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Adaptive Routing Strategies in IEEE 802.16 Multi-Hop Wireless Backhaul Networks Based On Evolutionary Game Theory Markos P. Anastasopoulos Student Member, IEEE, Pantelis-Daniel M. Arapoglou, Member, IEEE, Rajgopal Kannan, and Panayotis G. Cottis Abstract—The high frequency segment (10-66GHz) of the IEEE 802.16 standard seems promising for the implementation of wireless backhaul networks carrying large volumes of Internet traffic. In contrast to wireline backbone networks, where channel errors seldom occur, routing decisions in IEEE 802.16 networks are conditioned by wireless channel impairments rather than by congestion, exclusively. This renders a cross-layer routing approach between the routing and the physical layers more appropriate during fading periods. In this paper, an adaptive cross-layer routing scheme is presented based on the selection of the most reliable path in terms of packet error ratio (unipath routing). The paper argues that routing Internet traffic through wireless backhaul networks is modeled more realistically employing evolutionary rather than conventional game theory. The stability of the proposed routing algorithm is proven and the dependence of the speed of convergence on various physical layer parameters is investigated. Is is also shown that convergence may be further accelerated by increasing the amount of information from the physical layer, specifically the physical separation between the alternative paths provided to the routing layer. Index Terms—Evolutionary game theory, IEEE 802.16, multihop wireless backhaul networks, adaptive routing.
I. I NTRODUCTION
I
NTERNET based multimedia applications, such as voice and video, have been the locomotive for the recent growth of broadband access technologies. Although conventional backbone networks (e.g. copper or optical fiber cables) can afford large bandwidths, backhaul networks carrying the aggregate traffic from the access points to the backbone network through gateway stations remain the bottleneck of these multimedia applications [1]. Nevertheless, after the release of the IEEE 802.16 WirelessMAN WiMAX1 standard [2], multi-hop wireless backhaul networks (MHWBN) appear a cost effective solution to provide ubiquitous Internet services. This is especially true for the high frequency air interface of WiMAX in the 10-66GHz range, where bandwidth exceeding 1GHz is available. Hence, compared with existing backhaul solutions, MHWBN significantly reduce deployment costs and also enable fast and flexible network configurations by offering Manuscript received August 15, 2007; revised March 10, 2008. M. P. Anastasopoulos, P.-D. M. Arapoglou, and P. G. Cottis are with the Wireless & Satellite Communications Group, Division of Information Transmission Systems and Materials Technology, School of Electrical & Computer Engineering, National Technical University of Athens, Greece, GR15780 (email:
[email protected],
[email protected],
[email protected]). R. Kannan is with the Department of Computer Science, Louisiana State University (e-mail:
[email protected]). Digital Object Identifier 10.1109/JSAC.2008.080918. 1 Worldwide interoperability for microwave access
protection against link failures and by allowing adaptive rerouting strategies under adverse channel conditions. From the physical layer point of view, given that line-ofsight (LOS) operation is specified for IEEE 802.16 WirelessMAN in the frequency range 10-66GHz [3], two operating conditions exist: (a) Favourable channel conditions for high annual percentages corresponding to clear sky and (b) Unfavourable channel conditions for small annual percentages corresponding to rain fading, causing high error rates, thereby deteriorating transmission [4]. In the first case, typical routing algorithms may be employed to deal with congestion, whereas, in the second, a cross-layer approach between the routing layer and the physical layer seems more appropriate to take into account severe physical layer impairments [5]. As for routing IP traffic, typical routing algorithms which allow a single entity to take the routing decisions based on an overall network optimization criterion seem to be inefficient [6]. As a consequence, the current trend is to allow the network nodes themselves to adaptively choose their routing strategy. This allows nodes to optimize their own performance selfishly, that is without considering the impact on the overall network [7], [8]. In case the routing decisions are taken by a finite number of nodes, the appropriate optimization concept is based on the Nash Equilibrium (NE) [9]. In contrast, when each node carries a small portion of a large volume of traffic, the Wardrop Equilibrium (WE) [10] seems more appropriate. In conventional game theory, the objective of a player is to choose a routing strategy that maximizes its payoff. The game is played exactly once by fully rational players, all of them having complete and accurate knowledge about the details of the game. However, in modeling complex networks such as MHWBN, it is questionable whether these assumptions are valid, since players not only have limited rationality but, also, whenever rain fading exists, the network becomes a highly dynamic transmission environment. A more reasonable approach would be to let the players learn empirically how to select their own routing strategies during the game; this leads to the concept of evolutionary game theory (EGT) [11], [12]. The present study focuses on setting up flows that adaptively route information via the most reliable path employing EGT during periods when adverse channel conditions prevail. An EGT based adaptive routing scheme, which depends on the observed path latencies has been discussed in [13]. However, to the authors’ knowledge, no cross-layer work on this subject
