Which Protocol? Mutual Interaction ofHeterogeneous Congestion Controllers

Vinod Ramaswamy∗, Diganto Choudhury∗, and Srinivas Shakkottai∗∗Dept. of ECE, Texas A&M University, Email: {vinod83, dchoudhury, sshakkot}@tamu.edu

Abstract—A large number of congestion control protocolshave been proposed in the last few years, with all having thesame purpose—to divide available bandwidth resources amongdifferent flows in a fair manner. Each protocol operates onthe paradigm of some conception of link price (such as packetlosses or packet delays) that determines source transmissionrates. Recent work on network utility maximization has broughtforth idea that the fundamental price or Lagrange multiplierfor a link is proportional the queue length at that link, andthat different congestion metrics (such as delays or drops) areessentially ways of interpreting such a Lagrange multiplier. Wethus ask the following question: Suppose that each flow has anumber of congestion control protocols to choose from, which one(or combination) should it choose? We introduce a frameworkwherein each flow has a utility that depends on throughput,and also has a disutility that is some function of the queuelengths encountered along the route taken. Flows must choosea combination of protocols that would maximize their payoffs.We study both the socially optimal, as well as the selfish cases todetermine the loss of system-wide value incurred through selfishdecision making, so characterizing the “price of heterogeneity”.We also propose tolling schemes that incentivize flows to chooseone of several different virtual networks catering to particularneeds, and show that the total system value is greater, hencemaking a case for the adoption of such virtual networks.

I. INTRODUCTION

Recent years have seen the design of a large number ofcongestion control protocols for use on the Internet. Theirdesigns all revolve around the idea that link congestion isindicated by some notion of “price”, which the source canrespond to. Different congestion price metrics include packetloss, packet marks, packet delays or some combination thereof.However, the relative value of one protocol versus another isnot well understood. For example, it might be conjectured thata delay sensitive application would consider using a protocolthat has a delay-based congestion metric, and a throughputmaximizing application might favor a loss-based metric. Howshould applications choose the protocol to use?

An analytical framework for network resource allocationwas developed in seminal work by Kelly et al. [1]. If the flowi has a rate xi ≥ 0 and the utility associated with such a flowis represented by a concave, increasing function Ui(xi), theobjective is

max∑i∈N

Ui(xi) (1)

s.t. yl ≤ cl, ∀ l ∈ L (2)

Research was funded in part by NSF grants CNS-0904520, CNS-0963818,DTRA grant HDTRA1-09-1-0051 and the Google Faculty Research Awardsprogram.

where N is the set of sources, L the set of links, cl the capacityof link l ∈ L. Also let R be the routing matrix with Rli = 1if the route associated with source i uses link l. The load onlink l is yl =

∑r∈N Rlrxr. The problem can be solved using

ideas based on Primal-Dual system dynamics [1]–[5] to yielda set of controllers. At the source we have

Source: xi(t) = κi

(U ′

i(xi(t)) −∑l:l∈L

Rlixi(t)pl(t)

)+

xi

, (3)

where ki > 0 is constant, and the notation (φ)+ξ is used todenote the function

(φ)+ξ ={

φ ξ ≥ 0max{φ, 0} ξ = 0.

(4)

The controller in (3) has an attractive interpretation that thesource rate of flow i responds to feedback in the form of linkprices pl(t), with the end-to-end price being calculated as thesum of prices on all links that the flow traverses—somethingthat is common to all congestion control protocols. Source rateis always non-negative, which is enforced by the definition ofthe function in (4). The price pl(t) at link l is calculated using

Link: pl(t) = ρ(pl(t))

⎛⎝∑

j∈NRljxj(t) − cl

⎞⎠

+

pl(t)

. (5)

Each link has a buffer in which packets are queued. If thetotal load at a link l given by

∑j∈N Rljxj(t) is greater than

the capacity cl, the queue length increases, while if it is lessthan cl, the queue length decreases as seen in (5). The queuelength is always non-negative, as enforced by the definition in(4). The gain parameter ρ(pl) is any positive function. Thus,the link-price pl(t) can be identified with the queue length atlink l. It has been shown [1]–[5] that the above control schemeconverges to the optimal solution to the problem in (1).

While this framework indicates that the fundamental priceof a link is proportional to queue length, congestion controlprotocols use several different congestion metrics. For ex-ample, TCP Reno [6] uses packet drops (or marks) as itsprice metric, while TCP Vegas uses end-to-end delay [2].Other protocols include Scalable TCP [7] (that uses loss-feedback, and allows scaling of rate increases/decreases basedon network characteristics), FAST-TCP [8] (that uses delay-feedback, and is meant for high bandwidth environments), andTCP-Illinois [9] (that uses loss and delay signals to attainhigh throughput). However, drops, marks, and delays are all

This paper was presented as part of the main technical program at IEEE INFOCOM 2011

978-1-4244-9921-2/11/$26.00 ©2011 IEEE 2876

functions of the queue length. Thus, a key difference betweenprotocols is their way of interpreting queue length information.

A fall out of different price-interpretations is that whenflows choose distinct congestion control protocols, they donot obtain the same bandwidth on shared links. For example,studies such as [10]–[13] study inter protocol as well as intra-protocol fairness, while [14] considers a game of choosingbetween protocols, assuming that a certain bandwidth wouldbe guaranteed per combination. However, bandwidth share ob-tained by a flow does not completely capture the performanceof an application. For example, a delay sensitive applicationmight have a disutility for delay. Similarly, a loss sensitiveapplication might have a disutility for packet loss. Further, itis not clear whether one protocol would be better than anotherfor the system as a whole. We thus ask the following questions:Which protocol (or combination) should flows choose? Howwould their performance be impacted by others’ choices?

