moving data across networks

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Moving Data Across Networks

Arran Bedda0602158

April 9, 2010

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1 Introduction

In todays society, we find more networks than ever before, and with the coming

of telephony and the internet we find ourselves with very different kinds of con-cerns than in previous times. It is no longer just a case of trying to just make theshortest possible routes from one place to another. With the increases in trafficspeed and load, congestion and shaping of traffic flow is now a major concern.

Problems of congestion can be caused by an inefficient network design, and con-sidering the underlying mathematics can be an interesting, complex and counter-intuitive task. It is this that makes network design an area worthy of mathematicalstudy. We consider here a range of different network types, ranging from road andrail type networks to internet style considerations. Whilst seemingly different they

all share a large proportion of underlying mathematical ideas and structures. Anunderstanding of these, ideas and problems can help in the design of better, moreefficient networks.

Armed with better networks it is possible to drastically improve the quality ofservice that users receive without requiring the input of excessive capital. Minimis-ing expenditure is almost always desirable, and savings obtained by efficient useof current resources allows capacity expansion on a wider scale, and so is soughtby both the management and users of the network.

The aim here is to show some of the subtlety to be found within the art ofnetwork design and implementation, how the counter intuitive nature of theseproblems can lead you astray and that with large networks it can be almost im-possible to analyse some of these network properties in a good measure of time.We consider a couple of areas where there is still current mathematical researchongoing:

First we consider Braess paradox, where the addition of extra capacity toroutes can in fact worsen the performance of the network.

Secondly we will consider dynamical routing, trying to find an easily ap-plicable algorithm to route the traffic, in an attempt to create as reliable anetwork as possible and how, under certain switching algorithms even servercontrolled networks can suffer from similar issues to Braess paradox.

It is notable that Braess paradox was only shown in 1968, and as such this and

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the ideas of trunk reservation, are not just current but also recent, and that thereis much to still be understood about them. Together, their complexity, newnessand relevance to modern life make for a very interesting topic to study.

2 Braess Paradox

2.1 Motivation

Consider a heavily congested road network. If we want to ease the congestion itseems obvious that we want to add extra capacity. However, this is not as simpleas first apparent. With suitably low amounts of flow through the network, suchthat we do not encounter congestion, it is true that adding extra routes and ca-pacity can only help or at least not harm the network throughput.

On the other hand if the network is congested then this extra capacity can, infact, harm the network causing a detrimental effect on the quality of service. Thisis, in general caused by a lack of control over the users which results in them usingpersonally optimum, rather than socially optimum routes. Part of this idea canbe understood by considering Braess Paradox:

For each part of a road network, let there be a given number of cars start-ing from it, and the destination of the cars. If every driver takes the path most

favourable to him, then the resulting running times may not be optimal and fur-thermore an extension of the road network may cause a redistribution of traffic,resulting in longer individual running times. [10]

To understand this let us first rigorously define what we mean by mostfavourable. We consider this as an application of game theory, with each in-dividual user trying to minimise their individual costs. We will consider a directedgraph G(V, E), and give the graph latency functions over the edges of the graph,ce:e E. This will define the cost to traverse an edge. Now we define a set ofpaths Pin our graph with P P, where ifPa path betweem nodes s and t, then

P is of the form (s, va) =e0, e1, . . . , en = (vb, t), where ei E

Now for eachP P, we define our cost function for the path as cP =n

k=0 cek .In our definition of Nash Equilibrium below each player i will be trying to find apath Pi : cPi cPjPj P. That is minimising the sum of the latency functionsalong the path. Here we will deal only with positive linear cost functions of flowon the arcs. The positivity is to ensure loop free and therefore finite routes, whilst

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the linearity helps to simplify our model. However a brief mention will be madeto more general functions later.

This provides a fairly accurate model of road networks considering cost as thelength of time spent on a journey, and given a quicker route a driver will often takeit in an attempt to get to their destination earlier. However we assume that alldrivers know about the routes, and as such congestion may well follow when allthe routes again reach some form of equilibrium. We call this Nash equilibriumand it is defined below.

2.2 Nash Equilibrium

Definition 2.1. Nash EquilibriumNash equilibrium is a solution concept of a game involving two or more players, inwhich each player is assumed to know the equilibrium strategies of the other playerswhere no player has anything to gain by changing only their strategy. Formally:[?](S, F) a game with n players whereS=S1S2 . . .Sn. the strategy set forn and F the payoff function. Letxi be a strategy for all players except I, thenx

a Nash Equilibrium if:

i, xi S, xi=x : f(xi , x

i)f(xi, xi)

That is given a potential state of play at one instance, a player will changehis strategy, here their route, such that it is as minimal as possible given the otherplayers current strategies. We then consider this repeatedly until we reach a point,if it exists, where no player can further improve their return.

Now, we would like this equilibrium to exist order for us to obtain any inter-esting results. Uniqueness of this equilibrium is also desirable, as it ensures anyalgorithm we use will necessarily obtain a meaningful and sensible outcome. For

this we need to consider two different types of flow:

Non atomic flow, where there are a large number of users, each individu-ally negligible. As such we assume that we may split the traffic arbitrarilydown routes.

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Atomic flow, where we have a smaller number of players, and as suchwe must consider each player individually, as they control a non-negligibleamount of traffic. In this model we split the traffic into appropriately sized

blocks which must all follow the same route.

