Opportunism vs. Cooperation: Analysis of Forwarding Strategies in Multihop Wireless Networks with Random Fading

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1 1 Opportunism vs. Cooperation: Analysis o Forwarding Strategies in Multihop Wireless Networks with Random Fading Chi-Kin Chau, Anand Seetharam, Jim Kurose, Don Towsley Masdar Institute o Science and Technology, Abu Dhabi Department o Computer Science, University o Massachusetts, Amherst, MA 01003, USA ckchau@masdar.ac.ae,{anand, kurose, towsley}@cs.umass.edu Abstract A wide range o orwarding strategies have been developed or multi-hop wireless networks, considering the broadcast nature o the wireless medium and the presence o random ading that results in time-varying and unreliable transmission quality. Two recently proposed strategies are opportunistic orwarding, which exploits relay diversity by opportunistically selecting an overhearing relay as a orwarder, and cooperative orwarding, which relies on the synchronized transmissions o relays to reinorce received signal strengths. Although these strategies are well-known in the literature, there is no comprehensive comparative analysis o their network-level perormance in a realistic SINR signal-to-intererence-and-noise ratio setting with multiple network lows. In this paper, we develop Markovian models or these protocols in the case o multiple competing lows in a general network setting; we also provide recurrence relations or the special case o linear networks. We irst use these models to evaluate simple small-scale networks, and ind that opportunism oten outperorms cooperation a result corroborated by simulations in larger networks. We also present a ixed-point model to eiciently estimate the throughput o large networks using these models. We identiy the intererence resulting rom the larger number o transmissions under cooperative orwarding as a cause or mitigating the potential gains achievable with cooperative orwarding. I. INTRODUCTION Unlike wireline networks, the broadcast nature o wireless communication allows a much richer variety o approaches or orwarding packets between a source and destination than traditional hop-by-hop orwarding along pre-speciied paths. In particular, multiple nodes in addition to the intended next-hop recipient can overhear transmissions in a wireless network and serve as ad hoc relays to assist orwarding. Recently, two approaches have emerged that seek to exploit wireless channel characteristics when orwarding packets between source and destination in a multi-hop wireless setting: 1 Opportunistic Forwarding: Because o the broadcast nature o the wireless medium, several neighboring nodes may overhear transmissions, even i none o them is the intended next-hop or inal destination. A suitable relay can oten be selected opportunistically among these overhearing nodes to orward the packet downstream, until it reaches its inal destination [1] [8]. Cooperative Forwarding: In properly synchronized and coded networks, the signal strengths o multiple simultaneous transmissions o the same packet can be additive. Thus, i multiple nodes have received the same packet and can synchronize their orwarding transmissions o that packet, the signal strengths at downstream receivers will be increased, thus improving the reception probability at these downstream nodes [9] [11]. Although opportunistic orwarding and cooperative orwarding are well-known in the literature, their analysis and comparison in a network setting is rather limited. One o the challenges is to ind a simple and analyzable model that realistically captures important characteristics o the wireless medium, such as signal intererence strength and random ading. Most extant work either ocuses on link-level analysis in one-hop networks using a complex channel ading model e.g., [9], [10], or multi-hop network-level analysis using a very simple channel ading model e.g., [1], [1]. Moreover, it is also important that multiple competing lows and their interaction/intererence with each other be considered and understood. In this paper, we compare the perormances o idealized and representative opportunistic and cooperative orwarding strategies under common and realistic assumptions. We note that the opportunistic and cooperative orwarding strategies studied in this paper are simple and the perormance o both schemes can be enhanced by careul design decisions. We stress that our goal here is not to propose new protocols or investigate a speciic opportunistic or cooperative transmission protocol in detail. Instead, our more undamental goal is to characterize and understand the dierences between these two approaches to orwarding in various multi-hop wireless scenarios with multiple competing lows. Our contributions are as ollows: 1 We derive closed-orm ormulas or the packet reception probability in the presence o cooperative transmitters, interering transmitters, and random ading. These results are subsequently used to study the perormance throughput o opportunistic and cooperating orwarding strategies in multi-hop wireless networks with random ading. We then analyze a simple n-hop linear network supporting a single low e.g., a wireless network along a road under opportunistic and cooperative orwarding. We observe that in the single low case, cooperation outperorms opportunism. This result is intuitive; in the single low case there is no intererence among packets and as there are larger number o transmitters in cooperative orwarding, the downstream packet probability reception is greater than opportunistic orwarding. Studying a single low case in this special setting provides useul insights and helps us appreciate the results or multiple