II. MODEL AND MAIN RESULTS

We consider a system in which each flow i ∈ N has aso-called α−fair utility function [15],

Ui(xi) � wix1−αi /(1 − α), (6)

with α ≥ 1, and a disutility that depends on the vector of linkprices p as

Ui(xi, p) �∑l∈L

Rli(pl/Ti)βxi, (7)

where β > 0 is a constant. The overall payoff is the differenceof the two, given by

Fi(xi, p) � Ui(xi) − Ui(p). (8)

The idea is that flows prefer a large bandwidth, but arenegatively affected by the queue lengths at the links that theytraverse. α-fair utility can be motivated by the fact that thebehavior when α → 1 similar to TCP-Vegas, α = 2 is similarto TCP-Reno, while other values of α yield different fairnessmetrics. The threshold parameter Ti in (7) models the flow’ssensitivity to queue length, with a small value of Ti indicatinghigh sensitivity (e.g., delay sensitive applications need shortqueue lengths) and a large value indicating low sensitivity(e.g., loss sensitive applications are affected only by bufferoverflow). The total system value is the sum of all payoffs.

We define a set of protocols T , with cardinality T =|T |. Each protocol z ∈ T is associated with a price-interpretation function mz(pl) � (pl/Tz)β . Note that theseprice-interpretation functions take the same form as disutilities,and model the way in which a particular protocol z ∈ Tinterprets link prices1. Again, a delay-based protocol wouldhave a low value of Tz, while a loss-based protocol wouldhave a high value. However, while a flow i cannot change itsdisutility function parameterized by Ti it can choose to use a

1We will refer to “price-interpretation functions” and “protocols” inter-changeably.

combination of protocols as appropriate. A particular flow i’schoice could take the form

qi(p) �∑z∈T

εzi

L∑l=1

Rlimz(pl) (9)

where∑

z∈T εzi = 1, and εz

i ≥ 0. The convex combinationmodels the idea that a flow can choose to sometimes measureprice in one way (e.g., delay-based) and sometimes another(e.g., loss-based). εz

i can be thought of as the probability withwhich flow i uses protocol z. For example, this situation mightcorrespond to a flow using delay and loss measurements simul-taneously, and responding to congestion signals (loss or delay)probabilistically. We refer to the choice [ε1i , ε

2i , · · · εT

i ], madeby flow i as εi ∈ Ei � {εi :

∑z∈T εz

i = 1, εzi ≥ 0}. Further,

we denote aggregate choices of all flows by ε ∈ E � Πi∈NEi,and will refer to ε ∈ E as a protocol-profile.

We first show in Section III that for a given protocol-profile,the bandwidth allocations (and hence the payoffs) are unique.Further, a primal-dual type control will converge to this uniquebandwidth allocation. The result is essentially a consistencycheck that allows us to analytically determine the payoffs asa function of the protocol-profile chosen.

We show in Section IV that all bandwidth allocationsthat are attainable by a protocol-profile over T protocolswith m1(p) ≥ m2(p) ≥ · · · ≥ mT (p) are attainable bya protocol-profile over just the two protocols m1(p) andmT (p). The result has the appealing interpretation that whenmz(p) = (p/Tz)β , it is sufficient to only consider the“strictest” interpretation (smallest T z , which can be thought ofas delay-based feedback) and the most “lenient” (largest T z,associated with loss-based feedback). We next show that withtwo protocols with T1 < T2, the bandwidth allocation receivedby a flow i is decreasing in the weight it places on the strictprotocol. Although the proof is involved, the result is intuitivesince a strict protocol would always interpret p as a largercongestion than the lenient protocol. However, since payoffsare the sum of utility and disutility, it does not follow that allflows would choose the protocol with the higher threshold.

We show in Sections V and VI that in many cases, thetotal system value is maximized when all flows choose touse only m1(p) = (p/T1)β . On the one hand if flows haveprice-insensitive payoffs, the protocol-profile used does notmatter as long as all of them use the same profile. On theother hand, if there is a mix of flows, some of which havea large disutility function (price-sensitive) and others whichdo not (price-insensitive), using the strict price-interpretationm1(p) = (p/T1)β , ensures that the price does not become toolarge for all flows, which maximizes system value.

In Sections V and VI, we also consider the case flowsuse selfish optimizations to choose their protocol-profiles andstudy the Nash equilibrium. If all flows have price-insensitivepayoffs, then they all choose the lenient price-interpretationm2(p) = (p/T2)β . This case can be mapped to throughputmaximizing flows all choosing TCP Reno. If we have a mixof flow types, it turns out that the price-sensitive flows choose

2877

the strict price-interpretation m1(p) = (p/T1)β , regardless ofthe choice of others, while the others may employ mixedstrategies. The result is interesting since it suggests that adelay sensitive application cannot do any better in terms ofoverall payoff even if it chooses a more lenient protocol. Wealso characterize the ratio of system value in the game versusthe social optimum for the single-link case to determine anefficiency ratio, which can be quite high.

Finally, in Section VII we introduce virtual networks, eachof which is assigned a certain fraction of the capacity, andchooses a toll. Flows can choose a network and protocols.The idea is similar to Paris Metro Pricing (PMP) [16]–[18],and we show that the system value at Nash equilibrium canbe higher overall in-spite of tolling. The result suggests thatthe Internet might benefit by having separate tiers of servicefor delay-sensitive and loss-sensitive flows.

III. PROBLEM FORMULATION

We assume that for each link, there exists at least one flowthat uses only that link. The assumption implies that all linkshave a non-zero price. We hypothesize from (3) and (5) thatthe payoffs should be determined by the protocol-profile ε as

x∗i (p

∗, εi) = (U ′i)

−1

(T∑

z=1

εzi

L∑l=1

Rlimz(p∗l )

), (10)

with εi ∈ Ei and for all l ∈ L.