We consider these separately as whilst conceptually similar, different results aboutequilibrium hold about them and thus they require different analytical methods.As an example of their difference, consider the below diagram: [23]

Figure 1

Now, if we consider two players trying to route traffic from source s, to sinkt,with loads 1 and 2 respectively. Now if we label thae paths:

P1 = s t

P2 = s v t

P3 = s w t

P4 = s v w t

Then:

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If player 2 takes paths P1 or P2, then response by player 1 that minimisescost is the path P4

If player 2 takes paths P3 or P4, then response by player 1 that minimisescost is the path P1

If player 1 takes path P4, then response by player 2 that minimises cost isthe path P3

If player 1 takes paths P1, then response by player 2 that minimises cost isthe path P2

Thus we have shown that Nash Equilibrium does not exist here for our atomicflow. It is shown below, that the non-atomic case will give us a unique equilibriumpoint and in general for continuous, non-decreasing cost functions a Nash Equilib-rium will always exist for the non-atomic case.

2.3 Existence and Uniqueness of Nash Equilibrium in Atomic and Non-Atomic Flows [23]

In preparation we will now define some terminology used in the below proofs: (G,r,c) is an instance where:

Gis the graph, with the set Vof nodes and Eof edges

ris the rate variable, that is for our source-sink pairs (s1, t1), . . . , (sk, tk)withri the amount of traffic travelling betweensi and ti. Here we gen-erally consider only a single source-sink pair.

ca non-negative, continuous, non-decreasing cost function on each edgee, ce R+ R+

faflow, that is a distribution of our traffic between their respective sourcesand sinks, with fe representing the flow along an edge e

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We now consider some theorems about the existence and uniqueness of equi-librium flows in atomic and on-atomic networks.First on a non-atomic network:

Theorem 2.2. Let(G,r,c) be a non-atomic instance where the cost functions arecontinuous non-decreasing. Then:

(a) The instance(G,r,c) admits at least one equilibrium flow.

(b) Iff andf equilibrium flows for (G,r,c), then ce(fe) = ce(fe) for everyedgee.

Here we can note that (b) and our definition of a Nash Equilibrium flow to-gether imply that two equilibrium flows of a non-atomic instance have equal costand so implies the uniqueness of our equilibrium.To prove the theorem we use a potential function, where we take some functionthat is optimised precisely at our points of equilibrium.

Proposition 2.3. let (G,r,c) a non-atomic instance such that for every edgee,the functionxce(x) is convex and continuously differentiable. Letce denote themarginal cost function of the edgee, withce(x) = (xce(x))

=ce(x)+xce(x). Then

f is an optimal flow for(G,r,c)if and only if, for each commodityi {1, 2, . . . , k}and every pairP, P Pi ofsi ti paths withfp >0

cP(f) c

P(f).

Proof. This follows from the first order conditions of a convex optimisation prob-lem with non-negative constraints. The details of this proof are omitted.

Now note that from the definition of a Nash equilibrium and the proposition

that the following corollary holds:

Corollary 2.4. Let(G,r,c) be a non-atomic instance such that, for every edgee,the functionxce(x) is convex and continuously differentiable. Letc

e denote the

marginal cost function of edge e. Thenf is an optimal flow for (G,r,c) if andonly if, it is an optimal flow for(G,r,c).

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Now we construct our potential function. We need a function he such thathe(x) =e(x). Thus setting he(x) =

x

0 ce(y)dy for each edge e gives us the desired

function. Sincece continuous and non-decreasing for every edge e every function

he is both continuously differentiable and convex.As such, if we define the potentialfunction of our non-atomic instance (G,r,c)

(f) =eE

fe0

ce(x)dx

This satisfies the same conditions as in our definition, and thus the equilibriumflows are precisely those that minimise globally.

Proposition 2.5. Let (G,r,c) be a non-atomic instance. A feasible flow for

(G,r,c) is an equilibrium flow if and only if it is a global minimum of the cor-responding potential function given above.

Proof of Proposition and Theorem 2.2. First we note that, by definition, the setof feasible flows of (G,r,c) can be identified with a compact (closed and bounded)subset of|P |-dimensional Euclidean space. Since the edge cost functions are con-tinuous, the potential function is a continuous function on this set. Then, byWeirstrass theorem, achieves a minimum value on this set, and by the previousproposition, every point at which attains a minimum corresponds to an equilib-

rium flow of (G,r,c)

For (b) we see that each cost function is non-decreasing, amd hence each sum-mand on the right hand side is convex and so is a convex function.Now suppose that f and fare equilibrium flows for (G,r,c). By the proposition,both f and f minimise the potential . We consider all convex combinations off and f, that is all vectors of the form f+ (1 )f for [0, 1]. All of thesevectors are feasible flows. Since convex, a chord between two points on its graphcannot pass below its graph, ie. [0, 1]:

(f+ (1)f) (f) + (1)(f)

Since both f and f are global minima of , the inequality must hold for all oftheir convex combinations. Since every summand of convex, this can only occurif every summand

x0

ce(y)dy is linear between the values fe and fe. Similarly, this

implies that every cost function ce is constant between fe and fe.

Now we look at it again for an atomic instance:

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Theorem 2.6. Let (G,r,c) be an atomic instance in which every traffic amountri is equal to some common positive value R. Then (G,r,c) admits at least oneequilibrium flow.

Proof. We can assume without loss of generality that R = 1. We now make ourpotential function discrete and let

a(f) =eE

fei=1

ce(i)

for every feasible flow f. Where we have replaced thefe0

c(x)dx by the sum

fei=1 ce(i).

