Bounds on the tail distributions of Markov-modulated stochastic max-plus systems

Abstract

Considers a particular class of linear systems under the max-plus
algebra and derive exponential upper bounds for the tail distribution of
each component of the state vector in the case of Markov modulated input
sequences. The authors' results are then applied to tandem queues with
infinite buffers, Markov modulated arrivals and deterministic service
times

Full text

Page 1

?n
ZhenLIU
??
PhilippeNAIN
?y
allo
andDonTOWSLEY
?z
get
?
INRIA? B?P??????????SophiaAntipolisCedex?F rance
?
Departmentof ComputerScience? Universityof Massachusetts?Amherst?MA ??????USA
Abstract
Inthispap erweconsidera particularclassoflinear sys?
temsunderthemax?plusalgebraand derive exponen tialupp er
boundsforthetaildistribution ofeachcomponentofthestate
vectorinthecaseofMarkovmodulated inputsequences? Our
resultsarethen appliedto tandemqueueswithin?nite bu?ers?
Markovmodulatedarriv alsand deterministicservicetimes?
?In tro duction
Considerastochasticdiscrete?ev entsystemwith max?plusdy?
namics?
X
n???k
? max
j ?K
?X
n?j
?A
n?jk
?Y
n
??
?
??? ???
where?a?
?
?max???a?andK?f????????Kg?K????
ThestatevariablesX
n
??X
n?k
?k?K?areK?dimensional
nonnegativerandomvariables?r?v??s??X
n
??????
K
?The
sequence?Y
n
?
n
isa?nite?state?irreducible?aperiodicandho?
mogeneousMarkovchainonthe?nite setS?f????????Sg?
Foreachs?S?n???introducethematrixA
n
?s??
?A
n?jk
?s??
j?k?K
?Weassumethat
?A???A
n
????
n
??????A
n
?S??
n
formSm utuallyindependent
renewal sequencesofrandommatriceswith???
A
n?jk
?s???almostsurely?a?s???
?A??foreachs?S?thematrices?A
n
?s??
n
areindependentof
theMarkovchain?Y
n
?
n
?
Note?however?thatforanygivenn??ands?S?theen tries
ofthematrixA
n
?s?maybedependent r?v??s?Wefurther
assumethatX
?
??X
???
?????X
??K
?isanonnegativ e anda?s?
?niterandomvector?
Usingthe max?plus algebraoperators? ???canbewrittenin
vectorformasfollows
X
n??
?X
n
?A
n
?Y
n
?????n?????
whereX
n
??X
n??
?????X
n?K
?? In???the?plus?operator?
andthe?max?operator?replacetheusualmatrixmultipli?
cationandmatrixaddition?resp ectively?
Ourobjective istoderiveexp onentialupperboundsonthe
tail distributionof X
n?k
?namely?to?ndpositiveconstantsb
?
e?mailaddress?liu?sophia?inria?fr
