routing on a ring network - indian institute of...

Routing on a Ring Network

Ramya Burra, Joy Kuri, Chandramani Singh1, and Eitan Altman2

1 Department of ESE, Indian Institute of Science Bangalore, Indiaburra,kuri,[email protected]

2 Univ Cote d‘Azur, INRIA, France, and LINCS, Paris, [email protected]

Abstract. We study routing on a ring network in which traffic origi-nates from nodes on the ring and is destined to the center. The userscan take direct paths from originating nodes to the center and also mul-tihop paths via other nodes. We show that routing games with only oneand two hop paths and linear costs are potential games. We give explicitexpressions of Nash equilibrium flows for networks with any generic costfunction and symmetric loads. We also consider a ring network with ran-dom number of users at nodes, all of them having same demand, andlinear routing costs. We give explicit characterization of Nash eequilib-ria for two cases: (i) General i.i.d. loads and one and two hop paths,(ii) Bernoulli distributed loads. We also analyze optimal routing in eachof these cases.

Keywords: ring network, routing games, potential games, Nash equi-librium, optimal flow configuration

1 Introduction

Routing problems arise in networks in which common resources are shared by agroup of users. Examples of such scenario include flow routing in communicationnetworks, traffic routing in transportation networks, flow of work in manufac-turing plants etc. Each user incurs a certain cost (e.g., delay) at each link on itsroute, where the cost depends on the flows through the link. The routing prob-lems, when handled by a centralized controller, aim to optimize the aggregatecost of all the users, e.g., average network delay. However, a centralized solutionmay not be viable for several reasons. For instance, a very large network andits time varying attributes (e.g., traffic and link states in a communication net-work) could lead to excessive communication overhead for solving the problemcentrally. In other cases, the very premise of the network may be such that localadministrators control different portions of the network, e.g., different parts of atransportation network may be controlled by different depots. In either case, dis-tributed controllers may compete to maximize individual, and often conflicting,performance measures. It is imperative to assess the performance of distributedcontrol, especially how far it is from the global optimal.

Distributed control of routing has been widely modelled as noncoopera-tive games among self-interested decision makers. Nash equilibria of the games

2 Burra et al.

(Wardrop equilibria in case of nonatomic games) characterize the system-wideflow configuration resulting from such distributed control. Wardrop [10] intro-duced Wardrop equilibrium in the context of transportation networks, and Dafer-mos and Sparrow [3] showed that it can be characterized as a solution of a stan-dard network optimization problem. Orda et al [8] showed existence and unique-ness of Nash equilibrium in routing games under various assumptions on thecost function. They also showed a few interesting monotonicity properties of theNash equilibria. Cominetti et al [2] computed the worst-case inefficiency of Nashequilibria and also provided a pricing mechanism that reduces the worst-caseinefficiency. Altman et al [1] considered a class of polynomial link cost functionsand showed that these lead to predictable and efficient Nash equilibria. Hanwalet al [4] studied routing over time and studied a stochastic game resulting fromrandom arrival of traffic.

We study a routing problem on a ring network in which users’ traffic originateat nodes on the ring and are destined to a common node at the centre (see Fig. 1).Each user can use the direct link from its node to the centre and also a certainnumber of paths through the adjacent nodes, to transport its traffic. The usersincur two costs: (i) The cost of using a link between a node at the ring and thecentre, (ii) the cost of redirecting the traffic through adjacent nodes. The numberof users attached/connected to the node can be random. We characterize Nashequilibria of such routing games.

Scheduling problems are a class of resource allocation problems in whichresources are shared over time. In these problems, unlike simultaneous actionrouting problems, each user may see the system state that results from its pre-decessor’s actions. However, if we assume that such information is not availableto the users, our framework can also be used to analyze certain scheduling (or,temporal routing) problem.

In Section 2, we formally introduce our general framework and also illustratehow it can be used to model several problems arising in communication networks,transportation networks etc. In subsequent sections, we analyze special cases ofthis framework. Following is a brief outline of our contribution

1. In Section 3, we show that routing games with only one and two hop pathsand linear costs are potential games. We also give explicit expressions of Nashequilibrium flows for networks with any generic cost function and symmetricloads.