Now, since the atomic instance (G,r,c) has a finite number of players, eachwith a finite number of strategies, it is clear that there are only a finite number ofpossible flows. One of these, say f, is a global minimum of the potential functiona. We then claim that f is an equilibrium flow for (G,r,c).To prove it, assume for contradiction, the player i could strictly decrease its costby deviating from the path Pto the path P, yielding a new flow f, that is:

0> cP(f)cP(f) =

eP\Pce(fe+ 1)

eP\Pce(fe).

However, consider the impact of player is deviation on the potential function a:for edges in P\Pthe corresponding sum acquires the extra term ce(fe +1) and foredges inP\Psheds the termce(fe) and for edges inP Pthe sum stays the same.Thus a(f)(f) =

eP\Pce(fe+ 1)

eP\Pce(fe) < 0 thus contradicting

our choice off.

Corollary 2.7. The best response dynamic is guaranteed to converge to an equi-librium. That is if each player reacts to the current network state, on the currentbest possible route, then it will converge to a nash equilibrium.

Take for ExampleWithout AB ifadrivers go via Aand b viaB, then:

SA Etakes a/100 + 45minutes

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Figure 2

SB Etakes b/100 + 45minutes

Now if half of the 4000 drivers take each route we note that it takes each driver65 minutes, and if any driver changes route it will take them longer, therefore weare at a Nash Equilibrium, and by uniqueness the Nash Equilibrium.

With A B we note that every driver will clearly go S A B E asS A is always less than 40 minutes which is less thanSB and similarly willtake B E, then any driver changing route will take longer, and thus all driverstake 80 minutes.

This behaviour causing the loss of social optimisation is a result of the greedyand non-cooperative nature of the networks users, and the fact the Nash equilib-rium is not necessarily optimum.

We note that with traffic below 3250 that the extra route is beneficial to thetraffics running time, and that above 9000 the extra route is not, in fact, used.This again is a common behaviour, and occurs in general, because one route will

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be better with low flows and the other with high flows, and so with suitably highlevels of congestion, the high flow route is used solely. The values for this can beobtained using a convex linear programming.

2.4 Prevalence of Braess Paradox in Random Graphs [2]

One might question how common this phenomenon is, given its counterintuitivenature and seeming commonness of human interaction with these situations, how-ever it was shown by Steinberg and Zangwill that under reasonable assumptionsthat a randomly added route to a network was about as likely as not to be detri-mental to the flow. Some proof as to the validity of these assumptions is givenby the real life occurrences of the paradox, for example in Stuttgart, when after

a large road improvement program that only showed any positive improvement inthe traffic after one of the newly built road sections was closed again, or in NewYork City on Earth Day in 1990:New York Citys traffic commissioner decided to close 42nd street which as everyNew Yorker knows is always congested. Many predicted it would be doomsday.But to everyones surprise Earth Day generated no historic traffic jam and flowactually improved during the closure.[24]

[2] Now if we let N be our transportation network, with Tsource-sink pairs,w1, w2, . . . , wT. Let each source sink pair w be connected by a specified set of

allowable routes. Suppose that at equillibrium n of them have Positive flow.For source-sink pair wdesignate the routes with positive flow as Ri, i= 1, 2, . . . , n,

where route Ri consists of arcs labelled ei,1, e

i,2, . . . , e

i,s, . . .. Note that each arc

may have multiple labels. Further each arc ei,shas an associated user cost functionindependant of labelling, and given by:

ci,s(fi,s)a

i,sf

i,s+b

i,s (1)

where ai,s> 0, bi,s 0. Here f

i,s, the flow through arc e

i,s, is given by:

f

i,s=

T=1

nj=1

F

j (2)

where ei,s Rj and F

j is the flow along route R

j . Notice that f

i,s is the sum of

the total flow of all routes of all source-sink pairs that pass through ei,s.To proceed with Braess Paradox in the network N, let us now allow an additionalroute R to connect some source-sink pair. Without loss of generality we may

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assume that Rconnects s-s pair w1.Call the new network N. Assume that at equilibrium in Nthe original

n= n1+n2+. . .+nT (3)

routes are used. Then we say that the network N isBraessif after adding the newrouteR the user cost at equilibrium increases for all users travelling between thes-s pair w1 In order to state the theorem, we first define

,i,s,j as follows:

,1i,s,j =

1 ifei,s R , ei,s /R

1j

1 ifei,s /R , ei,s R

1j

0 otherwise

(4)

,i,s,j(=1)=

1 ife

i,s R

j

0 otherwise

To explain, consider the new route R and an old route R1j connecting the s-s pair

w1. The indicator ,1i,s,j tells is whether the new route Rand the old route R

1j use

the same arc. Consider an arc ei,s. If,1i,s,j = 1, then the new route R uses that

arc, but the old route R1j doesnt. If,1i,s,j = 1, then the new route R does not

use that arc, but the old route R1j does. If,1i,s,j = 0, then either both R and R

1j

use the arc, or neither does.Likewise, consider some other s-s pair w, = 1, and a route R

j connecting w.

Again, consider an arc ei,s. If,i,s,j = 1, then the route Rj uses the arc, but if,i,s,j= 0, then Rj does not use the arc. (see figure 3)

We now define:m,i,j =

s

,i,s,jai,s (5)

and then the nn matrices:

M, = [m,i,j ] = 1, 2, . . . , T , (6)

= 1, 2, . . . , T .

As will be demonstrated shortly, m,i,j can be interpreted as the increase in

user cost along routeRi resulting from a unit decrease in flow along route Rj Let

[X] =

XX...