2 competing lows. 3 We develop a Markovian model to determine the throughput achievable by opportunistic and cooperative orwarding or a general network with multiple competing lows. We analyze this model or a simple topology and show that opportunistic orwarding can achieve higher throughput than cooperative orwarding. We study larger-scale networks via simulation and observe that opportunism outperorms cooperation on average. The worse perormance o cooperative orwarding can be largely attributed to the higher intererence due to multiple competing lows. Lastly, we develop a ixedpoint model or eiciently, but approximately computing the throughput o the Markov model, allowing perormance comparisons in larger-scale networks. Together, our results indicate that the relatively simple and lower control overhead opportunistic orwarding strategies is preerable to more complex cooperative counterparts in large networks with multiple competing lows. Our results also highlight the importance o considering multiple lows, since insights gained rom single low scenarios do not always carry over to the more complex multi-low scenario, where intererence among lows becomes important. Organization: We describe the orwarding strategies in detail in Sec. III, and the wireless communication model in Sec. IV. We analyze a linear network supporting a single low in Sec. V. For multiple lows and general topologies, we present a Markovian model in Sec. VI, which we use to study a simple diamond topology, together with simulations on random topologies. In Sec VII, we provide the ixed-point iteration or solving the Markov model. II. RELATED WORK The irst work proposing opportunistic orwarding is [1]. Since then, several strategies have been proposed to improve the perormance o opportunistic orwarding [] [5]. Research eorts have also theoretically analyzed the beneits o opportunism, including [6], where the authors perormed a Markovian analysis to determine the expected number o network-wide link-layer transmissions needed to transer a packet rom source to destination in a wireless mesh network. [6] mostly assumes that link success probabilities are provided a priori and does not consider random ading in a SINR model, an important component o our models. Also, [1] provides a recursive relation or estimating the minimum number o required opportunistic transmissions. Similarly, [7] proposes an analytical model to study the perormance expected transmission count o opportunistic routing protocols. [8] quantiies the average end-to-end delay obtained by using opportunistic schemes and demonstrates that it is about hal that obtained using typical shortest path routing. None o these works consider a realistic SINR model with random ading. A considerable amount o research has also considered cooperation in wireless networks [9], [10], [13]. [10] and [9] summarize much o this prior work in cooperative diversity and demonstrate how cooperation improves network perormance. [14] is one o the ew papers that describes an implementation o cooperative orwarding. It demonstrates that by properly synchronizing sender transmissions to symbol boundaries, it is possible to outperorm opportunistic routing in the absence o intererence or a simple topology. Most past research on cooperation has been in the context o the physical layer, with only a ew eorts exploring how cooperation interacts with higher network layers [10] and in the presence o multiple interering lows. In [11] the authors discuss how to eectively schedule cooperative transmissions or multiple access scenarios by helping sources with poor channels to the destination use relays that have better channel quality. We note that our work diers rom prior work in that we address primarily the network-layer concern with multihop orwarding, with the goal o comparing opportunistic orwarding and cooperative orwarding using a simple model o SINR with random ading, and in a multihop setting. III. FORWARDING STRATEGIES This section describes the two orwarding strategies compared in this paper opportunistic orwarding and cooperative orwarding. We ocus on generic and representative instantiations o these strategies. a Opportunistic Forwarding: I the packet cannot reach the destination in one hop, it is relayed by the overhearing node closest to the destination 1. This proceeds in multiple steps, until the packet reaches the destination. In the literature, there are proposals [1], [15] to address implementation details, such as how to select the appropriate relay when multiple nodes overhear the transmission. We abstract away these details, and ocus on analyzing this idealized implementation in order to shed insight into the main advantage oered by opportunism - the ability to opportunistically select a relay that is closest to the destination. b Cooperative Forwarding: To exploit the additive property o wireless signals, multiple overhearing relays can transmit the packets towards the destination, when proper synchronization e.g. by GPS among multiple transmitters is easible. This is the key innovation introduced in a cooperative strategy. We assume that a low maintains a list o relays associated with it. When a node belonging to a list o relays o a particular low overhears the transmission rom the low, it will be assigned as a relay. In the case o multiple network lows, we do not assume that nodes are allowed to coordinate their transmissions with other nodes that receive packets rom other lows, as this would involve prohibitively high overhead. In this case, competing lows are essentially treated as intererence. A more sophisticated variant o cooperative orwarding is: c Selective Cooperative Forwarding: Although cooperation can reinorce signal strength, can also increase 1 In more sophisticated settings, it can be relayed by the node that has the best estimate channel condition in some metrics [5] to the destination. We ocus on the simplest setting or our analysis. One solution is to use a very low-data rate, reliable out-o-band control channel to transmit the ACKs among overhearing relays [1], while the relays can be selected in a way to ensure that they can overhear ACKs among themselves [5].

