One-Hop Throughput of Wireless Networks with Random Connections
Seyed Pooya Shariatpanahi, Babak Hossein Khalaj, Kasra Alishahi, Hamed Shah-Mansouri
aa r X i v : . [ c s . I T ] N ov One-Hop Throughput of Wireless Networks withRandom Connections
Seyed Pooya Shariatpanahi, Babak Hossein Khalaj, Kasra Alishahi, Hamed Shah-Mansouri A BSTRACT
We consider one-hop communication in wireless networkswith random connections. In the random connection model,the channel powers between different nodes are drawn from acommon distribution in an i.i.d. manner. An scheme achievingthe throughput scaling of order n / − δ , for any δ > , isproposed, where n is the number of nodes. Such achievablethroughput, along with the order n / upper bound derived byCui et al., characterizes the throughput capacity of one-hopschemes for the class of connection models with finite meanand variance. Keywords : Wireless Networks, Random Connection Model,Achievable Throughput.
I. I
NTRODUCTION
Wireless networks are subject to fundamental limitationsin establishing source-destination data sessions. Investigatingsuch limitations, along with discovering potential communica-tion capabilities of wireless networks is of vital importance indesigning efficient and practical algorithms for their operation.While Shannon’s approach ( [1] ) to mathematical analysis ofcommunication systems is the most powerful approach, it isnot easily extendable to wireless networks with large numberof nodes.The pioneering work of Gupta and Kumar ( [2] ) in 2000,ignited the efforts in characterizing the fundamental communi-cation limits and capabilities of wireless networks. Gupta andKumar’s work, along with subsequent papers ( [3], [4] and [5]), established the order of √ n achievable aggregate throughputfor wireless networks with multihop technology, where n isthe number of nodes. However, in another line of research,the linear upper bound of order n is derived for the capacityof wireless networks by exploiting information-theoretic max-flow min-cut discussions [6]. The notable work of ¨Ozg¨ur et al.in 2007 resolves such gap between the upper and lower bounds[7]. In fact, by not considering interference as being alwaysharmful, and by exploiting Multiple-Input Multiple-Output(MIMO) techniques, they propose a hierarchical cooperationscheme achieving linear throughput scaling.Many of papers on wireless networks capacity, use channelmodels based on distance between nodes, while others use S. P. Shariatpanahi, B. H. Khalaj and H. Shah-Mansouri are withthe Department of Electrical Engineering and Advanced CommunicationResearch Institute (ACRI). K. Alishahi is with the Department of Math-ematical Sciences, Sharif University of Technology, Azadi Ave., Tehran,Iran. Emails: [email protected], [email protected], [email protected],[email protected]. models based on random distributions. As also mentionedin [8], in many scenarios, a wireless channel model basedon randomness is a more appropriate choice than distance-based models. As an example of such scenario, we can pointto the case where randomly moving obstacles block signalpropagation, and the distance-based model cannot addresssuch issues. Also, when the network area size is small, thedominating factor in characterizing the channel propertiesbetween nodes is the random fluctuations due to fading, ratherthan the distance-based path-loss effect. In addition, in suchsituation, the network is strongly interference-limited, whichis best modeled by a random-based channel model. Moreover,many wireless systems employ a unit called Automatic GainControl (AGC) which compensates for the distance effect.Accordingly, in many scenarios, it is more suitable to use arandomness-based channel model which is called the “RandomConnection Model”. In such model, the channel power γ between each two nodes is drawn from a common parentdistribution f ( γ ) , and different links are independent.The first work considering the random connection model incommunication over wireless networks is by Gowaikar et al.