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Wireless QoSNC: A Novel Approach to QoS-based Network Coding in Wireless Networks Amir Hesam Salavati† , Babak Hossein Khalaj‡ , Pedro M. Crespo§ , Mohammad Reza Aref† † Information

Systems and Security Lab (ISSL) Sharif University of Technology E-mail: [email protected], [email protected] §

‡ Dept.

of Electrical Engineering Sharif University of Technology E-mail: [email protected]

CEIT and Tecnun (University of Navarra) E-mail: [email protected]

Abstract— The emergence of new multimedia services over wireless networks has generated much interest in QoS provisioning algorithms in such networks. Due to sensitivity of these services to network parameters such as delay and throughput, QoS provisioning algorithms are designed to provide guarantees on these metrics. In this paper, we introduce a novel distributed algorithm to provide QoS in wireless networks using network coding. The suggested algorithm identifies subgraphs that satisfy delay and rate constraints. Among all subgraphs with these properties, the one with minimum cost is selected as the final solution. Our algorithm is differentiated from earlier works in the sense that it guarantees QoS instead of only improving the quality of services in a network coding setup.

I. I NTRODUCTION Wireless communications have experienced a tremendous growth in recent years. Today mobile devices and wireless services offer TV broadcasting, video telephony and other sorts of media streaming. These multimedia services have their own challenges, the most important of which is their sensitivity to delay, jitter, and reliability issues. Consequently, algorithms for such systems need to directly take quality of service (QoS) issues into account. QoS routing methods have addressed this issue by finding the best route to the best serving node that provides the highest quality to the user. For surveys on QoS routing in wireless networks and their issues see [1] and [2]. While network coding was first applied to fixed networks by Ahlswede et. al. [3], applications of network coding in wireless networks have been gaining more attention in the past few years. Practical implementations and simulations [11]- [14] have shown that by using network coding, it is possible to send multimedia streams over wireless networks at a higher throughput, less delay and improved levels of reliability. In general, network coding has shown to improve resource allocation efficiency [5], reliability [6] and security [7] (For a survey on network coding see [8] and [9]). Moreover, Lun et. al. [10] have shown that using network coding makes it possible to find minimum cost subgraph in polynomial time that would otherwise be NP-hard. This work was partially supported by Iranian NSF under grant numbers 84.5193 and 84094/14 and was partially supported by Iranian Telecommunication Research Center (ITRC)

Although, devising network coding algorithms to improve QoS metrics in wireless networks have been addressed extensively, providing guarantees on such metrics seems to be an open issue. Introducing an algorithm that will provide such guarantees is the main concern in this paper. The algorithms proposed in the literature so far, do not directly addressed the problem of satisfying a given set of QoS constraints. For example, the Minimum Cost Multicast (MCM) algorithm proposed by Lun et. al [10] does not directly consider QoS, although it relies on optimization schemes to determine proper flow subgraphs that minimize a given cost function. There are other algorithms that maximize network utility instead of minimizing cost [16]. For example, the algorithm proposed in [17] finds flow subgraphs that guarantee a minimum rate and maximize a given utility function and in that sense, it can be regarded as the closest approach to the one proposed in this paper. We introduce an optimization-based algorithm that not only improves QoS, but also ensures that the final solution satisfies the given constraints on end-to-end delay and rate. In [18] we have addressed the problem of network coding based QoS provisioning in fixed networks.

The proposed algorithm determines flow subgraphs that satisfy delay and rate constraints. Among all subgraphs with these properties, the one with minimum cost is selected as the final solution. Comparison of our results with minimum cost approaches, like [10], shows that the proposed algorithm can guarantee quality of service at a total cost which is only slightly higher than the cost obtained through minimum cost techniques that do not take such guarantees into account. Moreover, as the main requirement of providing QoS in a network is differentiating between different types of flows, the proposed scheme provides the framework for such differentiation between different flows according to their quality class.

The rest of this paper is organized as follows. In section II, we explain the problem of QoS network coding in wireless networks. Section III compares our work with minimum cost multicast approaches. Finally, section IV concludes the paper and suggests several ideas for future work.