2. In Section 4 we consider networks with random loads and linear routing costs.We give explicit characterization of Nash eequilibria for two cases: (i) Generalload distribution and one and two hop paths, (ii) Bernoulli distributed loads.

2 System Model

Let us consider a ring network with N nodes and Mn users at each noden ∈ [N ] := {1, . . . , N}. Let us assume that the ith user at node n has a flowrequirement φin to be sent to the centre. Let c(z) represent the cost per unit offlow at any link where z is the aggregate traffic through this link. Throughout

Routing on a Ring Network 3

we assume that c(·) is positive, strictly increasing and convex. We assume thateach user can use the direct link to the centre and the K other links throughK adjacent nodes in the clockwise direction. For example, any user at at noden can use links (n, 0), (n+ 1, 0), . . . , (n+K, 0).3 We also assume that a user atnode n incurs kd additional per unit flow cost for any flow that it routes throughlink (n+ k, 0). Note that we assume no cost for using the links along the ring.

Fig. 1. A ring network with K = 2. For example, the users at node 1 can use paths(1, 0), (1, 2, 0) and (1, 2, 3, 0).

For each n ∈ [N ], i ∈ [Mn], l ∈ [n, n + K], let xinl be the flow of ith userat node n that is routed through link l. For brevity, we let xin denote the flowconfiguration of the ith user at node n, x denote the network flow configurationand xl denote the total flow through link l; xin = (xinl, l ∈ [n, n + K]), x =(xin, n ∈ [N ], i ∈ [Mn]) and xl =

∑n∈[l−K,l]

∑i∈[Mn] x

inl. Then the total cost

incurred by this user is

Cin(x) =∑

l∈[n,n+K]

xinl(c(xl) + (l − n)d), (1)

and the aggregate network cost is C(x) =∑n∈[N ]

∑i∈[Mn] C

in(x). Note that the

flows must satisfy ∑l∈[n,n+K]

xinl = φin (2)

for all i ∈ [Mn], n ∈ [N ] in addition to nonnegativity constraints.We now illustrate how this framework can be used to model a variety of

routing and scheduling problems.

1. We can think of this framework as modeling routing in a transportationnetwork in a city. The ring and the centre represent a ring road and thecity centre, respectively. We have sets of vehicles starting from various entry

3 Clearly, the addition here is modulo N .

4 Burra et al.

points, represented as nodes on the ring, all destined to the city centre. Thecosts here represent latency. We assume that the ring road has large enoughcapacity to render the latency along it independent of the load. On the otherhand, latency on the roads joining the ring to the centre is traffic dependent.Each node has a set of depots, each controlling routing of a subset of vehiclesstarting at this node, and interested in minimizing routing costs of only thosevehicles.

2. We can use this framework to model load balancing in distributed computersystems [5].

3. We can also use this framework to model scheduling of charging of elec-tric vehicles at a charging station. Here, the nodes represent time slots andplayers represent vehicles. The per unit charing cost in a slot depends on thecharge drawn in that slot. Each vehicle can wait up to K slots to be charged.Further, each vehicle would like to minimize a weighted sum of its chargingcost and waiting time. We assume that the vehicles do not know pendingcharge from the earlier vehicles when making scheduling decision.

3 Deterministic Loads

Nash Equilibrium: A flow configuration x is a Nash equilibrium if, for alli ∈ [Mn], n ∈ [N ],

Cin(x) = minyin

Cin(yin, x \ xin) (3)

subject to (2) and nonnegative constraints. Under our assumptions on c(·), therouting game is a convex game [9]. Existence and uniqueness of the Nash equi-librium then follows from [8]. It follows that the equilibrium is characterizedby the following Kuhn-Tucker conditions(using cost from equation (1)): for ev-ery i ∈ [Mn] there exists a Lagrange multiplier λin such that, for every linkl ∈ [n, n+K],

c(xl) + (l − n)d+ xinlc′(xl) ≥ λin (4)

with equality if xinl > 0. From this,

λin =

∑l:xinl>0

c(xl)+(l−n)dc′(xl)

+ φin∑l:xinl>0

1c′(xl)

, for all i ∈ [Mn], n ∈ [N ].