[8]. They propose a multihop scheme achieving linear scaling,for a specific case of parent distribution. Their scheme is basedon establishing routes in random graphs. Their subsequentpaper investigates a model which considers both the geometryand randomness effects [9]. Another work using the randomconnection model is the paper by Cui et al. [10]. In theirwork, one-hop and two-hop communication schemes are in-vestigated. It is shown that, in the class of parent distributionswith finite mean and variance, the one-hop throughput is upperbounded by order n / . Also, for two-hop schemes, theyprovide upper and lower bounds of order n / .While Cui et al. prove that in one-hop schemes, and in theclass of parent distributions with finite mean and variance, onecannot surpass the throughput scaling of order n / , they leavethe achievability part unanswered. In this letter, we solve thisopen problem and propose an scheme achieving the throughputscaling of order n / − δ , for any δ > and independentof n . Our proposed scheme is very simple and is based onestablishing the largest number of concurrent communications.The letter structure is as follows. In section II, the networkmodel is explained. In section III, we explain the proposedscheme and prove that it achieves throughput of order n / − δ .Finally, section IV concludes the letter.II. N ETWORK M ODEL
Consider a wireless network consisting of n nodes. Eachnode is capable of transmitting and receiving signals simulta-eously (i.e., full duplex communication). The nodes follow anon/off strategy. In such strategy, at each time slot, a subset ofnodes with m elements are “on” and transmit simultaneously,while other nodes do not transmit any signal. We call thesubset of active nodes S . Each node in the network is a sourceof data for exactly one destination, and also, it is destination ofdata for exactly one source. Thus, we have n sources (i.e., S , . . . , S n ), and n destinations (i.e., D , . . . , D n ). Each sourcenode S i wishes to transmit to the destination node D i for i = 1 , . . . , n . The signal received by D i at a specific time slotis: y i = X j ∈ S h j,i x j + n i (1)where x j is the signal transmitted by j th source node, and h j,i is the channel gain between S j and D i . We define γ j,i , | h j,i | to be the channel power, which is a random variabledrawn from the parent distribution f ( γ ) . In addition, all linksare independently and identically distributed (i.i.d.). Finally, n i is the additive white gaussian noise at each receiver whosevariance is N .The communication between S i and D i is successful, if andonly if the received Signal to Interference and Noise Ratio(SINR) at D i is above a given threshold level: SIN R i , γ i,i N + P j ∈ S ,j = i γ j,i > β (2)As explained earlier, such channel mode, also known as the“Random Connection Model” is a very appropriate model inmany network scenarios [8], [10].III. T HROUGHPUT A CHIEVABILITY OF O RDER n / We consider one-hop communication between sources anddestinations. At each time slot, the nodes belonging to theactive subset S broadcast their signals, and the rest of thenodes do not transmit. We define the one-hop throughput ofthe network as the expected number of successful receptionsat each time slot (similar to [10]). Cui et al. have proved thatthe throughput of such one-hop strategy, when f ( γ ) has finitemean and variance, is upper bounded by order n / . In thissection, we propose an achievable scheme which achieves thethroughput of order n / − δ for any δ > and independentof n . The main result of the letter is stated in the followingtheorem: Theorem 1
There exists a one-hop communication scheme achieving thethroughput of order n / − ǫ/ for any strictly positive ǫ . Theparent distribution resulting in this throughput is f ( γ ) = ǫ (1+ γ ) ǫ for γ > , which has finite mean and variance. Proof:
Consider source nodes S , . . . , S n and destination nodes D , . . . , D n . The channel power between S i and D i is γ i,i .Let us sort the source and destination pairs based on the powerof direct link between them (i.e., γ i,i ’s). Define S ( n − i +1) − D ( n − i +1) as the source-destination pair which have i th mostpowerful channel, γ ( n − i +1) , ( n − i +1) . Thus, we have: γ (1) , (1) γ (2) , (2) , . . . , γ ( n ) , ( n ) (3)In the proposed scheme, at each time slot, the first m strongest source-destination pairs (i.e., S ( k ) − D ( k ) , k = n − m + 1 , . . . , n ) are active, and other nodes are inactive. In otherwords, at each time slot, sources S ( k ) , k = n − m + 1 , . . . , n broadcast their signals simultaneously, and the correspondingreceivers D ( k ) , k = n − m + 1 , . . . , n attempt to decodetheir messages. If we define M as the number of successfulreceptions, by defining r , n − m + 1 , for the networkthroughput we have : E { M } = n X k = r P { SIN R ( k ) > β } (4) > m P { SIN R ( r ) > β } = m P { γ ( r ) , ( r ) > β ( N + n X j = r +1 γ ( j ) , ( r ) ) } > m P { γ ( r ) , ( r ) > β ¯ γm } P { β ( N + n X j = r +1 γ ( j ) , ( r ) ) < β ¯ γm } where ¯ γ , E { f ( γ ) } . The first inequality is due to the factthat S ( r ) − D ( r ) has the weakest direct channel power amongthe active pairs. The last inequality is due to the indepen-dence of γ ( r ) , ( r ) and β ( N + P nj = r +1 γ ( j ) , ( r ) ) . According toMarkov’s inequality we have: P { β ( N + n X j = r +1 γ ( j ) , ( r ) ) > β ¯ γm } β ( N + ( m − γ )2 β ¯ γm (5) ≃ for large m . From (4) and (5) we have: E { M } > m P { γ ( r ) , ( r ) > β ¯ γm } (6)At this stage of the proof, we need the following theorem dueto Falk [11]: Theorem 2
Suppose X , . . . , X n are n i.i.d. random variables with theparent distribution f ( x ) . Define X (1) , . . . , X ( n ) to be the orderstatistics of these random variables. Suppose F ( x ) is the cu-mulative distribution function (cdf) of the parent distribution,which is absolutely continuous, and for some α > we have(von Mises condition [12]): lim x →∞ x f ( x )1 − F ( x ) = α (7)Then, if i → ∞ and i/n → as n → ∞ , there exist sequences a n and b n > such that X ( n − i +1) − a n b n ⇒ N (0 , (8) E { . } and P { . } are the expectation operator and probability measurerespectively. here ⇒ stands for convergence in distribution, and N (0 , is the normal distribution with zero mean and unit variance.Furthermore, one choice for a n and b n is: a n = F − (1 − in ) (9) b n = √ inf ( a n ) Now, we are ready to apply Theorem 2 to the throughputanalysis of our scheme. In our scheme, we look for the statis-tical properties of γ ( r ) , ( r ) to analyze P { γ ( r ) , ( r ) > β ¯ γm } ,which appears in (6) (note that r = n − m + 1 , and m is the number of active sources.). Thus, we have the same“intermediate order statistics” problem as the one stated inTheorem 2. Consequently, we put: X ( n − i +1) = γ ( r ) , ( r ) (10)and i = m = n − δ (11)for any δ > and independent of n . Also, the probabilitydistribution function (pdf) of the parent distribution whichresults in the desired throughput is: f ( x ) = 2 + ǫ (1 + x ) ǫ , x > (12)where ǫ > is any small non-zero real number. Thisdistribution has finite mean and variance. Also, we observe thatthe corresponding cdf is absolutely continuous and satisfies thevon Mises condition: lim x →∞ x f ( x )1 − F ( x ) = 2 + ǫ > (13)Accordingly, due to theorem 2 we have: γ ( r ) , ( r ) − a n b n ⇒ N (0 , (14)where a n = F − (1 − in ) (15) ≃ n
13 2+3 δ ǫ = n where we have put δ = ǫ/ . Therefore, we have: P { γ ( r ) , ( r ) > β ¯ γm } = P { γ ( r ) , ( r ) > (2 β ¯ γn − δ ) n / } (16) > P { γ ( r ) , ( r ) > n / } = 12 where the inequality is valid for large-enough n , due to thefact that ¯ γ and β are independent of n . The last equality isa consequence of the result of Theorem 2, which is stated inequation (14). By putting (16) in (6) we will have: E { M } > m (17)where m = n / − δ , and Theorem 1 is proved. (cid:3) IV. C
ONCLUSION
In this letter, we have proved that the lower bound ofone-hop communication in wireless networks with randomconnection model, in the class of finite mean and variancechannel powers, is n / − δ , where δ > is independent of n . Our result, combined withmises the upper bound of n / derived by Cui et al., characterizes the throughput capacity ofsuch networks. V. A CKNOWLEDGEMENT
This work was supported in part by Iran National Sci-ence Foundation under Grant 87041174 and in part by IranTelecommunications Research Center.R
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