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II. Q UALITY OF S ERVICE N ETWORK C ODING FOR W IRELESS N ETWORKS

set of nodes K ⊂ J [10]. Define: P (c) biJK

A. Network Model and Notations Our wireless network model follows the same steps as in [5], where the broadcast nature of wireless medium and link losses are taken into account. We model the network by a directed and connected hypergraph, H(N,A), where, N denotes the set of nodes and A represents the set of hyperlinks. Hyperlinks are the generalization of links and are used to account for broadcast nature of the wireless medium. A hyperlink is shown by a pair (i, J) in which i is a member of N and J is a nonempty subset of N . We denote the start node and end nodes of the hyperlink by i and J, respectively. We further denote the ordinary links in the network by (i, j) where i and j are members of N . In this paper, we address the case of one multicast session. The source of the multicast session is shown by s. The set of multicast receivers is indicated by T and we consider M quality classes in the network. Each class has a minimum required rate, R(c) and a maximum tolerable delay constraint, D(c) . Multicast receivers are subscribed to receive several of these flows from s. The source s either refuses to send requested flows to the receivers or by accepting the request, it has to guarantee the quality of the flows, i.e. the end-to-end delay of each flow must be less than D(c) and the rate of flow must be at least R(c) . In order to measure the end-to-end delay of each flow, we assumed that each link (i, j) has the delay indicated by dij which is assumed to be constant during the optimization process. Moreover, it is assumed that before running the algorithm, each node has measured the delay of its outgoing links. As a result, dij could be viewed as the short term average delay of link (i, j). This includes propagation delay plus additional delay caused by other sources such as MAC layer. It should be noted that we assumed a perfect orthogonal model for MAC layer. Other types of delay, for example the delay caused by congestion, could be handled through inclusion in the cost function. We denote the flow toward the destination t, belonging (t)(c) to class c and passing through link (i, j) by xij . Since our algorithm is based on network coding, flows from the same class are encoded together in each node. Denote the (c) coded flow of class c on the hyperlink (i, J) by ziJ . In order to provide QoS, we have to discriminate between flows of different classes. Therefore, inter-class coding is not allowed; otherwise, flows from different classes are mixed together and all the classes experience the same quality. As a result, coded flows are sent across each link independently. The total flow passing through hyperlink (i, J) is indicated by ZiJ . In the (t)(c) remainder of derivations, we denote the set of all xij by (c) the vector x and the set of all ziJ by the vector z. In a wireless network, hyperlinks are usually lossy. In order to account for such losses, we assume that hyperlinks experience packet erasures and the effect of link losses is included in our model by assuming ziJK to be the rate at which the packets sent over hyperlink (i, J) are received by

L

=

(c)

L⊂J T K6=∅

ziJL

(1)

(c)

ziJ

(c)

Then, biJK is approximately equal to the fraction of the packets that are received by the set of nodes in K which belong to class c. B. Problem Definition We formulate the problem of providing quality of service for wireless multicast users in a network flow optimization framework. The solution of this problem would be flow subgraphs that satisfy delay and rate constraints. Among all subgraphs with these properties, the minimum cost subgraph is selected. The network cost is assumed to be the sum of hyperlink costs; a convex and non-decreasing function of total flow passing through the hyperlink (i, J), i.e. ZiJ . It should be noted that the main goal of the proposed algorithm is finding flow subgraphs that satisfy target properties and not to determine a specific network code. As shown in [10] code construction and finding the best subgraphs, could be done separately without affecting the optimality of the final solution. Therefore, one can run code construction algorithms, such as [19] or [20], on the subgraphs found by our proposed algorithm in order to obtain a network coded QoS provisioning method. The problem of QoS provisioning with network coding for wireless multicast users would then be as follows: X (2) min f (ZiJ ) (i,J)∈A

where ZiJ =

M X

(c)

ziJ

c=1

X

(t)(c) xij

6

(c) (c) ziJ biJK

∀(i, J) ∈ A, K ⊂ J, t ∈ T

(3)

j∈K

Subject to: ZiJ ∈ Z

∀(i, J) ∈ A

(4a)

c = 1, . . . , M

(4b)

