Discrete-event control of stochastic networks: multimodularity and regularity
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2003]
|
| Series: | Lecture notes in mathematics
1829 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XIV, 313 Seiten Illustrationen |
| ISBN: | 3540203583 |
| ISSN: | 0075-8434 |
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MARC
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| 100 | 1 | |a Altman, Eitan |d 1959- |0 (DE-588)171173171 |4 aut | |
| 245 | 1 | 0 | |a Discrete-event control of stochastic networks |b multimodularity and regularity |c Eitan Altman ; Bruno Gaujal ; Arie Hordijk |
| 246 | 1 | 3 | |a Discrete event control of stochastic networks |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c [2003] | |
| 264 | 4 | |c © 2003 | |
| 300 | |a XIV, 313 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1829 |x 0075-8434 | |
| 650 | 4 | |a Control theory | |
| 650 | 4 | |a Discrete-time systems | |
| 650 | 4 | |a Queuing theory | |
| 650 | 4 | |a Stochastic analysis | |
| 650 | 4 | |a Diskretes Ereignissystem - Stochastische Kontrolltheorie - Kombinatorische Wahrscheinlichkeitstheorie | |
| 650 | 4 | |a Control theory | |
| 650 | 4 | |a Discrete-time systems | |
| 650 | 4 | |a Queuing theory | |
| 650 | 4 | |a Stochastic analysis | |
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| adam_text |
Table
of
Contents
Introduction
. 1
Part
I Theoretical foundations
1
Multimodularity, Convexity and Optimization
. 11
1.1
Introduction
. 11
1.1.1
Organization of the chapter
. 12
1.2
Properties of multimodular functions
. 12
1.2.1
General properties
. 13
1.2.2
Multimodularity and convexity
. 14
1.3
The optimality of bracket policies for a single criterion
. 21
1.3.1
Upper Bounds
. 22
1.3.2
Lower Bounds
. 24
1.3.3
Optimality of the Bracket Sequences
. 25
1.4
The optimality of bracket policies for multiple criteria
. 26
1.5
Application of the optimization theorems
. 29
1.5.1
Applications in high-speed telecommunication systems
29
1.6
Clustering versus Smoothing
. 34
1.7
Appendix A: proof of Lemma
2 . 34
1.8
Appendix B: Proof of Lemma
5 . 36
1.9
Appendix C: average and weighted costs
. 37
2
Balanced Sequences
. 39
2.1
Introduction
. 39
2.1.1
Organization of the chapter
. 39
2.2
Balanced sequences and bracket sequences
. 39
2.3
Constant gap sequences
. 42
2.4
Characterization of balanced sequences
. 45
2.5
Balanceable rates
. 46
2.5.1
The case A'
= 2. 46
2.5.2
The case A"
= 3. 47
2.5.3
The case A'
= 4. 48
2.5.4
The general case
. 49
2.5.5
Extensions of the original problem
. 52
2.
G
Appendix
. 53
Table
of Contents
Stochastic Event Graphs
. 55
3.1
Introduction
. 55
3.1.1
Organization of the chapter
. 55
3.2
Stochastic
Petri
Nets
. 55
3.2.1
Properties
. 59
3.2.2
Event graphs
. 59
3.3
Dynamics of Event Graphs
. 60
3.3.1
State variables: the firing epochs
.
GO
3.3.2
The (max,plus) Semi-ring
.
G2
3.3.3
Evolution Equation in the (inax.-f) semi-ring
.
G3
3.4
Queuing networks
.