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ANASTASOPOULOS et al.: ADAPTIVE ROUTING STRATEGIES IN IEEE 802.16 MULTI-HOP WIRELESS BACKHAUL NETWORKS
has been carried out so far, particularly for IEEE 802.16 MHWBN. The large information flows accommodated by the MHWBN consist of packets –referred to as agents according to the EGT terminology– each carrying an infinitesimal load compared to the total traffic; this profile corresponds to IP traffic. The aim is to minimize the individual end-to-end packet error ratio (PER) without considering the impact on the overall network performance. The routing strategy of every agent is continuously updated by sampling all the alternative paths, taking advantage of the information obtained from the physical layer. The population of agents using each strategy is modeled based on the population dynamics model known as the replicator dynamics [14]. After a brief overview of EGT in Section II, its application to the proposed network model is presented in Section III, where the maximum reliability routing problem is formulated applying EGT. First, it is proven that the proposed routing algorithm is asymptotically stable. Also, the impact of several physical layer parameters on the speed of convergence is investigated. Motivated by the previous results, in Section IV, a simple algorithm –the path correlation coefficient (PCC) algorithm– is proposed to accelerate the convergence of the routing scheme. Finally, conclusions are drawn in Section V. II. E VOLUTIONARY G AME T HEORY: BASIC C ONCEPTS In a conventional game, the objective of a rational player is to choose the strategy which maximizes its payoff. Instead, in the frame of EGT, the game is played repeatedly by agents2 randomly drawn from a large population [11], [12]. In general, an evolutionary process combines two significant mechanisms: a mutation mechanism which provides varieties and a selection mechanism which favours some varieties over others. The role of mutation is highlighted by the notion of Evolutionary Stable Strategies (ESS) –which is a refinement of the NE–, while selection is associated to the replicator dynamics model, which assumes that a subpopulation grows (declines) when it plays strategies that are better (worse) than average. A. Evolutionary Stable Strategies ESS is a key concept in EGT. A population following such a strategy is invincible. Specifically, suppose that the initial population is programmed to play a certain pure or mixed strategy3 x (the incumbent strategy). Then, let a small population share of agents ∈ (0, 1) play a different pure or mixed strategy y (the mutant strategy). Hence, if an individual is drawn to play the game, the probabilities that its opponent plays the incumbent strategy x and the mutant strategy y are 1 − and , respectively. The payoff of such a game is the same as that of a game where the individual plays the mixed strategy w = y + (1 − )x. The payoffs of strategies x and y given that the opponent adopts strategy w are denoted by u(x, w) and u(y, w), respectively. Definition 1. A strategy x is called evolutionary stable if, for every strategy y = x, a certain ∈ (0, 1) exists, such that 2 Players
in EGT. terms of routing, a strategy is characterized as pure (or mixed) when a single (or multiple) path(s) is (are) selected to route the information to the destination. 3 In
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the inequality u[x, y + (1 − )x] > u[y, y + (1 − )x]
(1)
holds for all ∈ (0, ). A weaker notion of evolutionary stability also exists, called neutral stability. Instead of requiring that all mutant strategies gain less than the incumbent strategy x, in neutral stability it is required that no mutant strategy prospers, in the sense that it gains a higher payoff than the incumbent strategy. Definition 2. A strategy x is called neutral stable if, for every strategy y = x, a certain ∈ (0, 1) exists, such that inequality u[x, y + (1 − )x] ≥ u[y, y + (1 − )x]
(2)
holds for all ∈ (0, ). Using the linearity property of the payoff function, (1) yields (1 − )u(x, x) + u(x, y) > (1 − )u(y, x) + u(y, y) (3) If is close to zero, (3) yields either u(x, x) > u(y, x)
(4)
u(x, x) = u(y, x) and u(x, y) > u(y, y)
(5)
or Hence, it becomes obvious that an ESS must be a NE; otherwise, (4) does not hold. Furthermore, any strict NE is an ESS [15]. B. The Replicator Dynamics The replicator dynamics, first proposed by Taylor and Jonker [14], specifies how population shares associated with different pure strategies evolve over time. In contrast to evolutionary stability, in replicator dynamics agents are programmed to play only pure strategies4 . To define the replicator dynamics, we consider a large but finite population of agents all programmed to play pure strategy k ∈ K, where K is the set of strategies. At any instant t, let λk (t) ≥ 0 be the number of agents programmed to play pure strategy k. The total population of agents is given by λ(t) = k∈K λk (t). Let xk (t) = λk (t)/λ(t) be the fraction of agents using pure strategy k at time t. The associated population state is defined by the vector x(t) = [x1 (t), . . . , xk (t), . . . , xK (t)]. Then, the expected payoff of using pure strategy k given that the population is in state x is u(k, x) and the population average payoff, that is the payoff of an agent drawn randomly from K the population, is u(x, x) = k=1 xk · u(k, x). Suppose that payoffs are proportional to the reproduction rate of each individual and, furthermore, that a strategy profile is inherited. This leads to the following dynamics for the population shares xk x˙ k = xk · [u(k, x) − u(x, x)] (6) where x˙ k is the time derivative of xk . Equation (6) states that populations with better (worse) strategies than average grow (shrink). However, there are cases when even a strictly 4 In network terminology, this assumption is related to packets not being fragmented while traversing multiple hops.
