Abstract—In this paper we propose joint multipath routing and rate allocation under assumption of strategic behavior of users using mechanism design. We use decomposition techniques and pricing mechanism that generate efficient allocations, maximize net utilities and can be implemented as the distributed algorithm. The algorithm is fair and robust against manipulation from users who usually are selfish and behave strategically.
Index Terms—computer network, network utility maximization, rate allocation, routing, mechanism design, game theory
I. INTRODUCTION
N important approach to design a computer network system is by formulating the design as the aggregate maximization of the utilities of the nodes subject to physical and economic constraints in the network. This is referred to as Network Utility Maximization (NUM). Much research effort has been put in the design of distributed algorithms for NUM [10]-[13]. The main ingredient to obtain distributed algorithms is the decomposition techniques, widely used in optimization theory [6].
One approach to joint routing and rate allocation is to allow multi-path routing, i.e. a source can transmit its data on multiple paths to its destination. In this formulation, a decision is decomposed into two – how much traffic to send (rate allocation) and how to distributed it over the available paths (multi-path routing) – in order to maximize aggregate network utility.
The main disadvantage of existing solutions on resource allocation based on Kelly’s approach [8] is that they strongly depend upon the users declaring their resource demand in a truthful way. They lack robustness against strategic manipulation from users.
This paper presents a decomposition method for joint rate allocation and routing complemented by explicitly considering the strategic behaviors of users using mechanism design [1], [2], [3], [4], [5].
The paper is organized as follows: In Section II we present the flow model in computer network. In Section II we introduce the utility maximization framework for resource
Magdalena Turowska is with the Institute of Informatics, Wroclaw University of Technology, Wyb. Wyspianskiego, 30-570 Wroclaw (e-mail:
Magdalena.turowska@pwr.wroc.pl).
allocation. We formulate the problem of joint rate allocation and routing and propose the decomposition in Section III. In Section IV we present mechanism design for the subproblem dependent on the behavior of users. The conclusions are presented in Section V.
II. FLOW MODEL
We consider a data network with a topology which can be represented by a directed graph. A collection of nodes, labeled
= 1, … , can send, receive, and relay data across communication links. A communication link is represented by an ordered pair , of distinct nodes. The presence of a link , means that the network is able to send data from the start node to the end node . The links are labeled by integers
= 1, … . The network topology can be represented by a
matrix = ∈ , whose entry = 1 if ∈ ,
= −1 if ∈ and = 0 otherwise,where is the set of links that are incoming to node , and is the set of links that are outgoing from node .
We identify the flows in the network by their destinations, i.e., flows with the same destination are considered as one single commodity, regardless of their sources. The destination nodes are labeled by = 1, … , = 1, … , denotes index of transmission demand from source to destination , ! "
is data rate of -th demand/user with source and destination .
On each link , $ ≥ 0 is the amount of flow destined for node . At each node , flows satisfy the constraint
& ! "
'()
"*+
+ & $
∈-
= & $
∈.
, = 1, … , .
We assume finite capacity of links, thus
& $ ≤ 0
*+
, where 0 is the capacity of link .
This model describes the average behavior of data transmissions, i.e., the average data rates on the communication links, and ignores packet-level details of transmission protocols and forwarding mechanisms. The link capacity in practical communication systems should be defined appropriately, taking into account packet loss and
Maximization of Utility in Computer Network with Application of Game Theory
Magdalena Turowska
A
2012
retransmission, so the flow conservation law holds for the effective throughput or goodput [9].
III. NETWORK UTILITY MAXIMIZATION (NUM) Many network resource allocation problems can be formulated as constrained maximization of some utility function [7, 9]. In the Network Utility Maximization (NUM) framework, each transmission demand (or user) has its utility function and link bandwidths are allocated so that network utility (i.e., the sum of all users’ utilities) is maximized. A utility function can be interpreted as the level of satisfaction attained by a user as a function of resource allocation.
Efficiency of resource allocation algorithms can thus be measured by the achieved network utility. Utility functions can also be interpreted as the ‘knobs’ to control the tradeoff between efficiency and fairness.
To formulate the joint rate allocation and routing problem in communication network we need to consider the following network utility maximization problem
()4max, 5)6& & & 7 "
'()
"*+
*+,8 *+
! "
subject to constraints:
& ! "
'()
"*+
+ & $
∈-
= & $
∈.
! "≥ 0, $ ≥ 0
& $ ≤ 0
*+
over ! ", $ , for = 1, … ; , = 1, … ; = 1, … , where 7 " is the utility of k-th network transmission from source to destination (k-th user).
The objective function is differentiable and strictly concave and the feasible region is compact; hence a maximizing value
of ! ", $ exists and can be found by Lagrangian methods.
The Lagrangian of formulated maximization problem is defined as
: x, y, λ, β =
= & & & 7 "
'()
"*+
*+,8 *+
! " − & & & > ! "
'()
"*+
*+,8 *+
−
− & & λ $ + & & & > $
*+8
*+
*+
+ & λ
*+
0
where λ , β are the Lagrange’s multipliers (prices) associated with link capacity and network structure constraints, respectively.
The convex optimization (optimization a convex or concave function over a convex constraint set) has useful Lagrange duality properties, which lead to decomposability structures.
Lagrange duality theory links the original maximization problem, termed primal problem, with a dual minimization problem, which sometimes readily presents decomposition possibilities [9]. The basic idea in Lagrange duality is to relax the original problem by transferring the constraints to the objective in the form of a weighted sum.
IV. DECOMPOSITION
The main idea of decomposition is to decompose the original large problem into distributively solvable subproblems which are then coordinated by a high-level master problem by means of some kind of signaling [6].