N∑i=1

Rlix∗i (p

∗, εi) = cl p∗l > 0, (11)

Note that although we have denoted x∗ as depending on both εand p∗, the prices themselves depend on ε through x∗, and thesolution (x∗(ε), p∗(ε)) (if it exists) is solely a function of ε.We show that the equilibrium exists, and can be reached usingPrimal-Dual dynamics. We have the following proposition.

Proposition 1. Given any protocol-profile ε, Primal-Dualdynamics converge to the unique solution (x∗, p∗) of theconditions (10) and (11).

Proof: For price-interpretation functions of the form(p/Tz)β , the source dynamics in (3) can be re-written as

xi(t) = κi

(U ′

i(xi) −(

T∑z=1

εzi

(T1

Tz

)β)

L∑l=1

Rlim1(pl)

)+

xi

where m1(pl) = ( pl

T1)β . Let Ui(xi) = 1

γiUi(xi) where γi =∑T

z=1 εzi (

T1Tz

)β , and let κi = γi. Then the above equation canbe modified as

xi(t) = γi

(U ′

i(xi(t)) −L∑

l=1

Rlim1(pl(t))

)+

xi

. (12)

Now, in (5) choose ρ(pl) = 1m′1(pl)

, where m′1 is derivativeof m1. Then the price-update equation can be re-written as,

m1(pl(t)) =

(N∑

i=1

Rlixi(t) − cl

)+

pl

. (13)

Equations (12) and (13) correspond to the primal-dual dynam-ics of the following convex maximization problem

maxx>0

N∑i=1

Ui(xi)

subject toN∑

i=1

Rlixi ≤ cl, ∀l ∈ L.

The above is a convex optimization problem with a uniquesolution satisfying (10) and (11). Thus, by the usual Lyapunovargument [2]–[5] Primal-Dual dynamics converge to this so-lution. Note that our choice of price interpretation makes it aspecial case of the result in Appendix A Case-1 of [11].

We are now in a position to ask questions about what theflows’ payoffs would look like at such an equilibrium, and howthis would impact the choice of the protocol-profile. Recallthat the payoff obtained by a flow when the system state is atx∗(ε), p∗(ε) is given by

Fi(ε) = Ui(x∗i (ε)) − Ui(p∗(ε)). (14)

We define a system-value function V , which is equal to thesum of payoff functions of all flows in the network,

V (ε) =N∑

i=1

Fi(ε). (15)

Our first objective is to find an optimal protocol-profile thatmaximizes the system-value function.

Opt: maxε∈E

V (ε). (16)

Let ε∗S be an optimal profile vector for the above problem.Then we refer to VS = V (ε∗S) as the value of the socialoptimum.

An alternative would be for flows to individually maximizetheir own payoffs. However, such a proceeding might notnot lead to an optimal system state that maximizes the valuefunction (15). We characterize the equilibrium state of such aselfish behavior by modeling it as a strategic game.

Let G =< N , E ,F > be a strategic game, where N is theset off flows (players), E is the set of all protocol profiles(action sets) and F = {F1, F2, · · · , FN}, where Fi : E → R

is the payoff function of user i defined in (14). Define ε−i =[ε1, ε2, · · · , εi−1, εi+1, εN ], i.e., this represents the choices ofall flows except i. Then ε = [εi, ε−i]. For any fixed ε−i, flowi maximizes its payoff as shown below.

Game: maxεi∈Ei

Fi(εi, ε−i) ∀i ∈ N . (17)

The game is said to be at a Nash equilibrium when flows donot have any incentive to unilaterally deviate from their currentstate. We define ε∗G as a Nash equilibrium of the game G if

(εG)∗i = arg maxεi∈Ei

Fi(εi, (εG)∗−i), ∀i ∈ N

We refer to VG = V (ε∗G) as the value of the game. Finally,we define the “Efficiency Ratio (η)” as

η =VG

VS. (18)

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IV. BASIC RESULTS

We first show that a T -protocol network can be replacedwith an equivalent 2-protocol network. Consider a T -protocolnetwork with price interpretation functions [m1,m2, · · · ,mT ].Let ε ∈ ET be a profile state in the T -network. Then theequilibrium rate vector x∗(ε) and price vector p∗(ε) satisfythe equilibrium conditions (10) and (11). Now, consider a 2-protocol network with price interpretation functions m1 andmT . Note that m1 ≥ mz ≥ mT , z = 2, · · · , T −1. Let μ ∈ E2

be a profile state in the 2-protocol network.

Proposition 2. For any equilibrium (x∗(ε), p∗(ε)) in a T -protocol network, ∃ a protocol-profile μ s.t. (x∗(ε), p∗(ε)) isalso an equilibrium for the 2-protocol network.

Proof: For any given ε ∈ ET , let (x∗(ε), p∗(ε)) be anequilibrium pair that satisfies the equilibrium conditions (10)and (11), which are reproduced below for clarity.

x∗i (ε) = (U ′

i)−1(∑T

z=1 εitqz∗i

),∀i ∈ N ,

Rx∗(ε) = c, p∗l > 0,∀l ∈ L.

where qz∗i =

∑Ll=1 Rlim

z(p∗l (ε)). The fact that mT ≤ mz ≤m1, implies, qT∗

i ≤ qz∗i ≤ q1∗

i , ∀i ∈ N , Z ∈ T . Since bothm1 and mT are strictly increasing functions, there exists aunique μi ∈ [0, 1], such that,

T∑z=1

εzi q

z∗i = μiq

1∗i + (1 − μi)qT∗

i .

Now, we have

x∗i (ε) = (U ′

i)−1

(T∑

z=1

εzi q

z∗i

)

= (U ′i)

−1(μiq

1∗i + (1 − μi)qT∗

i

),∀i ∈ N ,

Rx∗(ε) = c, p∗l > 0,∀l ∈ L.

