Page 1
Memorandum COSOR 97-08, 1997, Eindhoven University of Technology
BOUNDSF OR
PERF ORMANCECHARA CTERISTICS?
ASYSTEMATICAPPROACH
VIACOSTSTRUCTURES
G?J?v anHoutum?W?H?M?Zijm
UniversityofTwente?
DepartmentofMechanicalEngineering?
Box????????AE?Enschede?
theNetherlands?
email??g?j?j?a?n?vanhoutum?w?h?m?zijm??wb?utwente?nl
I?J?B?F?Adan?J?Wessels
EindhovenUniversityofTechnology?
DepartmentofMathematicsandComputerScience?
Box???? ????MB ?Eindhoven?
theNetherlands?
email??iadan?wessels??win?tue?nl
ABSTRACT
Inthispaperwe presentasystematicapproachfor theconstructionofboundsforthe
averagecost inMarkovchainswithin?nitelymanystates?Thetechniquetoprovethe
bounds isbasedondynamicprogramming?MostperformancecharacteristicsofMarko?
viansystemscanberepresentedbytheaveragecostforsomeappropriatelychosencost
structure?Therefore?theapproachcanbeusedtogenerateboundsforrelevantperfor?
mancecharacteristics?Theapproachisdemonstratedfortheshortestqueuemodel?Itis
shownhowforthismodelsev eral boundsforthemeanwaitingtimecanbeconstructed?
Weincludenumericalresultstodemonstratethequalit yofthesebounds?
?
Page 2
?INTRODUCTION
InthispaperweconsideranirreducibleN?dimensionalMarkovchainwith states
m
?
?m
?
?????m
N
??whereeachm
i
isaninteger?andtransitionprobabilities p?m
?
n??Let?
denoteitsequilibriumdistribution ?whichwe assumetoexist??Ifc?m?isthe costper
periodinstate
m
?thentheaveragecostgisgivenby
g?
X
m
c?m???m??
Todetermineg?weneedthedistribution??whichinmanycasesisdi?cultto obtain
exactly?Therefore wetrytodevelopasystematicapproachfortheconstructionofbounds
forgwhichcaneasilyande?cientlybe computed?
Theapproachtriestoconstruct mo di?cationsoftheoriginalmodelwhichproduce
boundsforg? Andofcoursethesemodi?cationsshouldbeeasiertohandlethanthe
originalmodel? Themaincontribution of thepaperisthatit isshownhowsuchmodi?
?cationsma ybefoundsystematically?one should?rst try toidentifyprecedences?with
resp ecttocost?betweenstates?Based ontheseprecedencesit app earstobeeasyto
producesuitablemodi?cations?Infact?theapproac
horiginatesfromearlierwork in
???? ????????????????????andattemptstounifythetechniquesusedin thesereferences?
The techniqueused inthispapertoestablish computablebounds?isalsoapowerful
tool toprovequalitativeproperties?likee?g?monotonicitypropertiesinqueueingnetworks
?cf??????oroptimalityofroutingpoliciestoparallelqueues?cf???????
Manyperformancecharacteristics ofMarkoviansystems?suchas?e?g??meanqueue
lengthsandmeanwaitingtimesin queueingproblems?canberepresentedbytheaverage
costforsomeappropriatelychosencoststructure?Hence?theapproachcanbeusedto
generateboundsforrelevantperformancecharacteristics?
Wewilldemonstratetheapproachfortheshortestqueueproblem?Tokeepthepre?
sentationsimple?weonlyconsiderthesystemwithtwoqueues?But? in fact? thepow er
of the approachisthatitalsoworkswellformorethantwoqueues?becauseinthat
casethereisnoanalyticalsolutionavailable?Theproblemwithtwoqueues has been
extensiv elystudiedintheliterature?Exactanalyticalresultscanbefoundin??????? ??
Therearealsomanypapersanalyzingapproximationsfor theshortest queueproblem?see
??????????????????????????Itappearsthatthepresentapproachleadstoseveralmodels
producingboundsforperformancecharacteristicssuchas?e?g??themeanwaitingtime?
?