G5
3.4.1
The G/G/l queue
. 65
3.4.2
Queues in tandem
. 66
3.4.3
Kanban
systems
. 68
3.4.4
Window flow control
. 70
3.4.5
Leaky buckets
. 72
3.5
Liudley's equation for (max.+ j systems
. 73
Part II Admission and routing control
4
Admission control in stochastic event graphs
. 79
4.1
Introduction
. 79
4.1.1
Organization of the chapter
. 80
4.2
Mult ¡modularity and admission control
. 80
4.2.1
Admission policy: the time slot approach
. 80
4.3
The FIFO queue
. 81
4.3.1
Coupling of the service times with the customers
. 82
4.3.2
Multimodularity
. 85
4.4
(max.-b) systems with one input: multmiodularity
. 88
4.5
A dual policy: counting variable and waiting time
. 92
4.5.1
Waiting time
. 92
4.5.2
Coupling
. 92
4.5.3
Multimodularity
. 93
4.6
Optimal admission sequence
. 94
4.6.1
Dual bracket sequences
. 95
4.6.2
The time slot approach
. 96
4.6.3
The counting approach, the bounded case
. 97
4.6.4 1
he unbounded case
. 99
5
Applications in queuing networks
. 105
5.1
Introduction
. 105
5.2
'lopological assumptions
. 106
5.3
Stochastic assumptions
. 107
Table
of Contents XI
5.3.1
Independence between service times and inter-arrival
times
.108
5.3.2
Cross traffic
.109
Optimal routing
.
Ill
6.1
Introduction
.
Ill
6.1.1
Organization of the chapter
.112
6.2
Routing of customers in multiple queues
.112
6.2.1
Presentation of the model
.112
6.2.2
Optimal routing sequence
.113
6.3
Study of some special cases
.117
6.3.1
The case
К
= 2.117
6.3.2
The homogeneous case
.117
6.3.3
Two sets of identical servers
.118
Optimal routing in two deterministic queues
.119
7.1
Introduction
.119
7.2
Bracket words
.120
7.3
Expansion in continued fractions
.121
7.4
Average waiting time in a single queue
.125
7.4.1
Jumps
. 125
7.4.2
Formula of the average waiting time
. 127
7.4.3
Properties
. 132
7.5
Average waiting time for two queues
. 134
7.5.1
Presentation of the Model
.134
7.5.2
Optimal Policies
.134
7.5.3
Optimal Ratio
.135
7.5.4
Algorithm and computational issues
.138
7.6
Numerical experiments
.141
7.7
Appendix: proof of Theorem
29.142
7.8
Appendix: proof of Lemma
37.146
7.9
Appendix
7.9 :
proof of Theorem
30 .147
Part III Several extensions
8
Networks with no buffers
. 155
8.1
Introduction
. 155
8.1.1
Organization of the chapter
. 156
8.2
The admission into a single server
. 156
8.3
Properties of the cost and of policies
. 158
8.4
Time averages
. 161
8.5
Routing to several servers
. 162
8.6
The service assignment problem
. 163
8.7
The multimodularitv
. 164
XII Table of
Contents
8.8
MDP
formulation and new structural results
.166
8.9
Multimodularity of the global cost: two servers
.172
8.10
Examples of arrival processes
.174
8.11
Robot scheduling for web search engines
.179
9
Vacancies, service allocation and polling
.183
9.1
Introduction
.183
9.1.1
Organization of the chapter
.183
9.2
The generic control models and main results
.184
9.3
A single queue with service driven vacations
.186
9.4
An arrival-driven vacation model
.191
9.5
Arrival-driven polling model
.195
9.6
The potential vacation times are a renewal process
.196
9.6.1
A single queue
.196
9.6.2
The polling control problem
.199
9.7
1-gatod service
.199
9.7.1
A single queue
.200
9.7.2
The case of several queues
.201
9.7.3
Application to an ATM switch
.202
10
Monotonicity
of feedback control
.205
10.1
Introduction
.205
10.1.1
Organization of the chapter
.206
10.2
Monotonieity in initial actions
.206
10.3
Exogenous random variables and information patterns
.208
10.4
Monotonicity: full information case
.211
10.5
Monotonicity: general case and delayed information
.212
10.6
State representation
.213
10.6.1
General full information case
.214
10.6.2
Monotonicity of the switching curves
.216
10.6.3
State representation with delayed information structure
216
10.7
Relation between multimodularity. superconvexity and
submodularity
.218
10.8
Appendix: Key Lemma
.220
Part IV Comparisons
11
Comparison of queues with discrete-time arrival processes
229
11.1
Introduction
.229
11.2
On the Ross conjecture in discrete-time
.231
11.3
A comparison lemma and its applications
.232
11.3.1
Application
1:
two i.i.d. MAP-sources perform better
than
2
completely coupled Map-sources
.235
Table of
Contents XIII
11.3.2
Application
2:
а
fixed batch size is better than
random batch sizes
.235
11.3.3
Application
3:
Fluid scaling improves the performance 23G
11.4
A second comparison lemma
.237
11.5
A stochastic lower bound on the traveling times
.238
12
Simplex convexity
.243
12.1
Introduction
.243
12.1.1
Organization of the chapter
.243
12.2
Multimodular
Triangulations
.244
12.3
Multimodular Functions
.247
12.4
Sub-meshes
.251
12.5
Cones
.253
12.5.1
Minimization
.254
12.5.2
Cone ordering and monotonicity
.255
12.
