Cataloged from PDF version of article.Throughput model for non-persistent CSMA networks which was proposed by Kleinrock and Tobagi has been widely used, although it provides a loose lower bound when nodes are distributed in a large area because the analysis assumes that the propagation delay between each pair of users equals to the largest propagation delay in the network. We present a throughput analysis which considers the spatial distribution of nodes. We obtain a simple throughput expression which predicts throughput with an 8% maximum error whereas the earlier model results in a 44% error when the maximum propagation delay equals to the packet transmission time

Karasan, E.

Koseoglu, M.

IEEE Communications Letters

Throughput model for non-persistent CSMA networks which was proposed by Kleinrock and Tobagi has been widely used, although it provides a loose lower bound when nodes are distributed in a large area because the analysis assumes that the propagation delay between each pair of users equals to the largest propagation delay in the network. We present a throughput analysis which considers the spatial distribution of nodes. We obtain a simple throughput expression which predicts throughput with an 8% maximum error whereas the earlier model results in a 44% error when the maximum propagation delay equals to the packet transmission time. © 2014 IEEE

Bilkent University Institutional Repository

564 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 4, APRIL 2014Spatio-Temporal Analysis of Throughput forSingle-Hop CSMA NetworksMehmet Koseoglu and Ezhan Karasan, Member, IEEEAbstract—Throughput model for non-persistent CSMA net-works which was proposed by Kleinrock and Tobagi has beenwidely used, although it provides a loose lower bound when nodesare distributed in a large area because the analysis assumesthat the propagation delay between each pair of users equalsto the largest propagation delay in the network. We present athroughput analysis which considers the spatial distribution ofnodes. We obtain a simple throughput expression which predictsthroughput with an 8% maximum error whereas the earliermodel results in a 44% error when the maximum propagationdelay equals to the packet transmission time.Index Terms—Spatio-temporal analysis, single-hop CSMA.I. INTRODUCTIONA CSMA network with non-negligible propagation delay issensitive to spatio-temporal effects because the locationsof the nodes affect their time of exposure to the transmissionsof other nodes in the network. In the seminal analysis of non-persistent CSMA [1], these effects are ignored by assumingthat the propagation delay between each node in the networkequals to the largest propagation delay in the network. Thisassumption results in a loose lower bound for the networkthroughput [2]. We here propose a novel analysis of non-persistent CSMA without ignoring the spatial distribution ofthe nodes, which predicts the throughput much more accu-rately.The model proposed in [1] is still frequently referred in thestudies on the performance analysis of terrestrial networks [3]–[7] and on the performance analysis of underwater networkswhere the propagation delay is much higher than the terrestrialnetworks [8]. Despite the model’s widespread usage, it is notaccurate for scenarios where the nodes are distributed overa large area. We propose an exact model of non-persistentCSMA and, then, an approximate simpler model which resultsin an 8% estimation error whereas the model in [1] resultsin a 44% error when the maximum propagation delay in thenetwork equals to the packet transmission time.The rest of the letter is organized as follows: The nextsection describes the system model that we study, we explainthe main idea behind our analysis in Sec. III. Derivation of thethroughput expression is presented in Sec. IV and numericalresults are given in Sec. V.II. SCENARIO DESCRIPTIONWe consider uniformly distributed nodes over a circulararea which are transmitting to the base station located at theManuscript received October 11, 2013. The associate editor coordinatingthe review of this letter and approving it for publication was P. Serrano.The authors are with the Department of Electrical and Electronics En-gineering, Bilkent University, TR-06800, Bilkent, Ankara, Turkey (e-mail:{kmehmet, ezhan}@ee.bilkent.edu.tr).Digital Object Identifier 10.1109/LCOMM.2014.020414.132292center. We assume that a large number of nodes are spatiallydistributed according to a homogeneous Poisson point process.We only consider the traffic from the nodes to the base station,and we assume that the nodes always have a packet to send.We do not consider retransmissions; that is, each node hasalways a single packet to send. Nodes attempt to transmitaccording to the Poisson distribution where λ denotes the totalattempt rate in the network. We also assume that consecutivetransmissions are independent, i.e., we analyze a