R(c) 6 r(c) M X

r(c) 6 <

(4c)

c=1

D(t, c) =

(t)(c)

X

Dij (xij

) 6 D(c)

∀c, t ∈ T (4d)

(i,J)∈A,j∈J

where (t)(c)

Dij (xij X

X

(t)(c)

xij

J|(i,J)∈A j∈J

(t)(c)

σi

 )= −

(t)(c)

dij if xij > 0; 0 otherwise X (t)(c) (t)(c) xji = σi

i|(j,I)∈A,i∈I

 (c) if i = s;  r , = −r(c) , if i ∈ T ;  0, otherwise

(4e)

3

(t)(c)

0 6 xij

∀{(i, J) ∈ A, j ∈ J}, t ∈ T, c = 1, . . . , M (4f)

In the above equations, f (ZiJ ) is the cost of hyperlink (i, J). Z is assumed to be a convex subset of non-negative orthant. Delay of the link between the two nodes i and j is denoted by dij . R(c) is the minimum required rate and D(c) is the maximum tolerable delay of class c. < is the max-flow-mincut rate of the network and r(c) denotes the actual rate of class c flow. It should be noted that constraint (4a) implies that z must lie in the feasible region denoted by Z. This region puts some constraints on z including non-negativity and some potential constraints regarding capacity. Constraint (4b) states that the rate of class c must be greater than the minimum acceptable value, R(c) , which is determined by the users. Equation (4c) ensures that the total rate of all classes is less than the maxflow rate, which may or may not be the capacity of the network. Constraint (4d) is the delay constraint, ensuring that the delay of flows from class c is less than the maximum tolerable delay, D(c) . Flow conservation constraint is shown in (4e). Finally, constraint (4f) guarantees the non-negativity of x. C. Distributed Solution

By applying the subgradient method to the Lagrangian of (5) as shown in equation (8) we will get: X f (ZiJ ) L(z, λ, β) = (i,J)∈A

+

(i,J)∈A

(i,J)∈A

subject to: ZiJ ∈ Z ∀(i, J) ∈ A X

(t)(c)

xij

(c) (c)

6 ziJ biJK ∀(i, J) ∈ A, K ⊂ J, t ∈ T

(6a) (6b)

j∈K

min

X

x

f ? (x)

j∈K

(9)

subject to (6a). And: max L? (β)

(10)

β

where L? is the solution of problem (9). In order to achieve a distributed solution, (5) and (7) are decoupled into |A| subproblems, where |A| is the number of hyperlinks in the network. Each node then has to solve the following problems for its outgoing hyperlinks: M X XX K⊂J k∈K

(iJ) (c) (c)

βktc ziJ biJK ∀(i, J) ∈ A (11)

t∈T c=1

and ? max[f (ZiJ ) β

M X XX K⊂J k∈K

(iJ)

βktc (

t∈T c=1

(t)(c)

X

xij

(c) (c)

− ziJ biJK )]

j∈K

∀(i, J) ∈ A

(12)

In the algorithm, the subgradient method is used to solve (c) (11) and (12), iteratively [23]. At each iteration, ziJ and its corresponding Lagrange multipliers are updated as follows: X X (iJ) (c) (c) ziJ (τ + 1) = [ziJ (τ ) − α(τ )(∇fiJ − βktc ]z∈Z K⊂J k∈K

t∈T

(13) (iJ)

βktc (τ + 1)

=

(iJ)

[βktc (τ ) + α(τ )(

(t)(c)

X

xij

?(c) (c)

− ziJ biJK )]+

j∈K

(∀K ⊂ J, k ∈ K)

(14)

where [ ]+ indicates that β must be non-negative and [ ]z∈Z ensures that the updated z lies in the feasible region. ∇fiJ represents the (iJ)th member of ∇f (Z). α(τ ) is the algorithm step size, chosen such that convergence of the algorithm is guaranteed. Subproblem (7) can be solved in the same fashion as above. Define the Lagrangian equivalent of (7) as below: =

X

f ? (x) +

+

X (i,J)∈A

M X X

υ (t)(c) (D(t, c) − D(c) )

c=1 t∈T

(i,J)∈A

subject to constraints (4d), (4e) and (4f). In the above equation, f ? indicates the solution of subproblem (5).