ΰ
Application: periodic admission sequences in G/G/l/oc
tandem queues
.257
12.6.1
An example
.259
13
Orders and bounds for multimodular functions
.
2(Jl
13.1
Introduction
.201
13.2
The multhnodular order and tin1 coin- order
.
2t>2
13.2.1
Shift invariant counterparts
.
2(J(i
13.3
The graph order and the unbalance
.208
13.4
Relations and counterexamples
.272
13.4.1
The shift invariant cone order does not imply the
graph order
.272
13.4.2
The shift invariant inultiinodular order does not
imply the shift invariant cone order
.273
13.4.3
The graph order does not imply the shift invariant
multiniodular order
.274
13.5
Bounding the Difference in Waiting Times for One Queue
. . . 275
13.
ϋ
Routing to Parallel Queues
.278
14
Regular Ordering
.283
14.1
Introduction
.283
14.2
Preliminaries
.285
14.2.1
Gap Sequences
.285
14.2.2
Balanced Sequences
.285
14.2.3
Regularity and
Schur
Convexity
.287
14.3
Application
1:
Maximal Waiting Times in Networks
.289
14.3.1
The D/D/l Model
. 289
14.3.2
Characterization using regular preserving functions
. . . 290
14.3.3
The average4 waiting time is not regular preserving
. 291
14.3.4
The G/G/l Model
. 291
XIV Table of
Contents
14.3.5
The Event Graph
Model
.294
14.3.6
Routing
Problem
.297
14.3.7
Computational Problems
.298
14.3.8
A routing example
.299
14.4
Application
2:
Assignment to queues with no buffer with
redundancy
.299
14.5
Appendix: properties of the gap sequences
.301
14.6
Appendix: relations between regularity and multimodularity
. 303
References
.305
Index
.311
Opening new directions in research in both discrete event dynamic systems as well as
in stochastic control, this volume focuses on a wide class of control and of optimiza¬
tion problems over sequences of integer numbers. This is a counterpart of convex opti¬
mization in the setting of discrete optimization. The theory developed is applied to the
control of stochastic discrete-event dynamic systems. Some applications are admis¬
sion, muting, service allocation and vacation control in queueing networks. Pure and
applied mathematicians will enjoy reading the book since it brings together many
disciplines in mathematics: combinatorics, stochastic processes, stochastic control and
optimization, discrete event dynamic systems, algebra. |
| any_adam_object | 1 |
| author | Altman, Eitan 1959- Gaujal, Bruno 1966- Hordijk, Arie |
| author_GND | (DE-588)171173171 (DE-588)173657397 (DE-588)171331370 |
| author_facet | Altman, Eitan 1959- Gaujal, Bruno 1966- Hordijk, Arie |
| author_role | aut aut aut |
| author_sort | Altman, Eitan 1959- |
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| bvnumber | BV017622626 |
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| classification_tum | MATR 600f |
| ctrlnum | (OCoLC)249451791 (DE-599)BVBBV017622626 |
| dewey-full | 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV017622626 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T04:30:29Z |
| institution | BVB |
| isbn | 3540203583 |
| issn | 0075-8434 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010601431 |
| oclc_num | 249451791 |
| open_access_boolean | |
| owner | DE-824 DE-91G DE-BY-TUM DE-706 DE-703 DE-11 DE-83 DE-188 DE-634 |
| owner_facet | DE-824 DE-91G DE-BY-TUM DE-706 DE-703 DE-11 DE-83 DE-188 DE-634 |
