Exact solutions for first-passage and related problems in certain classes of queueing system

Exact solutions for first-passage and Exact solutions for first-passage and related problems in certain classes of related problems in certain classes of

queueing systemqueueing system

Michael J KearneySchool of Electronics and Physical Sciences

University of Surrey

June 29th 2006

Presentation outlinePresentation outline

Introduction to the Geo/Geo/1 queue Some physical examples Mathematical analysis

– Link to the Brownian motion problem

Further problems

Queueing schematicQueueing schematic

Buffer Server

Service protocol - First come, first served

Customers in Customers out

A discrete-time queueing systemA discrete-time queueing systemGeo/Geo/1Geo/Geo/1

a ddcbbaa

b b c c d

f

e

e

d

t

Nt

0P

P

P

y probabilit

(service) Departure

y probabilit Arrival

0

10

20

1 50

time

occupancy

Small scale queue dynamics

0

10

20

30

40

50

60

70

80

90

100

1 2000

Large scale queue dynamics

0

10

20

30

0 25 50 75 100

Time

Displacement

Brownian motion with drift

Some questions of interestSome questions of interest

Time until the queue is next empty– Busy period (first passage time) statistics– Probability that the busy period is infinite

Maximum queue length during a busy period– Extreme value statistics (correlated variables)

Cumulative waiting time during a busy period– Area under the curve

a ddcbbaa

b b c c d

f

e

e

d

t

Nt

condition Critical

17,3,10 ALT

Areas of applicationAreas of application

Abelian sandpile model Compact directed percolation Lattice polygons Cellular automaton road traffic model

Nagel and Paczuski (1995)

The link to road trafficThe link to road traffic

a e d c b

time

a a a a b b c d

d c c b b

e d

d c

e e

time t

N(t)

Cellular automaton model

Queueing representation

The critical scalingsThe critical scalings

3/43/1

0

2/32/1

21

2/1

2/32/1

~)(~)(

~~ time waitingCumulative

~)(~)(

~length queue Maximum

~)(~)(

periodbusy The

aaAPaaAP

TdttA

llLPllLP

TL

ttTPttTP

T

The busy period (first passage time)The busy period (first passage time)

pathMotzkin dissipatesit before queue

1-length a becomefirst must queue 2-length a that noting

)()1()1()(

3For

)2(;)1(

thusone,y probabilit with 0at happens arrival initial The

2

10

0

t

k

kTPktTPPtTPPtTP

T

PPTPPTP

t

xP

xPPxPxPxG

xxGPxxGPxPxG

xtTPxGt

t

2

4)1(1)(

)()()(

obtains then One

)()(

function generating theDefine

2200

20

Moments and ‘defectiveness’Moments and ‘defectiveness’

when

when )/(1/)(

oned)(unconditimoment first For the

when 1)(; when 0)(

)(lim1)(

is occuring queue infinitean ofy probabilit The

1

1

T

dxxdGT

PPTPTP

xGTP

x

x

The probability distributionThe probability distribution

polynomial Legendreth - theis )( and

1)/41(

where

)()()1(2

)(

spolynomial

Legendrefor identity function generating a Using

2/120

11

10

tzL

PPP

LLtP

PtTP

t

ttt

t

tttTP

t

zz

tO

e

e

t

tzL

t

z

zt

t

4exp

4

1~)(

as

line critical near the that so ,0)(cosh)( where

11

)1()1(

)(1)(

large andinteger For

2

2/3

1

2/1)(2

)()1(21

The maximum (extreme) lengthThe maximum (extreme) length

0

10

20

0 25 50 75 100

Time t

Len

gth

x(t

)

Maximum length L

Lifetime T

0)1,( and 1)0,( subject to

)1,(),()1,(),(

recursion backward simple a has oneThen

)(),( i.e. ,length fixed

somen larger tha be togrowsnever ,0length initial

whosequeue, ay that probabilit thebe ),(Let

}....,,2,1,0:)(max{ definitionBy

0

llPlP

nlPPnlPPnlPPnlP

nlLPnlPl

n

nlP

TttNL

1;)1(

1)(

1;)1)(1(

)1()(

that)1,1()1,()( since followsIt

1;1

1),(

1;1

1),(

1 and /for that so simple, issolution The

1

21

1

1

lllLP

lLP

nlPnlPlLP

l

nnlP

nlP

lPP

ll

l

l

nl

Two important consequencesTwo important consequences

O

llLlP

llLP

llLP

lll

l

l

1ln~L

)1)(1()1( )(L

momentfirst For the

exp~)(

line critical Near the .~)( line critical On the

11

12

1

2

2

Mapping onto staircase polygons – Mapping onto staircase polygons – the area problemthe area problem

a ddcbbaa

b b c c d

f

e

e

d

t

Nt

(D)

(A)

111 11 100000

1 1 11 1 1 00000

(D)

(A)

111 11 100000

1 1 11 1 1 00000

a

c

bb

aa

b b

c c

dddd

ee

f

(i)

(ii)

y

x

Arrivals

Departures

A functional equationA functional equation

others and (1991) Takacs (1919),Ramanujan following

),(),(),(),(

polygons staircase onto mapping on the Based

),(),(

function generatingparameter - twoa Introduce

0

,

yxyGyxxyGPyxxyGPxyPyxG

yxaAtTPyxGat

at

for )(

),1(A

although obtain, harder tomuch is ),1(However

before as )1,()1,()1,(

),()1,(

thatfollowsIt

21

20

,

y

at

t

y

yG

yG

xxGPxxGPxPxG

xaAtTPxG

1

0

0

222

2/)3(21 0

2

2/)1(222

)1();(

where

))1(1();();(

)(

);();(

)()(

),(ˆ

n

i

in

m

m

mm

mmmn m mm

mnmmmn

tqqt

xyyxyyyyx

yx

yyyyx

yyxyxx

yxG

Three-fold strategyThree-fold strategy

A scaling approach based on the dominant balance method, following Richard (2002)

