Exact solutions for first-passage and related problems in certain classes of queueing system
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Transcript of 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
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b b c c d
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d
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y probabilit
(service) Departure
y probabilit Arrival
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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
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b b c c d
f
e
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d
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condition Critical
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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
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2/1
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~~ time waitingCumulative
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The busy period (first passage time)The busy period (first passage time)
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Moments and ‘defectiveness’Moments and ‘defectiveness’
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spolynomial
Legendrefor identity function generating a Using
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The maximum (extreme) lengthThe maximum (extreme) length
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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
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Two important consequencesTwo important consequences
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momentfirst For the
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Mapping onto staircase polygons – Mapping onto staircase polygons – the area problemthe area problem
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Arrivals
Departures
A functional equationA functional equation
others and (1991) Takacs (1919),Ramanujan following
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polygons staircase onto mapping on the Based
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function generatingparameter - twoa Introduce
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thatfollowsIt
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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
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31
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0
and – functionsAiry
of in terms solved becan – which equation (Ricatti)
aldifferentilinear -non a obeys )(function scaling The
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The Brownian motion approachThe Brownian motion approach
0
10
20
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0 25 50 75 100
Time
Displacement
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The area distributionThe area distribution
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Time
Displacement
line critical near the )(8
where
2~)(
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function generating theproblem discrete For the
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ˆ ˆ 1ˆ AA
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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