X

and X= X, X, . . . X

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Figure 3

Also, ifAa matrix, specify its determinant by|A|. Then we have:

Theorem 2.8. Assume all n routes used in N are also used inN. If D2 = 0,thenNis Braess if and only if:

1. The new routeR is used, and

2. (1)n(D1/D2)> 0. where

D1 =

[1] M1,1 M1,2 M1,T

[1] M2,1

M2,2

M2,T

. . . ...

... ...

[1] MT,1 MT,2 MT,T

11

. . .

1

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D2=

[0] M1,1 M1,2 M1,T

[1] [0] M2,1 M2,2 M2,T

[1] ...

... ...

. . . ... ...

...

[0] ...

... ...

[1] MT,1 MT,2 MT,T

0 10 1

. . .

0 1

Proof. Denote the flows in N(The network before the addition of the new route)

alond the routesRi byFi , = 1, 2, . . . , T ,i = 1, 2, . . . , n. Denote the flows in N(the network after the addition of the new route) along routes Ri by

Fi , where:

Fi =Fi

i = 1, 2, . . . , T , (7)

i= 1, 2, . . . , n.

Here each i , the loss in flow along the route Ri , can be positive, negative, or

zero.Denote by F the flow along the new route R joining s-s pair w1. Because thenumber of users travelling between each s-s pair is fixed, for s-s pair w1 where therouteRwas added:

n1i=1

F1i +F=

n1i=1

F1i (8)

And for s-s pair w. = 1, where no route was added:

ni=1

Fi =n1i=1

Fi (9)

from (7), (8), (9) :n1i=1

1i = F (10)

n1i=

i = 0 (a= 1) (11)

Recall that fi,s denotes the flow along arc ei,s on N. Let

fi,s denote the flow

through ei,s inN.

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Specify i as the change in user cost along routeRi . Then:

i = s

ci,s(fi,s)

s

ci,s(fi,s)

By (1):

i =s

(ai,sfi,s+b

i,s)

s

(ai,sfi,s+b

i,s)

Hence:i =

s

ai,s(fi,sf

i,s) (12)

But by (2), in N the flow in arc ei,s is given by:

fi,s=

T=1

nj=1

Fj ei,s Rj

In Nsince the route R has been added:

fi,s=(T

=1

nj=1

Fj ) +F

ei,s Rj e

i,s R

Using (12)

i =s

ai,s(T

=1

nj=1

Fj +F

T=1

nj=1

Fj )

ei,s Rj e

i,s R e

i,s R

j

But since Fj =Fj

j This reduces then to:

i = s

ai,s(T

=1

n

j=1

j +F

ei,s Rj e

i,s R

But from (10), F =n1

j=1 1j , thus:

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i =s a

i,s(

T=1

nj=1

j +{

n1j=1

1

j})

ei,s Rj e

i,s R

=s

ai,s(n1

j=1

1j T

=2

nj=1

j +{n1

j=1

1j})

ei,s R1j e

i,s R

j e

i,s R

=s

ai,s(n1

j=1

[{1}+{1}]1j +T

=2

nj=1

{1}j )

ei,s R1j ei,s R ei,s Rj

Now, by (4)

i =s

ai,s(n1j=1

,1i,s,j1j +

T=2

nj=1

,i,s,jj )

= s ai,s(

T

=1nbeta

j=1 ,i,s,j

j )

Then, by changing the order of summation we have terms

s ,i,s,ja

i,s which by

(5) correspond to m,i,j , and so gives us:

i =T

=1

nbetaj=1

m,i,j j (13)

Now, by assumption, all routes Ri used inNare also used inN. So by Wardrops

principle, that is that at Nash Equilibrium that the length of any unused routes

used must be at least long as all of the used ones, user cost at equilibrium isidentical for thenroutes connecting s-s pairw, = 1, 2, . . . , T . Thus, thechangein user cost is identical for these routes. This means that for = 1, 2, . . . , T thereexists some such that:

i = i i= 1, 2, . . . , n (14)

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So, from (13), (14), (10), (11), we have the following system:

0 =1 +j

m1,1,j j (15)

...

0 =1 +

j

m1,n1,jj

0 =2 +

j

m2,1,j j

...

...

0 =T +

j

mT,nt,jj

0 =j

1j F

0 =j

2j

...

0 =j

Tj

The system (15) consists of n+ T linear equations in n+ T+ 1 unknowns,where n= n1+. . .+nT, the total number of allowable routes defined in (3). Tosolve this system, apply Cramers Rule and obtain:IfD2 = 0, then

F = (1)n(D1/D2)1 (16)

where D1 and D2 are as above.By definition, Nis Braess iff 1 >0. However an obvious necessary condition forNto be Braess is that the new route R be used; this is, that F > 0. Thus from(16), ifD2= 0 then the network is braess iff:

1. The new route is used, and

2. (1)n(D1/D2)> 0

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We note now that our starting conditions are fairly sound. Our linear cost func-tions are a good approximation to real conditions, with our constant representinga minimal time to traverse the edge when empty, and the linear part representing,

with a reasonable accuracy the effects of congestion. It is also reasonable to assumethat a route would only be added to a network if it were shorter than some other,and as sich we may wxpect it to be used. Given also that we are mainly looking atthe effects of congestion, it is unlikely that the new route will cause others to beabandoned because, as traffic migrates to our new route, then the time to traversethe old one is reduced.