transmissionin isolation from a previous transmission.The largest propagation delay normalized to the packettransmission time is denoted as T ; hence a node can havea maximum normalized propagation delay of T/2 to the basestation. A node ni is described by its polar coordinates (ri, θi)where ri ≤ T2 is the propagation delay between ni and thebase station, such that all distances are expressed in terms ofnormalized propagation delay.III. MAIN IDEAThe main difference between the proposed analysis ofCSMA and the traditional analysis of CSMA [1] is that weconsider the spatial distribution of nodes in our analysis. Dueto the spatial distribution of nodes, some nodes may stayexposed to a finished transmission for a longer time than othernodes or may be exposed to a starting transmission earlierthan other nodes. This results in a time-varying packet arrivalrate at the base station different than the time-homogeneousarrival rate assumption used in [1]. In this part, we explainthis phenomenon.A. Decreasing arrival rate after the start of a transmissionIn the CSMA protocol, at the beginning of a transmission,the transmission undergoes a vulnerable period of T duringwhich a colliding transmission may arrive. In the traditionalanalyses of CSMA, it is assumed that colliding transmissionsarrive at a fixed rate during this vulnerable period. However, inpractice, as time progresses within the vulnerable period, thearrival rate of colliding transmissions reduces as more nodesare exposed to the started transmission.A sample network where node n0 has started a transmissionis depicted in Fig. 1 and the timeline of the transmission isdepicted in Fig. 2. Assume n0 has started a transmission att = t0. Node n1 becomes exposed to this transmission att = t0 + d(n0, n1), where d(n0, n1) is the temporal distancebetween nodes n0 and n1. After t = t0 + d(n0, n1), node n1cannot start a transmission since it now senses the channelbusy. A transmission which is started by n1 just before itbecomes exposed, will arrive to the base station at time t =t0 + d(n0, n1) + r1. Since the transmission of n0 will arrive1089-7798/14$31.00 c© 2014 IEEEKOSEOGLU and KARASAN: SPATIO-TEMPORAL ANALYSIS OF THROUGHPUT FOR SINGLE-HOP CSMA NETWORKS 565n1:(r1, θ1)r0-d(n0,n1)r1d(n0,n1)n0:(r0, θ0)d(n0,n1)APFig. 1. A sample single-hop network scenario with marked distances. Grayedcircle denotes the exposure region for the transmission by node n0.t0+d(n0,n1) t0+r0 t0+r1+d(n0,n1)n0n1APr1+d(n0,n1)-r0t0r1+d(n0,n1)-r0Fig. 2. Timeline for a transmission by n0. Time difference between thearrival of n0’s transmission at the base station and the arrival of the latestpossible transmission from n1 is r1+d(n0, n1)−r0. Similarly, transmissionof n1 cannot arrive at the base station for a duration of r1 + d(n0, n1)− r0after the transmission of n0 is fully received at the base station.to the base station at time t = t0 + r0, n0’s transmissionstays vulnerable to n1’s transmission for a duration of (r1 +d(n0, n1)− r0) at the beginning.We define λstn(r1,θ1)→n0(t) as the rate of colliding transmis-sions from nodes located at (r1, θ1) to the transmission of n0where t denotes the time passed after the transmission of n0arrives to the base station. The superscript st is an abbreviationof start indicating the arrival rate after the transmission startsat the base station. Since nodes are uniformly distributedwithin the circular region, the attempt rate of nodes located at(r1, θ1) is λπT 2/4dA where dA = r1dr1dθ1. The latest timethat n0’s transmission can experience a collision from a nodelocated at (r1, θ1) is (r1 + d(n0, n1) − r0), so, the rate ofcolliding arrivals can be written as:(1)λstn(r1,θ1) →n0(t)={4λπT 2 r1dr1dθ1 if t < (r1 + d(n0, n1)− r0)0 if t > (r1 + d(n0, n1)− r0)whered(n0, n1) =√r20 + r21 − 2r0r1 cos(θ0 − θ1). (2)Total rate of colliding transmissions from all nodes after atransmission by node n0 is obtained by integrating over theentire circle:λst∪in(ri,θi)→n0(t) =∫ 2π0∫ T20λstn(r1,θ1)→n0(t). (3)To obtain the average arrival rate of colliding transmissionsafter a transmission by an arbitrary node in the network,λst(t), we take the average over all nodes, n0. Since eachnode has an equal probability of starting a new transmission,we obtainλst(t) =∫ 2π0∫ T204πT 2λst∪in(ri,θi)→n0(t)r0dr0dθ0. (4)B. Increasing arrival rate after the end of a transmissionA similar phenomenon occurs after the end of a trans-mission. A transmission from n1(r1, θ1) cannot arrive for aduration of (r1 + d(n0, n1)− r0) after the transmission of n0is fully received at the base station as depicted in Fig. 2.We define λendn(r1,θ1)→n0(t) as the rate of arrivals from nodeslocated at (r1, θ1) after the transmission of n0 finishes wheret denotes the time elapsed after transmission of n0 finishes atthe base station. This arrival rate is given by:(5)λendn(r1,θ1) →n0(t)={0 if t < (r1 + d(n0, n1)− r0)4λπT 2 r1dr1dθ1 if t > (r1 + d(n0, n1)− r0)The arrival rate of starting transmissions from all nodes in thenetwork is given by:λend∪in(ri,θi)→n0(t) =∫ 2π0∫ T20λendn(r1,θ1)→n0(t). (6)and the average arrival rate distribution can be found as:λend(t) =∫ 2π0∫ T204πT 2λend∪in(ri,θi)→n0(t)r0dr0dθ0. (7)IV. THROUGHPUT COMPUTATIONA. Linear approximation of the arrival rate function.Closed form expressions for the arrival rate functions givenby (4) and (7) cannot be obtained. However, simulation resultsand numerical computation of these arrival rate functionsdemonstrate that these functions can be approximated by thefollowing linear functions:λst(t) ={λT−tT if t ≤ T ,0 if t > T .