The above equations correspond to the equilibrium conditionsof a 2-protocol network with price interpretation functionsm1 and mT . Therefore, there exists a protocol-profile μ =[μ1, · · · , μN ] such that (x∗(ε), p∗(ε)) is an equilibrium pairof 2-protocol network.

The above proposition shows that any equilibrium state ofa T -protocol network can be obtained with an equivalent 2-protocol network. Therefore we restrict our study to 2-protocolnetworks with a “strict” price interpretation m1 = ( p

T1)β and a

“lenient” price interpretation m2 = ( pT2

)β , i.e., T1 < T2. Sincethere are only two protocols in the system, the protocol-profilefor the user i, is given by εi = [ε1i 1 − ε1i ] and we simplyuse εi in the place of ε1i .

We next show that the that the bandwidth allocation receivedby a flow i is decreasing in the weight it places on the strictprotocol m1(p) = (p/T1)β .

Proposition 3. Let x∗i (ε) be the equilibrium rate of flow i for

any ε ∈ E2. Then ∂x∗i

∂εi< 0,∀i ∈ N .

Proof: From the equilibrium conditions we have

U ′i(x

∗i ) =

L∑l=1

Rlim1(p∗l )

(ε1 + (1 − ε1)

(T1

T2

)β)

= q1∗i

(ε1 + (1 − ε1)

(T1

T2

)β)

. (19)

Define qi(ε) � q1∗i

(ε1 + (1 − ε1)(T1

T2)β)

. Then differen-

tiating (19) with respect to εj , and replacing U ′′i (x∗

i ) =−αx∗

iU ′

i(x∗i ), we obtain

U ′′i (x∗

i )∂x∗

i

∂εj=

∂qi

∂εj+

L∑l=1

∂p∗l∂εj

∂qi

∂p∗l

∂x∗i

∂εj=

1U ′′

i (x∗i )

∂qi

∂εj+

1U ′′

i (x∗i )

L∑l=1

∂p∗l∂εj

∂qi

∂p∗l

= Aij +L∑

l=1

∂p∗l∂εj

Bil, (20)

where

Aij =(1−(

T1T2

)β)(PLl=1 Rlim1(p

∗l ))

U ′i(x

∗i )

−x∗i

α δij

=(1−(

T1T2

)β)(PLl=1 Rlim1(p

∗l ))“

εi+(1−εi)(T1T2

)β”(PL

l=1 Rlim1(p∗l ))

−x∗i

α δij

=(1−(

T1T2

)β)

εi+(1−εi)(T1T2

)β

−x∗i

α δij

Bil =Rlim

′1(p

∗l )(εi+(1−εi)(

T1T2

)β)

U ′i(x

∗i )

−x∗i

α

= Rlim′1(p

∗l )PL

k=1 Rkim1(p∗k)

−x∗i

α ,

and δij = 1 if i = j, and zero otherwise. At equilibrium,∑Ni=1 Rlix

∗i (ε) = cl,∀l ∈ L. Now, differentiating this equa-

tion with respect to εj , we get

N∑i=1

Rli∂x∗

i

∂εj= 0 ∀l ∈ L. (21)

Replacing ∂x∗i

∂εjwith expressions from (20) we obtain

N∑i=1

Rli−x∗

i

α(∑L

k=1 Rkim1(p∗k)) L∑

k=1

Rkim′1(p

∗k)

∂p∗k∂εj

+ Rlj

(1 − (T1T2

)β)

εj + (1 − εj)(T1T2

)β

−x∗j

α= 0,

which implies that

L∑k=1

m′1(p

∗k)

∂p∗k∂εj

N∑i=1

RliRkix∗

i

α(∑L

l=1 Rlim1(p∗l ))

= Rlj

(1 − (T1T2

)β)

εj + (1 − εj)(T1T2

)β

−x∗j

α.

We can represent the above in a matrix form as

RWRT ζ = r,

2879

where

W = diag

(x∗

i∑Ll=1 Rlim1(p∗l )

)

ζ =1α

[m′

1(p∗1)

∂p∗1∂εj

· · ·m′1(pL)

∂p∗L∂εj

]T

r =−x∗

i (1 − (T1T2

)β)

α(εj + (1 − εj)(T1T2

)β)[R1j · · · RLj ]T .

Note that RWRT is a positive definite matrix. Now, we have

ζ = (RWRT )−1r. (22)

Let H = (RWRT )−1, where H is an L × L matrix. Let usrepresent its elements using hlm. Thus, from (22) we have

∂p∗l∂εj

= −∑L

k=1 Rkjhlk

m′1(pl)

(1 − (T1T2

)β)x∗i

(εj + (1 − εj)(T1T2

)β). (23)

Finally, from (20) and (23) we have

∂x∗j

∂εj=

−(1−(T1T2

)β)

(εj+(1−εj)(T1T2

)β)

x∗j

α

(1 − x∗

j

PLl=1

PLk=1 RljRkjhlk

(PLl=1 Rljm1(p∗

l ))

)(24)

∂x∗i

∂εj=

(1−(T1T2

)β)

(εj+(1−εj)(T1T2

)β)

x∗i x∗

j

α

PLl=1

PLk=1 RliRkihlk

(PLl=1 Rlim1(p∗

l )) (25)

We observe that since RWRT is positive definite, H is as well.Let νj =

∑Ll=1

∑Lk=1 RljRkjhlk. Then positive definiteness

of H implies that νj ≥ 0,∀j ∈ N . (since hll > 0 andhll ≥ 2hlk = 2hkl). Suppose that νj = 0, then from equation(25), ∂x∗

i

∂εj= 0,∀i ∈ N , i �= j. Then in order to satisfy the

equilibrium condition∑N

i=1∂x∗

i

∂εj= 0,

∂x∗j

∂εjshould be equal to

zero. But from equation (24), it is clear that when νj = 0the above derivative term does not go to zero, which is acontradiction. Thus, νj �= 0,∀j ∈ N . Since νj > 0, itis evident from (25) that ∂x∗

i

∂εj> 0, ∀i �= j, ∀j. Since∑N

i=1∂x∗

i

∂εj= 0, we conclude that

∂x∗j

∂εj< 0,∀j.