Essentially then, the theorem states that the network will be Braess if and onlyif !!(2) holds, namely:

(1)n(D1/D2)> 0

This ratio is about as likely to be positive as nonpositive. Thus we see that, undera variety of circumstances, Braess Paradox is about as likely to occur as not.

2.5 The Damage that Braess Paradox can Cause [3]

Now in order to understand the importance of the effect of selfish routing, it isdesirable to know how much damage it can cause, or how far from the social op-timum it take us.

Lemma 2.9. (a) a non-atomic flowf is at Nash equilibrium inG if and only iffor each source-sink pairi andP, P Pi withfP >0,

eP

aefe+beeP

aefe+be

(b) a flow f is (globally) optimal in G if and only if for each source-sink pairi andP, P Pi withf

P >0,

eP

2aefe +be

eP

2aefe +be.

Corollary 2.10. LetG be a network in which each edge latency functionce is ofthe form ce(x) = aex. Then for any rate vector r, a flow feasible for (G,r,c) isoptimal if and only if it is at Nash equilibrium.

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Proof. This follows as for feasible flow for such an instance (a)(b) asbe= 0eG.

Lemma 2.11. Suppose (G,r,c) has linear latency functions and f is a flow atNash equilibrium. Then,(a) the flowf /2 is optimal for(G,r/2, c)(b) the marginal cost of increasing the flow on a pathPwith respect to f /2 equalsthe latency ofPwith respect to f.

Proof. For part (a) we consider that iffsatisfies the conditions for lemma 4.1 (a)then f /2 satisfies the conditions for lemma 4.1 (b) (G,r/2, l).Now for part (b) recall that for edgee with latency functionce(x) =aex + be thenehas marginal cost functionce(x) = 2aex + be, and thus ce(fe/2) =ce(fe) for eachedge e and so cP(f /2) =cP(f) for each pathP P.

Lemma 2.12. Suppose (G,r,c) is an instance with linear latency functions forwhichf is an optimal flow. LetCi(f

) be the minimum marginal cost of increas-ing flow on ansi ti path with respect to f

. Then for any >0, a feasible flowfor the problem instance(G, (1 +)r, c) has cost at least

C(f) +k

i=1

Ci(f)ri.

Proof. First, note that if eachCi is nondecreasing inri, then routingriadditionalunits of flow fromsitoti would cost at leastC

i(f

)riand the lemma would thenfollow easily by summing over si ti pairs. Although it is intuitively plausiblethat marginal costs are increasing in the amount of flow (it is certainly true foreach edge individually), the proof requires a little work.Formally, fix > 0 and suppose f is feasible for (G, (1 +)r, x). In general femay be larger or smaller than fe . For any edge e E, convexity of the functionxce(x) =aex2 +beximplies that

ce(fe)fe ce(fe )fe + (fef

e )c

e(f

e ).

In essence, this inequality states that estimating the cost of changing the flow valueon edgeefromfe tofeby (fef

e )c

e(f

) (i.e., by the marginal cost of flow increaseatfe times the size of the perturbation) only underestimates the actual cost of anincrease (when fe> f

e ) and overestimates the actual benefit of a decrease (when

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fe< fe ). We may thus derive:

C(f) =eE

ce(fe)fe (17)

eE

ce(fe )f

e +

eE

(fefe )c

e(f

e ) (18)

=C(f) +k

i=1

PPi

lP(f)(fPf

P). (19)

Since we have Ci(f) cP(f

) for each i and each P Pi and equality holdsunless fP= 0 (see Lemma 4.1(b)), we obtain:

C(f) C(f) +k

i=1

Ci(f)PPi

(fPfP) (20)

=C(f) +k

i=1

Ci(f)ri, (21)

completing the proof.

We remark that Lemma 4.4 and its proof remain valid in much more generalsettings; all that is required is convexity of the function xce(x) for each edge e(which holds when, for example, each edge latency function le is convex). We nowdefine (G,r,c) as the ratio between the cost of flow at Nash Equilibrium to thecost of an optimum flow. We are now prepared to prove the main theorem.

Theorem 2.13. If(G,r,c) has linear latency functions, then(G,r,c) 43

.

Proof. Let f be a flow in G at Nash equilibrium. Let Ci(f) be the latency of

an si ti flow path, so that C(f) =

i Ci(f)ri (see Lemma 2.3). By Lemma4.3(a), f /2 is an optimal solution to the instance (G,r/2, c). Moreover, by Lemma4.3(b), Ci(f /2) = Ci(f) for each i (in words, marginal costs with respect to f /2and latencies with respect to f coincide); this establishes the necessary connectionbetween the cost of augmenting f /2 to a flow feasible for (G,r,c) and the cost ofa flow at Nash equilibrium.Taking= 1 in Lemma 4.4, we find that the cost of any flowf feasible for (G,r,c)

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satisfies:

C(f) C(f /2) +k

i=1

Ci(f /2)fracri2 (22)

=C(f /2) +1

2

ki=1

Ci(f)ri (23)

=C(f /2) +1

2C(f). (24)

Finally, its easy to lower bound the cost off /2:

C(f /2) =e

1

4aef2

e +

1

2befe

1

4e

aef2

e +befe =

1

4 C(f) (25)

and thus C(f) 34

C(f).

We note that the analysis of this section can easily be extended to prove thatin any instance (G,r,c) where for some p, ce(x) =aex

p +be (with ae, be 0) foreach edge e, (G,r,c) (1p(p+ 1)(p+1)/p)1 = ( p

lnp).

