Queueing models with admission and termination control

198 Pages·2017·0.93 MB·English

Queueing models with admission and termination control :
monotonicity and threshold results
Citation for published version (APA):
Brouns, G. A. J. F. (2003). Queueing models with admission and termination control : monotonicity and
threshold results. [Phd Thesis 1 (Research TU/e / Graduation TU/e), Mathematics and Computer Science].
Technische Universiteit Eindhoven. https://doi.org/10.6100/IR570263
DOI:
10.6100/IR570263
Document status and date:
Published: 01/01/2003
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Queueing models with admission
and termination control
Monotonicity and threshold results
CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN
Brouns, Gido A.J.F.
Queueing models with admission and termination control | Monotonicity
and threshold results / by Gido A.J.F. Brouns. - Eindhoven : Technische
Universiteit Eindhoven, 2003.
Proefschrift. - ISBN 90-386-0732-6
NUR 919
Keywords: Markov decision processes / dynamic programming /
queueing theory
2000 Mathematics Subject Classiflcation: 90C40, 90C39, 60K25, 90B22
Printed by Universiteitsdrukkerij Technische Universiteit Eindhoven
Copyright (cid:176)c 2003 by Gido A.J.F. Brouns, The Netherlands
All rights reserved
Queueing models with admission
and termination control
Monotonicity and threshold results
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
Rector Magniflcus, prof.dr. R.A. van Santen, voor
een commissie aangewezen door het College voor
Promoties in het openbaar te verdedigen op
donderdag 30 oktober 2003 om 16.00 uur
door
Gido Antonius Johannes Fransiscus Brouns
geboren te Amsterdam
Dit proefschrift is goedgekeurd door de promotoren:
prof.dr.ir. J. van der Wal
en
prof.dr. K.M. van Hee
\I hope I can make it across the border. I
hope to see my friend and shake his hand.
I hope the Paciflc is as blue as it has been
in my dreams.|||I hope."
{Red, The Shawshank Redemption

Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Onset of a mathematical model formulation . . . . . . . . . . 3
1.3 Queueing terminology . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Dynamic control of queues . . . . . . . . . . . . . . . . . . . . 8
1.5 Outline of literature on the dynamic control of queues . . . . 10
1.6 Termination control . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 21
2 Two routing control problems 25
2.1 Literature on routing to parallel queues . . . . . . . . . . . . 26
2.2 Model description Model I . . . . . . . . . . . . . . . . . . . . 28
2.3 Main results for Model I . . . . . . . . . . . . . . . . . . . . . 31
2.4 Model description Model II . . . . . . . . . . . . . . . . . . . 42
2.5 Main results for Model II . . . . . . . . . . . . . . . . . . . . 44
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
vii
viii Contents
3 An MjE j1 queue with admission and termination control 51
N
3.1 Model description. . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Proof of the Key Proposition . . . . . . . . . . . . . . . . . . 60
3.4 Inflnite time horizon . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Convex rewards . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Extensions of the MjE j1 model 77
N
4.1 Extension to batch Poisson arrivals . . . . . . . . . . . . . . . 78
4.2 Extension to Erlang arrivals . . . . . . . . . . . . . . . . . . . 80
4.3 Extension to Markov feed-forward routing . . . . . . . . . . . 85
4.4 Translation to deterministic decision epochs . . . . . . . . . . 95
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5 A multi-server extension of the MjE j1 model
N
Computational issues and a near-optimal heuristic 105
5.1 Model description. . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Cutting down on the action space . . . . . . . . . . . . . . . . 109
5.3 Properties of the optimal policy . . . . . . . . . . . . . . . . . 112
5.4 Optimal control versus heuristics . . . . . . . . . . . . . . . . 118
5.5 The MjE „ij1 model . . . . . . . . . . . . . . . . . . . . . . . 120
N
5.6 Description of the heuristic . . . . . . . . . . . . . . . . . . . 122
5.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.8 Comparison of two heuristics via simulation . . . . . . . . . . 136
5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Contents ix
6 A two-class preemptive priority queue with admission
and termination control 143
6.1 Model description. . . . . . . . . . . . . . . . . . . . . . . . . 144
6.2 Overview of the results . . . . . . . . . . . . . . . . . . . . . . 149
6.3 Proof of the Key Proposition . . . . . . . . . . . . . . . . . . 153
6.4 General multi-server model . . . . . . . . . . . . . . . . . . . 161
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7 Conclusions and outlook 171
7.1 Design and control of work(cid:176)ow processes . . . . . . . . . . . . 171
7.2 Structural results for the dynamic control of queueing
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
References 177
Samenvatting (Summary in Dutch) 183
About the author 187