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dominated strategy may gain more than average. Hence, it is not a priori clear whether such strategies get wiped out in the replicator dynamics. The following theorem answers this question [11]: Theorem 1: If a pure strategy k is strictly dominated then ξk (t, x0 )t→∞ → 0, where ξk (t, x0 ) is the population at time t and x0 is the initial state. On the other hand, it should be noted that the ratio xk /x of two population shares xk > 0 and x > 0 increases with time if the strictly dominated strategy k gains a higher payoff than the strictly dominated strategy . This is a direct result of (6) and may be expressed analytically via d xk xk (7) = [u(k, x) − u(, x)] dt x x From (7) it is evident that even suboptimal strategies could temporarily increase their share before being wiped out in the long run. However, there is a close connection between the steady states of the replicator dynamics, that is states where the population shares do not change their strategy over time, and NE. Thus, since in NE all strategies have the same average payoff, every NE is a steady state. The reverse is not always true: Steady states are not necessarily NE, e.g., any state where all agents use the same pure strategy is a steady state, but, it is not stable [11]. III. A PPLICATION TO M ULTI -H OP W IRELESS BACKHAUL N ETWORKS A. Network Model Next, the general theoretical framework outlined in the previous section is adapted to the routing problem under consideration. Specifically, consider an IEEE 802.16 MHWBN represented by an undirected graph G(N, L), where N is the number of nodes and L is the number of links. It is assumed that the information generated at the source node S must be transmitted to the destination node D. For example, S and D may represent the gateway stations connecting the MHWBN to the Internet backbone at the two ends of the metropolitan area network. The set of all available paths (strategies) connecting the source and the destination nodes is denoted by K. The network information flow is accommodated by a total population of λ = k∈K λk packets (agents), where λk is the number of packets following path k. The normalized number of packets xk = λk /λ programmed to use each path is collected in the vector x = [x1 , . . . , xk , . . . , xK ] (population state). The objective is to find a positive flow vector λ = [λ1 , . . . , λk , . . . , λK ] that routes information via the paths k ∈ K with the minimum end-to-end PER, under the flow conservation condition 0 ≤ λk ≤ Λk ,
k∈K
(8)
where Λk is the capacity of path k. A similar formulation may be found in [16]-[18], where multiple parallel paths are used to route information from a single source node to a single destination node. In the present work, PER is used as a figure of merit for the data transfer reliability. However, for LOS wireless links operating at frequencies above 10GHz, rain attenuation
A is the appropriate physical layer metric. If the modulation scheme, code rate, power level and packet size are known, PER may be expressed as an increasing function of A, that is P ER = f (A). Consequently, the data transfer reliability increases as A decreases and the most reliable path is the one suffering from the minimum rain attenuation. Therefore, as indicated by the following cross-layer analysis, the decisions related to routing with maximum reliability should be based on rain attenuation, as determined from the inverse relationship A = f −1 (P ER). B. Rerouting Dynamics Originally, all agents select one of the possible routing paths at random. Since every agent wishes to minimize its own endto-end PER from the source node S to the destination node D, it reconsiders periodically its routing strategy by randomly sampling different paths and comparing the corresponding PER with its own PER. When a lower value is found, the agent switches to a new path; otherwise, the current routing strategy remains unchanged. It is evident that one of the basic elements of the replicator dynamics is the rate at which agents revise their strategy. This rate depends on the performance of the agents current strategy and on the population state x. In the following, the average review rate