An example can show that this result is tight, consider a network with 2 nodes,

with 2 distinct edges connecting them, say e1, e2. If we set ce1 = 1, cee2 = x theflow along the edge. We will try to route 1 unit of non-atomic flow through thenetwork. Now in Nash equilibrium all the traffic flows on the bottom edge, therebyhaving latency 1. However if we route half the traffic along each edge, then wenote that our system latency is 3

4, thus here 4

3

Now we consider an example. If we randomly generate a graph on an NxN gridrandomly generated vertices on the intersections, with randomly generated con-necting edges, and randomly generated affine costs, under varying atomic loads ona route between distinct randomly generated source and sink.We use Dijkstras algorithm iteratively to generate the traffic between the nodes,and then again on each path in until we obtain Nash equilibrium. We then removean edge and rerun the program to see how flow through the network is affected.We then replace it and remove another, until we have tried every edge in the net-work. Below is an example network the edges colour coded by their effect on theaverage users cost. Those which are coloured RED are those which show Braessparadox, and cause an improvement in the flow.

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We now consider a similar problem on a network where each user individuallygenerates its source and sink points.

So given how selfish routing affects a large number of networks, we ask if thereis a possible method of solving this. As such we now consider one technique calledagtunecrypted marginal cost taxation in which we charge the user for the cost itspresence causes on the other users on its path.We will consider this for a non-atomic instance.

Theorem 2.14. Let (G,r,c) be a non-atomic instance such that, for every edgee, the function x ce(x) is convex and continuously differentiable. Let f be anoptimal flow for(G,r,c) and let e =f

e c

e(f

e ) denote the marginal cost tax for

edgee with respect to f

. Thenf

is an equilibrium flow for(G,r,c+).

Proof. Obvious by (2.4)

This then induces the optimum flow through the network, however this maybe difficult to implement in some systems (road networks etc.)

2.6 More General Cost Functons [1]

Proposition 2.15. Letfbe a feasible flow for the instance(G,r,c) for a vetexvinG and a commodity i, letdi(v) denote the length, with respect to edge lengthsce(fe), of a shortestsi path inG. Then f is at Nash equilibrium iff for everypairv, w of vertices inG, every commodityi, and everyv pathP:

1. di(w)di(v)

ePce(fe)

2. iffie >0 for every edgee P, thendi(w)di(v) =

ePce(fe).

We omit the proof.

Theorem 2.16. For any instance(G,r,c) with |V(G)|= n, the trivial algorithmreturns a solution of value at mostn

2times that of the optimal solution.

Proof. Let f and f be flows at Nash equilibrium for (G,r,c) and (H,r,c), re-spectively, with H a subgraph of G containing a s t path. We may assumethat f is acyclic. PutC=C(G,r,l) and C =C(H,r,l); We wish to prove that

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C n2

C.

Now if we defined(v) withv V(G) as the length (with respect to edfge lengths

ce(fe)) of a shortest s v path. Assume for simplicity that n is odd and everyvertex of G is incident to an edge e withfe> 0; extending the following argumentto the general case is straightforward. Order the verticess = v0, v1, . . . , vn1 = taccording to nondecreasing d(v)-value. If there is an edgee= (v, w) with fe >0and ce(fe) = 0 , (and so d(v) = d(w)) then break the tie by placing v before win the ordering (possible as f is acyclic). This ordering is then a topological onewith respect to the flow f - that is, whenever fe > 0, e is a forward edge withrespect to our ordering. Our proof approach will be to show, by induction on ithat d(v2i) iC

; the base case i= 0 is trivial.

Before considering the inductive step, we require a definition and a claim. Callan edgee light iffe f

e andf

e >0 (in particular,e must be present in H). Light

edges are useful to us because they have latency at most C with respect tof (asevery flow path off has latency C) and hence latency at most C with respecttof (since latencies are nondecreasing); thus, vertices ofG that are adjacent via alight edge differ ind-values by at most C. The next claim assures us of a healthysupply of light edges: everys t cut consisting of a set of consecutive vertices(with respect to our topological ordering) contains a light edge (see Figure 5.2).

Claim 2.17. LetS={v0, . . . , vk}for somek {0, 1, . . . , n 2}. Then some light

edge has its tail inSand head outside ofS.

Proof. Let +(S) denote the edges with tail inside S and head outside S, and(S) the edges with head inside Sand tail outside S. Since S is a s t cutand f is an s t flow of value r with no flow on edges in (S) (as the verticesare topologically sorted according to f),

e+(S) fe =r . Since S is an s t cut

and f is an s t flow,

e+(S) fe r. Hence, fe f

e for some e

+(S) withfe >0.

Now suppose i {1, . . . , (n1)/2} and d(v2(i1)) (i1)C. Let k be thelargest integer such that there is a path of light edges from vj to vk for somej 2(i1); we will show that k 2i. The previous claim immediately impliesthat k is well defined with k > 2(i 1) (consider the head of a light edge in+({v0, . . . , v2(i1)})). To see that k 2i, observe that ifk = 2i1 then all lightedges in+({v0, . . . , v2(i1)}) (and there must be one) have head v2i1 and no lightedge in+({v0, . . . , v2i1}) has tail v2i1 (otherwise we would append such an edge

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to our maximal path), contradicting that +({v0, . . . , v2i1}) must contain a lightedge.We have established the existence of a path Pof light edges from vj to vk with

j 2(i 1) and k 2i. Inductively, we have d(vj) d(v2(i1)) (i 1)C;since d(v2i) d(vk), we can finish the inductive step and the proof by showingthat d(vk)d(vj) C

(informally, d(v2(i1)) and d(v2i) are sandwiched betweend(vj) and d(vk), so it suffices to upper bound the gap between the latter pair ofnumbers). Lettingd(v) denote the length of a shortest s v path in H withrespect to edge lengthsce(f

e ), we can apply Proposition 2.15 tof

inHto obtain0 =d(s) d(vj) d

(vk) d(t) = C. ByProposition 2.15, this implies that

the latency ofP with respect f is at most C; since all edges ofPare light, itfollows that the latency ofPwith respect to f is at most C. A final applicationof Proposition 2.15 then yields d(vk)d(vj) C, completing the inductive step

and the proof.

