The asymptotic behavior of the price of anarchy

14 Pages·2017·0.17 MB·English

The asymptotic behavior of the price of anarchy
RiccardoColini-Baldeschi1,RobertoCominetti2,
PanayotisMertikopoulos3,andMarcoScarsini1
1 DipartimentodiEconomiaeFinanza,LUISS,VialeRomania32,00197Roma,Italy
2 FacultaddeIngenieríayCiencias,UniversidadAdolfoIbáñez,Santiago,Chile
3 Univ.GrenobleAlpes,CNRS,Inria,LIG,F-38000GrenobleFrance
Abstract. Thispaperexaminesthebehaviorofthepriceofanarchyasafunc-
tion of the traffic inflow in nonatomic congestion games with multiple origin-
destination(O/D)pairs.Empiricalstudiesinreal-worldnetworksshowthatthe
priceofanarchyiscloseto1inbothlightandheavytraffic,thusraisingtheques-
tion:cantheseobservationsbejustifiedtheoretically?Wefirstshowthatthisis
notalwaysthecase:thepriceofanarchymayremainboundedawayfrom1for
allvaluesofthetrafficinflow,eveninsimplethree-linknetworkswithasingle
O/Dpairandsmooth,convexcosts.Ontheotherhand,foralargeclassofcost
functions(includingallpolynomials),thepriceofanarchydoesconvergeto1in
bothheavyandlighttrafficconditions,andirrespectiveofthenetworktopology
andthenumberofO/Dpairsinthenetwork.
1 Introduction
Almosteverycommuterinamajormetropolitanareahasexperiencedthefrus-
tration of being stuck in traffic. At best, this might mean being late for dinner;
at worst, it means more accidents and altercations, not to mention the vastly
increased damage to the environment caused by huge numbers of idling en-
gines. To name but an infamous example, China’s G110 traffic jam in August
2010 brought to a standstill thousands of vehicles for 100 kilometers between
Hebei and Inner Mongolia. Not caused by weather or a natural disaster, this
massive 10-day tie-up was instead laid at the feet of a bevy of trucks swarm-
ing on the shortest route to Beijing, thus clogging the G110 highway to a halt
(while ironically carrying supplies for construction work to ease congestion).
This,therefore,raisesthequestion:howmuchbetterwouldthingshavebeenif
alltraffichadbeenroutedbyasocialplannerwhocouldcalculate(andenforce)
theoptimumtrafficassignment?
Ingame-theoreticterms,thisquestionboilsdowntotheinefficiencyofNash
equilibria.Themostwidelyusedquantitativemeasureofthisinefficiencyisthe
so-calledpriceofanarchy(PoA):introducedbyKoutsoupiasandPapadimitriou
(1999)andsodubbedbyPapadimitriou(2001),thePoAistheratiobetweenthe
social cost of the least efficient Nash equilibrium and the minimum achievable
cost. By virtue of this simple definition, deriving worst-case PoA bounds has
given rise to a vigorous literature at the interface of computer science, eco-
nomicsandoperationsresearch,withmanysurprisingresults.
In the context of network congestion, Pigou (1920) was probably the first
tonotetheinefficiencyofselfishrouting,andhiselementarytwo-roadexample
withaPoAof4/3isoneofthetwoprototypicalexamplesthereof.Theotheris
duetoBraess(1968),andconsistsofafour-roadnetworkwheretheadditionof
azero-costsegmentmakesthingsjustasbadasinthePigoucase.Thesetwoex-
ampleswerethestartingpointforRoughgardenandTardos(2002)whoshowed
thatthepriceofanarchyin(nonatomic)routinggameswithaffinecostsmaynot
exceed4/3.Ontheother hand,ifthenetwork’scostfunctionsare polynomials
of degree at most d, the price of anarchy may become as high as Θ(d/logd),
implyingthatselfishroutingcanbearbitrarilybadinnetworkswithpolynomial
costs(Roughgarden,2003).
Atthesametimehowever,theseworst-caseinstancesareusuallyrealizedin
networkswithdelicatelytunedtrafficloadsandcosts;ifanetworkoperatesbe-
yondthisregime,itisnotclearwhetherthepriceofanarchyisstillhigh.Indeed,
using both analytical and numerical methods, a recent study by O’Hare et al.