3 3 the intererence level to other simultaneous lows. A more reined strategy is not to assign all nodes as relays, but to instead select only a small subset o nodes that are closest to the destination or have advantageous wireless channel conditions when transmitting towards the destination. This is essentially a hybrid strategy o both opportunistic and cooperative orwarding. For convenience o analysis, we ocus on a simple selective cooperative orwarding strategy that only assigns two nodes as relays that are closest to the destination among the overhearing nodes in the list o potential relays. IV. WIRELESS COMMUNICATION MODEL In order to compare the perormance o dierent orwarding strategies, we use the ollowing channel model to account or SINR and random ading. We proceed in multiple steps. Let us assume that there are C nodes in the network. a Single Transmission: Let us irst consider the simplest case with a single transmission between node i and node j i, j 1 to C, i j. Denote by S the signal-to-noise ratio rom transmitter i to receiver j: S x 1 N 0 where N 0 is a constant background noise, x is the Rayleigh ading coeicient the lat ading channel is modeled as a Gaussian random process x [16], d is the distance between i and j, α is the path loss exponent, and P is the transmission power at i. Note that d 1 and α. We assume that parameters N 0, α, P, d are constants either known or measured a priori. On the other hand, x is a random variable, assumed to be exponentially distributed 3 with normalized mean 1. We also assume that x is a collection o i.i.d. random variables. In this work we also assume that x is i.i.d in dierent time slots. We discuss how to relax this assumption in Sec. VIII. The probability that S s where s > 0 is P{S s} exp sn0 We model the physical layer coding scheme by assuming that a received packet can be decoded successully when S β or a certain threshold β. An important quantity is the packet reception probability that j can successully receive the packet rom i, denoted by: P P{S β} 3 b Cooperative Synchronized Transmissions: We next consider a set o cooperative transmitters T {i 1,..., i m } that can synchronize their transmissions such that the signal-tonoise ratio at receiver j is the sum o the individual signal-tonoise ratios rom the transmitters see Fig. 1 a. Note that j cannot belong to T. The total signal-to-noise ratio S T,j rom transmitters T to j is: r T S T,j x 4 N 0 3 In narrowband Rayleigh ading channel, the power o a signal with envelope as Rayleigh distribution is an exponential random variable [17]. Since the individual signal-to-noise ratio is an exponential random variable, the total signal-to-noise ratio is the sum o exponential random variables. Let T,j s be the probability density unction o S T, which is a convolution o unctions i1,js,..., im,js, deined by: T,j s i1,js im,js 5 where i1,js N0 sn exp 0 is the probability density unction o individual signal-to-noise ratio S. The probability that j can successully receive the packet rom a set o transmitters T is given by P T,j P{S T,j β}. Deriving a general ormula or P T,j or an arbitrary set o transmitters T is challenging. Hence, in this paper we assume that d i,j d or every pair o transmitters i, i and any node receiver j. This signiicantly simpliies the proos on the convolution o exponential distribution unctions see Lemma 1. This mild assumption, likely satisied by real topologies, is useul to simpliy the expression o P T,j. Lemma 1: Denote m s as the probability density unction o the sum o m independent exponential random variables with distinct means λ 1,..., λ m. m m m s λ r r1 r1 exp sλ r m r 1:r r λ r λ r Proo: See [18]. We remark that the general case with non-distinct values λ r are called hypoexponential random variables [19]. There are ormulas in [0], [1] or hypoexponential random variables, which appear too complicated or the analysis o networklevel perormance. Hence, we will rely on Lemma 1 under the assumption o distinct values o λ r. Lemma : The probability that j can successully receive the packet rom a set o transmitters T is: P T,j exp βn0 r T 1 d r T \{r} Proo: Based on Lemma 1, see []. i i 1 i 3 Synchronized Transmission j k 1 i k j 6 d r,j α 7 Interering Transmission a b Interering Transmission Fig. 1. a Cooperative synchronized transmission vs. b Competing interering transmission. c Competing Interering Transmissions: Lemma only considers the case o cooperative synchronized transmitters. To address the case o competing interering transmissions e.g., Fig. 1 b, let I be the set o simultaneously competing transmitters. The signal-to-intererence-and-noise ratio S I rom transmitter i to receiver j in the presence o a set o interering transmitters I is deined as: S I x N 0 + k I x k,j 8 k,j

4 4 It is clear that i j and that i and j cannot belong to I. The probability that j can successully receive the packet rom transmitter i is given by P I P{SI β}, which can be obtained rom the ollowing lemma. Lemma 3: P I exp βn0 k I 1 + β d α d k,j 1 d k,j α 9 d k,j k I\{k} Proo: Omitted due to space constraints. See []. d Mixed Cooperative & Interering Transmissions: Last, we consider the general case with a set o cooperative transmitters T and a set o simultaneously competing transmitters I. The signal-to-intererence-and-noise ratio S I T,j rom a set o cooperative transmitters T to j in the presence o a set o interering transmitters I is: S I T,j r T x N 0 + k I x k,j k,j 10 Note that T I and j cannot belong to T or I. The probability that j can successully receive the packet rom a set o cooperative transmitters T in spite o interering transmitters I is given by PT,j I P{SI T,j β}, which can be obtained by the ollowing lemma, derived using Lemmas -3. Lemma 4: PT,j I exp βn 0 d r T k I 1+β d k,j α d 1- d α d r 1- k,j,j d α k,j r T \{r} k I\{k} V. NETWORKS WITH SINGLE FLOW Sec. IV provides single-transmission/reception models or wireless networks with random ading. We now use this model to construct simple recurrence relations or source-destination paths and compare the perormance o opportunistic to cooperative orwarding strategies in a linear network. We observe that in the single low case where there is no network intererence, the throughput provided by cooperative orwarding is greater than that provided by opportunistic orwarding. In the ollowing, we consider the single packet case 4 - the source sends no new packet until the packet reaches the destination. Fig.. p α p nα p n-1α p s r 1 r r n-1 t d d d An n-hop linear network. A. Opportunistic Forwarding In Fig., we consider only one low in a n-hop linear network, where s is the source, t is the destination, and s r 0, r 1,..., r n 1, r n t are the relays. Assume that the distance between r i 1 and r i i, j 1 to n in the linear network is d, and denote by p exp βn 0 Pd the packet α reception probability or a transmission over one hop. Hence, 4 We will discuss the multiple packet case in Sec.VIII the probability that i can successully transmit packets to j when they are n hops apart is given by: P exp βn 0 α p n Pnd α 11 For convenience o analysis, we assume α is an integer. Suppose the distance between i and j is nd, and the distance between j and interering transmitter k is m d. P k exp βn0 Pnd α 1 + β n m α p nα 1 + β n m α 1 There are two quantities o interest. One quantity is the throughput o the linear network. Denote by N op [n] the expected number o transmissions required by opportunistic orwarding to reach the destination rom the source in a n- hop linear network. We obtain: N op [1] 1 p n 1 N op [n] p nα + + n i1 ji+1 1 p jα p iα 1 + N op [n i] n 1 p jα 1 + N op [n] j1 13 To write a recursive equation or N op [n] 13, we have to consider three cases: 1 With probability p nα, the source can reach the destination in one transmission. With probability n ji+1 1 pjα p iα, the source can reach the node that is n i hops away rom the destination in one transmission, rom which the expected number o transmissions to reach the destination is N op [n i]. 3 Otherwise, with probability n j1 1, the source cannot reach any other nodes. pjα Because there is only a single low, the throughput is: 1 T op [n] 14 N op [n] Another quantity o interest denoted by H op [n] is the average number o hops traversed in one transmission, given that the destination is n hops away. We obtain the recurrence equation: H op [1] p 15 H op [n] np nα + 1 p nα H op [n 1] We can solve H op [n] in closed orm. Lemma 5: n H op [n] jp jα When n, j1 n lj+1 1 p lα 16 H op [n] p + p α p 1+α + 3p 3α + Op 1+3α 17 Proo: Omitted due to space constraints. See []. Theorem 1: The throughput is upper bounded by: T op [n] 1 n H op[n] 18 Proo: Omitted due to space constraints. See []. In general, we observe that the upper bound is tight see Fig. 3.