2.7 Conclusion

We can conclude from this chapters results then that selfish routing on networks,which whilst easy to implement, can cause a network running significantly belowthe social optimum. There are various ways to solve these issues from taxation,to route closure to route augmentation. These solutions work to varying degrees,and their main advantages / disadvantages lie in the easiness to compute and

implement. We have shown the difficulty of some graph analysis, here with regardsto Braess Paradox. And how even a minor alteration of the model can turn aproblem that whilst troublesome, but is at least bounded, into an unbounded one.

3 Dynamic Routing

We now consider server controlled loss networks, first we define what a loss networkis:

Definition 3.1. Formally we study an integer capacity resource C, suppose thatthe finite setI indexes the calls offered to the resource. For i I calls of type iarrive as a Poisson stream of rateki. When a call arrives it is a control decisionas to whether the call should be accepted by the resource. An arriving call must berejected and lost from the system if the resource has insufficient capacity to acceptit. It may also be rejected in other situations.

A call of type i requires the use of Ai units of capacity (Ai Z+) and, if

accepted, will use this amount of resource for the holding time of the call, after

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which this capacity is released for use by other calls. The holding time of a typeicall is exponentially distributed1i . All call arrival streams and holding times areindependent. An accepted call of type i earns reward at ratewi per unit time in

progress. We may assume thatgcd(A1, . . . , AN) = 1(as this ensures an irreduciblestochastic process).

Given this the aim is to design a control policy in order to maximise the ex-pected reward per unit time, for these policies in general we assume knowledge ofall previous call arrivals and departures and of the acceptance decisions. Howeverthey may not have future knowledge or allow a call in progress to be terminatedprematurely.

First we consider theMinimum Hop Only policy (MHO). This is, given a callfrom one node to another, it will take a route using the minimum numberof nodes possible, and if none have sufficient free capacity, then it will beblocked.

Advantages:

This is a very simple routing policy, and as such it is possible toimplement with a fast and highly efficient routing algorithm

Under high network load, with large amounts of call blocking, thismethod is fairly efficient, as by taking the shortest possible routesit does not use unnecessary resources as a result of diversion.

Disadvantages:

However, under lighter loads it can be wasteful, as it does not takeadvantage of other, free network resources. As such, routes, es-pecially busy ones, will suffer from a very poor Quality of Service

(QoS) due to call blocking.

Next we consider theMinimum Hop Policy (MH). Given a call from one nodeto another, it will use a minimum node using route from the set of routeswith sufficient capacity to take it, and if no routes have sufficient capacity,then it will be blocked.

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Advantages:

This policy again has very efficient algorithms for route finding,and so excessive resource will not be needed to ensure fast enoughrouting.

This policy is also fairly efficient under high loads as it does notconsume unnecessary resources by taking extended diversions.

We also can note that it solves some of the low load QoS issues thatthe MHO policy gives us.

Disadvantages:

On the other hand, it will still cause premature congestion on theshorter routes, and issues may be exacerbated if the diversionaryroutes also become congested.

Now we consider a somewhat different approach to the problem, that of theLeast Loaded algorithm (LLA). This looks at the paths available and usesthe least congested routes thus balancing the traffic across the network.

Advantages:

In doing this, the LLA minimises congestion, and so, especially onlower loads, will give a higher QoS due to reduced call blocking.

Disadvantages:

This system, however suffers under heavy traffic as it uses unnec-essary resources, which will clearly be detrimental to the system.

This implies then that we want to balance Minimum Hop routes at heavyloads with load balancing at low loads, in order to achieve a system efficientat a larger range of traffic flows. A first attempt at this may be made usingTrunk reservation, described below:

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The form of a trunk reservation policy is: given of call type i, describing itssource and sink, capacity demands, and other such information (pertainingperhaps to priority etc.). We then test paths, in increasing order of conges-

tion, until a resource accepts the call, or until all routes have been exhausted,in which case it is lost. The call will be accepted by a resource if and only ifthe resource currently has at least ri units of free capacity. Where we defineri as the trunk reservation parameter used against calls of type i.

It is usually implemented such that if the capacity of a link is less than someri it will only accept the call if it is a minimum hop route.

Advantages:

Trunk reservation is fairly effective at low loads, as the load bal-ancing ensures a lower amount of congestion along the shorter butbusier route, thereby giving a reasonable level of QoS

At high loads also, the minimum hop policy enacted by the re-served capacity, ensures that overly large amounts of resource arenot wasted by the diversionary routes, and as such, congestion isavoided and network throughput is increased.

Disadvantages:

However trunk reservations can oscillate between choosing shortestand alternate paths and some overutilization of the shortest pathscan cause some call loss to occur, even if not necessary.