(2016) suggests that the PoA is usually close to 1 for very high and very low
traffic,anditfluctuatesintheintermediateregime.Inasimilarsetting,Monnot
et al. (2017) recently used a huge dataset on commuting students in Singapore
to estimate the so-called “empirical” PoA (a majorant of the ordinary price of
anarchy); their observations yield a value between 1.11 and 1.22, suggesting
thattheactualvalueofthepriceofanarchyisevenlower.
Allthisleadstothefollowingnaturalquestions:
1. UnderwhatconditionsdoesthePoAconvergeto1inlightorheavytraffic?
2. Dotheseconditionsdependonthenetworktopology,itscosts,orboth?
3. Can general results be obtained only for networks with a single origin-
destinationpairordotheyextendtonetworkswithmultiplesuchpairs?
1.1 Ourresults
Ourfirstresultisacautionarytale:weshowthatthepriceofanarchymayoscil-
latebetweentwoboundsstrictlygreaterthan1forallvaluesofthetrafficinflow,
eveninsimpleparallel-linknetworkswithasingleorigin-destination(O/D)pair
(cf.Fig.1).Thecostfunctionsinourexampleareconvexanddifferentiable,so
neitherconvexitynorsmoothnessseemstoplayamajorroleintheefficiencyof
selfish routing. Moreover, our construction only involves a three-link network,
sosuchphenomenamayariseinanynetworkcontainingsuchathree-linkcom-
ponent.
c (x)=(cid:2)1+1/2sin(logx)(cid:3)x2 ��������������������������������������
1 ○○○○ ○○○○
○ ○ ○ ○
����� ○ ○ ○ ○
○ ○ ○ ○
o c2(x)=x2 d ������ ○ ○ ○ ○
c3(x)=(cid:2)1+1/2cos(logx)(cid:3)x2 ������� ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ ○○○○○○○○○○○
�����
��-� ��� � �� ���
�����(�)
Fig.1:Anetworkwhereselfishroutingremainsinefficientforbothlightandheavytraffic.
Heuristically,thereasonforthisbehavioristhatthenetwork’scostfunctions
exhibithigher-orderoscillationswhichpersistatanyscale,forbothhighandlow
traffic.Thus,toaccountforsuchpathologies,wetakeatwo-prongedapproach:
– In the low congestion limit, we focus on cost functions that are real ana-
lytic, i.e., they are equal to their power series expansion near 0. Under this
regularity assumption, we show that the PoA converges to 1, no matter the
networktopologyorthenumberofO/Dpairsinthenetwork.
– At the other end of the spectrum, to tackle the high congestion limit, we
introduce the concept of a benchmark function. This is a regularly varying
function c(x) that classifies edges into fast, slow or tight, depending on the
growth rate of the cost along each edge;4 paths are then classified as fast,
slowortight,basedontheirslowestedge.5 Wethenestablishthefollowing
general result: if the “most costly” O/D pair in the network admits a tight
path,thenetwork’sPoAconvergesto1underheavytraffic.
Among other classes of functions, polynomials satisfy all of the above re-
quirements,leadingtothefollowinggeneralprinciple:
Innetworkswithpolynomialcostfunctions,
thepriceofanarchybecomes1underbothlightandheavytraffic.
Inotherwords,abenevolentsocialplannerwithfullcontroloftrafficassignment
would not do any better than selfish agents in light or heavy traffic. Only if the
traffic falls in an intermediate regime can there be a substantial gap between
optimumandequilibriumstates.
4Regularvariationmeansherethatlim c(tx)/c(t)∈(0,∞)forallx>0(cf.Section4.2).
t→∞
5Asanexample,ifallthenetwork’scostfunctionsarepolynomialsofdegreed,alledges,paths
andO/Dpairsaretightwithrespecttothebenchmarkfunctionc(x)=xd.
1.2 Relatedwork
Much of the literature on congestion games is devoted to the study of bounds
for the price of anarchy under different conditions. Roughgarden and Tardos
(2002) proved a bound of 4/3 in the case of affine costs, independently of the
network topology. This bound is sharp in that, for every M > 0, there exists
a network with traffic inflow M and affine costs such that the PoA equal to
4/3. Importantly, ouranalysis shows that the order ofthe quantifierscannot be
exchanged: in any network as above, the PoA gets arbitrarily close to 1 if the
trafficinflowissufficientlylarge.