Dual decomposition methods is used correspond to a resource allocation. The master problem sets the price for the resources to each subproblem, which has to decide the amount of resources that can be used depending on the price.
The Lagrange dual problem associated with the primal problem under consideration is given by
minA,B λ, β
(2) subject to constraints:
λ > 0, where
λ, β = x, β + 5 y, λ, β + & λ
*+
0
and
x, β = max
()4 & & & 7 "
DEF G*+
H8I
! " −> ! "
(3) 5 y, λ, β
= max5
)6 { − & & λ $ − & >
H8I
$ }, (4) for = 1, … ; , = 1, … ; = 1, … ; where x =
! " *+,…, *+,…,
"*+,…,DEF
and y = $ *+,…
*+,… .
It is well known that for a convex optimization, a local optimum is also a global optimum and solving the problem
after decomposition is equivalent to solving the original (primal) problem.
For solving dual problem we propose to use three-level decomposition presented in Fig. 1.
Fig. 1. A hierarchical dual decomposition used to solving NUM problem (L- number of links, P- number of paths).
The first (upper) level solves problem (2), the second (middle) level solves problems (3) and (4), which can be decomposed.
On the third (the lowest) level each source and each link can independently compute !∗ " > and $∗ ∑H8I > ;λ as
min
()4 7 " ! " − > ! " (5) and
min5
)6 λ $ − & >
8 *+
$
respectively. On the upper level the problem we can formulate as
minA
6,O() & & & 7 "
'()
"*+
*+,8 *+
!∗ " >
− & & & > !∗ " >
'()
"*+
*+,8 *+
−
− & & λ $∗ P& > ;
H8I
λ Q
+ & & & > $∗ & > ;
H8I
λ
*+8
*+
*+
+ & λ
*+
0
for = 1, … ; , = 1, … ; = 1, … , where
!∗ " > and $∗ ∑H8I > ;λ are the optimal !∗ " and
$∗ for the given R and > – the results obtained on the lower
level. R and > are usually interpreted as users (demands) and links prices.
V. MECHANISM DESIGN
As results of the decomposition we have received to solve subproblems A which can be resolved by users (sources) independently and subproblems B which links can solve independly.
In order to prevent selfish behaviors of users, because such behaviors make impossible maximization of the network utility, we introduce game-theoretic mechanism for subproblems A. We refer to the class of noncooperative games derived from such mechanism as auction, in which Nash equilibrium is usually assumed as a solution concept. A Nash equilibrium captures the notation of a stable solution and is the solution from which no user (demand/player) can individually improve his utility by deviating.
In this paper we assume that each user is restricted to communicate his demand to the network with the price that user is willing to pay for the resource. We present distributed and dynamic process that converges to Nash equilibrium.
Initialization: We assume an initial resource allocation
! "=∑∈. 0 − ∑ S*+, S8 ∑∈- $
∑ S*+, S8
for each user (demand) and initial price
> "='+
(). Step 1:
Each user receives information from the network:
>̅ "=∑ S*+, S8 ∑'"*V()U> S"− > "
∑ S*+, 8 S
which represents the average price per unit of resource from the other users, and
W "= & 0
∈.
− & & $
∈-
−
S*+, 8
− & & ! S"+ ! "
'()U
"*V S*+, S8
which is the excess resource demand excluding the demand from -th user with source and destination .
We define the cost of using resource:
X " ! ", > " ≜ ! " >̅ "+
+ { > "− >̅ " 1 − Z W "− ! " − −max {0, Z W "− ! " }}[
(7) (6)
which represents the price that user pays for ! " amount of resource corrected by penalty that user pays due to the mismatch of its price to the average price of the other users and by the term that is introduced to prevent the solution from reaching an inefficient Nash equilibrium.
Step 2:
Each user calculate ! " and > " as
max ()4,O()4 7 " ! " − X " ! ",> " , that means
()4max,O()4 7 " ! " − ! " >̅ "+
+{− > "+ >̅ " 1 − Z W "− ! " + +max {0, Z W "− ! " }}[.
The maximization of the utility can only occur when
> "=>̅ " 1 − Z W "− ! " +
+ max{0, Z W "− ! " }, where Z is a parameter of algorithm.
The draft of algorithm for joint rate allocation and routing with application of game-theoretic mechanism design is following:
1) Each source nodes locally computes >̅ " and W ", and transmit they to the users.
2) Each user based on received information calculates
> " and ! " >̅ " .
3) Each network node calculate locally
$ ∑H8I ∑'"*V()>̅ ", λ , using current value of λ and >̅ ".
4) Each link (node where the link begin) updates R . 5) Return to Step 1.
VI. PRICE OF ANARCHY
The price of anarchy of the game is the most popular measure of efficiency and is defined as the ratio between the worse objective function value of an equilibrium of the game and that of an optimal outcome [1].
The mechanism presented in Section V implements in Nash equilibrium problem described in Section III. It can be shown that Nash equilibrium of mechanism presented in Section V satisfy the optimality conditions of problem from Section III.
The necessary and sufficient conditions for the efficient allocations can be determined from Karsh-Kuhn-Tucker (KKT) conditions. In order to show that the mechanism in Section V implements in Nash equilibrium allocation problem presented in Section III, it can be shown that the Nash allocation satisfies the KKT conditions for primal problem.
This result indicates that the price of anarchy equals 1, which
means that the Nash equilibrium and the optimal solution are the same.
VII. CONCLUSION
In this paper we study the problem of joint rate allocation and routing in the network with competing and selfish users.
The important feature of proposed mechanism is indicated: it gives the optimal solution of allocation problem in spite of autonomous of users. The simulation and implementation for the proposed framework are part of future research.
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