(c) (c)

− ziJ biJK )

z

(7)

(i,J)∈A

xij

min L(z, β)

L0 (x, υ)

and

t∈T c=1

(t)(c)

X

subject to (6a). Subsequently, subproblem (5) will be equivalent to:

z

z

K⊂J k∈K

(iJ)

βktc (

(8)

min f (ZiJ ) −

In order to solve (2), a primal-dual decomposition method is used. To simplify the algorithm, we assume that r(c) = R(c) , for all the classes, therefore, guaranteeing quality of service for the users. As a result, the constraint (4b) can be relaxed. Moreover, since r(c) is now fixed, the constraint (4c) becomes a feasibility condition which should be checked before the algorithm is run. It is assumed that the source checks this feasibility condition and if the condition is not met, the users’ request is withdrawn. Primal decomposition method is used to break (2) into two subproblems [21]. It is first assumed that x is constant and the problem is solved over z. Then, the resulting cost function is minimized over x. Each subproblem is solved using dual Lagrange method [21]. The result is a simple and distributed solution. Since we have assumed that the cost function is convex, problem (2) is a convex optimization problem and the duality gap of decomposition method will be zero (see section 5.2.3 of [24]). The two primal subproblems are as follows: X min f (ZiJ ) (5)

M X XX

X

M X XX K⊂J k∈K

t∈T c=1

(iJ)

βktc (

X

(t)(c)

xij

?(c) (c)

− ziJ biJK )

j∈K

(15)

4

Then, solving subproblem (7) is equivalent to solving the following subproblems: min L0 (x, υ)

(16)

x

subject to (4f) and (4e) . In addition: max L0? (υ)

(17)

υ

where L0? is the solution of (16). Similar to the earlier case, the subgradient method is used to solve (16) and (17). In each iteration, x and its Lagrange multiplier are updated according to (18) and (19) as follows: (t)(c)

(t)(c)

(τ + 1) = [xij (τ ) − X (iJ) ? α(τ )(∇fiJ + βktc + υ (t)(c) dij )]x∈X (18) xij

K⊂J k∈K,j∈J

and

assuming such linear cost functions, each of |A| subproblems (11) further decouple into M subproblems, one for each class: X X (iJ) (c) (c) (c) βktc ziJ biJK ∀(i, J) ∈ A, c = 1, . . . , M min f (ziJ )− z

K⊂J k∈K

t∈T

(21) Consequently, each node must solve (21) for its outgoing hyperlinks, update x and Lagrange multipliers according to subgradients method using (14), (18) and (19). Subsequently, nodes have to exchange the updated multipliers with their neighbors and continue this process until convergence is achieved. The above procedure leads to a distributed and simple solution in which each node i must solve at most M.|T |.outdegree(i) where |T | is the number of multicast receivers and outdegree(i) is the number of hyperlinks leaving i. E. Lossless Networks

υ (t)(c) (τ + 1) = [υ (t)(c) (τ ) + α(τ )(D(t, c) − D(c) )]+ (19) ? is the (iJ)th member of ∇f (Z) for which where ∇fiJ (t)(c) (c) xij = zij and x ∈ X forces updated x to lie in the feasible x region, determined by flow conservation equation (4e) and non-negativity constraint (4f). Any distributed mapping method could be used to ensure feasibility of updated x. In our algorithm, we have chosen a diminishing step size α(τ ) = 1/τ in order to guarantee convergence of the subgradient method. Other diminishing step-size may also be used (see section 6.3.1 of [23]). If a constant step-size is used, as is more convenient for distributed algorithms, the gradient algorithm converges to the optimal value provided that the step-size is sufficiently small (assuming that the gradient is Lipschitz) [23]. To summarize, each node has to perform the following steps to solve (2): 1) Initialization 2) Solve (13) for its outgoing hyperlinks 3) Update Lagrange multipliers according to (14) 4) Update x according to (18) 5) Update delay related Lagrange multipliers according to (19) 6) Repeat until convergence is reached