We observe that the equilibrium flow configuration is the solution of the followingsystem of equations.

xinj = max

1

c′(xj)

∑l:xinl>0

c(xl)−c(xj)+(l−j)dc′(xl)

+ φin∑l:xinl>0

1c′(xl)

, 0

, (5)

for all i ∈ [Mn], j ∈ [n, n+K], n ∈ [N ].


Optimal Solution: Recall that the aggregate routing cost is

∑n∈[N ]

xnc(xn) + d∑

l∈[n,n+K]

(l − n)∑

i∈[Mn]

xinl

.

Clearly, for each n ∈ [N ], we only need to optimize the total requirement ofthe users [Mn] which flows through link l ∈ [n, n + K], denoted as xnl. Indi-vidual user’s flows can always be chosen to match the aggregate optimal flowrequirements. We can thus restate the optimal routing problem as

min∑n∈[N ]

xn

c(xn) + d∑

l∈[n,n+K]

(l − n)xnl

,

subject to∑

l∈[n,n+K]

xnl = φn :=∑

i∈[Mn]

φin, for all n ∈ [N ],

and nonnegativity constraints. The optimal flows satisfy the following Kuhn-Tucker conditions: for every n ∈ [N ] there exist a Lagrange multiplier λn suchthat, for every link l ∈ [n, n+K],

c(xl) + (l − n)d+ xlc′(xl) ≥ λn

with equality if xnl > 0. From this,

λn =

∑l:xnl>0

(c(xl)+(l−n)d

c′(xl)+ x−nl

)+ φn∑

l:xinl>01

c′(xl)

, for all n ∈ [N ],

where x−nl is the total flow through link (l, 0) that has not originated from noden. We observe that the optimal flow configuration is the solution of the followingsystem of equations.

xnj = max

1

c′(xj)

∑l:xnl>0

(c(xl)−c(xj)+(l−j)d

c′(xl)+ x−nl − c′(xj)x−nj

c′(xl)

)+ φn∑

l:xinl>0

1c′(xl)

, 0

, (6)

for all j ∈ [n, n+ k], n ∈ [N ].We can elegantly obtain Nash equilibria and optimal solutions in special

cases. In the following we consider two such cases, the first allowing only one-hop and two-hop paths to the centre and linear costs, and the second havingsame number of users, all with identical requirements, at all the nodes.

3.1 Maximum Two Hops and Linear Costs (K = 1 and c(x) = x)

Here, using xinn + xin(n+1) = φin, from (5),

xinn =

[c(xn+1)− c(xn) + d+ c′(xn+1)φin

c′(xn) + c′(xn+1)

]φin0

6 Burra et al.

for all i ∈ [Mn], n ∈ [N ].4 For linear costs, substituting c(x) = x for all x,

xinn =

[xn+1 − xn + d+ φin

2

]φin0

. (7)

Further, using xn =∑j∈[Mn] x

jnn +

∑j∈[Mn−1](φ

jn−1 − x

j(n−1)(n−1)) and xn+1 =∑

j∈[Mn](φjn − xjnn) +

∑j∈[Mn+1] x

j(n+1)(n+1),

xinn =

[2φin +

∑j∈Mn\i(φ

jn − 2xjnn)

4

+

∑j∈Mn+1

xj(n+1)(n+1) −∑j∈Mn−1

(φjn−1 − xj(n−1)(n−1)) + d

4

]φin0

. (8)

Notice that flow configuration of ith user at node n is completely specified byxinn. The above equation can be seen as the best response of this user.

Lemma 1. If the players update according to(8), round-robin or random updateprocesses converge to the Nash equilibrium.

Proof. The routing game under consideration is a potential game with potentialfunction

V (x) =1

2

∑n∈[N ]

x2n +

∑i∈[Mn]

((xinn)2 + (xin(n+1))2 + 2xin(n+1)d)

.

Hence it exhibits the improvement property and convergence as stated in thelemma [7].