Consider the singularity structure of the generating function G(1,y) as y tends to unity, following Prellberg (1995)

Consider the equivalent problem for Brownian motion, following Kearney and Majumdar (2005)

The scaling approachThe scaling approach

32

31

2

0

and – functionsAiry

of in terms solved becan – which equation (Ricatti)

aldifferentilinear -non a obeys )(function scaling The

)1()1(~),(

1 and 0for form scaling a

),(),(),(),(

provided

F

y

xFyyxG

yAssume

yxyGyxxyGPyxxyGPxyPyxG

The The qq-series approach-series approach

)(Li2)(Li2ln)ln(4

3

where

))1((Ai

))1((iA)1(

2~),1(

esn techniquintegratiocontour using

),1(for tion representa series- formal a ngmanipulatiBy

22

3/23/2

3/23/23/1

3/1

y

yy

PyG

yGq

The Brownian motion approachThe Brownian motion approach

0

10

20

30

0 25 50 75 100

Time

Displacement

TsA

T

tdtxsdAexAPxsP

xtxtdtxAtdWdttdx

00

00

0

0

)(exp),(),(~

transformLaplace for theThen

)0(;)();()(

equationLangevin heConsider t

)2(Ai

)22(Ai),(

~

solution thehaswhich

0),(~),(

~),(

~

2

1

equationPlanck -Fokker backward a

derivecan one step lincrementaan gconsiderinBy

3/23/22

3/23/220

3/13/1

0

000

02

0

02

0

s

sxsexsP

xsPsxx

xsP

x

xsP

x

The area distributionThe area distribution

Ax

x

eA

xxAP

s

sxsexsP

9/23/4

0

313/2

3/1

0

3/23/22

3/23/220

3/13/1

0

30

0

3

2),(

give to0 when inverted becan

)2(Ai

)22(Ai),(

~

transformLaplace The

0

10

20

30

0 25 50 75 100

Time

Displacement

line critical near the )(8

where

2~)(

as that implies

))1((Ai

))1((iA)1(

2~),1(

function generating theproblem discrete For the

2

3

3/13/13/43/1

3/23/2

3/23/23/1

3/1

AfAP

AP

A

y

yy

PyG

3

2exp

)(6

1

2

1~)(

as 3

16exp

3

41~)(

moments theofstudy aon

Based entire. be shown to bemay )(function scaling The

3

1)0(;

2~)(

2/1

2/32/14/3

2/3

4/74/1

2/32/1

4/74/1

313/2

3/13/13/43/1

AAAP

ssssf

sf

fAfAP

AP

ˆ ˆ 1ˆ AA

2/14/1 exp2

1~)ˆPr( xxxA

1

ln)1()1(2ˆ2 2/1

Taking the continuous time limit (but discrete customers)

The M/M/1 queueThe M/M/1 queue

Guillemin and Pinchon (1998)

3/1

3/131

3/1

ˆ3

~)ˆPr(

xxA

1

ˆ/ˆ

x y

time0obPr

xpobPr

ypobPr

1obPr

Rules

Compact directed percolationCompact directed percolation

y

x x y

yp xq

Critical condition

0 01 yx pp

Making the connection …

Summary of key CDP resultsSummary of key CDP results

Probability that the avalanches are infinite– critical condition

Distribution of avalanches by duration (perimeter)

Distribution of avalanches by size (area)

1 yx pp 2/1 cpp

)exp(~

ADAPAcpp

APA ~cpp

3/4 2/1 4/3

2/12

0

011

1

)4(;)()(

)1(2

PPP

PLL

TP TTT

T

T

Dhar and Ramaswamy (1989)

Rajesh and Dhar (2005)

0

10

20

30

0 25 50 75 100

Time

Displacement

Brownian motion

00max /1ln~ xxx

0~

xT

20

2~

x

A

ConclusionsConclusions

New results for discrete and continuous-time queues, and possibly deeper results

Large area scaling behaviour for CDP determined exactly at all points in the phase diagram

Exact solution for the v = 1 cellular automaton traffic model of Nagel and Paczuski

A solvable model of extreme statistics for strongly correlated variables

N = 5

T = 7 Time

Queuelength

Time

Departures

Partition polygon queuesPartition polygon queues

N

ii

A

AAN z

zzPzG

1 )1()(),(ˆ

t

t

MMP

t

t

NNP

State dependent queues (balking)State dependent queues (balking)

a ddcbbaa

b b c c d

f

e

e

d

t

Nt

)()()(

1 or

e.g. dependent, state is where

00

ttxdt

tdx

ne n

nn

n

0

10

20

30

0 25 50 75 100

Time

Displacement

Some referencesSome references

On a random area variable arising in discrete-time queues and compact directed percolation– M J Kearney 2004 J.Phys. A: Math. Gen., 37 8421

On the area under a continuous time Brownian motion– M J Kearney and S N Majumdar 2005 J.Phys. A: Math.

Gen., 38 4097

A probabilistic growth model for partition polygons and related structures– M J Kearney 2004 J.Phys. A: Math. Gen., 37 3749