The above proposition is intuitive in that a strict protocolwould force the flow to cut down its rate for the same priceas a lenient protocol. We now study different mixes of flowtypes in order to understand the system value in each case.

V. FLOWS WITH PRICE-INSENSITIVE PAYOFF

We associate each flow i ∈ N to a class, based onits disutility function of the form

∑l∈L Rli(pl/Ti)βxi. We

begin by considering a system of flows that have a price-insensitive payoff, i.e., Ti = ∞ ∀i ∈ N . This meansthat payoff is solely a function of bandwidth, and we haveFi(ε) = Ui(x∗(ε)). However, even in this situation, flows mustemploy congestion control, i.e., they must choose a protocol-profile. From Section (IV), recall that since we only have twoprotocols, the flow i’s choice of protocol profile is defined bya scalar value εi. Also note that T z �= ∞ for each protocolz = 1, 2. The system-value is equal to the sum of user payoffs,V (ε) =

∑Ni=1 Ui(x∗(ε)). We then have the following result.

Proposition 4. The system-value is maximized when theprotocol choices made by all users are the same. Thus, if

ε∗S = arg maxε∈E V (ε), and (ε∗S)i is used to denote theprotocol choice made by-profile of user i, then (ε∗S)i =(ε∗S)j ,∀i, j ∈ N .

Proof: We first derive an upper bound for system-valueV (ε) and then show that the upper bound is achieved when allsources choose the same protocol. Suppose that X = {x|Rx =c}. Let x = arg maxRx=c

∑Ni=1 Ui(xi). Note that equilibrium

rate x∗(ε) ∈ X , since Rx∗ = c. Then the value of∑N

i=1 Ui(x)evaluated at x∗(ε) satisfies

V (ε) =N∑

i=1

Ui(x∗i (ε)) ≤

N∑i=1

Ui(xi).

We showed in Proposition 2 that the equilibrium ratex∗(ε), is the unique maximizer of the convex problemmaxx>0,Rx=c

∑Ni=1

1γi

Ui(xi), where γi = εi +(1− εi)(T1T2

)β .Then, x∗(ε) can be made equal to x, the optimal point in setX , by choosing γi = γj ∀i, j ∈ N . Such a choice means that

γi = γj ⇒ εi + (1 − εi)(T1

T2)β = εj + (1 − εj)(

T1

T2)β ,

⇒ εi = εj .

Thus, if ε∗S = arg maxε∈E V (ε) ⇒ (ε∗S)i = (ε∗S)j ,∀i, j ∈ N .

We next consider the game in which flows are allowed tochoose their protocols selfishly.

Proposition 5. Let G =< N , E ,F > be a strategic gamewith payoff function of user i is given as Fi(ε) = Ui(x∗

i (ε)).Then there exists a Nash equilibrium for game G, and theequilibrium profile for any user i ∈ N is (ε∗G)i = 0.

Proof: Differentiating Fi w.r.t εi, and using Proposition 3

∂Fi

∂εi= U ′(x∗

i (ε))∂x∗

i (ε)∂εi

< 0

Hence, Fi(ε) is maximized when εi = 0. Therefore, (ε∗G)i =0,∀i ∈ N .

Efficiency Ratio: We showed in Proposition 4 that the valuefunction is maximized when all flows pick the same protocol-profile. In Proposition 5 we saw that when each flow selfishlymaximizes its own payoff, there exists a Nash equilibriumunder which every source chooses the lowest priced protocol,i.e., the protocol with the higher value of T. Such a profile isa special case of all flows choosing the same protocol-profile.Thus, value of the social optimum and the value of the gameare identical and Efficiency Ratio (η) is unity.

Example-1: Consider the case in which a single link withcapacity c = 10 is shared by 2 price-insensitive flows. Usershave α-fair utility functions with α = 2, w1 = 100 and w2 =100. We use price-interpretation functions (p

2 )2 and (p5 )2. In

Figure (1) we show the system value for different choicesof protocol profiles. The plot illustrates that system value ismaximized when both flows choose the same profile. Figure(2) shows how the payoff function of a flow varies with itsprotocol profile. We find that regardless of the value of theprotocol profile chosen by the other flow, the payoff functionis maximized when it picks the lower price protocol.

2880

0 0.2 0.4 0.6 0.8 1−50

−49

−48

−47

−46

−45

−44

−43

−42

−41

−40

Sys

tem

Val

ue

System Value Vs. Protocol Profile

ε2

ε1 = 0ε1 = 1

ε1 = 0.5ε1 = 0.25

Fig. 1. System Value with price-insensitive flows as a function of theprotocol-profile. We observe that the system value is maximized when bothflows choose the same protocol-profile.

0 0.2 0.4 0.6 0.8 1−40

−35

−30

−25

−20

−15

−10

Pay

off U

ser

1

Payoff User 1 Vs. Protocol Profile for User 1

ε1

ε2 = 0ε2 = 1

ε2 = 0.5

Fig. 2. Payoff of a price-insensitive flow as a function of its protocol-profile.We observe that payoff is maximized when the flow chooses the more lenientprice interpretation, regardless of the other flow.

VI. MIXED ENVIRONMENT

We now consider the case where a network is sharedby flows with different disutilities. We identify the optimalprotocol profile that maximizes the system value, and compareit with and the Nash equilibrium. We first study the case of anetwork consisting of a single link.