of an agent employing strategy k will be denoted as rk (x), k ∈ K. Significant elements of the problem are the switching probabilities of a reviewing agent: The probability that a reviewing agent switches from strategy (path) k to strategy (path) is denoted by pk (x). Hence, the number of agents switching from path k to path is x k · rk (x) · pk (x). Consequently, the outflow from path k is ∈K,=k xk ·rk (x) · pk (x) = xk · rk (x) · k p (x) = x · r (x) 1 − p (x) , while the inflow k k k k ∈K,=k k to path k is ∈K,=k x ·r (x)·p (x). Subtracting the outflow from the inflow, the following differential equation comes up x˙ k = x · r (x) · pk (x) − xk · rk (x) · 1 − pkk (x) ∈K,=k
=
x · r (x) · pk (x) − xk · rk (x)
(9)
∈K
which satisfies the replicator dynamics model introduced in (6). The procedure of sampling strategy by an agent using strategy k is based on the reported values of the rain attenuation Ak and A along the corresponding paths k and , respectively. The reviewing agent switches to the sampled strategy if and only if the observed rain attenuation difference is positive, i.e. Ak > A . The difference between the random variables Ak and A is also a random variable with a continuously differentiable cumulative distribution function φ : R → [0, 1]. The conditional probability that an agent will switch to strategy , given that its current strategy is k, is φ(Ak − A ) = Pr{Ak − A > 0|Ak > 0, A > 0}
(10)
The above probability is determined analytically in the Appendix, where a pure propagation approach is presented concerning the joint statistics of rain fading over two paths taking into account their spatial correlation properties. Moreover, since the probability of an agent who samples path is
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ANASTASOPOULOS et al.: ADAPTIVE ROUTING STRATEGIES IN IEEE 802.16 MULTI-HOP WIRELESS BACKHAUL NETWORKS
x (strategy profiles are inherited), the resulting conditional switching probability is given by x φ(Ak − A ) if k = pk (x) = (11) 1 − i=k,i∈K xi φ(Ak − Ai ) if k = Assuming for simplicity that rk (x) = 1 reviews per unit of time, ∀k ∈ K, the substitution of (11) into (9) yields the following selection dynamics x [φ(A − Ak ) − φ(Ak − A )] (12) x˙ k = xk ∈K, k=
Setting for brevity nk = φ(A − Ak ) − φ(Ak − A )
(13)
and substituting into (12), the following state equation results x˙ k = xk x · nk (14) ∈K,k=
C. Stability Analysis In this part of the analysis, the stability of the proposed routing algorithm is examined. It is proven that, if the system operates under rain fading conditions, unipath routing is an evolutionary stable strategy. Therefore, two paths k, ∈ K connecting node S to node D cannot exhibit the same performance with regard to reliability during data transfer. Lemma 1: Under long term rain fading conditions, two paths cannot suffer from the same level of rain attenuation. Proof: The proof of the Lemma is straightforward: Since Ak and A are random variables in certain real number ranges, the probability that the relevant paths suffer equally from long term rain attenuation tends to zero. Next, it is shown that under rain fading conditions, all data are routed via a single path (unipath routing). The proof is based on Lyapunov’s first method. Theorem 2: Under long term rain fading, if the flow requirements do not exceed the path capacity, unipath routing is ESS. Proof: The set of differential equations in (14) must satisfy the constraints xk ≥ 0, k ∈ K, xk = 1 k∈K
The critical points of the system are determined by setting (14) equal to zero and solving the resulting system of algebraic equations. At this point, one may assume that A1 < . . . < Ak < . . . < AK . Then, based on Lemma 1 and taking into account the constraints previously stated, the unique solution of the system corresponds to the population state vector x∗ = [1 0 0 · · · 0] of dimension K. Substituting x1 = 1−x2 −x3 −· · ·−xK , the previous set of differential equations (14) is rewritten in the downsized version (15), where by X(t) = [X1 (t), . . . , Xk (t), . . . , XK−1 (t)] the corresponding downsized population vector of dimension (K − 1) is denoted which is almost linear around the equilibrium point X∗ = [0 0 · · · 0]. Then, employing the