y
e?mailaddress?nain?sophia?inria?fr
z
e?mailaddress?to wsley?gonzo?cs?umass?edu?Thisauthorwas
supportedinpartbyNSFundergrantNCR????????
and?suchthatP?X
n?k
?x??bexp???x?forallx???
k?K?n???
Thereisnowasubstan tialbo dy ofwork?????? ???????
?????? ??????????????? ???amongothers?ontheproblem
ofderivingexact?approximateorasymptoticboundsforthe
taildistributionoftheworkload?queue?length?anddelayin
aqueueinisolation?Boundsofthis typehaveprov edtobe
helpfulforanalyzingATMmultiplexersfedbydi?erent classes
oftra?csuchasdata?voice? video?etc??seee?g? ??????
Thispaperis oneofthe?rstattempts?seealso????? ?????
???forrelatedwork?to deriveboundsfor moregeneralstruc?
tures?includingacyclicqueueingnetw orks?Boundingquanti?
ties such as theprobabilitydistribution oftheend?to?end delay
seen by individual sessionsalongtheirrouteinthe network will
w network designerstoab etterunderstandingofthe
networkbeha viorandto comeupwithmoree?cientadmis?
sioncontrol schemes as opposedtoschemesbased onlyonthe
performanceatisolatednodes?
Asin????we de?nethecommunic atinggraph G??V?E? of
thesystem asfollows? Thesetof v erticesisV?K?F oreach
pairofverticesj?k?K? thereisanarc fromj tokif and only
ifthereisat leastone states?S suchthatA
n?jk
?s?????
withstrictlyp ositive probability?LetG?fG
?
?G
?
?????G
g
?
g
be themaximal decompositionof G intostrongly connected
subgraphs?where G
i
??V
i
?E
i
????i?g?are stronglycon?
nectedsubgraphs ofG? Withoutlossof generality?we assume
thatverticesof G
i
have smallerindices than thoseofG
i??
?
Undersucha decomposition?the matrices A
n
?s??s?S?
ha vetheform?
A
n
?s??
?
B
B
B
B
?
B
n??
?s?R
n????
?s????R
n???g??
?s?R
n???g
?s?
??B
n??
?s????R
n???g??
?s?R
n???g
?s?
???? ???R
n???g ??
?s?R
n???g
?s?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???????B
n?g ??
?s?R
n?g ???g
?s?
???? ????? B
n?g
?s?
?
C
C
C
C
A
???
whereB
n?i
?s???A
n?jk
?s??
j?k?V
i
andR
n?i
?
?i
?
?s??
?A
n?jk
?s??
j?V
i
?
?k?V
i
?
???i?g? and??i
?
?i
?
?g?
LetP??p
ij
?be thetransition matrixof theMarkovchain
?Y
n
?
n
andlet???? ??????????K??be its inv ariant measure?
De?ne?
n
?i??P
?
?
?Y
n
?i? fori?S?n??? and let?
?
??
?
?????????
?
?K??betheinitialprobabilitydistributio n?In
thefollowingweshall dropthesubscript?
?
inP
?
?
?