Page 3
Thesemodelscovertheonesin???????????????Noneofthesereferences?however?rig?
orouslyproves thatthesemodelsindeedpro ducebounds?Butthis isdonein thepresent
paper? Animportantprop ertyof theb oundspresentedhereisthattheirqualitycanbe
controlledbysomethreshold?or truncation? parameter?Thelargerthis parameter?the
moreaccuratetheboundswillbe?butalsothe moree?ort ittakestocomputethem?
The paper isorganizedasfollows?Inthenextsection?wedescribetheshortestqueue
model?Thismodelwillbeused throughoutthepap erto demonstratetheconcepts and
techniques?InSection??we introduceamodi?cationoftheoriginalMarkovchain?and
subsequentlywecomparetheaveragecostsofthe modi?edandoriginal chaininSection
??Section?dealswiththeproofofprecedences?InSection??wepresentnumerical
resultstodemonstrate thequality of theb oundsproducedforthemeanwaitingtimeof
theshortest queuemodel?Finally?Section?is devoted toconclusions?
? BASICEXAMPLE?SHORTEST QUEUEMODEL
Theshortestqueuemodelischaracterizedasfollows? Therearetwoidenticalparallel
servers?eachwithitsownqueue ?seeFigure???Theservicetimesareexponentialwith
rate??JobsarriveaccordingtoaPoissonstreamwithrate?andjointheshortestqueue?
ThissystemcanbedescribedbyaMarkovprocesswithstates
m
??m
?
?m
?
?wherem
?
andm
?
arethelengthoftheshortestandlongestqueue?resp??som
?
?m
?
??Without
lossofgeneralitywemaytake??????and assumethattheserversalwayswork ?also
when thereisno job??But? service completiononly leads toa departure ifthere isa job
in thequeue?otherwiseit isfake?and?fakejobs willbeinterrupted as soonasa realjob
arriv es??Thearti?cialassumptionofworking onfak ejobs impliesthat ineach state the
outgoingtransitionratesaddup to??i?e??we uniformizedthe Markov process??The?ow
diagram issho wninFigure?? Ascost ratewetakethenumberofwaitingjobs?so
c?m??maxfm
?
????g?maxfm
?
????g????
Thentheaveragecostyieldsthemeannumberofw aitingjobsinthesystem?andby
Little?s la w?themeannormalizedwaiting timeW?Here? thenormalizedwaiting timeis
de?nedastheratio ofthew aiting timeandthe meanservice time???????
Theprocessobserv edatjumps is aMarkovchain?andsince themean timebetw een
jumps isalwa ys??it hasthesame equilibrium distributionas the originalMarkovpro cess?
Ifwetakec?m?ascostperperiod?thenitalsohasthesameaveragecost?Fromnow
?
Page 4
λ
μ
μ
Figure??Theshortestqueuemodel
m2
m1
?
??
???
????
?????
??????
???????
λ
2μ
λ
μ
μ
λμ
μ
2μ
λ
Figure??Flowdiagramfortheshortestqueuemodel
onweonlyconsidertheMarkovchain?atjumps? insteadofthecontinuous?timeMarkov
process?NotethattheratesinFigure?correspondtotransition probabilitiesinthejump
process?
?THEMODIFIEDMODEL
Weconsiderthefollowingmodi?cationoftheoriginalmodel?Markovchain? introduced
inSection ??Insomestates
m
weredirectoneormoreoutgoingtransitions?Thismeans
thata transitionfrom
m
to
n
?say?isredirectedto anotherstate
?
n
? Thenthenew
transitionprobabilityto
n
iszeroandto
?
n
itequalsp?m
?
n
??p?m
?
?
n
?? Denotethe
newtransitionprobabilitiesby?p?m?
n??Thecostsperstateare notaltered?Weassume
thatthemodi?edchainisunichained?somestates may nowbetransient? and that its
equilibriumdistributionexists? Theav eragecost isdenotedby?g?
Ofcourse?therearemanypossibilities tomodifythesystem?Whichtransitionsshould
beredirectedandhow?Whendoesitleadtoanupperorlowerboundforg?Howto
provethis?Inthenextsectionswewillattempttoanswerthesequestions?
?
Page 5
Example????Modi?c ationsof theshortestqueuemodel?