In the earlier sections, the distributed solution of (??) for lossy wireless networks has been proposed. It is possible to simplify the solution further if the network is lossless, i.e., (c) biJK = 1 for all nonempty K ⊂ J for all classes. In a lossless network, the same QoS provisioning approach is applicable by replacing equation (3) with (22). X (t)(c) (c) (22) xij 6 ziJ ∀(i, J) ∈ A, t ∈ T j∈J

Also, in the case of separable cost functions, each node has to solve the following subproblems for its outgoing hyperlinks: X (iJ) (c) (c) (c) ziJ (τ + 1) = [ziJ (τ ) − α(τ )(∇fiJ − βtc ]z∈Z t∈T

(iJ) βtc (τ

+ 1) =

(iJ) [βtc (τ )

(23) X (t)(c) ?(c) + α(τ )( xij − ziJ )]+ (24) j∈J

(t)(c)

xij −

(t)(c)

(τ + 1) = [xij

?(c) α(τ )(∇fiJ

+

(τ )

(iJ) βtc |j∈J

+ υ (t)(c) dij )]x∈X (25)

and υ (t)(c) (τ + 1) = [υ (t)(c) (τ ) + α(τ )(D(t, c) − D(c) )]+ (26)

D. Separable Cost Function Up to this point, convexity is the only assumption that we have made in the process of solving (2). While the proposed solution is distributed, it can be decomposed further if the cost function is also separable. Linear functions are one class that lead to such further decomposition: f (ZiJ ) = eiJ ZiJ =

M X c=1

(c)

eiJ ziJ =

M X

(c)

f (ziJ )

(20)

c=1

where the cost function (20) represents energy consumption. In many wireless scenarios, energy consumption is the main issue of concern as battery lifetime should be maximized. By

F. Some Notes on QoS Network Coding in Wireless Networks It should be noted that in general, network coding could achieve higher rates and lower delays than routing. A similar situation also arises when QoS network coding and QoS routing are compared. For example, it can be shown that there are cases in which QoS routing is unable to deliver flows with desired quality whereas QoS network coding can provide such quality [18]. However, QoS and minimum cost network coding solutions should be compared with care. Since QoS network coding requires differentiation between flows and involves delay and rate constraints, there could be scenarios in which minimum

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TABLE I (1,1)

d

D ELAY OF Q O S N ETWORK C ODING VS . M INIMUM C OST M ULTICAST (1,1)

(1,3)

t1

a (1,1) (2,1)

(1,1)

s

c (1,1)

(2,1)

(1,1)

b

t2

(1,3) (1,1)

(1,1)

TABLE II D ELAY OF Q O S N ETWORK C ODING VS . M INIMUM C OST M ULTICAST

e

Fig. 1. Extended butterfly network. Links are marked with their (delay,cost).