Page 2

?A????f?????
jk?s
?????for all?j?k?s??K?K?Sg
isanonemptyset?Thistechnicalassumption issatis?ed
inmostcasesofpracticalinterestwhichincludesr?v??swith
phase?typedistributions?
Intro duceh
jk?st
?????
jk ?s
???p
st
?LetH
jk
????
?h
jk?st
????
s?t?S
beS ?by?Smatrices?andletH???betheKS?
by?KSmatrixde?ned asH?????H
jk
????
j?k?K
?
Then?inview of????H???canberewrittenas
H????
?
B
B
B
B
?
J
?
???J
???
??????J
??g??
???J
??g
???
?J
?
matrix
??????J
??g
there
??
???J
??g
p
???
?????J
??g??
???J
??g
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?????J
g??
???J
g???g
???
??????J
g
???
?
C
C
C
C
A
???
whereJ
i
?????h
jk?st
????
j?k?V
i
?s?t?S
andJ
i
?
?i
?
????
?h
jk?st
????
j?V
i
?
?k?V
i
?
?s?t?S
for??i
?
?i
?
?g?
Let us introducesomeadditional notation?We willsaythat
avectorv ispositive?resp?nonnegativ e?if itscomp onents
areall largerthan ?resp? largerthan orequalto? zero? The
notation v???resp?v??? willindicatethatthevectorvis
positive?resp? nonnegative??Moregenerally?foranypairof
vectorsvandw?thenotationv?wwillindicatethatvisless
thanor equalto wcomponentwise?
SincethetransitionmatrixPoftheMarkovchain?Y
n
?
n
isirreducible?thematricesJ
i
???areall irreducible? Thus?
according toP erron?F robeniusTheorem ????Theorem ???????
the spectralradius?
i
???ofmatrixJ
i
??????i?g?isa
simpleeigenvalueoftheandisaositivevector
z
i
?????z
k ?t
????
k?V
i
?t?S
suchthat
z
i
???J
i
?????
i
???z
i
???????
F oreach??i?g?wenormalizetheleft?eigen vectorz
i
???so
that
X
k?V
i
?t?S
z
k ?t
?????? ???
Let????be the spectralradiusofH?? ??Sincethematrix
H???isblocktriangular?itsspectrumistheunion ofthesp ec?
traofmatrices J
?
????????J
g
????whichin turn impliesthat
?????max
??i?g
?
i
???????
It then follows????Theorem??????that????isalsoan eigen?
valueofH???and thatthereis anonnegativ ev ectory????
?y
k ?t
????
k?K?t?S
suchthat
y???H????????y???????
n?k
Weare nowin positiontostatethe mainresultsof this
work?
? Exponential UpperBounds
In this sectionwederive exponential upperbounds for thetail
distributionofX
n?k
under Assumptions?A????A???
Proposition? ?ExponentialUpper Bound I?Assume
that ????If???????y??????andifforallk?K?t?S?
P?X
??k
?x? Y
?
?t??b???y
k?t
???e
??x
??x??????
then?foralln???k?K?x???
P?X?x??b???e
??x
????
where
b????sup
x??
n??
k?K
t?S
X
j ?K?s?S
p
st
?
n
?s????F
jk?s
?x??
X
j?K?s?S
p
st
y
j?s
???
Z
?
x
e
??u?x?
dF
jk ?s
?u?
???
????
????
?
It isworthnoting thatcondition???? isautomaticallysat?
is?ed?inparticular?ifX
??k
?? a?s?
Proof? Fix??? such that?????? andy???? ??Let
f?
j?s
?x??j?K?s?Sg??
j?s
????????????? bea setof
functionssatisfying
X
j?K
s?S
p
st
?
Z
x
??
?
j?s
?x?u? dF
jk ?s
?u???? ?F
jk ?s
??x??
n
?s?
?
??
k ?t
?x??????
The?rststepofthe proof consistsinprovingthat
P?X
m?k
? x?Y
m
?t???
k ?t
?x?
forallm???x???k?K?t?S?F or this?weuse an
induction argumentonm?Notethat???? holdsform??
from????andassumethat????istrueform?????????n?
Letussho wthat ???? isstill trueform?n? ??
F orx? ??wehave
P?X
n???k
?x?Y
n??
?t?
?P
?
max
j?K
?X
n?j
?A
n?jk
?Y
n
???x? Y
n??
?t
?