A. Single Link Case

Consider a single link system with capacity c sharedby N flows. The payoff of user i ∈ N is Fi(ε) =

Ui(x∗i (ε))−

(p∗(ε)

Ti

)β

x∗i (ε). Then, the system value is V (ε) =∑N

i=1

(Ui(x∗

i (ε)) −(

pTi

)β

x∗i (ε)

).

Proposition 6. The system- value is maximized when allusers pick the protocol with lowest threshold, i.e., if ε∗S =arg maxε∈E V (ε), then (ε∗S)i = 1,∀i ∈ N .

Proof: (Sketch) We can show through straightforward dif-ferentiation that Ui(εi) is a monotonically decreasing functionof εi. Now, the value function V is maximum when U(ε) ismaximized and U(ε) is minimized. We already know fromProposition 4 that U(ε) is maximized when all flows choosethe same protocol-profile. Coupling this result with the fact

that Ui(εi) is decreasing in εi, we see that system value ismaximized when εi = 1,∀i ∈ N .

We now study the strategic game in which users individuallymaximize their payoff as in (17). We show that there exists aNash equilibrium and characterize the protocol-profile.

Proposition 7. Let G =< N , E ,F > be a strategic gamewith payoff of user i is Fi(ε) = Ui(x∗

i (ε)) − (p∗(ε)Ti

)βx∗i (ε).

Then there exists a Nash equilibrium (NE) for Game G. AtNE, flows with greatest sensitivity to price choose the strictprotocol, i.e., if Ti = T1, then εi = 1.

Proof: We will show that Fi(ε) is quasi-concave, anduse the Theorem of Nash to show existence of the NE.Differentiating Fi w.r.t εi,

∂Fi

∂εi= (U ′

i(x∗i ) − di(p∗))

∂x∗i

∂εi− d′i(p

∗)x∗i

∂p∗

∂εi,

where di(p∗) = (p∗

Ti)β and d′i(p

∗) is its derivative. Now,

substituting for ∂x∗i

∂εiand ∂p∗

∂εiby using (20) and (23)

∂Fi

∂εi= (U ′

i(x∗i ) − di(p∗))

∂x∗i

∂εi− d′i(p

∗)x∗i

∂p∗

∂εi

= −(U ′i(x

∗i ) − di(p∗))

×(

x∗i

α(1 − x∗

i

c)

1 − (T1T2

)β

εi + (1 − εi)(T1T2

)β

)

+ d′i(p∗)x∗

i

x∗i p

∗

cβ

1 − (T1T2

)β

εi + (1 − εi)(T1T2

)β

= −x∗i

α

1 − (T1T2

)β

εi + (1 − εi)(T1T2

)β

×(

(U ′i(x

∗i ) − di(p∗))(1 − x∗

i

c) − αd′i(p

∗)x∗

i p∗

βc

).

Now, using U ′i(x

∗i ) =

(p∗

T1

)β

(εi +(1−εi)(T1T2

)β), di(p∗) =

(p∗

Ti)β and di(p∗) = β

p∗ (p∗

Ti)β we obtain

∂Fi

∂εi= −x∗

i

α

1 − (T1T2

)β

εi + (1 − εi)(T1T2

)β

×(((

p∗

T1

)β(

εi + (1 − εi)(

T1

T2

)β)

−(

p∗

Ti

)β)(

1 − x∗i

c

)

− α

(p∗

Ti

)βx∗

i

c

).

We show that if the above expression has a root, then it isunique. The roots are characterized by(

p∗

T1

)β(

εi + (1 − εi)(

T1

T2

)β)

−(

p∗

Ti

)β

=α(p∗

Ti)β x∗

i

c

(1 − x∗i

c ).

Dividing by ( p∗

T1)β on both sides,

εi + (1 − εi)(T1

T2)β − (

T1

Ti)β = (

T1

Ti)β α

x∗i

c

1 − x∗i

c

(26)

2881

0 0.2 0.4 0.6 0.8 1−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

Sys

tem

Val

ue

System Value Vs. Protocol Profile

ε2

ε1 = 1ε1 = 0ε1 = 0.25ε1 = 0.5

Fig. 3. System value against protocol choices (εi): Two flows sharing a link.

First observe that the left side of the above expression isstrictly increasing in εi (since T1 < T2). Since ∂x∗

i

∂εi< 0

(from Proposition 3), the right side of the above expression isstrictly decreasing. Therefore, the set of roots of the equation,∂Fi

∂εi(x) = 0 is a singleton or null set. Thus, Fi is unimodal or

monotonic in εi for any fixed ε−i and hence quasi concave.Since εi ∈ [0, 1] is a non-empty compact convex set, by the

theorem of Nash, the quasi concavity of Fi(εi, ε−i) guaranteesthat there exists a ε∗G, such that for all i = 1, · · · , N ,

(ε∗G)i = arg maxεi∈[0,1]

Fi(εi, (ε∗G)−i).

Hence, the first part of the proof is complete.Now, consider a flow with disutility (per unit rate) ( p

T1)β .

Replacing Ti with T1 in (26), we see that both terms insidebrackets are negative, and the overall expression is positive.Therefore, payoff is maximized when εi = 1.

Example-2: We consider a link with capacity c = 10 sharedby two flows with disutilities (p

2 )2 and (p5 )2, respectively, and

w1 = w2 = 1. The other parameters are unchanged fromExample-1. We show the system value for different choices ofprotocol-profiles in Figure 3. The value is maximized whenboth flows choose the strict protocol. Figure (4) shows how thepayoff of each flow varies with its choice of protocol profile,given other’s is fixed. We find that for the first (sensitive)flow, the payoff function is maximized when it chooses thestrict protocol, regardless of the other flow. But the payoffof the second (less-sensitive) flow is maximized for somecombination of protocols. The results validate our findings.