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transformation x = X + u, (15) is transformed into the linear matrix equation du = Qu (16) dt where Q is a (K − 1) × (K − 1) matrix. Its elements are the partial derivatives (17) evaluated at the equilibrium state X∗ . Equation (17) easily leads to the evaluation of Q ⎞ ⎛ n21 0 ··· 0 ⎜ 0 n31 · · · 0 ⎟ ⎟ ⎜ (18) Q=⎜ . . .. ⎟ .. ⎝ .. . ⎠ 0 0 · · · nK1 From (18), it is deduced that n21 , n31 , . . . ,nK1 are the eigenvalues of Q. Since for every k, k = 2, . . . , K, nk1 = −φ(Ak − A1 ) < 0
(19)
∗
the solution X = [0 0 · · · 0] is asymptotically stable and, consequently, all information is routed via path 1. Given that a strategy is evolutionary stable if and only if it is asymptotically stable [19], this unipath routing strategy is also evolutionary stable. The above analysis leads to the conclusion that, if the flow requirements do not exceed the capacity of the most reliable path, all information will be routed via this path. Otherwise, if the capacity of the most reliable path cannot accommodate the information flow is required, a part of the flow will remain routed via the most reliable path and the rest of the flow will be routed via the second, the third, and so on, next reliable paths. D. Convergence Analysis To investigate the speed of convergence of the proposed routing algorithm, the MHWBN topology depicted in Fig.1 is considered: Five multi-hop paths connect the source to the destination node. For simplicity, the horizontal and vertical distances d between adjacent nodes are assumed equal. Also, P ER1 < P ER2 < P ER3 < P ER4 < P ER5 , that is A1 < A2 < A3 < A4 < A5 is assumed. Since the focus is on a cross-layer routing scheme, the speed of convergence will be examined solely as a function of physical layer parameters, which, in turn, determine the extent of rain fading and the relevant levels of PER. For instance, in Fig.2 the dependence of the routing scheme on the operational frequency f of the IEEE 802.16 system is examined. The unipath routing effect is established after about 100 path samplings. The algorithm converges faster for f =25GHz than for f =20GHz. In the numerical calculations, a node distance d = 1km has been assumed. The path sampling periods of the figures correspond to multiples of the S-D round trip time (RTT). Similar conclusions are obtained from Fig.3, where the evolution of the population shares is shown for two different node distances d. It is observed that the algorithm converges faster for larger values of d. This is attributed to the fact that large values of d result in decorrelation of rain fading over the various paths. Hence, the differences in the performance of the corresponding links are larger and a greater variety between the strategies (routing paths) becomes available; this accelerates the convergence.
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X˙ k = Xk nk1 (1 − Xk ) +
K
Xi · (nki − rk1 ) , k = 2, . . . , K
(15)
i=2,i=k
⎤ K ∂Xk nk1 (1 − Xk ) + i=2,i=k Xi · (nki − rk1 ) ⎦ =⎣ ∂X ⎡
qk
k, = 2, . . . , K
Fig. 1.
X=X∗
(17)
MHWBN topology: Five parallel multi-hop paths connecting the source node S to the destination node D.
Fig. 2. Dependence of the speed of convergence of the proposed cross-layer routing algorithm on frequency.
Fig. 3. Dependence of the speed of convergence of the proposed cross-layer routing algorithm on the node distance (f =20GHz).
Fig. 4. Dependence of the speed of convergence of the proposed cross-layer routing algorithm on the climatic conditions (f =20GHz).
Finally, the dependence of the speed of convergence on the climatic conditions of the area where the MHWBN is deployed is illustrated in Fig.4. In this example, the proposed routing procedure is applied for two climatic regions: Mediterranean and Central European. The routing algorithm converges faster in Central Europe, where rainfall is more intense, leading to higher attenuation differences among the various paths. Furthermore, note that in Figs.2-4, the number of agents that use path 2, although being a strictly dominated strategy (i.e. vanishes in the long run according to Theorem 1), tends to increase its share during the first few path samplings, where its payoff is higher than average.