5 5 Fig. 3. ThroughputTopn Hopnn 0.4 Topn n 0 p 0.8 The throughout T op[n] is tightly bounded by 1 n Hop[n]. B. Cooperative Forwarding Next, we consider cooperative orwarding using all overhearing relays to transmit the packets to the destination. We consider the idealized case o perect cooperative orwarding, where we can use all the relays between the source and arthest overhearing relay in the linear network or cooperative orwarding. Thus, in Fig., i r k overhears the packet, then we assume all relays r 1,..., r k 1 also overhear the packets. We aim at bounding the gap between opportunistic orwarding and cooperative orwarding. Hence, it suices to consider perect cooperative orwarding in order to establish an upper bound on this gap. Assuming perect cooperation, let H co [n] be the average number o hops reached in one transmission by cooperative orwarding, given that the destination is n hops away. Theorem : The expected number o hops a packet can reach in one time slot by cooperative orwarding is related to that o opportunistic orwarding by: H co [n] O n H op [n] 19 Proo: Omitted due to space constraints. See []. In this section, we have seen that cooperative orwarding provides higher perormance at most sub-linear, i.e., n improvement than opportunistic orwarding in the singlelow, linear-network case. As we will see in subsequent sections, where we consider multiple competing lows within the network, transmission intererence among lows which is not present in the single low case becomes a critical actor. This will mitigate the advantages o cooperative orwarding ound in this section, suggesting that the relative advantages o opportunistic and collaborative orwarding depend strongly on network topology and assumptions about traic lows. VI. NETWORKS WITH MULTIPLE FLOWS Having studied the single low case or a linear network in the previous section, we next consider a general setting with an arbitrary network topology and multiple lows. The major dierence between the single low and the multiple low case is the increased intererence due to competing lows. We ormulate Markovian models or analyzing the dierent orwarding strategies in the multiple low scenario, using the packet reception probabilities rom Sec. IV. Using these models, we study a simple topology, and show that opportunistic orwarding can outperorm cooperative orwarding in the absence o inter-low cooperation. For more general networks we use simulation and observe that opportunism outperorms cooperation on average. Thus, we conclude that intererence mitigates the potential gains o cooperative orwarding. A. Opportunistic Forwarding First, we present the Markov model or opportunistic orwarding in a general network topology and multiple lows; we then evaluate this model or a simple diamond network. We denote a set o lows by F. Each low F has a list o participating nodes denoted by P v s, v 1,..., v d, where v d is the destination and v s is the source. Each succeeding node in P e.g., v i has a higher priority than its preceding nodes i.e., v j or all j < i or orwarding the packet, until the packet reaches v d. Formally, we denote v i v j to represent that v i has a higher priority than v j in P. We denote the state o the network as r r P : F, where r is the active relay or low or the next orwarding operation. Recall that PT,j I is the packet reception probability at j rom a set o cooperative transmitters T in the presence o interering transmitters I. We denote by P r,r the state transition probability rom state r to state r, where r r or r r and r v d, or at least one low F. Let r {r : F \{}}. We obtain: P r,r F P r r,r v P :v r 1 P r r,v 0 Namely, P r,r is the packet reception probability that every low can receive a packet rom r to r, subject to the condition that the set o succeeding nodes {v P : v r } that cannot receive the packet. Recall that we assume that the low s source transmits a new packet ater the successul delivery o a packet to the the low s destination. Thereore when a low reaches state r v d i.e., the packet reaches the destination, the state transition in the next time step will correspond to the states reachable rom the source with their respective probabilities. We remark that Eqn. 0 contains two prohibited cases: 1 two packets cannot be received by the same receiver at the same time; and a node cannot be receiving and transmitting at the same time. Because we assume β > 1, Eqn. 0 will give zero transition probability or the above two cases. The stochastic behavior o the network is characterized by the Markov chain deined by state transition probability P r,r or every pair o states r, r. We then can evaluate the stationary distribution πr or each state o the network r, which satisies the ollowing balance equation: πr P r,r πr P r,r 1 r r subject to r πr 1. The throughput o each low, T op, is given by: T op πr r:r v d B. Cooperative Forwarding For cooperative orwarding, we denote the state o the network as R, where R P is a set o cooperative transmitters o low. Let R F\{} R. The state