Page 3

j?K?s?S
?
X
j ?K?s?S
p
st
?
n
?s?
?
Z
x
??
?
P?X
n?j
?x?ujY
n
?s?dF
jk?s
?u?
???F
jk?s
?x?
?
?
X
j ?K?s?S
p
st
?
Z
x
??
?
j?s
?x?u?dF
jk?s
?u?
??
n
?s????F
jk?s
?x??
?
????
??
k?t
?x?
where????followsfromtheinductionh ypothesisandthelatter
inequality follows fromthede?nitionofthefunctions?
j?s
given
in????? Thisconcludesthe pro of of?????
Beforegettingto thesecondstepoftheproof?letus?rst
show thatb?????if y??????Indeed?we have inthiscase
b????sup
x??
n??
k?K
t?S
X
s?Sj?K
p
st
?
n
?s????F
jk?s
?x??
X
s?S?j?K
p
st
y
j?s
??
condition
??? ?F
jk?s
?x?
yields
????
?sup
n??
j?K
s?S
?
n
?s?
y
j?s
???
thelatterquantityb eing?niteify?????sincebothsets
KandSare?nite?weusedtheinequality
R
?
x
exp???u?
x??dF
jk ?s
?u????F
jk?s
?x? toderive??????
The secondstepofthe proofconsistsincheckingthat the
functions f?
j?s
?b???y
j?s
???exp???x??j?K?s?Sgsatisfy
?????This proofisanalogoustotheproofofProposition?in
???? andisthereforeomitted?Letussimplypointoutfor later
usethatthis proofonlyusesthepropertythaty???H????
y???if???????whic hfollowsfrom???? anddoesnotrequire
theidentit y????
Substitutingnow?
k?t
?x? in ???? forb???y
k?t
???exp???x??
thensumming upbothsidesoftheinequalityov erall theval?
uestinSandusingthenormalizing????????
Theconditionthaty?????in Proposition? maybe
dropp edin thecasethat ???????Moreprecisely?we have
thefollowing result?
Prop osition? ?Exponen tialUpp erBoundII?Assume
that???with ???????Then?thereexist strictlyposi?
tiveconstantsa
?
?????a
g
suchthatthepositivevectorv????
P?X
??k
?x?Y
?
?t??c???v
k?t
???e
??x
??x??????
then?
Let
for
us
al
no
l
w
n???k?
the
K?x???
P?X
n?k
?x??c???e
??x
????
where
c??
i??
?? sup
x??
n??
k?K
t?S
X
j?K?s?S
p
st
?
n
?s? ???F
jk?s
?x??
X
j ?K?s?S
p
st
v
j?s
???
Z
?
x
e
??u?x?
dF
jk ?s
?u?
???
????
?
Proof? Assume ?rstthe existenceofthevector v?? ?? Then?
thepro ofof ????is identicaltothe pro ofof???? aftersub?
stitutingy???forv?? ?? Thisfollo wsfromtheobservation?cf?
step ?in the proofofProp osition ??thatonly theinequality
y???H????y??? andthe
j
factthat
j
y????? are usedinthe
proofof?????
address existenceofconstan tsa
?
?????a
g
such
thatv???H????v?? ?? Byusingthe de?nitionoftheeigen?
vectorsz
i
???? ??i?g? itisseen thatthe vectorinequality
v???H????v???translates intotheset of inequalities
j??
X
a
i
z
i
???J
i?j
????a
j
??
j
???? ??z
j
??????for ??j?g?
????
Since?
i
?????foralli?????????gunderthe assumptionthat
???????cf? ?????thefollo wing simpleprocedurecanbede?
riv ed from???? togenerate p ositiveconstan tsa
?
?????a
g
?pick
an arbitrarya
?
???thenpic ka
?
???????a
g
??successively
sothat
a
j
? max
l
?
P
j??
i??
a
i
?z
i
???J
i?j
?? ??
l
?????????z?? ??
l
?
? for??j?g
where?v?
l
denotes thel?th componen tof anyv ectorv?The
pro of is concludedbynormalizing a
?
?????a
g
sothat thecom?
p onents ofv??? sumup to ??
? ApplicationtoTandemQueues
Consider an op entandemqueueingnetwork consistingofK
single?serv erqueues within?nite bu?ers?We assume that cus?
tomers may only enterthenetwork fromtheoutside atno de?
andthat they allleave thenetw ork up oncompletion of their