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IV. A N I MPROVED S AMPLING A LGORITHM T O ACCELERATE C ONVERGENCE A. Algorithm Description So far it has been assumed that agents select randomly their routing strategies from the set of available paths connecting the source to the destination node. In this section, motivated by the results of the previous convergence analysis, a simple algorithm to accelerate convergence is presented which takes advantage of the spatial correlation of the rainfall medium in wireless channels operating at frequencies above 10GHz. Actually, this property is exploited by cell-site diversity to mitigate rain fading and increase the availability of broadband wireless access networks [20]. The performance of a spatial diversity scheme depends mainly on the physical separation between the alternative paths. To achieve maximum reliability during data transfer, a new metric, the path correlation coefficient P CCk , is introduced to indicate the overall physical separation between the paths. If sk is the surface defined by the trajectories of paths k and as they link the source node S to the destination node D (see Fig.5), P CCk is defined by P CCk = exp(−sk )
(20)
If two paths have a large physical separation, the respective value of the P CC is close to zero. Therefore, a significant attenuation difference exists between the two paths with low P CC. To take advantage of this geometrical metric, during rain fading, each agent performs the following actions: 1) First, it selects a routing path k ∈ K at random. 2) For every ∈ K, = k, the relevant P CCk values are evaluated. Given that MHWBNs comprise fixed nodes, the PCC values may be evaluated a priori and stored at the network nodes to reduce the computational load. 3) In the next path sampling period, to decide whether a possible switching to a new strategy is required, the path with the smallest P CCk is examined. If this path exhibits a lower PER, the agent modifies its strategy accordingly. Otherwise, the agent’s strategy k remains unchanged. 4) The agent repeats Step 3 in the next path sampling periods examining the paths with the second, third, and so on, smaller P CC, until all alternative paths available have been sampled without finding a strategy outperforming the current one. It is clear that the PCC based algorithm affects only the sampling probability x of any path . However, from (13), the eigenbalues nk do not depend on x . Hence, even after the application of the PCC algorithm, the eigenvalues of matrix Q remain the same, all having negative values, and Theorem 2 holds ensuring evolutionary stability. B. Performance Evaluation Valuable insight concerning the performance of the proposed algorithm to accelerate convergence may be obtained by implementing a physical channel tool known as the rain field. Rain fields are stochastic models that capture the properties of the rainfall rate R (in mm/h) over a specific area both in space
Fig. 5. Geometrical definition of the surface sk bounded by the trajectories of the paths k and .
and time. How a rain field is generated lies outside the scope of the present paper. Anyway, the physical and mathematical basis of the model are well described in [21]. In Fig.6, a snapshot of such a rain field is superimposed over the area of the MHWBN introduced in Fig.1. The next step is the calculation of rain attenuation over each of the alternative paths. To relate rainfall rate to rain attenuation, the specific rain attenuation γR (in dB/km) [22] is integrated over the corresponding path lengths. Referring again to Fig.6, paths 1, 3, 4 and 5 are strictly dominated (see Theorem 1) by the performance of path 2, that is, the population of agents following these strategies shrinks to zero and all information is routed via the most reliable path 2. Fig.7 demonstrates the convergence acceleration of the routing protocol after the application of the PCC algorithm. It is observed that the lower part of the 10-66GHz frequency range benefits more when the routing layer is made aware of this additional physical layer information. V. C ONCLUDING R EMARKS The release of the IEEE 802.16 WiMAX standard enables the utilization of the 10-66GHz air interface as a wireless backhaul network to aggregate Internet traffic from the access points to the Internet backbone. As explained in the main body of the paper, under rain conditions (on average about 5%8% of a typical year), routing is dictated by the performance aggravation caused by rain fading and not by congestion. This necessitates a cross-layer analysis where the routing algorithm is directly related to the physical link parameters. In the present work, because of the large number of players with bounded rationality who learn about the network status empirically, the concept of EGT is invoked, which seems to reproduce the real situation concerning IP backhaul traffic better than conventional game theory. The corresponding evolution of population shares adopting a specific routing strategy follows the replicator dynamics model. Having proven that the proposed EGT based routing scheme is unipath and evolutionary stable, the speed of convergence is investigated. It is shown that convergence is accelerated if more information from the physical layer is transfered to the routing layer. In the particular routing problem under study, convergence acceleration has been achieved by taking into account the physical separation between the fixed paths of the IEEE 802.16 MHWBN.
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Fig. 6.
IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 26, NO. 7, SEPTEMBER 2008
Routing strategies under rain fading for the MHWBN of Fig.1. Snapshot of a rain field model.
To derive (11) and (12), the conditional probability φ(Ak − A ) = Pr{Ak − A > 0|Ak > 0, A > 0}, which is a function of two random variables Ak and A , is needed. To this end, the joint bivariate lognormal distribution is employed, that is +∞ Ak φ(Ak − A ) = dAk dA pAk A (Ak , A ) (22) 0
0
where
1 1 pAk A (Ak , A ) = exp − F (Ak , A ) 2πSAk SA Ak A 2 (23) is the joint bivariate lognormal pdf and
F (Ak , A ) = Fig. 7. Convergence acceleration of the proposed routing algorithm employing the PCC coefficient method.