6 6 transition probability P R,R, where R R or at least one F and v d R, is given by: P R,R P R R,v R 1 P R,v 3 F v R \R v P \R Similarly, Eqn. 3 also contains the prohibited cases, to ensure that 1 two packets cannot be received by the same receiver at the same time; and a node cannot be receiving and transmitting at the same time. The stationary distribution is denoted by πr, and the throughput o low, T co, is given by: T co πr P R,R 4 R,R :v d R C. Selective Cooperative Forwarding Selective cooperative orwarding only assigns two closest nodes to the destination that currently have a copy o the packet as relays. We again denote the state o the network as R, where R P is a set o potential transmitters o low that have received the packet. The two nodes r 1, r R are selected, such that r 1 r r or all r R \{r 1, r }. Hence, we denote the two selected relays by a set rr. Let I rr F\{} rr. The state transition probability P R,R, where R R or at least one F and v d R, is given by: P R,R F v R \R P I rr rr,v v P \R D. Analysis o a Simple Diamond Network I 1 P rr rr,v Using the Markov models deined in Secs. VI-A and VI-B, we compare the perormance o opportunistic orwarding and cooperative orwarding with two lows in the diamond network depicted in Fig. 4 a. There are two lows: s 1 and s t. Relay r can contribute to either low, depending on i it can overhear the low. We assume β > 1, and by Eqn. 10 a node can receive a packet rom only one low at a time. The orwarding operations or opportunistic orwarding and cooperative orwarding respectively generate the Markov chains in Fig. 5. Each state corresponds to a subset o transmitters that can orward the packet. In opportunistic orwarding, relay r can orward the packet or a low during a time slot, provided that it received a packet previously. In cooperative orwarding, both source and relay will participate in orwarding. Hence, both Markov chains have the same structure, but dierent state transition probabilities. Fig. 4. Simple Diamond Network a Opportunistic Forwarding: The stationary distribution is: Fig. 5. t s 1 r s 1-p c 1-p a -p d 1-p b p c p d t p a t s 1 r s s 1 r s p b π A π B π C a Opportunistic s 1-p c 1-p a -p d 1-p b t t t p c p a s 1 r s s 1 r s s 1 r p d p b π A π B π C b Cooperative Markov chains or the orwarding operations in a diamond network. p b p d p b p c π A, π B p a p c + b c + p b p d p a p c + p b p c + p b p p d a p c π C p a p c + p b p c + p b p d 5 where the state transition probabilities can be expressed in terms o the packet reception probabilities: p a 1 Ps s 1, Ps s 1,r p d p a 6 p b Pr,t s 1 p c p b In Table I, we compute all the packet reception probabilities using Lemmas -4. Consider α. The throughput o low s 1 using opportunistic orwarding is given by: Top s1 t1 π A Ps r 1, + π B Ps s 1, + π C Pr,t s 1 p p+3β+β +41+3β+β p 3 +β p +4β 1+β1+β p 3 +β+p+3β+β 7 b Cooperative Forwarding: The state transition probabilities can be expressed in terms o the packet reception probabilities by Lemmas -4 see Table I: p a 1 Ps s 1, Ps s 1,r p d p a 8 p b P s {s 1,r}, p c p b The throughput o low s 1 using cooperative orwarding is given by: T s1 t1 co π A P {r,s} s 1, + π B P s s 1, + π C P s {s 1,r}, p 3 + p 4β 1 + β p 31 + β 9 Lemma 6: The throughput o opportunistic orwarding is superior to that o cooperative orwarding: T s1 t1 op T s1 t1 co 30 Proo: See the Appendix. We also plot the throughput o a low using opportunistic orwarding and cooperative orwarding in the case o two competing lows in Fig. 6. This corroborates our intuition that cooperation can degrade perormance due to the increased amount o intererence generated by the larger number o simultaneous transmitters. E. Simulations or Random Networks Having studied a simple diamond network via a Markovian analysis, we next use simulation to study larger network settings. As we will see, the insights gained in the small

7 7 t s 1 r s s 1 r s α P s s 1, p 1+β s P r, p 1+β α t s 1 r s Ps r 1, p t s 1 r s s P s 1,r α 1+β α t s 1 r s p 1+β s 1 r s P {s,r} s 1, P s {s 1,r}, p α 1+β1+ α p α 1+β1 α + p 1+β α 1 α TABLE I PACKET RECEPTION PROBABILITIES FOR TRANSMISSIONS IN BOLD ARROWS, UNDER THE INTERFERENCE OF THE TRANSMISSIONS IN DASHED ARROWS. Throughput Coop.Β Oppo.Β 3 Coop.Β 3 0. Oppo.Β Coop.Β 10 p Fig. 6. The throughput o opportunistic orwarding and cooperative orwarding in the simple diamond network. Oppo.Β 1 scale setting generally apply in this more general setting. We consider 50 nodes uniormly distributed in a area. We select 4 distinct source-destination pairs reerred to as a coniguration at random rom the 50 nodes. We simulate the link quality between dierent pairs o nodes or every time slot using the Rayleigh ading channel model. The simulation begins by all 4 sources transmitting packets. A node is able to receive a packet i the SINR between the transmitter and itsel is above a threshold. The opportunistic and cooperative routing protocols govern the nodes that transmit packets in the next time slot. When a packet reaches the corresponding destination, the source starts transmitting a new packet. We conduct this simulation or 5000 time slots and keep track o the number o packets received at the destination to calculate the throughput. We reer to the simulation o a given coniguration as a run. Recall that our earlier results revealed that intererence among dierent lows played an important actor in determining perormance. Thus, when presenting throughput comparisons in this section, we would like to quantiy such intererence among packets lowing rom source to destination along paths between given sets o source-destination pairs. Note, however, that there is no well-deined notion o a deterministic path along which packets low or either opportunistic or cooperative orwarding. Thus we characterize the level o intererence among lows by taking 10 points equidistantlyspaced between a low s source and destination or each low. Let L denote the set o these 10 points or low. We consider the distance between all pairs o such points i, j in the ollowing manner: Int-Metric d α 31 F F\{} i L j L Int-Metric provides a coarse measure o the intererence as caused by the nearness o potentially interering nodes or dierent lows. Higher value o Int-Metric indicates a greater amount o intererence in the network. In Fig. 7 we plot the dierence in throughput between the opportunistic and cooperative strategies versus the Int-Metric or a β o 4. The igure is obtained by conducting 100 runs, each time with a dierent coniguration. Points above the line drawn at Throughput Dierence0 indicate the cases where opportunism perorms better than cooperation while the points below the line depict the opposite. The results in Fig. 7 indicate that on average the perormance o opportunism is better than that 