Worst-case PoA bounds have been obtained for larger classes of cost func-
tions.Forpolynomialcostswithdegreeatmostd,Roughgarden(2003)showed
that the worst possible instance grows as Θ(d/logd). Dumrauf and Gairing
(2006)providedsharperboundsformonomialsofmaximumdegreedandmin-
imum degree q, while Roughgarden and Tardos (2004) provided a unifying re-
sultforcoststhataredifferentiablewithxc(x)convex,whileCorreaetal.(2004,
2008)consideredlessregularclassesofcostfunctions.Forasurvey,thereader
isreferredtoRoughgarden(2007).
Thedifferencebetweenthemeanvalueofthepriceofanarchyanditsworst
valuehasbeen studiedinthecontextof cognitiveradionetworksbyLaw etal.
(2012). Youn et al. (2008) studied the difference between optimal and actual
system performance in real transportation networks, focusing in particular on
Boston’s road network. They observed that the price of anarchy depends cru-
cially on the total traffic inflow: it starts at 1, it then grows with some oscil-
lations, and ultimately returns to 1 as the flow increases. González Vayá et al.
(2015)studiedoptimalschedulingfortheelectricitydemandofafleetofplug-
in electric vehicles: without using the term, they showed that the PoA goes to
1 as the number of vehicles grows. Cole and Tao (2016) showed that in large
Walrasian auctions and in large Fisher markets the price of anarchy goes to
oneasthemarketsizeincreases.Finally,Feldmanetal.(2016)tookadifferent
asymptotic approach and considered atomic games where the number of play-
ers grows to infinity. Applying the notion of (λ,µ)-smoothness to the resulting
sequence of atomic games, they showed that the price of anarchy converges to
thecorrespondingnonatomiclimit.
Fromananalyticstandpoint,theclosestantecedenttoourpaperistherecent
workofColini-Baldeschietal.(2016)whostudiedtheheavy-trafficlimitofthe
priceofanarchyinparalell-linknetworkswithasingleO/Dpair.Theanalysisof
Colini-Baldeschietal.(2016)identifiedthatregularvariationplaysanimportant
part in heavy traffic; however, it offered no insights into the light traffic regime
ortheheavy-trafficlimitofthePoAinnon-parallelnetworkswithmorethanone
O/D pair. Our paper provides an in-depth answer to these questions: we show
that (a) the light-traffic analogue of regular variation is real analyticity; (b) the
topologyofthenetworkdoesn’tmatter;and(c)theadventofseveralO/Dpairs
doesn’tmatteraslongastheyadmitacommonbenchmark(whichisalwaysthe
caseifthenetwork’scostsarepolynomial).
Finally,ontheempiricalside,ourworkshouldbecomparedtothatofMon-
notetal.(2017)whoperformedanempiricalstudyofthepriceofanarchybased
on data from thousands of commuting students in Singapore. Focusing on the
network’s empirical price of anarchy (a PoA majorant), they showed that rout-
ing choices are near-optimal and the price of anarchy is much lower than tra-
ditional worst-case bounds would suggest. Interestingly, the study of Monnot
etal.(2017)alsosuggeststhattheSingaporeroadnetworkisoftenlightlycon-
gested: as such, their results can be seen as a practical validation of the light
traffic results presented here (and, conversely, our results provide a theoretical
justificationfortheirempiricalobservations).
2 Modelandpreliminaries
2.1 Networkmodel
Following Beckmann et al. (1956) and Roughgarden and Tardos (2002), the
basiccomponentofourmodelisafinitedirectedmulti-graphG ≡ G(V,E)with
vertex set V and edge set E. We further assume there is a finite set of origin-
destination (O/D) pairs i ∈ I, each with an individual traffic demand mi ≥ 0
whichhastoberoutedfromanoriginoi ∈ V toadestinationdi ∈ V viaG.
