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

λ α λ

λ α λ

λ α

=

−

= +

= +

= −

∑

, ,M M M MPµ α ×

( )( )

( )

1

'

( ') , ( ')

M

i i j j j ij

p

P

LCP I P I P

λ α λ µ

λ α λ µ

α µ

=

= + ∧

= + ∧

− − −

∑

iµiα

Traffic Equations (Stable Case):

Traffic Equations (General Case):

i jp

1µ

Mµ

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)( )Bλ γ λ µ= + ∧

00( ) '( ) 0

Bλ γ δδ λδ µ

λ δ µ δ

= +≤ ≤≤ ≤− − =

,w zλ δ µ δ= − = −

( ( ) , )LCP I B I Bγ µ− − −

δ λ µ= ∧Find :λ

Classic Product Form Results Jackson 1957, Goodman & Massey 1984

( )( )

( )

1

'

( ') , ( ')

M

i i j j j ij

p

P

LCP I P I P

λ α λ µ

λ α λ µ

α µ

=

= + ∧

= + ∧

− − −

∑

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

iµiαExact Traffic Equations:i jp

Mµ

11

M

i jij

p p=

= −∑

Problem Data:

, , , ,M M M M M M MP K Qµ α × ×

Explicit Solutions:

Generally NoiK

MK1

1M

i jij

q q=

= −∑

i jq

1µ1K

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 λ µ≈ ∧overflow rate ( )λ λ µ λ µ +≈ − ∧ = −

N

N

N

NN K

α α

µ µ

= Ν

=

Κ =

1,2,...N =

For K big:

Limiting Traffic Equations

( ) ( )1 1

M M

i i j j ji j j jij j

p qλ α λ µ λ µ+

= =

= + ∧ + −∑ ∑

limiting out rate λ µ= ∧

limiting overflow rate ( )λ µ += −

( )' '( )P Qλ α λ µ λ µ += + ∧ + −

or

( )1 1( ') ( ( ') ) , ( ') ( ')LCP I Q I P I Q I Pα µ− −− − − − −

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> ⇒ = > ⇒ =

It’s all about Choosing a Subset…For {1,..., } denote by ( ) a matrix withcollumns taken from (identity matrix)and collumns {1,..., } \ taken from .

n BI

n M

α αα

α

⊆

−

is about finding and 0such that

( )In this case:

LCP x

B x q

α

α

≥

=

0, .

0i

i ii

ix iw z

x iiαααα

∈∈ = = ∉∉

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

α

α

α

α

+ =

− + = −

− + = −

− − + = − −

=

=

=

=∅

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

qq

{1}C

{2}C

{ : ( ) , 0}C y y B u uα α= = ≥

C∅

Immediate naïve 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• Lemeke’s 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 xη η= + −

( ) ( ) 0Q x Q xη − ≥ ( ' ) '( ) ( ) ' ( )

2c Dx x x x x D x xη−+ − ≥ − −

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,η< <Let be feasible.x

( ' ) '( ) 0c Dx x x+ − ≥

( ' ) ' ( ' ) 'c Dx x c Dx x+ ≥ +

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 i

i

Kµ

∈ ≈

{1,..., }

{ 1,..., }

F s

F s M

=

= +

i i

i i

for i S

for i S

λ µ

λ µ

> ∈

< ∈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

µλ

µλ

µλ

≈

≈ −

≈ −

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,Ns s sS DPH T τ× ×⇒1,i

i

K i Fµ = ∈

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

αµ µ

=

∑= = ==

= =

11

1

1 iqµλ

−

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

µλ

11

1

1 qµλ

−

4 ip

4

1j ji

jaα

=∑

4

01

j jj

aα=∑

DPH distribution (hitting time of 0)transitions based on “Fast” chain

The DPH Parameters (Details)

1~ ( , )s s sS DPH T τ× ×

{1,..., }, { 1,..., }F s F s M= = +

1P( ) 1 1ksS k Tτ ×≤ = − ⋅ ⋅

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

µλ

µλ

× −× × − ×

−

− ×

−

= ⋅ + ⋅ −

1

10

0

0

M ss

s

M s s

B

µλ

µλ

×

− ×

=

1( )M sA I C B−× = − ⋅

0s s s s M sT I P A× × − = ⋅ ⋅ 1

1

1 Ts M

jj

Aτ αα

×

=

= ⋅

∑

“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.