| physical | XIV, 313 Seiten Illustrationen |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Altman, Eitan 1959- (DE-588)171173171 aut Discrete-event control of stochastic networks multimodularity and regularity Eitan Altman ; Bruno Gaujal ; Arie Hordijk Discrete event control of stochastic networks Berlin ; Heidelberg Springer [2003] © 2003 XIV, 313 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1829 0075-8434 Control theory Discrete-time systems Queuing theory Stochastic analysis Diskretes Ereignissystem - Stochastische Kontrolltheorie - Kombinatorische Wahrscheinlichkeitstheorie Diskretes Ereignissystem (DE-588)4196828-1 gnd rswk-swf Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd rswk-swf Stochastische Kontrolltheorie (DE-588)4263657-7 gnd rswk-swf Diskretes Ereignissystem (DE-588)4196828-1 s Stochastische Kontrolltheorie (DE-588)4263657-7 s Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 s DE-604 Gaujal, Bruno 1966- (DE-588)173657397 aut Hordijk, Arie (DE-588)171331370 aut Erscheint auch als Online-Ausgabe 978-3-540-39705-2 Lecture notes in mathematics 1829 (DE-604)BV000676446 1829 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010601431&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010601431&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Altman, Eitan 1959- Gaujal, Bruno 1966- Hordijk, Arie Discrete-event control of stochastic networks multimodularity and regularity Lecture notes in mathematics Control theory Discrete-time systems Queuing theory Stochastic analysis Diskretes Ereignissystem - Stochastische Kontrolltheorie - Kombinatorische Wahrscheinlichkeitstheorie Diskretes Ereignissystem (DE-588)4196828-1 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd Stochastische Kontrolltheorie (DE-588)4263657-7 gnd |
| subject_GND | (DE-588)4196828-1 (DE-588)4132446-8 (DE-588)4263657-7 |
| title | Discrete-event control of stochastic networks multimodularity and regularity |
| title_alt | Discrete event control of stochastic networks |
| title_auth | Discrete-event control of stochastic networks multimodularity and regularity |
| title_exact_search | Discrete-event control of stochastic networks multimodularity and regularity |
| title_full | Discrete-event control of stochastic networks multimodularity and regularity Eitan Altman ; Bruno Gaujal ; Arie Hordijk |
| title_fullStr | Discrete-event control of stochastic networks multimodularity and regularity Eitan Altman ; Bruno Gaujal ; Arie Hordijk |
| title_full_unstemmed | Discrete-event control of stochastic networks multimodularity and regularity Eitan Altman ; Bruno Gaujal ; Arie Hordijk |
| title_short | Discrete-event control of stochastic networks |
| title_sort | discrete event control of stochastic networks multimodularity and regularity |
| title_sub | multimodularity and regularity |
| topic | Control theory Discrete-time systems Queuing theory Stochastic analysis Diskretes Ereignissystem - Stochastische Kontrolltheorie - Kombinatorische Wahrscheinlichkeitstheorie Diskretes Ereignissystem (DE-588)4196828-1 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd Stochastische Kontrolltheorie (DE-588)4263657-7 gnd |
| topic_facet | Control theory Discrete-time systems Queuing theory Stochastic analysis Diskretes Ereignissystem - Stochastische Kontrolltheorie - Kombinatorische Wahrscheinlichkeitstheorie Diskretes Ereignissystem Kombinatorische Wahrscheinlichkeitstheorie Stochastische Kontrolltheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010601431&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010601431&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT altmaneitan discreteeventcontrolofstochasticnetworksmultimodularityandregularity AT gaujalbruno discreteeventcontrolofstochasticnetworksmultimodularityandregularity AT hordijkarie discreteeventcontrolofstochasticnetworksmultimodularityandregularity AT altmaneitan discreteeventcontrolofstochasticnetworks AT gaujalbruno discreteeventcontrolofstochasticnetworks AT hordijkarie discreteeventcontrolofstochasticnetworks |