Belowwe ?rstdescribesomemo di?cationsof the shortestqueuemodel?Ineachofthese
modi?cationswe useathresholdparameterT?which issome?xed?butarbitrarypositive
in teger?Themodi?cations ?forT? ??are depicted inFigure ??
FiniteBu?ers?FB??Thesimplestmodi?cationisobtainedbyredirectingonlyone
transition in onlyonestate? namelyby redirectingthetransitionfrom?T?T?to?T?T???
?an arrival?to state?T?T??rejectthe newjob??Notethatstateswithm
?
orm
?
?T are
nowtransient?Thismodelcorrespondstothesituationwherebothservershavea?nite
bu?erofsizeT?Ithasbeenanalyzede?g?in???????
CentralBu?er?CB??Forall states?m
?
?m
?
???withm
?
?Tthetransitionto
?m
?
???m
?
????adeparture?isredirectedto?m
?
?m
?
??Thismodelhasthefollowing
interpretation?Bothservershavea ?nitelocal bu?erofsizeT?and thereis a central
bu?erwith in?nite capacity? Onarrivalajobissenttothecentralbu?erifthere isno
roominthelocalbu?ers?Assoonas thereisroomagaina jobisreleasedfromthecentral
?ifthereisany?tothelocalbu?er?Hence?state?T???T???meansthatbothlocal
bu?ersarefulland?jobsarewaitinginthecen
tralbu?er?
ThresholdJock eying?TJ??Forallstates?m
?
?m
?
?T?withm
?
??thetransition
to?m
?
???m
?
?T? ?adepartureintheshortestqueue? isredirected to?m
?
?m
?
?T????
Thismeansthatajobswitchesfromthelongesttotheshortestqueue as so onas the
di?erencebetween thequeue lengthsexceedsT?Foran analysis ofthismo delwereferto
??? ??????
OneIn?nite Bu?er?OIB?? For allstates?T?m
?
?withm
?
?Tthe transitionto
?T???m
?
?isredirectedto ?T?m
?
???? Thismodelcorrespondstothesituation whereone
serverhasa?nitebu?erofsizeTandtheotherserv erhasanin?niteone?Onarrivala
jobjoinstheshortestqueueif thereisroom?andotherwisethelongestone?inthein?nite
bu?er??A matrix?geometricanalysisofthis model canbefoundin?????
ThresholdKilling?TK??Forallstates?m
?
?m
?
?T?withm
?
??thetransitionto
?m
?
???m
?
?T?is redirectedto?m
?
???m
?
?T????Sowhenthedi?erenceinqueue
lengthsexceedsTduetodepartureintheshortestqueue?thenthejobinserviceinthe
longestqueueisdirectlykilled?andremoved??
ThresholdBlocking?TB??Forallstates?m
?
?m
?
?T?withm
?
??thetransition
to?m
?
???m
?
?T?isredirectedto?m
?
?m
?
?T??Thismeansthatwhenajobiscompleted
intheshortestqueueanditsdeparturewouldleadtoadi?erenceinqueue lengthsgreater
?
Page 6
m2
m1
Two Finite Buffers
?
?
?
?
?
?
?
?
??
λ
m2
m1
Central Buffer
?
?
?
?
?
?
?
?
??
??
?
μ
μ
m2
m1
Threshold Jockeying
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
μ
μ
μ
μ
m2
m1
One Infinite Buffer
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
λ
λ
λ
m2
m1
Threshold Killing
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
μ
μ
μ
μ
m2
m1
Threshold Blocking
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
μ
μ
μ
μ
Figure??Flowdiagramsformodi?cationsoftheshortest queuemodel
?
Page 7
thanT?thenitsdeparture is block ed andthejobisserved oncemore?withanewservice
time??
Foreach ofthemodi?cationsdescribedabov e? theresulting statespaceofrecurrent
statesis?nite?theFBmodel?orin?niteinatmostone direction?the othermodels??
Thereforethesemodi?cationsaremucheasier tohandlethanthe originalmodel?In fact?
theFB model canbesolvedbyastandardnumericalprocedureandthe otheronescan
be e?cientlysolv edusingthe matrix?geometric approachof Neuts?????e?g??byapplying
thealgorithmin??????It maybein tuitively clearwhichmodi?cationsleadtoanupper
b oundandwhichonestoalowerboundforthemeannormalizedwaitingtime?Inthe
next section?wepresentatechniquetoprove thisandwedevelopasystematicapproach
toconstructsuchmodi?cationsleadingtobounds?
