Stability in Adversarial Queueing Theory Panagiot is Tsaparas

67 Pages·1998·3.82 MB·English

Stability in Adversarial Queueing Theory
Panagiot is Tsaparas
,A thesis submitted in conformity with the requirements
for the degree of Master of Science
Graduate Department of Corn puter Science
University of Toronto
@ Copyright by Panagiotis Tsaparas 1997
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Abstract
-4 queueing network is a set of interconnected servers. Customers are injected continu-
ously in the system. inducing some workload for each semer. A fundamental issue that arises
in this context is that of stability: will the total workload in the sÿstem remain bounded
over tirne:' in t his thesis, we investigate the question of stability within the model defined
in [BKRf 961. where workload is induced by an adversarÿ. Prior work in the adversarial
model considers only packet routing networks. a speciai case of queueing networks, where ail
custoniers require unit service tirne. We estend this mode1 to include more general queueing
networks, where the customers may require different service times at different servers of the
network. We show that the queueing poiicies proven to be universally stable for packet rou t-
ing networks, are universally stable in the new adversarial mode1 as well. Llnlike the packet
routing networks. we prove that in the new model. the unidirectional ring is not universally
stable. Furtherrnore. we define four new queueing policies for the new model. which depend
upon the service times of the customers. and we esarnine their stability. Finally. following
the outline in [BEiSWSG]. we provide a detailed proof that the Nearest-To-Go queueing
policy can become unstable when workload is injected at arbitrarily small constant rate.
Acknowledgements
This thesis would not have been possible without the help of my supervisor .-\Ilan
Borodin, who supported me in every possible way. His ent husiasrn was alwaÿs a source
of motivation. while his humor and class made working with him an enjoyable esperience.
Mÿ greatest thanks also to my second reader Charlie Rackoff. who devoted a lot of tirne
and care in reading my thesis. and whose comments improved substantially its final form.
1 would also like to thank Jon Kieinberg and Madhu Sudan who generously offered
me help, advice and information in the early stages of my thesis and Demetris .khlioptas
for many helpful suggestions and discussions. Lydia Kavraki had always been a mentor
throughout my undergraduate and graduate years.
1 owe a lot to my friends here in Toronto who helped me keep my life in perspective. In
no particular order. Theodoulos Garefalakis. Slichalis and Petros Faloutsos, Vasilis Tzcrpos,
Demetris Achlioptas, Mimi Vista, Stergios Anastasiades, Nikos Iioudas, Themis Palpanas.
'iick Zahariades. Vaso Bartzoka. Florine Tseu. Melanie Balko. and Lucia Moura. They
al1 added something new to my life. -4 special thanks to Kathy fin. the best graduate
secretary ever.
1 would also like to thank rny friends from Greece. Lefteris, Aris. -4ntwnhs Tzo, -4ntwnhs
Fou. Atalanti, Mitsos. k'iannis Spanakis. Although miles away. I feel that ttiey somehow
t rave1 wit h me.
There are too many things 1 should thank Thalia for. to sum them up here: she always
managed to give me ent husiasm to seize life.
Last but certainly not least. 1 would like to thank my family which has alwaÿs been a
strong reference point in rny life, and always will be.
Contents
1 Introduction 1
1.1 TheProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
. . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Adversarial Queueing Theo- 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Ourresults 4
2 Previous Work 6
2.1 Classical Queueing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
.
2.1.1 Single-ciass networks . . . . . . . . . . . . . . . . . . . . . . . . . . .
I
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2-12 ,L iulticlass networks S
. . . . . . . . . . . . . . . . . . .
2.1.3 Instabilities in multiclass networks 10
2.2 .Applications to packet routing . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 The New Adversarial Mode1 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Mode1 definition 1.5
. . . . . . . . . . . . . . . . . . . . .
3.2 Universal stability of queueing policies 18
3.2.1 SISisuniversallystable . . . . . . . . . . . . . . . . . . . . . . . . . 19
. . . . . . . . . . . . . . . . . . . . . . . . .
3 . 2 LIS is universally stable 'LO
3.2.3 FTG is universally stable . . . . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Instability of the ring 24
3.3.1 Instability of NTG . LTTG and MSD on the ring . . . . . . . . . . . 25
. . . . . . . . . . . . . . . . . . . . .
3.3.2 InstabiIity of FIFO on the ring 21
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 The MTTG queueing policy 30
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Stability of MTTG 31
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Instability of MTTG 33
4 Instability of NTG for constant rate injections 36
4.1 Mode1 description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.3 System properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 The Proof of Instability 47
4.5 InstabiIity of packet routing networks . . . . . . . . . . . . . . . . . . . . . -19
5 Conclusions 53
List of Figures
4 1 The4x4network.t'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 The 4 x 4 network -1;' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Chapter 1
Introduction
The Problem
Qiieueing networks are important models of compiex systems. such as packet routing net-
works. rnanufacturing systems. and time shared cornputer systems. In this nork we analyze
the behavior of queueing networks where customers are continuously injected into the sys-
tem. Each customer induces some workload in the network. A crucial question arising in
this contest is that of stability - will the total workload in the sÿstem remain bounded
as the systern runs for an arbitrarily long period of time*? The answer to this question typi-
cally depends on the rate at which tvorkload is injected into the system and the queueing
policy that is used at the nodes of the network.
