Overflow Queueing Networks: Limiting Traffic Equations, Trajectories and Sojourn Times

Stijn Fleuren, Yoni Nazarathy, Erjen Lefeber

Open Problem SessionEURANDOM

October 28, 2010

* Supported by NWO-VIDI Grant 639.072.072

Overview• Overflow queueing networks• Large buffer fluid scaling• Limiting: traffic equations, trajectories, sojourn times

(conjectures 1, 2, 3)• Some items for discussion (problem session):

– Related work? Where to take this?– Approaches for the limit proofs?– Generalizing DPH distributions?– “Almost Discrete” Sojourn Time Phenomenon

Disclaimer: Conjectures 1,2,3 are rough…

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

1

1

'

( ')

M

i i j j ij

p

P

I 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

1

1M

i jij

p p

Problem Data:

Assume: open, no “dead” nodes

Modification: Finite Buffers and Overflows Wolff, 1988, Chapter 8 & references there in & after

ii

Exact Traffic Equations:

i jp

M

1

1M

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)

When K is Big, Things are “Simpler”

out rate overflow rate ( )

for big,K

Large Buffer Fluid Scaling

N

N

N

N

N K

1,2,...N

And maybe scale space and initial conditions when needed

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

Properties of the Limiting Traffic Equations

Proposition: Unique solution exists under certain non-singularity assumptions of P and Q

Proposition: An algorithm in at most iterations (as opposed to )

Conjecture 1: Under general processing time and arrival assumptions

lim N

N

' '( )P Q

2M2M

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

Conjecture 2: Under general assumptions,

( )lim sup ( ) 0

N

tN

X tx t

N

( )x t

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

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!!!

Conjecture 3: If, ,

1,Ns s sS DPH T 1,i

i

K i F

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

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

“Almost Discrete” Sojourn Time PhenomenonTaken from seminar of Avi Mandelbaum, MSOM 2010 (slide 82):

Discussion– Related work? Where to take this?– Approaches for the limit proofs?– Generalizing DPH distributions?– “Almost Discrete” Sojourn Time Phenomenon:

modeling call-centers using overflow networks and variations