Having considered some of the technical aspects of server routing, we now lookat an interesting tie in with the previous section. Consider the below example:

Given out graph (figure 4), if we consider each edge, capacity 3. First we willconsider what happens under MH, MHO and LLA routing, with a load of 6 x1 unit calls, without edge e = (A, B). We note then, that due to the symmetryof the graph then that half the calls go viaAand half byB.Hence none are blocked.

Now if we add in edge e we see that under our policies we get the followingresults:

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Figure 4

MHO. It is clear here that the minimum hop route is S A B E,

and that after 3 calls the route is blocked, and hence 3 calls are blocked

MH. Similarly the shortest route here is S A B E, and then allroutes are blocked, and so again 3 calls are lost.

LLA. To start no edges are being used, so well pick SA B E forour first call. Then by symmetry our next 2 calls S A 1 2 EandS3 4 B E, and the same again for our nex 2 calls. Then SAand B Eare both blocked, so all routes are, and thus 1 call is lost.

In fact if our policy routes any traffic along the added route, then it will besuboptimal for loads of 6 or above, if our quality of service measure is in the num-ber of calls lost.

Note the similarity to Braess paradox in user controlled networks, where theadding of a link can decrease the efficiency of a system with high traffic, and how

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it affects things in the same manner here.

3.1 Conclusion

This shows some of the difficulty of server controlled switching, in that if insuffi-cient care is taken when choosing a switching protocol then it can result in eitherover usage of certain resources, rejection of calls or large amounts of congestionacross the network, all resulting in a poor Quality of Service for the users. Wehave also shown that under certain switching protocols, we obtain something akinto Braess Paradox. Together these show the potential complexity of the systemsthat we are examining, and that obvious methods of routing may not be the cor-rect ones.

References

[1] Tim Roughgarden,Selfish Routing. 2002.

[2] Richard Steinberg and Willard I. Zangwell, Prevalence of Braess Paradox.Transportation science, 1983.

[3] Tim Roughgarden and Eva Tardos, How Bad is Selfish Routing?. Journal ofthe ACM, 2001.

[4] Henry Lin, Tim Roughgarden, Eva Tardos and Asher Walkover, BraesssParadox, Fibonacci numbers, and Exponential Inapproximability. Lecturenotes in Computer Science, 2005.

[5] Tim Roughgarden,On the Severity of Braesss Paradox: Designing Networksfor Selfish Users is Hard. Journal of Computer and System Sciences, 2006.

[6] Shao-Shan Chiang, Chih-Hung Huang and Kuang-Chiung Chang, A Mini-mum Hop Routing Protocol for Home Security Systems Using Wireless SensorNetworks. Consumer Electronics, 2007.

[7] Shao-Shan Chiang, Chih-Hung Huang and Kuang-Chiung Chang, Routingand Capacity Allocation in Networks with Trunk Reservation. Mathematicsof Operations Research, 1990.

[8] Anna Nagurney, David Parkes and Patrizia Daniele,The Internet, evolution-ary variational inequalities, and the time-dependent Braess paradox. Compu-tational Management Science, 2007.

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[9] Leslie Arthur Keith Bloy,An investigation into Braess paradox. Unisa ETD,2007.

[10] Dietrich Braess, Anna Nagurney and Tina Wakolbinger, On a Paradox ofTraffic Planning. Transportation Science, 2005.

[11] Henryk Fuks and Anna T. Lawniczaka, Performance of data networks withrandom links. Mathematics and Computers in Simulation, 2001.

[12] Hyejin Youn, Michael T. Gastner and Hawoong Jeong, Price of Anarchy inTransportation Networks: Efficiency and Optimality Control. Physical ReviewLetters, 2008.

[13] Richard Arnott, Andre de Palma and Robin Lindsey, Properties of Dynamic

Traffic Equilibrium Involving Bottlenecks, Including a Paradox and Metering.Transportation Science, 1991.

[14] E. Altmana, T. Boulognea, R. El-Azouzia, T. Jimenezb and L.Wynterc, Asurvey on networking games in telecommunications. Computers OperationsResearch, 2006.

[15] P.J. Hunt and T.G. Kurtz,Large loss networks. Stochastic Processes and theirApplications, 1994.

[16] Saewoong Bahk and M El Zarki,Congestion control based dynamic routing inATM networks. Computer Communications, 1994.

[17] Antonios F. Atlasis, Evangelos D. Baltatzis, George I. Stassinopoulos andIakovos S. Venieris, A linear-based trunk reservation routing algorithm forATM networks. International Journal of Communication Systems, 1999.

[18] Joseph S. Kaufman, Blocking in a Shared Resource Environment. Interna-tional Journal of Communication Systems, 1981.

[19] Valeri Naoumov, Normal-type approximation for multi-service systems withtrunk reservation. Telecommunication Systems, 1995.

[20] P. J. Hunt and C. N. Laws,Optimization via Trunk Reservation in Single Re-source Loss Systems under Heavy Traffic. The Annals of Applied Probability,1997.

[21] Joel E. Cohen and Frank P. Kelly, A Paradox of Congestion in a QueuingNetwork. Journal of Applied Probability, 1990.

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[22] N. G. Bean, R. J. Gibbens and S. Zachary, Asymptotic Analysis of SingleResource Loss Systems in Heavy Traffic, with Applications to Integrated Net-works. Advances in Applied Probability, 1995.

[23] Noam Nisan, Tim Roughgarden, Eva Tardos and Vijay V. Vazirani,Algorith-mic Game Theory. Cambridge University Press, 2007.

[24] Gina Kolata, What if They Closed 42nd Street and Nobody Noticed?. NewYork Times, 1990.

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