ACKNOWLEDGEMENT The authors would like to thank Prof. Athanasios V. Vasilakos for originally proposing the EGT based routing approach. A PPENDIX The majority of studies concerning atmospheric propagation above 10GHz agree that rain attenuation A (in dB) and its underlying physical process of rainfall rate R (in mm/h) are well approximated by the lognormal distribution [23] with probability density function (pdf) 2 ln X − mX 1 √ exp − X = A or R pX (X) = √ 2πSX X 2SX (21) where mX and SX are the statistical parameters of the lognormal distribution.
1 1 − ρ2n
(ln Ak − ln mAk )2 − 2 SA k
(ln Ak − ln mAk )(ln A − ln mA ) + S Ak S A (ln A − ln mA )2 (24) 2 SA
2ρn
A key element of both the original and the improved PCC routing algorithms is the cross-correlation ρn of rain attenuation between the available routing paths. This coefficient is obtained according to the methodology described in the Appendix of [24]. R EFERENCES [1] M. Cao, X. Wang, S.-J. Kim, and M. Madihian, “Multi-hop wireless backhaul networks: A cross-layer design paradigm,” IEEE J. Select. Areas Commun., vol. 25, no. 4, pp. 738-748, May 2007. [2] IEEE 802.16, “IEEE Standard for Local and Metropolitan Area Networks - Part 16: Air Interface for Fixed Broadband Wireless Access Systems,” Oct. 2004. [3] C. Eklund, R.B. Marks, K.L. Stanwood, S. Wang, “IEEE Standard 802.16: A technical overview of the WirelessMAN air interface for broadband wireless access,” IEEE Commun. Mag., vol. 40, no. 6, pp. 98-107, Jun. 2002.
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ANASTASOPOULOS et al.: ADAPTIVE ROUTING STRATEGIES IN IEEE 802.16 MULTI-HOP WIRELESS BACKHAUL NETWORKS
[4] R.K. Crane, Propagation Handbook for Wireless Communication System Design, CRC Press, 2003. [5] L. Iannone, R. Khalili, K. Salamatian, S. Fdida, “Cross-layer routing in wireless mesh networks,” in 1st Int. Symp. Wirel. Commun. Sys., pp. 319323, Mauritius, September 20-22, 2004. [6] H.Tangmunarunkit, R. Govindan, S. Shenker, D. Estrin, “The impact of routing policy on Internet paths,” in Proc. IEEE INFOCOM, vol.2, pp. 736-742, 2001. [7] T. Roughgarden and E. Tardos, “How bad is selfish routing?,” J. ACM, vol. 49, no. 2, pp. 236-259, March 2002. [8] L. Qiu, Y. Richard, Y. Yin Zhang, S. Shenker, “On selfish routing in Internet-like environments,” IEEE/ACM Trans. Networking, vol. 14, no. 4, pp. 725-738, Aug. 2006. [9] J. Nash, “Non-cooperative games,” The Annals of Mathematics, vol. 54, no. 2, pp. 286-295, 1951. [10] J. Wardrop, “Some theoretical aspects of road traffic research,” Proceed. Instit. Civil Engineers, PART II, vol. 1, pp. 325-378, 1952. [11] J.W. Weibull, Evolutionary Game Theory, MIT Press, 1996. [12] M.A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Harvard University Press, 2006. [13] S. Fischer and B. V¨ocking, “Evolutionary game theory with applications to adaptive routing,” DELIS-TR-0194, Project Number 001907, 2005. [14] P. Taylor and L. Jonker, “Evolutionary stable strategies and game dynamics,” Mathematical Biosciences, vol. 16, pp. 455-484, 1978. [15] D. Fudenberg, D.K. Levin, The Theory of Learning in Games, MIT Press, 1998. [16] S. Orda, R. Rom, S. Shimkin, “Competitive routing in multiuser communication networks,” IEEE/ACM Trans. Networking, vol. 1, no. 5, pp. 510-521, Oct. 1993. [17] R. J. La, V. Anantharam, “Optimal routing control: Repeated game approach,” IEEE Trans. Automat. Contr., vol. 47, no. 3, pp. 437-450, March 2002. [18] T. Alpcan, T. Basar. R. Tempo, “Randomized algorithms for stability and robustness analysis of high-speed communication networks,” IEEE Trans. Neural Networks, vol, 16, no. 5, pp. 1229-1241, Sep. 2005. [19] B. Thomas, “On evolutionarily stable sets,” J. Math. Biology, vol. 22, pp. 105-115, 1985. [20] A.D. Panagopoulos, P.-D.M. Arapoglou, G.E. Chatzarakis, J.D. Kanellopoulos, P.G. Cottis, “LMDS diversity systems: A new performance model incorporating stratified rain,” IEEE Commun. Lett., vol. 9, no. 2, pp. 145-147. Feb. 2005. [21] L. Feral, H. Sauvageot, L. Castanet, J. Lemorton, F. Cornet, K. Leconte, “Large-scale modeling of rain fields from a rain cell deterministic model,” Radio Sci., vol. 41, no. 2, Apr. 2006. [22] ITU-R Recommendation P.838-3, Specific attenuation model for rain for use in prediction methods, Geneva, Switzerland, 2005. [23] C.-Y. Chu, K.S. Chen, “Effects of rain fading on the efficiency of the Ka-band LMDS system in the Taiwan area,” IEEE Trans. Veh. Technol., vol. 54, no. 1, pp. 9-19, Jan. 2005. [24] A.D. Panagopoulos, P.-D.M. Arapoglou, J.D. Kanellopoulos, P.G. Cottis, “Intercell radio interference studies in broadband wireless access networks,” IEEE Trans. Veh. Technol., vol. 56, no. 1, pp. 3-12, Jan. 2007.