Toroutethistraffic,thei-thO/DpairemploysasetPiof(simple)pathsjoin-
ing oi to di, each path p ∈ Pi comprising a sequence of edges that meet head-
to-tail in the usual way.6 For bookkeeping reasons, we also make the standing
assumption that (a) M ≡ (cid:80) mi > 0 (so there is a positive amount of traffic in
i
thenetwork);and(b)thesetsPi aredisjoint(whichholdsinparticularwhenall
pairs(oi,di)aredifferent).Then,writingP ≡ (cid:83) Pi fortheunionofallsuch
i∈I
paths,thesetoffeasibleroutingflows f = (f ) inthenetworkisdefinedas
p p∈P
F = (cid:110)f ∈ (cid:146)P : (cid:80) f = mi foralli ∈ I(cid:111). (2.1)
+ p∈Pi p
Inturn,aroutingflow f ∈ F inducesaloadoneachedgee ∈ E asx = (cid:80) f ,
e p(cid:51)e p
andwewrite x = (x ) forthecorrespondingloadprofileonthenetwork.
e e∈E
Given all this, the delay (or latency) experienced by an infinitesimal traffic
element in order to traverse edge e is determined by a nondecreasing, nonzero
6Tobeclear,wedonotassumeherethatPiisthesetofallpathsjoiningoitodi,butonlysome
subsetthereof.Thisdistinctionisimportantforpacket-switchednetworkswhereonlypaths
withalowhopcountareused.
continuouscostfunction c : [0,∞) → [0,∞).Specifically, if x = (x ) isthe
e e e∈E
loadprofileinducedbyafeasibleroutingflow f = (f ) ,thentheincurredde-
p p∈P
layonedgee ∈ E isc (x ).Hence,withaslightabuseofnotation,theassociated
e e
costofpath p ∈ P isgivenby
(cid:88)
c (f) ≡ c (x ). (2.2)
p e e
e∈p
Putting together all of the above, the tuple Γ = (G,I,{mi} ,{Pi} ,{c } )
i∈I i∈I e e∈E
willbereferredtoasa(nonatomic)routinggame.7
2.2 Equilibrium,optimality,andthepriceofanarchy
In this setting, the notion of Nash equilibrium is captured by Wardrop’s first
principle: at equilibrium, the delays along all utilized paths are equal and no
higher than those that would be experienced by an infinitesimal traffic element
goingthroughanunusedroute(Wardrop,1952).Formally,aroutingflow f∗ is
saidtobeaWardropequilibrium(WE)ofΓ if,foralli ∈ I,wehave
cp(f∗) ≤ cp(cid:48)(f∗) forall p,p(cid:48) ∈ Pi suchthat fp∗ > 0. (2.3)
BytheworkofBeckmannetal.(1956),itiswell-knownthatWardropequi-
libria can be characterized equivalently as solutions of the (convex) minimiza-
tionproblem:
(cid:88)
minimize C (x ),
e e
e∈E
(cid:88) (WE)
subjectto x = f , f ∈ F,
e p
p(cid:51)e
where C (x ) = (cid:82) xec (w)dw denotes the primitive of c . On the other hand, a
e e 0 e e
sociallyoptimum(SO)flowisdefinedasasolutiontothetotalcostminimization
problem:
(cid:88)
minimize L(f) = f c (f),
p p
p∈P (SO)
subjectto f ∈ F.
Toquantifythegapbetweensolutionsto(WE)and(SO),wewrite
Eq(Γ) = L(f∗) and Opt(Γ) = min L(f), (2.4)
f∈F
7Forsimplicity,whenthereisasingleO/Dpair,wewilldropIandtheindexialtogether.
where f∗ is a Wardrop equilibrium of Γ. As Beckmann et al. (1956) showed,
L(f∗) has the same value for all equilibria f∗. The game’s price of anarchy
(PoA)isthendefinedas
Eq(Γ)
PoA(Γ) = . (2.5)
Opt(Γ)
Obviously,PoA(Γ) ≥ 1withequalityifandonlyifWardropequilibriaarealso
sociallyefficient.Ourmainobjectiveinwhatfollowswillbetostudytheasymp-
toticsofthisratiowhen M → 0or M → ∞.
3 Anetworkwhereselfishroutingisalwaysinefficient
We begin by constructing a three-link network where the price of anarchy os-
cillates between two values strictly greater than 1, no matter the traffic inflow
M.Tothatend,letΓ beanonatomicroutinggameconsistingofasingleO/D
M
pairwithtrafficinflow M.Thistrafficistoberoutedoverthethree-linkparallel
graphofFig.1withcostfunctions
(cid:104) (cid:105)
c (x ) = xd 1+ 1 sin(logx ) , (3.1a)
1 1 1 2 1
c (x ) = xd, (3.1b)
2 2 2
(cid:104) (cid:105)
c (x ) = xd 1+ 1 cos(logx ) , (3.1c)
3 3 3 2 3
where d is an integer. It is easy to see that the cost functions (3.1) are convex
and differentiable on [0,∞) for all d ≥ 2. Furthermore, the functions x c (x )
e e e
arestrictlyconvex,sotheproblem(SO)admitsauniqueoptimumtrafficdistri-
bution. Hence, the only way for the game’s price of anarchy to be equal to 1 is
if the game’s (also unique) Wardrop equilibrium coincides with the network’s
sociallyoptimumflow.