Remarkthatthecentralbu?erandthethreethresholdmodelsexploittheproperty
thatmostoftheprobabilitymassintheshortestqueuemodelisconcentratedaround the
diagonal of the state space? So one might expectthatthesemodelsproducetightbounds
foralreadysmallvalues ofthethresholdT?
?
?COMPARISONOFAVERAGE COSTS
Weno w returntothegeneralmodel?Supposewewanttoshowthat?g?g?Todosowe
study theexp ectedcostovera?nitenumberofperiods?De?neu
n
?m
?andv
n
?m?asthe
expectedcostovernperiodsforthemodi?edandoriginalmodel?resp??whenstartingin
m?De?ning
u
?
?v
?
??we tryto provebyinductionthatforallnandallrecurrent
states
m
inthemodi?ed model
u
n
?m??v
n
?m
?? ???
Thenit followsthattheav eragecostsareorderedinthe sameway?Toestablish ??? we
?rst needpr ece dencesb etw eenstates?We saythat state
m
hasprecedenceover?oris
more attractive thanstate
n?if
m
and
n
satisfythefollowingprecedencerelation?
v
n
?m??v
n
?n?foralln??????
Inwords?startingin
m
yieldslo wertotalexpectedcost thanin
n
?Nowthe?rst?and
crucial step isthecharacterization ofasetPofpairs?m?
n
?satisfying????Thesepairsare
calledprecedencepairs? Usuallyprecedencepairsareintuitivelyobvious?andofcourse?
theydependontheone?periodcostc?setn??in?????Theproofofthesepairsisthe
?
Page 8
topicofthenextsection?Oncea su?cien tlyrich setP hasbeenc haracterized?theproof
of ???iseasyaswillbe sho wnb elow?
Theproof of??? followsbyinduction?For n??inequality???triviallyholds? As?
suming ???holdsfornwe trytoproveitforn???Theexpectedcostovern??p eriods
satis?es
v
n??
?m
??c?m
??
X
n
p?m
?
n
?v
n
?n??
Itfollo wsthatv
n??
?m??u
n??
?m
?providedwe haveconstructed themo di?ed modelby
redirecting transitionstomoreattractivestates ?i?e?atransitionto
n
isredirectedto
?
n
onlyif?n
?
?
n??P??Namely?thenwehave
v
n??
?m??c?m
??
X
n
?p?m
?
n
?v
n
?n?
?c?m
??
X
n
?p?m
?
n
?u
n
?n?
?u
n??
?m??
wherethesecondinequalityfollowsfromtheinductionassumption?These?ndingsare
summarizedinthe followingtheorem?
Theorem???Providedthemodi?edchainhasbeenc onstructedby redirectingtr ansitions
tomoreattractivestates?itholdsthat?g?g?
Theimportantconclusion isthatbasedonthe setPweareabletoconstructupper
andlowerbounds?Redirectingtransitionstomore?less?attractivestatesyieldsalower
?upper?boundmodel?Also?thericherPthemore?exibilityonehastoconstructbounds?
Beforeturningto theproof ofprecedencepairs?we?rstpresentprecedencepairsforthe
shortestqueuemodel?
Example??? ?Precedencerelationsfortheshortest queue model?
Itma ybesho wnthat state
m
??m
?
?m
?
? ismore attractivethan allstates
n
??n
?
?n
?
?
satisfyingn
?
?n
?
?m
?
?m
?
andn
?
?m
?
? Thismeansthatitis preferableto have less
jobsin thesystemand?ortohavemorebalanceinqueuelengths?Inparticular? denoting
theunityvectors?????and?????by
e
?
and
e
?
?resp??itfollows thatstate
m
ismore
attractivethanitsneighboringstates
m
?
e
?
?
m
?
e
?
and
m
?
e
?
?
e
?
?providedthese
neighborsare inthestatespace??Thelatter precedencesare illustratedin Figure??Note
thattheyimplytheprecedencesmentioned?rst?Thisasp ect willbeexploited inthenext
section?
?
Page 9
m2
m1
?
??
???
????