0.06 o cooperation and this is true or a wide range o Int-Metric values. Fig. 7. Throughput Dierence Opp Coop Intererence Metric Int Metric Simulation results o throughput dierence in random networks. VII. FIXED-POINT MODEL Sec. VI provided a Markovian model o multiple lows in a general network setting. However, the number o states increases exponentially with the number o lows in this model. Hence, evaluating the stationary distribution o the model quickly becomes intractable. Thus, in this section we introduce a ixed-point model to simpliy the evaluation. We will show that the throughput obtained rom the ixed-point model is a lower bound to the actual throughput o the Markov model. In order to establish this model as a lower bound, we assume that the sets o participating nodes among distinct lows are disjoint i.e., P P. Our approach is to model each low independently and capture their dependence by accounting or intererence between lows within each low model. This gives rise to a set o ixedpoint equations or the stationary distributions, which can be obtained via eicient iterative methods. A. Opportunistic Forwarding The state o a low F is speciied by the active relay r P that overhears the transmission and has the highest priority among the overhearing participating nodes. We next describe a set o ixed-point equations or individual lows. First, suppose that the stationary distribution o a node j to become an active relay is given by ˆπ j. Then, the P

8 8 expected intererence to j rom all other lows w.r.t. stationary distributions ˆπ {ˆπ : F \{}} becomes: Î j ˆπ ˆπ j E[ x ] F\{} i P F\{} i P ˆπ j 3 where the ading coeicient x is an i.i.d. exponential random variable with normalized mean 1. Note that Î j ˆπ does not depend on ˆπ, but only on {ˆπ : F \{}}. Suppose that the intererence rom other lows remains stationary and has distribution ˆπ. Then the packet reception probability that j can successully receive the packet rom i in low w.r.t. ˆπ is given by: { ˆP x ˆπ } β N0 P N 0 + Î j ˆπ β + exp Î j ˆπ 33 Next, we ocus on the Markov model o individual low. In such a model, we denote by ˆP r,r ˆπ the state transition probability rom an active relay r P to another active relay r P such that r r or r r and r v d, w.r.t. ˆπ : ˆP r,r ˆπ ˆP r,r ˆπ 1 ˆP r,vˆπ 34 v P :v r Denote the stationary distribution o this model as ˆπ. It satisies the ollowing balance equations or all r P : ˆπ r ˆP r,r ˆπ ˆπ r ˆP r,r ˆπ 35 r P r P subject to v P ˆπ v 1. The state transition probability P R,R rom R P to R Eqns orm a set o ixed-point equations or P such that R R and v d R, is deined by: ˆπ : F. Solving the ixed-point ˆπ : F ˆP R,R can be achieved by an iterative method. We irst assume a ˆπ ˆP r R,v ˆπ 1 ˆP r R,v ˆπ certain distribution ˆπ 0 : F. Then we obtain ˆπ1 rom v R \R v P \R Eqns w.r.t. ˆπ 0, or all F. We repeat the process or t steps, until ˆπ t has a small deviation rom ˆπt 1. The throughput o the ixed-point model is deined by: ˆT op ˆπ v d 36 Theorem 3: The throughput obtained rom the ixed-point model is a lower bound to the actual throughput o the Markov model: T op ˆT op 37 Proo: See the Appendix. B. Cooperative Forwarding The ixed-point model or cooperative orwarding is similar to that o opportunistic orwarding. But the state o a low is speciied by the set o cooperative transmitters R P. By Lemma, the packet reception probability that j can successully receive the packet rom the set o cooperative transmitters R in a low is given by: ˆP R,j ˆπ r R exp r R\{r} βn0+î j ˆπ 1 d α 38 d r,j Note that the state o low is R, a subset o cooperative transmitters. Then the stationary distribution o a node j P is given by: ˆπ j ˆπ R 39 R P :j R In the Markov model o an individual low, the state transition probability P R,R rom R P to R P such that R R and v d R, is deined by: ˆP R,R ˆπ ˆP R,v ˆπ 1 ˆP R,v ˆπ v R \R v P \R 40 To solve the ixed-point ˆπ : F, we rely on a similar iterative approach as or the case o opportunistic orwarding. The throughput obtained rom the ixed-point model can be shown as a lower bound to the actual throughput o the Markov model, using the same argument as in Theorem 3. C. Selective Cooperative Forwarding The case o selective cooperative orwarding is similar to basic cooperative orwarding, except with a modiication to consider the two best relays instead all relays. Speciically, let the two best relays be r 1, r R or low, such that r 1 r r or all r R\{r 1, r }. We denote the two selected relays by a set r R. The packet reception probability that j can successully receive the packet rom the set o cooperative transmitters R in a low is given by Pr ˆπ R,j. And the stationary distribution o a node j P is given by: ˆπ j ˆπ R 41 R P :j r R 4 To solve the ixed-point ˆπ : F, we rely on a similar iterative approach as or the case o opportunistic orwarding. The throughput obtained rom the ixed-point model can be shown as a lower bound to the actual throughput o the Markov model, using the same argument as in Theorem 3. D. Comparison o Fixed Point and Simulation We compare the perormance o our models with the simulation results. For these simulations we consider a 5 5 grid topology and consider opportunistic orwarding and selective cooperative orwarding. The simulation procedure is similar to the one outlined in Sec. VI-E. For the model we iteratively solve the ixed point equations in Sec. VII-A and VII-C or the opportunistic and selective cooperative orwarding strategies. Results in Fig. 8 are obtained considering 5 parallel lows, each moving vertically downwards in the grid. Flows are given ids ranging rom 1 to 5 starting rom one end o the grid to the other. As expected, we ind that or both schemes Flows 1 and 5 have maximum and comparable throughput as they experience the minimum amount o intererence rom other lows. Flow 3 has the minimum throughput because it is situated in the middle and experiences maximum intererence.