Optimal Solution: From (6),

xnn =

[c(xn+1)− c(xn) + d+ c′(xn+1)(xn+1 − xn(n+1) + φn)− c′(xn)(xn − xnn)

c′(xn) + c′(xn+1)

]φn0

for all n ∈ [N ]. For linear costs, substituting c(x) = x for all x,

xnn =

[2xn+1 − 2xn − xn(n+1) + xnn + d+ φn

2

]φn0

.

Further, using xn = xnn + φn−1 − x(n−1)(n−1) , xn+1 = φn − xnn + x(n+1)(n+1)

and xn(n+1) = φn − xnn,

xnn =

[x(n−1)(n−1) + x(n+1)(n+1)

2+

2(φn − φn−1) + d

4

]φin0

. (9)

4 [x]ba := min{max{x, a}, b}.


Notice that the aggregate flow configuration of the users at node n is completelyspecified by xnn. As for the game problem if we update according to(9), round-robin or random update processes converge to the optimal solution.

In fact, we can provide a more explicit characterization of the Nash equilib-rium when Mn = M for all n ∈ [N ] and xinl > 0 for all i ∈ [M ], l ∈ {n, n+1} andn ∈ [N ]. Also, we can show that, when Mn = M , there cannot be an optimalsolution with xnl > 0 for all l ∈ {n, n + 1} and n ∈ [N ]. This analysis is alongthe lines of [6] and is presented in Appendix A.

3.2 Symmetric Loads (Mn = M and φin = φ)

Here, we can restrict to symmetric flow configurations owing to symmetry of theproblem. We can express any symmetric network flow configuration as a vectorβ = (β0, β1 . . . , βK),

∑Kj=0 βj = φ where βj := xin(n+j) for all i ∈ [M ], n ∈ [N ]

and j ∈ [0,K].

Theorem 1. If Mn = M and φin = φ for all i ∈ [Mn], n ∈ [N ], then the uniqueNash equilibrium is

βj =

{φ

k∗+1 + (K∗−2j)d2c′(Mφ) if l ∈ [n, n+K∗]

0 otherwise.(10)

where K∗ = min{max{k : k(k + 1) < 2φc′(Mφ)d },K}

Proof. From the Karush-Kuhn-Tucker conditions for optimality of β (see (4)),

c(βj + x−in(n+j)) + jd+ βjc′(βj + x−in(n+j)) ≥ λ (11)

where x−in(n+j) is the total flow on link (n+ j, 0) except that of ith user at node

n. Note that, for β to be a symmetric Nash equilibrium, βj + x−in(n+j) = Mφ.

Hence (11) can be reduced to

c(Mφ) + jd+ βjc′(Mφ) ≥ λ

with equality if βj > 0. So, we see that

βj = max

{1

c′(Mφ)(λ− c(Mφ)− jd), 0

}(12)

Note that βj is decreasing in j. Let us assume that βk > 0 for all k ≤ K ′ for

some K ′ ≤ K, and 0 otherwise. Then, using∑K′

i=0 βi = φ in (12),

λ(K ′)− c(Mφ) =φc′(Mφ)

(K ′ + 1)+K ′d

2, (13)

where we write λ(K ′) to indicate dependence of λ on K ′. Substituting the aboveexpression in back in (12),

βj =φ

1 +K ′+d(K ′ − 2j)

2c′(Mφ), j ∈ [0,K ′]. (14)

8 Burra et al.

To complete the proof, we claim that K ′ equals K∗ where

K∗ = min{max{k : k(k + 1) <2φc′(Mφ)

d},K}.

Let us first argue that K ′ cannot exceed K∗. We only need to consider the casewhen K∗ < K. In this case, from the definition of K∗, for any K ′ > K∗,

1

K ′ + 1− dK ′

2φc′(Mφ)≤ 0,

which contradicts the defining property of K ′ that βk > 0 for all k ≤ K ′. Thiscompletes the argument. Now we argue that K ′ cannot be smaller than K∗,again by contradiction. Let K ′ < K∗. Then, from (13),

λ(K ′)− λ(K∗) =φc′(Mφ)(K∗ −K ′)(K∗ + 1)(K ′ + 1)

− (K∗ −K ′)d2

= φc′(Mφ)(K∗ −K ′){

1

(K∗ + 1)(K ′ + 1)− d

2φc′(Mφ)

}> 0,

where the inequality follows from definition of K∗. Hence, from (12),

βK∗ ≥1

c′(Mφ)(λ(K ′)− c(Mφ)−K∗d)

>1

c′(Mφ)(λ(K∗)− c(Mφ)−K∗d) > 0.