?????
??????
???????
Figure ?? Precedencerelationsfortheshortestqueuemodelwithone?periodcostgiv en
by ????Eacharro wp ointstoamoreattractive state
It iseasilyveri?ed?byusingthe precedence pairs?thattheOIBandTKmodel give
upperbounds forthemean normalizedwaitingtimeand theothersgivelowerb ounds?In
fact?weonlyneedtheprecedencesb etweenneighboringstatesdepicted inFigure ??And
toestablishthebounds fortheTKandTB model?we even need less?Forthesemodels
weonlyusethat
m
ismoreattractivethat
m
?
e
?
and
m
?
e
?
? resp?
?
?ESTABLISHINGPRECEDENCEPAIRS
Letusconsiderthegeneralmodelagain?SupposethatwehaveasetPofpairs?m?
n?
and thatwew ant toprove for alln??that?cf?????
v
n
?m
??v
n
?n? forall?m
?
n
??P? ???
Thepro of of??? is donebyinduction over n?Takingn??in??? directlyleadstothe
conclusionthattheordering
c?m
??c?n????
should hold forall?m
?
n
??P?Letussupposethatthisisindeedthecaseandconsider
theinductionstep?Assume???holdsforn?To establishitforn??itisconvenient
toexploitthat?istransitive?if?m?
n
? and?n
?
l
? satisfy ???forn??? thensodoes
?m?
l
? forn???So?there maybeasmallsubsetofPwiththe prop ertythatifinequality
???holdsfor thepairsinthatsubset?then italso holdsforall pairs inPbyvirtueof
transitivityof?? Denotesucha subsetbyP
?
?possiblyP
?
is thesame asP?? Hence? to
?
Page 10
prove inequality ???forn??forallpairsinP?it su?cesto dosoforthepairsinthe
smaller setP
?
?
Example??? ?SetP
?
for the shortest queuepr oblem?
Inthe previous sectionwe intro ducedthesetPforthe shortestqueue problem?Thisset
includesallpairs?m?
n? satisfyingn
?
?n
?
?m
?
?m
?
andn
?
?m
?
?LetP
?
betheset
ofpairs?m
?
n
?forwhich
n
isequalto
m
?
e
?
?
m
?
e
?
or
m
?
e
?
?
e
?
?ClearlyP
?
isa
subset ofPanditiseasilyseenthattheinequalities???forthepairsinP
?
generate?by
usingtransitivity?theonesforall pairsinP?
?
Toestablish???for n??? we have to show for each?m?
n
??P
?
that
v
n??
?m??c?m??
X
i
p?m
?
i?v
n
?i??c?n
??
X
j
p?n?
j
?v
n
?j
??v
n??
?n?????
By???? itsu?cestoshowthatthesumsare alsoordered?A common approach isto
comparesimilartermsinthetwosums?i?e?termscorrespondingtothesameevent?such
asanarrivalordeparture??Further?itisusuallysu?cienttodistinguishafewcasesfor
?m?
n?only? It depends onthe applicationonhand which terms are similarand which
cases have tobeconsidered? Below itis sho wn how thisworksfor theshortest queue
problem?
Example????Pr o ofof prece dencepairsforthe shortest queue model?
Let usillustrate for the shortestqueue modelho w ??? maybepro v
edfor the pairs?m?
m
?
e
?
??Wedistinguish fourcases?namely
m
? ??? ???
m
? ???m
?
?withm
?
???
m
?
?m
?
?m
?
???withm
?
??and?nally
m
??m
?
?m
?
?withm
?
??andm
?
?m
?
???In
the thirdcasewe have
v
n??
?m
??c?m???v
n
?m
?
e
?
???v
n
?m
?
e
?
???v
n
?m
?
e
?
?????
v
n??
?m
?
e
?
??c?m
?
e
?
???v
n
?m
?
e
?
?
e
?
???v
n
?m
???v
n
?m
?????
Nowcomparetheright?handsidesof???and????Thedirectcost in ??? isless thanthe
onein????Thesecondterms?bothcorrespondingtoanarrival?areorderedaccordinglyby
theinductionassumption?Thesameholdsforthethirdandfourthterms?Sov
n??
?m??
v
n??
?m
?
e
?