B. Network Case

We consider a system of flows with log utility functions,which is a special class of an α-fair utility function withα → 1. The payoff of flow i ∈ N is Fi(ε) = wi log(x∗

i (ε))−∑Ll=1 Rli(

p∗l (ε)Ti

)βx∗i (ε). Then the system-value is V (ε) =∑N

i=1 wi log(x∗i (ε)) −

∑Ll=1 Rli

(p∗

l (ε)Ti

)β

x∗i (ε).

Proposition 8. The System-Value function is maximized whenall flows pick the higher priced protocol, namely m1. Let ε∗S =arg maxε V (ε), then (ε∗S)i = 1,∀i = 1, · · · , N ,

Proof: The proof is similar to that of Proposition 6.

0 0.2 0.4 0.6 0.8 1−8

−6

−4

−2

0

Pay

off C

lass

1 U

ser

Payoff User Class 1 Vs. Protocol Profile

0 0.2 0.4 0.6 0.8 1−0.8

−0.6

−0.4

−0.2

0

Pay

off C

lass

2 U

ser

Payoff User Class 2 Vs. Protocol Profile

ε1

ε2

ε2 = 1ε2 = 0ε2 = 0.5

ε1 = 1ε1 = 0ε1 = 0.5

Fig. 4. Payoff against protocol choice (εi): Two flows sharing a link.

We now consider a game with two types of flows: price-insensitive flows with zero disutilities, and price-sensitiveflows with disutility (per unit rate) ( p∗

l

T1)β . In Proposition 5

we saw that price-insensitive flows pick the lenient protocolat Nash equilibrium. We will now show that price-sensitiveflows pick the strict protocol at Nash equilibrium.

Proposition 9. Any flow i with with disutility (per rate)(p/T1)β picks εi = 1 at Nash equilibrium.

Proof: The proof is similar to that of Proposition 7.

C. Efficiency Ratio

We now characterize the loss of system value at Nashequilibrium, as compared to the value of the social optimum.We focus on the case of a single link with capacity c, andassume that all flows have an α-fair utility function, withwi = 1 for all i. We categorize flows into 2 classes basedon their disutility functions. The disutility of Class-1 flows ismodeled using the function ( p

T1)β . Class-0 flows are price-

insensitive and hence their disutility function is zero. Weassume that there are Ns flows in Class-s and denote ns = Ns

N .In Proposition 6, we showed that the social optimum is

achieved when all flows select the protocol with lowest thresh-old. Let xs and p be the rate of a flow in Class-s and link priceat the social optimum. Then, xs = (U ′

s)−1(m1(p)) = ( p

T1)

−βα .

At equilibrium the rates of all flows add up to capacity c. i.e,∑1s=0 Nsxs = c ⇒

∑1s=0 Ns( p

T1)

−βα = c

xs = cN p = T1(N

c )αβ .

Then, the social optimum is

VS = N0(x0)

1−α

1−α + N1

((x1)

1−α

1−α − ( pT1

)βx1

)= (N

c )(α−1) N1−α (1 + (α − 1)n1) (27)

We next derive an expression for value of the game VG. Weshowed in Propositions 5 and 7 that at Nash equilibrium class-0 flows pick the ‘lenient’ protocol and class-1 flows choosethe ‘strict’ protocol. Let the equilibrium rate of a flow in classs be xs and the price of link be p. Then,

x0 =cT

βα2

N0Tβα2 + N1T

βα1

, x1 =cT

βα1

N0Tβα2 + N1T

βα1

. (28)

2882

0 0.2 0.4 0.6 0.8 11

2

3

4

5

6

7

8

9

Fraction of Class−0 Users

Effi

cien

cy R

atio

Efficiency Ratio − Mix of Class−0 and Class−1 Users

T2/T1 = 4

T2/T1 = 6

T2/T1 = 8

Fig. 5. Efficiency Ratio (η) in the single link case, plotted against the fractionof Class-1 flows for different ratios of T2/T1. Since VS and VG were negativein this example, a higher ratio is worse.

Now, we can write the value of the game VG as

VG = N0(x0)

1−α

1−α + N1

((x1)

1−α

1−α −(

pT1

)β

x1

)

Also, (p/T1)β = x1

−α. Hence, we can re-write the value ofthe game VG as

VG = N0(x0)

1−α

1−α + N1α(x1)

1−α

1−α

Replacing x with expression in (28),

VG = (N0(T2)βα +N1(T1)

βα )α−1

c(α−1)(1−α)

(N0

T( β

α)(α−1)

2

+ αN1

T( β

α)(α−1)

1

)

=N( N

c )(α−1)(n0+n1

“T1T2

”( βα

))α−1

“n0+n1α(

T2T1

)(β(1− 1α

))”

(1−α)

Finally, the Efficiency Ratio (η) is given by

VG

VS=

(n0 + n1

(T1T2

)( βα )

)α−1(n0 + n1α(T2

T1)(β(1− 1

α )))

(1 + (α − 1)n1)

Note that since the value of both the social optimum and thegame are negative when α > 1, η will be greater than 1 insuch cases. Therefore, the system performance degrades withincrease in η in such cases. If α = 1, the efficiency ratio is

VG

VS=

log(

c

n0+n1(T2T1

)β

)− n1(1 − β log(T2

T1))

log( cN ) − n1

.

Example-3 We consider N0 Class-0 flows and N1 Class-1flows. In Figure 5, we illustrate the Efficiency Ratio (η) as afunction of the fraction of Class-0 flows (n0) in the system.We have chosen α = 2 and β = 3. It can be observed that as(T2T1

) increases, η increases significantly.