This contradicts K ′ < K∗ which would imply βK∗ = 0.

Optimal Routing: Observe that the optimal strategy of any user will be β0 = φand βj = 0, 1 ≤ j ≤ K.

4 Random Loads

We now consider the scenario where the numbers of users at various nodes, Mn,are i.i.d random variables with distribution (p1, p2 . . . , pM ). Such a case may beused to depict random arrival of electric vehicles at the charging station. Weassume that a user knows the number of collocated users but only knows thedistribution of users at the other nodes. Throughout this section we restrict toequal flow requirements for all the users, i.e. φin = φ for all n ∈ [N ], i ∈ [Mn], andlinear per unit flow cost, i.e., c(x) = x. In the following we analyze two specialcase of this routing problem, the first assuming the users can only use one-hopand two-hop paths to centre, and the second having Bernoulli user distribution.


4.1 Maximum Two Hops (K = 1)

We consider symmetric flow configurations where all the users with equal num-ber of collocated users adopt same flow configuration. We can then express thenetwork flow configuration as a vector γ = (γ(1), γ(2), . . . ) where γ(m) repre-sents the flow that a user with m collocated users redirects to its two-hop path.Let us define

Pm = 1−M∑

l=m+1

lpll + 1

and Qm =

m∑l=0

lpl,

for all 0 ≤ m ≤M .

Theorem 2. The unique Nash equilibrium is given by

γ(m) =

{0, if 1 ≤ m ≤ mαφ2 −

(d−α)2(m+1) , otherwise,

where mα = min{min{m :d

φPm+QmPm

< m+ 2},M}

and α = d− d

Pmα− Qmαφ

Pmα.

Proof. Let us consider a user i with m collocated users. Let us fix the strategiesof all other users in the network to γ = (γ(1), γ(2), . . . ). Then the best responseof user i, say γ′(m), is the unique minimizer of the cost function

(φ− γ′(m))((m− 1)(φ− γ(m)) + φ− γ′(m) +∑

lplγ(l))

+ γ′(m)((m− 1)γ(m) + γ′(m) +∑l

lpl(φ− γ(l)) + d).

γ′(m) must satisfy the following optimality criterion

− 2(φ− γ′(m))− (m− 1)(φ− γ(m))−∑

lplγ(l)

+ 2γ′(m) + (m− 1)γ(m) +∑

lpl(φ− γ(l)) + d ≥ 0

with equality if γ′(m) > 0. For γ to be a symmetric Nash equilibrium, settingγ′(m) = γ(m) in the above inequality,

−(m+ 1)φ+ 2(m+ 1)γ(m) +∑

lplφ− 2∑

lplγ(l) + d ≥ 0

yielding

γ(m) = max{φ2

+2∑lplγ(l)− φ

∑lpl − d

2(m+ 1), 0}

10 Burra et al.

Clearly, the above should hold for all m ∈ {0, 1, . . . ,M}. Setting,

α = 2∑

lplγ(l)− φ∑

lpl, (15)

and mα = bd− αφ− 1c, (16)

we get

γ(m) =

{φ2 + α−d

2(m+1) , if m > mα

0, otherwise.(17)

We now show how to obtain α and mα. From (17),

mpm(2γ(m)− φ) =

{(α−d)mpm

m+1 , if m > mα

−mpmφ, otherwise.