??Theothercasescanbeprovedsimilarly?Tocompletetheinductionstep
we also havetoprove???for thecombinations?m?
m
?
e
?
?and?m?
m
?
e
?
?
e
?
??resp?
?cf? Section?in?????
?
??
Page 11
Itisalsopossibletoestablish ??? moresystematically?whichmaybe usefulincomplex
models??Belowwewillshowthatthis problem canbetranslated in toa transportation
problem ?see e?g??????
Toprove forapair?m?
n
??P
?
thatthesums in???are orderedwemayof course
restrict thesumsto states
i
and
j
forwhichp?m
?
i?andp?n?
j
??resp?? arepositiv e?
Denote the sets ofthesestatesbyV?m
?andV?n??Ifweintroducenonnegativevariables
a?i?
j
?satisfying
p?m
?
i??
X
j
?V
?n?
a?i?
j
??p?n?
j
??
X
i?V
?m?
a?i
?
j
?? ???
thenwe maywrite
X
i
?V
?m
?
p?m
?
i?v
n
?i??
X
j
?V
?n
?
p?n?
j
?v
n
?j
??
X
i?V
?m?
X
j
?V
?n?
a?i?
j
??v
n
?i??v
n
?j
???
????
Intransp ortationterminology?the states
i
aresupplyno deswithsupplyp?m?
i
?andthe
states
j
aredemandnodeswithdemandp?n?
j
??A solutionasatisfying???isanallocation
?therealwaysexistsonesincethesupplyanddemandbothaddupto???Ifthereexists
anallo cationaforwhicha?i
?
j
??? forallpairs?i?
j
???P? thenwe canconcludefrom
the inductionassumption thattheright?handside in????is lessthan orequal to??Suc h
anallo cationiscalled feasible? Hencetheproofof???fora pair?m?n??P
?
has now
been reducedtothat of?ndinga feasibleallo cationfora corresp ondingtransportation
problem? Thistransportationproblemisdenotedby
TP
?m
?
n
??
Theorem???Pr ovided
?i?c?m??c?n
?for all?m
?
n
??P?
?ii?Thetr ansportationpr oblem
TP
?m?
n
?hasa feasiblealloc ationfore ach?m?
n
??
P
?
?whereP
?
isasetofpairsgeneratingP?
itholdsforalln??thatv
n
?m??v
n
?n
?forall?m?
n??P?
Example ????Transportationproblemfor theshortestqueue mo del?
Wewillillustratefortheshortest queue model how ???canbe prov ed forthe pairs?m?
m
?
e
?
?bysolvingthecorrespondingtransportationproblem? Asbeforewedistinguishfour
cases? In case
m
??m
?
?m
?
??? withm
?
??thesupplynodesare
m
?
e
?
?
m
?
e
?
and
m
?
e
?
withsupply??? and??resp?The demandnodes are
m
and
m
?
e
?
?
e
?
with
??
Page 12
demand??and?? resp?Afeasibleallo cationmayonlytransshipsupplybetween pairs
ofnodesinP?i?e??bet w eenthepairs?m
?
e
?
?
m
???m
?
e
?
?
m
?
e
?
?
e
?
???m
?
e
?
?
m??
?m
?
e
?
?
m
?
e
?
?
e
?
? and?m
?
e
?
?
m
?
e
?
?
e
?
??Thistransportationproblemisillustrated
inFigure??The arrowsindicatetowhich no destransshipmentsareallow ed?
?
m−e1μ
?
m−e2μ
?
m+e1λ
? m
2μ
? m+e1+e2
λ
Figure ??T ransportation problem for thepair?m?
m
?
e
?
?? Thearrowsindicatetowhich
nodestransshipments areallowed?
It is easilyveri?ed thattheallocationa?m
?
e
?
?
m????a?m
?
e
?
?
m???and
a?m
?
e
?
?
m
?
e
?
?
e
?
???isa solution to thetransp ortation probleminFigure??
?
Remark????Existenceoffeasibleallocations?
Asimple conditionfortheexistenceofa feasibleallocation is thefollowingone?cf?????????
Thereexistsafeasibleallocationifandonlyif
X
i?U
p?m
?
i??
