

12

22

mm

11

21

mm

1

2

{1}C

{2}C

{ : ( ) , 0}C y y B u u = =

C

Immediate nave algorithm with complexity 3 32 2n nn or n+

Existence and UniquenessDefinition: A matrix, is a P-matrix if thedeterminants of all (2 1) principal submatrices are positive.

n n

n

M

Theorem (1958): ( , ) has a unique solutionfor all if and only if is a P-matrix.n

LCP q Mq M

11 22 11 22 12 21e.g.for 2 : 0, 0, 0n m m m m m m= > > >

P-matrix means that the complementary cones "parition" n

P-Matrixes

Symmetric MatrixesPD Matrixes

Relation of P-matrixes to positive definite (PD) matrixes:

Reminder(PD) :' 0 0x Mx x>

Reminder(PSD) :' 0x Mx x

Computation (Algorithms) Naive algorithm, runs on all subsets alpha (intractable) Generally, LCP is NP complete Lemekes Algorithm, a bit like simplex If M is PSD: polynomial time algorithms exists PD LCP equivalent to QP Special cases of M, linear number of iterations Note: Checking for P-Matrix is NP complete, checking for PD is

polynomial time For our special case we have an algorithm with a quadratic

number of iterations(Still have not done: proven uniqueness using LCP theory).

How does LCP generalize LP and QP?

Linear Programming (LP)

min '. .

0

c xs t Ax b

x

max '. . '

0

b ys t A y c

y

Primal-LP: Dual-LP:

Theorem: Complementary slackness conditions

min '. .

, 0

c xs t Ax b v

x v =

max '. . '

, 0

b ys t u c A y

y u=

Assume , , , are feasible for primaland dual:0, 0 Theyareoptimalsolutionsi i i i

x v y ux u y v= =

0 ',

0c A

LCPb A

0 '0

u A x cv A y b

=

, , , 0u v x y

' 0u x = ' 0v y =

The LCP of LPFind:

Such that:

And (complementary slackness):

Quadratic Programming1min ( ) ' '2

. .0

Q x c x x D x

s t Ax bx

= +

Lemma: An optimizer, , of the QP also optimizes min ( ) '. .

0

c Dx xs t Ax b

x

+

Proof:( )x x x x = +

( ) ( ) 0Q x Q x ( ' ) '( ) ( ) ' ( )

2c Dx x x x x D x x+

x

QP-LP:

QP-LP gives a necessary condition for optimality of QP in terms of an checking optimality of an LP

QP:

0 1,<

The Resulting LCP of QP

',

0c D A

LCPb A

Allows to find suspect points that satisfy the necessary conditions: QP-LP

Theorem: Solutions of this LCP are KKT (Karush-Kuhn-Tucker) points for the QP

Corollary: If D is PSD then x solving the LCP optimizes QP.

Proof: Write down KKT conditions and check.

Note: When D is PSD then M is PSD. In this case it can be shown that the LCP is equivalent to a QP (solved in polynomial time). Similarly, every PSD LCP can be formulated as a PSD QP.

Back To Our Problem:The Fluid Network

Limiting TrajectoriesIn similar spirit to the traffic equations, limiting trajectories, , may be calculated

( )lim sup ( ) 0N

tN

X t x tN

=

( )x t

a.s.

We think:

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 ii

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

Sojourn Times Scale to a Discrete Distribution!!!

We think: ( )1,N s s sS DPH T 1,ii

K i F =

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

ja

=

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 1k

sS 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

Mechanism of nonlinear flow pattern selection in moderately non-Boussinesq mixed convection

Yoni Nazarathy, Sergey Suslov,

John Beynon, William Phillips

Swinburne Applied Mathematics Seminar,

April 1, 2011.

Finite Buffer Queueing/Fluid Networks with OverflowsBackground Jackson Networks and LCPSlide Number 3The Linear Complementarity Problem (LCP)Min-Linear Equations (Using LCP)Slide Number 6Our Contribution:Finite Buffers with OverflowsSlide Number 8Slide Number 9Slide Number 10Digression:The Linear Complementarity Problem (LCP)The Linear Complementarity Problem (LCP)Its all about Choosing a SubsetIllustration: n=2Existence and UniquenessComputation (Algorithms)How does LCP generalize LP and QP?Linear Programming (LP)The LCP of LPQuadratic ProgrammingThe Resulting LCP of QPBack To Our Problem:The Fluid NetworkSlide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Mechanism of nonlinear flow pattern selection in moderately non-Boussinesq mixed convection