(8)λend(t) ={λ tT if t ≤ T ,λ if t > T .(9)Fig. 3 depicts the numerical computations along with thesimulation results for a 1000-node network and the linearapproximation for λ = 0.2 and λ = 1 at T = 0.9.Numerical computations approximate the arrival rate functionsvery closely and the linear approximation provides a closeestimation while allowing mathematical tractability.Reduction in the accuracy at λ = 1 is due to increaseddependency between consecutive transmissions at a higher λ.566 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 4, APRIL 20140 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91λst for λ=0.2λend for λ=0.2λst for λ=1 λend for λ=1TimeArrival rate  Linear approx.Numerical computationSimulationFig. 3. Comparison of the linear approximation against the numericalsolutions of (4) and (7) and simulation results λ = 0.2 and λ = 1 at T = 0.9.If a new transmission starts immediately after a previous trans-mission, some of the nodes in the network are already exposedto the previous transmission which results in a reduced arrivalrate of colliding transmissions for the new transmission. Thiseffect is not captured in our model which assumes consecutivetransmissions are independent.B. ThroughputThroughput of a CSMA network can be analyzed byinspecting the packet receptions at the base station. Fig. 4demonstrates a series of packet receptions at the base station.The timeline consists of renewal intervals where each renewalinterval corresponds to the time between the start of consec-utive transmissions. The busy period of the renewal intervalincludes either a single successful transmission or a number ofcollided transmissions. If the busy period consists of a singletransmission, its length is one. If a colliding transmissionarrives after the start of the busy period, the length of thebusy interval gets longer depending on the arrival time of thecolliding transmission. The idle period in the renewal intervalis the time required for a new transmission to arrive after theend of a busy period. Hence, the throughput of the CSMAnetwork can be written asσTCSMA =PsE[tbusy ] + E[tidle](10)where tbusy and tidle denote the lengths of the busy andidle periods, respectively and Ps represents the probability ofsuccess for the busy period, i.e., the probability that a singlepacket arrives in the busy period.To derive the probability distribution of lengths of theseperiods, we refer to a branch of statistics called survival orreliability analysis [9]. The varying arrival rate of a collidingtransmission is analogous to the hazard rate in this literatureand probability of success is analogous to the survival proba-bility up to the duration of a packet.1) Probability of success: For a given hazard rate function,the probability of survival up to time t is analogous to theprobability of no collisions up to t, which is written as [9]:Sst(t) = exp{−∫ t0λst(t)dt}. (11)Busy period Idle periodRenewal intervalBusyperiod Idle periodRenewal interval1 xxFig. 4. Illustration of renewal intervals and busy periods. The length ofthe busy period is 1 + X where X is the arrival time of the last collidingtransmission.Using (8), the approximate probability of no collision duringthe vulnerable period can be obtained as follows:Ps = Sst(T ) = exp{−∫ T0λst(t)dt} = e−T2 λ. (12)2) Expected duration of the idle period: For a givensurvival function, the expectation of life can be obtained as[9]μ =∫ ∞0S(t)dt. (13)which is analogous to the expected time till the arrival of anew transmission. Using (9), the survival function is given by(14)Send(t) = exp{−∫ t0λend(τ)dτ}={exp{−λt22T } if t ≤ T ,exp{−(t− T2 )λ} if t > T .Then, the mean duration of the idle period can be found as(15)E[tidle] =∫ ∞0Send(t)dt=√πT2λerf(√λT2)+e−λT2λ.3) Expected duration of the busy period: Duration of thebusy period is one for successful transmissions which occurswith probability Ps. If one or more colliding transmissionsarrives in the vulnerable period, then the length of the busyperiod is 1+X where X is the arrival time of the last collidingtransmission as it can be seen from Fig. 4.To obtain E[X ], we first find the probability of no arrivalsoccurring between t and T .FX(t) = exp{−∫ Ttλst(τ)dτ} = e−λt2−2λtT+λT22T (16)Then the expected value of X can be found as:E[X ] =∫ T0(1− FX(t))dt (17)The expected length of the busy period is given by:(18)E[tbusy] = 1 + E[X ]= 1 + T −√πT2λerf(√λT2)KOSEOGLU and KARASAN: SPATIO-TEMPORAL ANALYSIS OF THROUGHPUT FOR SINGLE-HOP CSMA NETWORKS 5670 0.5 1 1.5 200.050.10.150.20.250.30.35Total attempt rate, λThroughput  SimulationNumerical IntegrationLinear ApproximationKleinrock−Tobagi ModelFig. 