Markos P. Anastasopoulos (S’08) was born in Athens, Greece, on February 1982. He received the Diploma degree in electrical and computer engineering from the National Technical University of Athens (NTUA), Zografou, Greece, in 2004 and the M.Sc. in techno-economics in 2006. He is currently working toward the Dr. Ing. degree at the same university. From September 2005, he acted as a technical consultant for the Spectrum Management Division of the Hellenic Ministry of Transport and Communication, where he was involved with the coordination of the HELLAS-SAT satellite network series. For his academic progress he has been awarded by the Kyprianides, Eugenides and Propondis foundations. His research interests include applications of game theory in wireless networks, routing and resource allocation issues for ad-hoc, sensor and satellite networks. Mr. Anastasopoulos is a student member of the IEEE and of the Technical Chamber of Greece (TEE)
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Pantelis-Daniel M. Arapoglou (S’04–M’07) was born in Athens, Greece, on October, 1979. He received the Diploma degree in electrical and computer engineering and the Dr. Engineering degree from the National Technical University of Athens (NTUA), Greece, in 2003 and 2007, respectively. From January 2004 until December 2005 he was a Research Assistant at the School of Pedagogical and Technological Education (ASPETE). From September 2005, he acted as a technical consultant for the Spectrum Management Division of the Hellenic Ministry of Transport and Communication, where he was involved with the coordination of the HELLAS-SAT satellite network series. Since August 2007, he serves his military duty in the Electronic Warfare Corps of the Hellenic Army. His research interests include physical and link layer issues for satellite communication and fixed broadband wireless access networks. Dr. Arapoglou is a member of the IEEE and of the Technical Chamber of Greece (TEE). In 2004 he received the ”Ericsson Award of Excellence in Telecommunications” for his diploma thesis and in 2005 the URSI General Assembly Young Scientist Award. He is also an active delegate of Greece in the Study Group 3 of the ITU-R.
Rajgopal Kannan obtained his B.Tech in Computer Science and Engineering from IIT-Bombay in 1991 and the Ph.D in Computer Science from the University of Denver in 1996. He is currently an Associate Professor in the Computer Science department at Louisiana State University. His areas of interest are in algorithmic aspects of wireless sensor networks, game and information theory, data security, interconnection networks, optical networks and routing and multicasting protocols. He has published extensively and won several best-paper awards. He is an Associate Editor of IJAACS and has organized/co-organized several conferences. His research work has been funded by agencies such as NSF, DARPA, AFRL and DOE.
Panayotis G. Cottis was born in Thessaloniki, Greece, in 1956. He received the Dipl. Ing. degree in mechanical and electrical engineering and the Dr. Eng. degree from the National Technical University of Athens (NTUA), Zografou, Greece, in 1979 and 1984, respectively, and the M.Sc. degree from the University of Manchester (UMIST), Manchester, U.K., in 1980. In 1986, he joined the School of Electrical and Computer Engineering, NTUA, where he is currently a Professor. From September 2003 to September 2006, he has been the Vice Rector of NTUA. He has published more than 120 papers in international journals and conference proceedings. His research interests include microwave theory and applications, wave propagation in anisotropic media, electromagnetic scattering, powerline and wireless and satellite communications.
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