Using this in (15),

α =−d∑m>mα

mpmm+1 − φ

∑mαm=0mpm

(1−∑m>mα

mpmm+1 )

= d− d

Pmα− Qmαφ

Pmα,

and hence, from (16),

mα = b d

φPmα+QmαPmα

− 1c

Let us now turn to the expression of mα in the statement of the theorem. Clearly,mα >

dφPmα

+QmαPmα

− 2. Also,

d

φPmα−1+Qmα−1

Pmα−1≥ mα + 1,

implying

d

φPmα+QmαPmα

≥ mα + 1,

or,d

φPmα+QmαPmα

− 1 ≥ mα.

So, the two expressions of mα are equivalent, and γ(m)s in the statement of thetheorem indeed constitute a Nash equilibrium. Also note that existence of anoptimal γ ensures existence of at least one (α,mα) pair satisfying (15)-(16). Itremains to establish uniqueness of (α,mα) pair satisfying (15)-(16). We do thisin Appendix B.


Optimal Routing: The expected total routing cost will be N times the sumof expected routing costs on links (n− 1, n) and (n, 0) for an arbitrary n. In thefollowing, we optimize the latter to get the optimal flow configuration.

Theorem 3. The unique optimal flow configuration is given by

γ(m) =

{0, if 0 ≤ m ≤ mαφ2 −

(d−α)4m , otherwise,

where mα = min{min{m :d

2φPm+QmPm

< m+ 1},M}

and α = d− d

Pmα− 2Qmαφ

Pmα.

Proof. The optimal flow configuration γ is the unique minimizer of the followingcost.∑

mn,mn−1

pmnpmn−1{(mn−1γ(mn−1) +mn(φ− γ(mn)))2 +mn−1γ(mn−1)d}

For all 0 ≤ m ≤M , γ(m) must satisfy the following optimality criterion.

∑l 6=m

2pl(2mγ(m)− 2lγ(l) + (l −m)φ+d

2) + pmd ≥ 0,

with equality if γ(m) > 0. Equivalently,

4mγ(m)− 2mφ− 2∑

pl(2lγ(l)− lφ) + d ≥ 0

yielding

γ(m) = max{φ2

+4∑lplγ(l)− 2φ

∑lpl − d

4m, 0}.

Setting

α = 2∑

pll(2γ(l)− φ),

and mα = bd− α2φc,

we get

γ(m) =

{φ2 + α−d

4m , if m > mα

0, otherwise.

The remaining steps are similar to those in the proof of Theorem 2 and areomitted for brevity.

12 Burra et al.

Distribution γ(1) γ(2) γ(3) γ(4) γ(5)

Uniform{0,5} Optimal 0 0 0 0.125 0.2

Game 0 0.106 0.203 0.263 0.275

Binomial(5,0.5)Optimal 0 0 0 0.0821 0.166

Game 0 0.117 0.2125 0.27 0.308

Table 1:Optimal and game solutions in various distributions forφ = 1, d = 2,M = 5,K = 1 .

4.2 Bernoulli Loads (p0 + p1 = 1)

We again focus on only symmetric flow configuration. As in Section 3.2, we letxinl = βn−l for all i ∈ [M ], n ∈ [N ] and l ∈ [n, n+K].

Theorem 4. The unique Nash equilibrium is given by

βi =

{φ

K∗+1 + d(K∗−2i)2(2−p) , if 0 ≤ i ≤ K∗

0, otherwise,

where K∗ = min{max{k : k(k + 1) < 2φ(2−p)d },K}.

Proof. Let us consider a node i. This node may have a user with probabilityp, in which we refer to this user as user i. Let all other users in the networkhave flow configuration β′j , 0 ≤ j ≤ K. Then the following optimization problemyields the best response of user i.

min

j=K∑j=0

{βj(βj + jd+∑l 6=i

pβ′l)}

subject to

K∑j=0

βj = φ,

βj ≥ 0, j ∈ [0,K].

From the Karush-Kuhn-Tucker conditions for optimality of β, there exists aLagrange multiplier λ such that

2βj + jd+ p∑l 6=j

β′l ≥ λ, for all 0 ≤ j ≤ K, (18)

with equality if βj > 0. Note that, for β to be a symmetric Nash equilibrium,β = β′. Hence (18) can be reduced to

(2− p)βj + p+ jd ≥ λ.