X
j
?R?U
?
p?n?
j
?forallU?V?m
??
whereR?U?are thestates whichmayreceivesupplyfrom U? i?e? states
j
?V?n?for
whichthereisan
i
?Usuch that?i?
j
??P?
?
Remark????Costfunctions?
It willbeclearthat theprecedence relationsare validfor any costfunctioncsatisfying
ordering ???forall ?m?
n??P?Soforappropriatede?nitionsofcwemayestablish
b oundsforv ariousperformancemeasures?F or instance?fortheshortest queuemo del?if
wetakec?m
???forall states
m
withm
?
?Mand?otherwise?thengcorresponds to
theprobabilitythatthe shortest queueislongerthanM?Thiscostfunction? how ev er?
onlysatis?esordering???forthepairs?m?
n?satisfyingn
?
?m
?
andn
?
?m
?
?butnot
for pairslike?m?
m
?
e
?
?
e
?
??Fortunately?we do notneed thelatterfor the TKand
TB mo del? andhencewe ?ndthatthese mo delsstillproducea lowerand upperbound?
??
Page 13
resp??forthe probabilitythattheshortestqueue is longerthanM?So? thelengthofthe
shortestqueueintheTK?TB? modelisstochastically smaller?larger? thanin theoriginal
model?Asaconsequence? thew aitingtimeis alsostoc hasticallysmaller?larger??
?
?NUMERICAL RESULTS
Inthis section?wepresentfortheshortestqueuemodelsomenumericalresultsforthe
boundsonthemean normalizedwaitingtime W?Recallthateach of the modelsin tro?
duced inSection?canbe e?cientlysolved?seetheremarksat theendofthatsection?
Thekeyparameterineachof thesemodels isthe thresholdT?ThelargerTthemore
accuratetheb ounds willb e? butalsothegreaterthe e?orttocompute them?
The e?ectofTontheaccuracyofthe b ounds isdemonstrated inTable ??F orserver
utilization?????? where?is de?nedas?????welistforincreasingv aluesofTandfor
eachmodelthedi?erenceofthe boundand the trueW? whichis?????? Notethatthe
servicecapacityisnote?cien tly usedtheOIBandTBmodel? whichexplainswhy these
modelsare notstable forT???TheresultsinTable?showthatthebounds are tight
foralreadysmallv aluesofT?exceptforthe ?rstmodel? whichperformspoorly?
TFBCB TJOIBTKTB
???????????????????????????
???????????????????????? ???????????
??????????????????? ????????????????
? ???????????? ??????????? ???????????
? ???????????? ??????????? ?????? ?????
? ???????????? ????????????????? ?????
??????? ?????? ??????????? ?????? ?????
????????????? ?????? ????????????????
Table ??Di?erences of theb oundsand thetrueW?????? for?????
Thesystem withtwoqueues may of coursebe solv ed exactlyandv ery e?ciently ?see
?????F orlargersystems?however? noexactanalyticalresultsareavailaible?Thepower
ofthepresent approachisthat alsow orkswellforlargesystems?In ????? ?extensive
numerical materialcanbefounddemonstrating that the CB?TJ and TB modelsproduce
accuratebounds forthemeanwaitingtime insystems withup to??queues?
??
Page 14
?CONCLUSIONS
Inthispaperwe have presen teda systematicapproachfortheconstructionofb oundsfor
theav eragecost in anin?nite stateMarkovc hain?Theessence of theapproachis thatone
should ?rst try toidentifyprecedencesbetw een statesof the Markovchain?Basedonthese
precedencesitappearstobeeasytoformulatesuitableMarkovchainsproducingbounds
fortheaveragecost?Itisoftenpossibletoconstructasequenceofb oundsconv ergingto
thetrueav eragecost?But? ofcourse?the moreaccuratetheb ound? themoree?ortit
takestocomputeit?
Manyqueueingor inventorysystemscanbedescribedbyMarkovian modelsandrel?
ev antperformancecharacteristicsinthesesystems? such as? e?g?? meanw aiting timesor
meanleadtimes? mayberepresentedin theMarkovianmodelbytheaveragecostfor
someappropriatelychosen cost structure?Therefore?theapproachpresentedinthispa?
permaybe appliedtomany systemstogenerateb oundsfor therelevantp erformance
characteristics?
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