For a given value of the total inflow M = x + x + x , the load profile
1 2 3
x = (x ,x ,x )isaWardropequilibriumifandonlyifc (x ) = c (x ) = c (x ),8
1 2 3 1 1 2 2 3 3
i.e.,ifthenormalizedprofilez = x/M satisfies
(cid:104) (cid:105) (cid:104) (cid:105)
zd 1+ 1 sin(logMz ) = zd = zd 1+ 1 cos(logMz ) . (3.2)
1 2 1 2 3 2 3
Likewise,afterdifferentiatingandrearranging,thecorrespondingconditionsfor
thenetwork’ssociallyoptimumfloware
(cid:104) (cid:105)
zd 1+ 1 sin(logMz )+ 1 cos(logMz )
1 2 1 2(d+1) 1
(cid:104) (cid:105)
= zd = zd 1+ 1 cos(logMz )− 1 sin(logMz ) . (3.3)
2 3 2 3 2(d+1) 3
8Sinceanunusededgealwayshasacostofzero,allpathsareusedatequilibrium.
A simple algebraic argument shows that Eqs. (3.2) and (3.3) never admit
a common solution; since Eqs. (3.2) and (3.3) are periodic in logM, it also
followsthatthegame’spriceofanarchyoscillatesperiodicallyatalogarithmic
scale.Thus,focusingontheperiod1 ≤ M ≤ e2π,weconcludethat
inf PoA(Γ ) = min PoA(Γ ) > 1, (3.4)
M M
M≥0 1≤M≤e2π
i.e., Wardrop equilibria in the network of Fig. 1 remain strictly inefficient no
matterthevalueof M.
4 NetworkswithasingleO/Dpair
The example of the previous section shows that the price of anarchy may be
bounded away from 1 for all values of the traffic inflow, even in a three-link
parallel network with a single O/D pair. That being said, the behavior of the
cost model (3.1) at both ends of the congestion spectrum is fairly irregular, so
the question remains: is selfish routing bad under light/heavy traffic for more
“reasonable”classesofcostfunctions?
4.1 Thelighttrafficlimit
A key observation regarding the counterexample (3.1) is that the “topologist’s
trig” terms sin(logx) and cos(logx) are highly pathological: their oscillations
become dense near 0, so the corresponding cost functions do not admit deriva-
tives of all orders at 0. To exclude such singularities, we will instead focus on
functionsthataresmoothenoughtoadmitafaithfulTaylorexpansionat0:
Definition1. Afunctiong: (cid:146) → (cid:146)iscalled (real)analyticat x ifthereexists
0
anopenneighborhoodU of x andrealnumbersg ,k = 0,1,...,suchthat
0 k
(cid:88)∞
g(x) = g (x−x )k forall x ∈ U. (4.1)
k 0
k=0
All polynomials are analytic, as are exponential, trigonometric, and most
special functions (like the gamma function). Remarkably, under this mild reg-
ularity requirement, we have the following general result for lightly congested
networkswithasingleO/Dpair:
Theorem1. Let Γ be a nonatomic routing game with a single O/D pair and
M
trafficinflow M.Ifthenetwork’scostfunctionsareanalytic,wehave
lim PoA(Γ ) = 1. (4.2)
M
M→0+
Despite appearances, Theorem 1 is fairly surprising. Indeed, at first sight,
onewouldexpectthatwhenM → 0,trafficissolightthatitdoesn’treallymatter
howitisrouted.Thisisindeedthecaseif,forinstance,allpathsinthenetwork
exhibitapositivecostfor M = 0.However,ifthecostofanemptypathiszero,
this is no longer the case: the optimum and equilibrium assignments could be
fairlydifferent(evenforlowtraffic),sothereisnoapriorireasonthattheprice
of anarchy should converge to 1 as M → 0 (the example of Section 3 clearly
illustrates this phenomenon). Theorem 1 shows that all that is needed for this
tooccurisforthenetwork’scostfunctionstobefaithfullyrepresentedbytheir
Taylor series. When this regularity condition is met, optimum and equilibrium
costsnolongerfluctuatebut,instead,theyconvergetothesamevalue.