VII. PARIS METRO PRICING

We partition a single-link into 2 virtual sub-networks eachhaving its own queueing buffer, and independent price (queue-length) dynamics. We represent the subnetworks using S0 andS1. Let c0 and c1 = c − c0 be the capacities allocated toS0 and S1 respectively. Let S1 be a tolled network with a

fixed entrance toll τ. The other subnetwork has zero toll. Thisscheme is similar to Paris Metro Pricing (PMP) with twoclasses [16]. We assume there are large number number flowsin the system, with N0 and N1 flows in Class-0 and Class-1, respectively. Also let N0 > N1. We have showed earlier inPropositions 4 and 7, that in selfish games, the price insensitive(Class-0) flows pick the protocol with higher threshold (m2),and price-sensitive flows (Class-1) flows pick the one withlower threshold m1.

We use x∗in and F ∗

in to represent the rate and payoffreceived by a Class-i flow in subnetwork-Sn. A flow thatseeks to maximize its payoff picks a subnetwork that yieldsthe maximum payoff. Thus, if n is the subnetwork chosen byflow i,

n = arg maxn∈{0,1}

Fin i = 0, 1.

A Nash equilibrium (NE) here is a state from which none ofthe flows has an incentive to deviate from its current choiceof subnetwork. Note that we already know the flow’s choicesof protocols in either network so no deviations in protocol arepossible. The desired NE is one in which all Class-0 flowsselect S0 and all Class-1 flows select S1. Our objective is todecide the subnetwork capacity allocation (c0) and entry fee(τ ) for a given number of flows in each class (N0 and N1).

Assume that the system is at the desired equilibrium. NowClass-0 flows use S0 and they share the network capacityequally. i.e x∗

00 = c0N0

. Similarly, each Class-1 flow gets a rateequal to x∗

11 = c−c0N1

. We assume that there are large numberof flows in the system and therefore entry of a Class-0 flow toS0 does not change its price (p) significantly. Then a Class-0

flow anticipates that it would receive x∗01 = c−c0

N1

(T2T1

) βα

if itshifted to S1. Similarly a Class-1 flow’s anticipated rate in S0

is x∗10 = c0

N0

(T1T2

) βα

.Now, the payoffs obtained by a Class-0 flow by choosing

S0 or S1 are, respectively,

F00 =(

c0N0

)1−α1

1−α , F01 =(

c−c0N1

(T2T1

) βα

)1−α

11−α − τ,

and the payoff of a Class-1 flow are, respectively,

F10 = (x∗10)

1−α α1−α =

(c0N0

(T1T2

) βα

)1−α

α1−α ,

F11 =(

c−c0N1

)1−αα

1−α − τ.

In order to ensure that no flow has an incentive to deviate,we require F00 > F01 and F11 > F10. Let

LT (c0) =[(

N0c0

)α−1

−(

N1c−c0

(T1T2

)βα

)α−1]

1α−1 ,

and

UT (c0) =[(

N0c0

(T2T1

)βα

)α−1

−(

N1c−c0

)α−1]

αα−1 .

Then the toll τ , should satisfy LT (c0) ≤ τ ≤ UT (c0). If tollturns out to be zero, then PMP is not viable, and system returns

2883

0.5 0.6 0.7 0.8 0.9 11

1.5

2

2.5

3

3.5

Fraction of Class−0 Users

Effi

cien

cy R

atio

Efficiency Ratio− Single Tier Network Vs Two Tier Network

Single Tier NetworkTwo Tier Network

Fig. 6. Comparison of Efficiency Ratio (η) between PMP scheme and Gamein a network with price-insensitive flows and delay sensitive flows. Since VS

and VG were negative in this example, a higher ratio is worse.

to a single network regime. It is straightforward to show thatthe condition needed is

c0

c − c0≤ N0

N1(T2

T1)

βα .

Now, we have to make optimal choices for PMP parameters(c0, τ) that maximizes the system-value with tolling VT ,

VT = N1F11 + N0F00

= N1

(c−c0N1

)1−αα

1−α − N1τ + N0

(c0N0

)1−α1

1−α .

The optimal choices (c0, τ) maximizes the below problem.

max0<c0<c, τ>0

VT ,

subject toc0

c − c0≤ N0

N1(T2

T1)

βα ,

LT (c0) < τ < UT (c0).

Proposition 10. Assume T2T1

>> 1 , N0 > N1 and β α−1α > 1.

Let c0 and τ be the maximizers of the above problem, thenc0 = c

1+N1N0

(αN0

N )1α

and τ = LT (c0).

Proof: The proof follows from straightforward calculus.

We define Efficiency Ratio (η) here as the ratio of System-Value with tolling (VT ) to Social optimum (VS). Let n0 =N0/N and n1 = N1/N . . From (27) and Proposition 10

η =VT

VS=

nα−10 (1 + n1

n0(αn0)

1α )α

(1 + (α − 1)n1).

Note that η does not scale with (T2T1

). Further, for n0 < 1,VT > VG and hence the System-Value with tolling is higher.But when the assumption β α−1

α > 1 in Proposition (10) isviolated, the System-Value with tolling may fall back to thatof Game. It can be shown that this is indeed true for the caseα = 1.

Example-4: In Figure (6), we have compared η attainedusing the PMP scheme versus that of a single-tier. We choseα = 2, β = 4 and (T2

T1) = 4. We observe that in-spite

of tolling, the PMP scheme always performs better than thesingle-tier scheme.

VIII. CONCLUSION

In this paper we examined the consequences of the idea thata protocol is simply a way of interpreting Lagrange multipliers.We showed that flows could choose the interpretations, basedon criteria such as delay or loss sensitivity. We determinedthe socially optimal protocol, as well as the choice that wouldresult by flows taking their own selfish decisions. We showedthat the social good is maximized by using the strictest possi-ble price interpretation. However, based on different mixesof flow types a mix of interpretations could be the Nashequilibrium state. We characterized the loss of efficiency forsome specific cases, and showed that a multi-tier networkwith tolling is capable of achieving superior system value.The result suggests the consideration of multiple tolled virtualnetworks, each geared towards a particular kind of flow. Inthe future we propose to explore the idea of virtual, tolledsubnetworks further.

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