9 Throughput per Flow 9 Moreover the throughput o opportunism is greater than cooperation. This is primarily due to increased intererence in the cooperative case because o greater number o transmitters. We note that although the throughput obtained by our ixed-point model is lower than that obtained by simulation the relative ordering between opportunistic and cooperative orwarding is preserved. Fig Flow 1 Flow Flow 3 Flow 4 Flow 5 Opp Simu Coop Simu Opp Model Coop Model Comparison between simulation and ixed-point model VIII. CONCLUSION AND FUTURE WORK This paper has used modeling and analysis to investigate the perormance beneits o using opportunism and cooperation orwarding in wireless networks. Rather than proposing new protocols or investigating the perormance o speciic opportunistic or cooperative transmission protocols, our goal instead was to compare the perormance o idealized and representative opportunistic and cooperative orwarding strategies using generic models and under common realistic assumptions. We began with a single low linear network, and observed that cooperation outperorms opportunism. We then considered the case o more general network topologies with multiple low and observed that unlike the linear network case, opportunism outperormed cooperation on average. We identiied the intererence resulting rom the larger number o transmissions under cooperative orwarding as a cause or mitigating the potential gains achievable with cooperative orwarding. There are numerous avenues or uture research. In this paper we considered the single packet case i.e., there is only one packet in transit or each source-destination pair. Our current research is aimed at extending our analysis to cases where multiple packets are pipelined or each low. Here we believe our single-packet throughput analysis can orm the basis or a pipelined analysis or protocols that control transmission to provide a guard zone around each packet within a low, ensuring that intra-low intererence among packet transmissions does not occur. In this case, the model reduces to serial, pipelines instances o the models considered in this paper. A potential complexity here will be to determine the size o the guard zone. Too large guard zone will decrease the pipelining eiciency, whereas too small guard zone would result in high intererence. An on-going extension is to optimize the guard zone and study perormance o dierent orwarding strategies. We assumed that ading is i.i.d distributed in dierent time slots. This assumption will hold true only or ast ading where the coherence time is smaller than the duration o a time slot. In case o slow ading where the i.i.d assumption will not hold, cooperation will have additional beneits over opportunism because o multiple transmitters. The ading correlation o x is a Bessel unction [16] and we plan to incorporate the eect o correlation into the Markov model in our uture work. A longer term challenge is to compare opportunistic and cooperative orwarding in the presence o competing lows, with optimized centrally or distributed scheduling. The practical, but important, question o the overhead needed to achieve this coordination in practice, and whether this additional complexity is warranted by the increase in perormance is also a question or uture research. REFERENCES [1] S. Biswas and R. Morris, EXOR: Opportunistic routing or wireless networks, in Proc. ACM SIGCOMM, 005. [] M. Zorzi and R. R. Rao, Geographic random orwarding GeRaF or ad hoc and sensor networks: multihop perormance, IEEE Trans. Mobile Computing, vol., no. 4, pp , October 003. [3] M.-H. Lu, P. Steenkiste, and T. Chen, A theoretical model or opportunistic routing in ad hoc networks, in Proc. ACM MobiCom, 009. [4] S. Chachulski, M. Jennings, S. Katti, and D. Katabi, Trading structure or randomness in wireless opportunistic routing, in Proc. ACM SIGCOMM, 007. [5] E. Rozner, J. Sheshadri, Y. A. Mehta, and L. Qiu, SOAR: Simple opportunistic adaptive routing protocol or wireless mesh networks, IEEE Trans. Mobile Computing, vol. 8, no. 1, pp , December 009. [6] M.