4.2 Theheavytrafficlimit
In the heavy traffic limit, Taylor expansions are no longer meaningful so we
require a different criterion to rule out pathological oscillations. We do so by
meansofthenotionofregularvariation:
Definition2. Afunctiong: [0,∞) → (0,∞)issaidtoberegularlyvaryingif
g(tx)
lim isfiniteandnonzeroforall x ≥ 0. (4.3)
t→∞ g(t)
In words, regular variation means that g(x) grows at the same rate when
viewedatdifferentscales.Standardexamplesofregularlyvaryingfunctionsin-
cludeallaffine,polynomialand(poly)logarithmicfunctions.Theconceptitself
dates back to Karamata (1933) and has been used extensively in probability
and large deviations theory (see e.g. de Haan and Ferreira, 2006; Jessen and
Mikosch, 2006; Resnick, 2007); for a comprehensive survey, the reader is re-
ferredtoBinghametal.(1989).
With all this at hand, we will discard growth irregularities (such as those
observedinSection3)bypositingthateachcostfunctionc (x)canbecompared
e
asymptotically to some regularly varying function c(x). Specifically, given an
ensembleofcostfunctionsC = {c } ,wesaythataregularlyvaryingfunction
e e∈E
c: [0,∞) → (0,∞)isabenchmarkforC ifthe(possiblyinfinite)limit
c (x)
α = lim e (4.4)
e
x→∞ c(x)
existsforalle ∈ E.
When it exists, this limit will be called the index of edge e, and e will be
calledfast,slow,ortight(relativetoc)ifα isrespectively0,∞,orin-between.
e
Since bottlenecks in a path are caused by the slowest edges, we also define the
indexofapath p ∈ P as
α = maxα , (4.5)
p e
e∈p
and we say that p is fast, slow, or tight based on whether α is 0, ∞, or in-
p
between.Finally,wesaythatthenetworkistightiftheindexofthenetwork
α = minα (4.6)
p
p∈P
isfiniteandpositive(0 < α < ∞).
In words, a path is fast (resp. tight/slow) if its slowest edge is fast (resp.
tight/slow), and the network is tight if its fastest path is tight. In particular,
tightness guarantees that the network admits a path whose cost grows asymp-
totically as a multiple of some regularly varying benchmark function c(x). The
importance of this growth requirement is illustrated by the counterexample of
Section 3: if we slightly relax the tightness concept by asking that the network
admitsapathwhosecostgrowsasΘ(c(x)),thepriceofanarchymaybebounded
awayfrom1forallvaluesof M.Instead,undertightness,wehave:
Theorem2. Let Γ be a nonatomic routing game with a single O/D pair and
M
trafficinflow M.Ifthenetworkistight,then
lim PoA(Γ ) = 1. (4.7)
M
M→∞
Inotherwords,ifthefastestpathinthenetworkistight,selfishroutingbecomes
efficientinthehighcongestionlimit.
Asanimmediatecorollary,wethenhave:
Corollary1. InnetworkswithpolynomialcostsandasingleO/Dpair,wehave
PoA(Γ ) → 1as M → ∞.
M
Proof. Let d be the degree of c , set d = max d , and let d = min d .
e e p e∈p e p∈P p
Thenetworkisclearlytightwithrespecttoc(x) = xd,soTheorem2applies. (cid:4)
Combining Theorem 1 and Corollary 1, we conclude that selfish routing
becomesefficientunderbothlightandheavytrafficinnetworkswithpolynomial
costs.Beyondthepolynomialcase,Theorems1and2showthatanalyticityand
regularvariationcanbeseenasdifferentsidesofthesamecoin:theybothensure
asymptotic regularity and they both exclude pathological oscillations (at zero
and infinity respectively). As such, our results for light and heavy traffic are
chieflysetapartbythenotionoftightness(whichonlyappliesforheavytraffic).
The reason for this qualitative difference is that costs might diverge to infinity
at very different rates when the traffic inflow grows large; by contrast, all costs
arefinitewhenthereisnotraffic,sothenotionoftightnessisredundantthen.