Page 4

no dek???k?K?andlet?
n
?Z
n
?betheinterarrivaltime
inqueue?betweencustomersnandn???Inotherwords?
weconsiderasystemwheretheinterarrival andservicetimes
are modulatedbythe Markovchain ?Z
n
?
n
?De?neX
n?k
as
thecumulatedwaiting time?excludingtheservicetimes?of
customer n throughitssojourninqueues????????k?
??
??
??
??
?
?
????
K
?
n??
?Z
n
??
n?K
?Z
n
?
?
?
n
?Z
n
?
Figure??Tandemqueues
De?ne
U
n?j
?z
?
?z
?
??
j
X
i??
?
n?i
?z
?
??
j??
X
i??
?
n???i
?z
?
???
n
?z
?
??
It iseasy tocheck that
X
n???k
?max?X
n?k
?U
n?k
?Z
n
?Z
n??
??X
n???k??
?
?
????
?max
??j?k
?X
n?j
?U
n?j
?Z
n
?Z
n??
??
?
????
for alln??andk?K?byconventionX
n????
??in??????
Thestochastic recursion ????canbewrittenintheform???
byde?ningtheMarkovchain?Y
n
?
n
andthematrix A
n
?s?as
follows?
Y
n
??Z
n
?Z
n??
?????
A
n?jk
?s??
?
U
n?j
?z
?
?z
?
?for??j?k
??fork?j?K
????
withs?S?f?i?j?jq
ij
???i?j?Tg?
Inthiscase?B
n?k
?s?andR
n?k ?j
in???areall??by?? matri?
cesgiv enbyB
n?k
?s??U
n?k
?z
?
?z
?
?fork?KandR
n?k?j
?s??
B
n?k
?s?forj?k?foralls??z
?
?z
?
??respectively?Inparticu?
lar?the matrixJ
i
????i?????????K?isgivenby
J
i
?????E?exp??U
n?i
?z
?
?z
?
???p
st
?
s??z
?
?z
?
??S?t??z
?
?z
?
??S
withp
st
?q
z
?
z
?
ifz
?
?z
?
andp
st
??ifz
?
??z
?
?forall
s??z
?
?z
?
??S?t??z
?
?z
?
??S?
Inorder toapplytheresults obtainedin Section?tothe
comp onentsofthevectorX
n
wemustensure thatAssump?
tions?A????A??aresatis?ed?Theseassumptionswillhold?in
particular?if theservicerequirementsaredeterministic ?but
notnecessarilyallequal?? namely?forevery?xedk?K?z?T?
?
n?k
?z?is constantforalln??? if??
n
????
n
???????
n
?T??
n
are
that Propositions and?notto queues
?give exponentialupperboundsonthe taildistributionofthe
cumulated backlogs?
Inparticular?the uniquepositive solution?
?
ofthe equation
??????giv estheb estexp onentialdeca yfor P?X
n?k
?x?if
y??
?
?is positiv e?
Itisworth notingthatthe sequencesof matrices
??A
n
?s??
n
?
s?S
arenot
mutuallyindep endent under only the
assumptionthat theserviceandin terarriv altimes areall
m utuallyindep endentr?v??s? Thisfollo ws fromthefact
thatmatrices A
n
?s?andA
n??
?t?bothdependon the r?v??s
?
n????
?z
?
???????
n???K??
?z
?
?inthecasethat s??z
?
?z
?
??S
andt??z
?
?z
?
??S?Furthermore? ifq
zz
??? thenthe
matrices A
?
?s??A
?
?s?????are notmutually indep endent for
s??z?z?for thesamereason?Thisimplies? inparticular?
?doapply M?M??in
series?
Note thatcertainformsof non?deterministicservice times
arepermittedbyaugmen ting the statespaceof theoriginal
Markovchain?Z
n
?
n
?
? ConcludingRemarks
Inthis pap er?we haveextended ourw orkforsingle queues
???? ??? tostoc hasticlinear systems under themax?plus al?
gebra? asde?ned by????and have derived exponential upper
boundsonP?X
n?k
?x??The max?plusstructurede?nedin
???appears tobea particularcase ofthe structureconsidered
byChangin ????Web elieve thatourmethodbased onan
extensionofKingman?s method forb oundingthetail of the
waiting time distributioninaGI?GI??queue? gives sharp er
upp erboundsthanthecorresp ondingbounds in????based on
Cherno? ?sbound?We alsopointoutthat the i?i?d?assump?
tionplacedontheinput sequence in??? isstronger thanours?
Ourapproach willalso allow us toderiveexponential low er
boundsfor the taildistribution ofX
n?k
? Thelatterresultswill
be rep ortedinaforthcomingpap er?
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