cost network coding will lead to a solution which is not in feasible range of QoS network coding problem. Naturally, these cases are a result of a more stringent delay or rate constraint. For example, consider the network shown in fig. 1. It is clear that minimum cost multicast with rate 1 is feasible in this network. Yet, QoS network coding with the same rate and a delay constraint of 4 units does not lead to any feasible subgraph. Such result is due to the fact that providing flows whose delay is less than 6 units is not possible in the given network. Therefore, the results of these two types of algorithms should be compared with such insight in mind. This issue will be further elaborated in the following sections. III. C OMPARISON WITH M INIMUM C OST M ULTICAST A PPROACHES In this section, we investigate the performance of the algorithm through simulations where the results are specifically compared with minimum cost multicast (MCM) technique [10]. MCM is a powerful tool in finding subgraphs with minimum cost in coded multicast networks. Lun et. al. [10] have shown the advantages of network coding-based MCM over traditional routing based techniques to find minimum energy multicast trees in wireless networks. While this problem is NP-hard in the routing framework, MCM is able to find the minimum energy multicast subgraph in polynomial time. It is not hard to see that the proposed algorithm is actually a generalization of MCM and the results of the two approaches will coincide if link delays and differentiation of flows into different quality classes are not taken into account. However, when providing guarantees on such metrics are taken into account, the proposed algorithm provides quality of service for the users, at a total cost which is only slightly higher than the MCM solution. In order to investigate the performance of the algorithm, we have run QoSNC and MCM over several random lossless networks with different number of nodes and multicast sinks.

Nodes were randomly placed over a 10×10 square according to a uniform distribution. The maximum transmission radius of a node was 5. All nodes within this radius receive the transmitted symbol correctly and were considered as neighbors of the transmitter. We also assumed that each link had a delay of one unit. The node i has to consume eiJ units of energy to transmit a symbol over the hyperlink (i, J). It is assumed that eiJ is proportional to `2 where ` is the distance between i and its furthest neighbor j where j ∈ J. In our simulations we have only focused on the energy minimization and have relaxed the constraint (6a). We have simulated the problem for the case of two quality classes each at rate 1 and also assumed that multicast sinks have requested one flow from each class. Simulation results are shown in tables I and II. Table I shows that the final cost of the algorithm is at most 14.18% higher than the cost obtained by the MCM. However, the proposed algorithm guarantees QoS, whereas MCM is unable to provide such guarantee in some cases as shown in table II. From table II, it is clear that using QoSNC, the delay for each receiver is less than the maximum tolerable value, D(c) . Nevertheless, using MCM, some multicast sinks will receive the flow with a delay more than the required maximum value (marked in bold italic form). Finally, as mentioned earlier, in a case where there is a minimum cost subgraph that also achieves the delay and rate constraints, both algorithms lead to the same solution. We have also performed simulations

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negotiate with the operator to get services with lower qualities and lower costs. Especially, the negotiation and pricing process should be designed carefully when QoS is provided using network coding. V. ACKNOWLEDGMENT We would like to thank Mr. Hamed Shah-mansouri, Mr. Pooya Shariatpanahi, Mr. Hadi Goudarzi and Mr. Amir Mehdi Khodaiian at Information Systems and Security Lab (ISSL) for their helpful discussions. R EFERENCES

Fig. 2.

Average energy of QoSNC versus iteration number

over a lossless wireless network with 30 node and 2 sinks. Fig. 2 illustrates the cost of algorithm versus iteration number. As shown in this figure, after about 150 iterations, the cost of the proposed algorithm is quite close to the cost obtained without considering the QoS constraints. IV. C ONCLUSIONS AND F UTURE W ORK In this paper, a new distributed approach for providing quality of service using network coding in wireless multicast networks is proposed. Our algorithm yields subgraphs that satisfy certain quality measures, namely delay and rate constraints. Among all subgraphs with these properties, the algorithm selects the one with minimum cost as the final solution. The proposed algorithm can handle any convex cost function in addition to user-defined delay and throughput requirements. However, since minimizing energy consumption is of great importance in wireless networks, we have focused on linear cost functions. Several options are available for future works. Our model does not include noise, interference, or network dynamism in the wireless medium. Fairness and admission control are other issues of importance for further research. In this work we have assumed that if the multicast source is unable to provide QoS according to required constraints for all users, it withdraws the request. While this provides some form of uniform fairness in the network, it may not constitute the best strategy. There may be cases that the source could provide QoS for some users at the price of not servicing less important users so admission control should be addressed in such scenarios. Another issue of high practical importance is including pricing and negotiation processes. In QoS provisioning, users make contracts with operators so that the quality of their services is guaranteed. There are cases where network is unable to provide services that satisfy all the required metrics. Then, the user has to either look for another operator or

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