βj = max

{λ− p− jd

2− p, 0

}. (19)

The remaining steps are similar to those in the proof of Theorem 1 (see the stepsafter (12)) and are omitted for brevity.

Optimal Routing: The expected total routing cost will be N times the sumof expected routing costs on links (n− 1, n) and (n, 0) for an arbitrary n. In thefollowing, we optimize the latter to get the optimal flow configuration.

Theorem 5. The unique optimal flow configuration is given by

βj =

{φ

K∗+1 + d(K∗−2j)4(1−p) , if 0 ≤ i ≤ K∗

0, otherwise,

where K∗ = min{max{k : k(k + 1) < 4(1−p)φd },K}.

Proof. The optimal routing is the solution of the following optimization problem.

min∑

δ∈{0,1}K+1

p∑δj (1− p)(K+1)−

∑δj

(K∑l=0

δjβj

)2

+ pd

K∑j=0

jβj

subject to

K∑j=0

βj = φ,

βj ≥ 0, j ∈ [0,K].

From the Karush-Kuhn-Tucker conditions for optimality of β there exits a La-grange multiplier λ such that, for all 0 ≤ j ≤ K,

2∑

δ∈{0,1}K+1

p∑δl(1− p)(K+1)−

∑δlδj

(K∑l=0

δlβl

)+ pjd ≥ λ,

with equality if βj > 0. Equivalently,

2

K∑l=0

βl

∑δ∈{0,1}K+1

p∑δl(1− p)(K+1)−

∑δlδjδl

+ pjd ≥ λ,

which reduces to

2p

βj + p∑l 6=j

βl

+ pjd ≥ λ. (20)

Using βj = 1−∑l 6=j βl, (20) further reduces to

2p(1− p)βj + 2p2 + jpd ≥ λ,

14 Burra et al.


βj = max

{λ− 2p2 − jpd

2p(1− p), 0

}. (21)

The remaining steps are similar to those in the proof of Theorem 1 (see the stepsafter (12)) and are omitted for brevity.

5 Conclusion and Future Work

We studied routing on a ring network. We studied both, non-cooperative gamesbetween competing users and network optimal routing. We considered severalspecial cases of networks with deterministic and random loads. We provided char-acterization of Nash equilibria and optimal flow configuration in these cases (seeTheorems 1-5).

Our future work entails extending this analysis to more general cases. Wewould like to study price of anarchy, and also pricing mechanisms (tolls) thatinduce optimality.

Acknowledgments:

References

1. Altman, E., Basar, T., Jimenez, T., Shimkin, N.: Competitive routing in networkswith polynomial cost. IEEE Transactions on Automatic Control 47(1), 92–96 (Jan-uary 2002)

2. Cominetti, R., Correa, J.R., Stier-Moses, N.E.: The impact of oligopolistic compe-tition in networks. Operations Research 57(6), 1421–1437 (June 2009)

3. Dafermos, S.C., Sparrow, F.T.: The traffic assignment problem for a general net-work. Journal of Research of the National Bureau of Standards, Series B 73B(2),91–118 (1969)

4. Hanawal, M.K., Altman, E., El-Azouzi, R., Prabhu, B.J.: Spatio-temporal con-trol for dynamic routing games. In: International Conference on Game Theory forNetworks. pp. 205–220. Shanghai, China (April 2011)

5. Kameda, H., Altman, E., Kozawa, T., Hosokawa, Y.: Braess-like paradoxes in dis-tributed computer systems. IEEE Transactions on Automatic Control 45(9), 1687–1691 (September 2000)

6. Karouit, A., Haddad, M., Altman, E., Matar, A.: Routing game on the line: Thecase of multi-players. In: International Symposium on Ubiquitous Networking. pp.14–24. Casablance, Morocco (May 2017)

7. Monderer, D., Shapley, L.S.: Potential games. Games and Economic Behavior14(1), 124–143 (May 1996)

8. Orda, A., Rom, R., Shimkin, N.: Competitive routing in multi-user communicationnetworks. IEEE/ACM Transactions on Networking 1(5), 510–521 (October 1993)