-H. Lu, P. Steenkiste, and T. Chen, Video transmission over wireless multihop networks using opportunistic routing, in Proc. Packet Video Workshop, 007. [7] A. S. Cacciapuoti, M. Calei, and L. Paura, Design, implementation and evaluation o an eecient opportunistic retransmission protocol, in Proc. ICUMT, 009. [8] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, On the perormance o time-space opportunistic routing in multihop mobile ad hoc networks, in Proc. WiOPT, 008. [9] J. N. Laneman, Cooperative diversity: Models, algorithms and architectures, in Cooperation in Wireless Networks: Principles and Applications. Springer, 006. [10] A. Scaglione, D. L. Geockel, and J. N. Laneman, Cooperative communications in mobile ad-hoc networks: Rethinking the link abstraction, IEEE Signal Processing Mag., vol. 3, no. 5, pp. 18 9, September 006. [11] Z. Ding, D. Kin. K. Leung, L. Goeckel, and D. Towsley, A relay assisted cooperative transmission protocol or wireless multiple access systems, IEEE Trans. Communications, 008. [1] S. Chachulski, Trading structure or randomness in wireless opportunistic routing, Master s thesis, Massachusetts Institute o Technology, 005. [13] A. Ozgur, O. Leveque, and D. Tse, Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks, IEEE Trans. Inormation Theory, vol. 53, no. 10, pp , October 007. [14] H. Rahul, H. Hassanieh, and D. Katabi, Sourcesync: A distributed wireless architecture or exploiting sender diversity, in Proc. ACM SIGCOMM, 010. [15] H. Liu, B. Zhang, H. Moutah, X. Shen, and J. Ma, Opportunistic routing or wireless ad hoc and sensor networks: Present and uture directions, IEEE Comm. Magazine, pp , December 009. [16] M. Zorzi, R. Rao, and L. Milstein, On the accuracy o a irst-order markov model or data transmission on ading channels, in Proc. o ICUPC, [17] D. Wong and D. Cox, Estimating local mean signal power level in a rayleigh ading environment, IEEE Trans. Vehicular Technology, vol. 48, no. 3, pp , May [18] M. Balázs, Sum o independent exponential random variables with dierent parameters, balazs/sumexp.html. [19] Hypoexponential distribution, Hypoexponential distribution. [0] S. V. Amari and R. B. Misra, Closed-orm expression or distribution o the sum o independent exponential random variables, IEEE Trans. Reliability, vol. 46, no. 4, pp , 1997.

10 10 [1] S. Favaro and S. G. Walker, On the distribution o sums o independent exponential random variables via Wilk s integral representation, Acta Appl. Math., vol. 109, pp , 010. [] Opportunism vs. cooperation: Analysis o orwarding strategies in multihop wireless networks with random ading technical report, anand/paper.pd. A. Proo o Lemma 6 IX. APPENDIX Proo: By substitution, we obtain: T s 1 op T s 1 co p 31+βp+3β+β +41+3β+β p 3 +β p +4β 3+p 4β1+β p 3 +β+p+3β+β Because 0 p 1, p + 3β + β β + β p 3 + β p + 4β 1 + β p 3 + β + p + 3β + β β + 7β p 1 + β 1 + 6β + 7β p 31 + β 3 + p 4β Jensen s inequality, 3 p + 5β 1 + 5β Thereore, we obtain Ts 1 op T s 1 co B. Proo o Theorem 3 1 Proo: First, we consider the actual Markov chain o multiple lows deined in Sec. VI-A. We recall that the stationary distribution o the actual Markov chain o multiple lows is π over the set {r : r P }. Note that π is the ixedpoint to Eqns Let R be the random set o active relays that are transmitting or the lows. Let the random total intererence level to node j be: I j R r R\{j} x 43 Let P be the packet reception probability rom node i to j, where j belongs to low. Since the set o active relays is random, P is a random variable, with expected value { E[P [P ] E x }] N 0+I jr β [ β ] N 0+I jr E exp 44 The throughput o a low can be obtained by averaging over time. By the ergodicity o the Markov model, it is equivalent to averaging over stationary distribution. Let the stationary distribution o each state j o low be π j r:r j πr, and 1r v d be the indicator unction that there is a state transition rom r to v d or low at a timeslot. ] T op π r E π [1r v d r P \{v d } r P \{v d } r P \{v d } π r E π [P r,v d π r E π [P r,v d ] v P :v r v P :v r ] 1 Pr,v 1 E π [Pr,v] The last equality is due to the assumption o independent random ading among pairs o nodes i.e., ading coeicient x is an i.i.d. random variable or any pair o nodes i, j. Second, recall that ˆπ : F is the ixed-point to Eqns. 3-35, which is also a ixed-point to Eqn. 35 and the ollowing equation: ˆP r,r ˆπ ˆP r,r ˆπ 1 ˆP r,vˆπ F v P :v r 45 Note that Eqns. 45 & 35 are comparable to Eqns. 0-1 or stationary distribution π. The throughput can be given by: ˆT op ˆπ r P r,v d ˆπ r P \{v d } r P \{v d } ˆπ r ˆP r,v d ˆπ v P :v r 1 ˆP r,vˆπ Next, we compare E π [P r,r ] and ˆP r,r π under a certain distribution π over r. Since exp x is a convex unction, by E π [P ] E π [ β N 0+I jr exp β exp N 0+E π[i jr] ] 46 Then, by the deinition o P π, we have E π [I j R] Î j π. This implies E π [P ] P π 47 I E π [ ˆP ] ˆP π or any pair o nodes i, j and any distribution π, then every orwarding operation carries a lower packet reception probability in the latter case. Hence, the latter case must have decreased throughput under the respective ixed-point: T op ˆT op 48