A queueing network is a set of iriterconnected servers. We assume tliat eacli server
is associated with a single queue of infinite size. and each queue is serviced by a sirigte
server. Customers arrive at the servers of the network. When the service of some customer
at some server is completed the custorner rnoves to another server or it esits the system.
IF more than one customer wishes to be serviced at some server S. the queueing policy
chooses one of the customers to be serviced nest: the rernainder of these customers wait
in the queue of server S. ..\ queueing policy is work-conserving if a server is never idle
whenever there is at least one customer in the queue of the server. A queueing policy is
called non-preemptive if the service of some customer at some server is never interrupted
until it is completed.
In general, a queueing network is characterized by the following parameters.
The process that injects the custorners in the network.
0
The routing of the customers in the network.
0
The service times of the custorners at the servers.
0
The queueing policy.
It is obvious that if the rate at which workload arrives at some server is greater than the
rate at which the workload is serviced at that server. then the system becomes unstable.
.A natural condition for stability, referred to as the rate condition. is that the rate at
which workload is induced at any server is strictly Iess than the service rate of the server.
Intuitively, it seems that the rate condition should be sufficient for stabiIity of any queueing
network. This is true for man. systems [liIei5. Iie179], but it is not true in al1 cases
[LI<91, RS92. DW96. Bra94a. Bra9lb. SeiS-l]. The question of necessary and sufficient
conditions for stability is one of the rnost interesting questions in the area of queueing
networks.
Queueing networks are the object of study of queueing theory. Typical assumptions
in queueing theory are that the customers are injected according to a time invariant dis-
tribution (often a Poisson process), and that the service time of a custonier, or customer
type at some server is a random variable following a tirne invariant distribution ( often an
esponential distribution). In this thesis we work within a model proposed by Borodin et
al. [BIiR+96]. in which probabilistic assumptions are replaced by worst-case in pu ts. The
underlying goal is to determine whether it is feasible to prove stability results even when
customers are injected by an adversary, rather than a randomized process: the adversary
deterrnines the route of the customers in the system and the service tirne they receive at
each server. For systems that are not stable, we seek to establish instability within the
weakest possible adversarial model.
1.2 Adversarial Queueing Theory
.Adversarial Queueing Theory was introduced by Borodin et al. [BKR+96]. for the study of
packet routing networks. .A packet routing network is a special case of a queueing network
where packets (i-e. customers) move on the edges (Le. servers) of a directed graph. Al1
packets take unit time to cross any edge in the network. .Any packet routing network
1.2. ADVERS4 RI-4 L Q UEUEING THEORY 3
can be easily transformed into a queueing network. simply by transforming edges into
servers. The resulting queueing network is stable. if and only if the packet routing network
is stable. However the converse does riot hold: there exist queueing networks that can not
be transformed into packet routing networks.
In the adversarial model the packets are injected by an adversary. rather than some
independent randornized process. The mode1 considers time evolution in the system as a
game between an adversary and a queueing policy. Tirne proceeds in discrete steps. In cach
time step, the adversary injects a set of packets at some nodes; for each packet it specifies a
path of edges that it must traverse, after which the packet is absorbed. .A queueing policy
is said to be stable against an adversary if the total number of packets in the system is
bounded over time.
.A crucial parameter of the adversary is the rate. In [BKR+96] a request by the adversary
is defined to be a set of packets requesting edge-disjoint paths: in their terminology an
adversary injects at rate r. if for al1 t, no more than rrtl requests are made in any interva1
of t steps. We cal1 this mode1 the adversarial path-packing model. .A more general
model was suggested in [BKRf 961, and was fully developed in [.LAF+96], in which the rate
of the adversary is specified by a pair (u?.r ), where w is a naturd number, and O < r < 1.
The requirement of the adversary is the following: of the packets that the adversary injects
in any interval of w steps at most LrwJ can have paths that contain any one edge. iVe will
refer to this mode1 as the adversarial packet routing model.
In [.L-\F+96] a universally stable queueing policy is defined as a queueing policy
that is stable against any adversary of rate less than 1, on everÿ network. A universally
stable graph is defined as a graph G. such t hat every queueing policy is stable against
every adversary of rate less than 1 on G'.
Borodin et al. [BIiR+96] showed that directed acyclic graphs (DAGs) are universally
stable in the adversarial packet routing model. Ir* [A..\F+96] the unidirectional ring is shown
to be universallÿ stable in the same model. Furthermore, Andrews et al., prove that in t his
mode1 the queueing policies Furthest-To-Go (FTG), Longest-In-System (LIS) and
Shortest-In-System (SIS) are universally stable. The SIS queueing policy gives prece-
dence to the customer most recently injected in the network; the LIS queueing policy gives
precedence to the customer injected the earliest in the network; the FTG queueing policy
gives precedence to the customer whose distance to its destination is maximal. However, two