9. Rosen, J.B.: Existence and uniqueness of equilibrium points for concave N-persongames. Econometrica 33(3), 520–534 (July 1965)

10. Wardrop, J.G.: Some theoretical aspects of road traffic research. Proceedings ofthe Institute of Civil Engineers 1(3), 325–378 (May 1952)


Appendix A

Let us recall the discussion on Nash equilibrium in Section 3.1. When Mn = Mfor all n ∈ [N ] and xinl > 0 for all i ∈ [M ], l ∈ {n, n+ 1} and n ∈ [N ],

−xn+1 + xn + 2xinn = d+ φin, (22)

and − xn−1 + xn + 2xi(n−1)n = −d+ φin−1.

Adding the above two equations for all i ∈ [M ],

−xn−1 + 2

(1 +

1

M

)xn − xn+1 =

1

M

∑i∈[M ]

(φin−1 + φin).

Further, multiplying the above by un and adding over n ∈ [N ],

−u(F (u) + xN − uNxN ) + 2

(1 +

1

M

)F (u)− F (u)− x1u+ x1u

N+1

u= ψ(u),

where

F (u) =∑n∈[N ]

xnun

and ψ(u) =1

M

∑n∈[N ]

∑i∈[M ]

(φin−1 + φin)

un.

Rearranging the terms in the above equation

−F (u)

(u2 − 2

(1 +

1

M

)u+ 1

)+ u2(1− uN )

(x1

u− xN

)= uψ(u).

Let u1, u2 denote the two solutions of u2 − 2(1 + 1

M

)u+ 1 = 0:

u1 = 1 +1

M(1−

√1 + 2M), u2 = 1 +

1

M(1 +

√1 + 2M).

Substituting u = u1, u2 in the above equation we obtain the following two linearequations in x1 and xN .

x1

u1− xN =

ψ(u1)

u1(1− uN1 ).

x1

u2− xN =

Ψ(u2)

u2(1− uN2 ).

These together yield

x1 =ψ(u1)u2(1− uN2 )− Ψ(u2)u1(1− uN1 )

(u2 − u1)(1− uN1 )(1− uN2 ),

xN =ψ(u1)(1− uN2 )− Ψ(u2)(1− uN1 )

(u2 − u1)(1− uN1 )(1− uN2 ).

16 Burra et al.

We can now use (22) to obtain the remaining xns and (7) to obtain all xinns.Next, let us focus on optimal routing. We provide via contradiction that an

optimal solution cannot have xnl > 0 for all l ∈ {n, n+ 1} and n ∈ [N ]. Assumethis to be the case. Then, from (9),

2xnn − x(n−1)(n−1) − x(n+1)(n+1) =2(φn − φn−1) + d

2

Adding these for all n ∈ [N ] gives d = 0, which contradicts our hypothesis thatd > 0.

Appendix B

We establish uniqueness via contradiction. Let (α,mα) and (α′,mα′) be twopairs satisfying (15)-(16). We assume mα′ > mα without any loss of generality.

Recall that α = d− dPmα

− φQmαPmα

and d−αφ < mα + 2, implying

d

φ< (mα + 2)Pmα −Qmα . (23)

Similarly, α′ = d− dPm

α′− φQm

α′Pm

α′and d−α′

φ ≥ mα′ + 1, implying

d

φ≥ (mα′ + 1)Pmα′ −Qmα′ . (24)

We argue that (mα′+1)Pmα′−Qmα′ ≥ (mα+2)Pmα−Qmα , and hence both (23)and (24) cannot hold simultaneously. Indeed note that

(mα′ + 1)Pmα′ − (mα + 2)Pmα

= (mα′ + 1)(Pmα′ − Pmα′−1) + (mα′ + 1)Pmα′−1 − (mα + 2)Pmα

= mα′pmα′ +

mα′−1∑m=mα+1

{(m+ 2)Pm − (m+ 1)Pm−1}

≥ mα′pmα′ +

mα′−1∑m=mα+1

(m+ 1)(Pm − Pm−1)

=

mα′∑m=mα+1

mpm

= Qmα′ −Qmα

This completes the argument.

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