5. Throughput as a function of attempt rate in the network for T = 1.4) Throughput of CSMA: Using (12), (15) and (18), we canobtain the throughput expression given by (10) as follows:σTCSMA =λe−λT2λ(T + 1) + e−λT2(19)It should be noted that the proposed throughput function hasthe same form with the expression in [1] but the maximumdelay is replaced by half of the maximum delay.V. NUMERICAL RESULTSIn this section, we evaluate the proposed throughput analy-sis against simulation results. Fig. 5 depicts the throughput ver-sus the total attempt rate in the network for T = 1 and Fig. 6plots the change in the maximum throughput (both in absolutevalue and percentage error with respect to simulations) asT increases. Both figures include the throughput obtainedby exact computation of λst and λend by using numericalintegration in (4) and (7), throughput expression obtained bythe linear approximation as given by (19) and the throughputexpression given in [1]. The discrepancy between the proposedmodels and the simulated throughput increases as the totalattempt rate increases since this increase results in an increaseddependency between consecutive transmissions as discussedin Sec. IV-A. The estimation errors for all three models alsoincrease as T increases. The numerical integration provides theclosest result to the simulation as it estimates the maximumthroughput with a 3% error. The linear approximation ofthe arrival rate function results in a 8% underestimationwhereas the model given in [1] underestimates the maximumthroughput with a 44% error.VI. CONCLUSIONWe proposed an analysis of the throughput for a non-persistent CSMA network in which we incorporate the spatio-temporal effects caused by large propagation delay by mod-eling the arrival rate of colliding transmissions after the startof a transmission and after the end of a transmission. Wecomputed the throughput using an exact computation and withan approximate model which results in the simple throughput0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.10.20.30.40.50.6Maximum ThrougputPropagation Delay, T  01020304050Percentage ErrorSimulationNumerical IntegrationLinear ApproximationKleinrock−Tobagi ModelFig. 6. The maximum throughput and the percentage error as T increases.Dashed lines indicate the percentage error with respect to simulations.expression (19). In estimating the maximum throughput, themodel with numerical integration deviates from simulationresults with 3% error and the linear approximation modelresults in an 8% error whereas the Kleinrock-Tobagi modelresults in an error of 44% when the maximum propagationdelay equals to the packet transmission time. The simpleexpression given by (19) can be used for accurate estimationof throughput for CSMA networks in emerging applicationswith large propagation delays.ACKNOWLEDGMENTThis work was supported by the European Commission inthe framework of the FP7 Network of Excellence in WirelessCommunications NEWCOM#.REFERENCES[1] L. Kleinrock and F. Tobagi, “Packet switching in radio channels—part I: carrier sense multiple-access modes and their throughput-delaycharacteristics,” IEEE Trans. Commun., vol. 23, no. 12, pp. 1400–1416,Dec. 1975.[2] R. Rom and M. Sidi, “Multiple access protocols: performance andanalysis,” 1990.[3] D. Chan, T. Berger, and L. Tong, “Carrier sense multiple access commu-nications on multipacket reception channels: theory and applications toIEEE 802.11 wireless networks,” IEEE Trans. Commun., vol. 61, no. 1,pp. 266–278, 2013.[4] B. Nardelli and E. Knightly, “Closed-form throughput expressions forCSMA networks with collisions and hidden terminals,” in Proc. 2012IEEE INFOCOM, pp. 2309–2317.[5] L. Dai, “Toward a coherent theory of CSMA and Aloha,” IEEE Trans.Wireless Commun., vol. 12, no. 7, pp. 3428–3444, 2013.[6] M. Koseoglu and E. Karasan, “Throughput modeling of single hop CSMAnetworks with non-negligible propagation delay,” IEEE Trans. Commun.,vol. 61, no. 7, pp. 2911–2923, 2013.[7] R.-H. Gau, “Modeling the slotted nonpersistent CSMA protocol forwireless access networks with multiple packet reception,” IEEE Commun.Lett., vol. 13, no. 10, pp. 797–799, 2009.[8] Y. Zhu, Z. Jiang, Z. Peng, M. Zuba, J.-H. Cui, and H. Chen, “Towardpractical MAC design for underwater acoustic networks,” in Proc. 2013IEEE INFOCOM, pp. 683–691.[9] G. Rodrı́guez, “Lecture notes on generalized linear models,” 2007.Available: http://data.princeton.edu/wws509/notes/.

Spatio-temporal analysis of throughput for single-hop CSMA networks

564 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 4, APRIL 2014Spatio-Temporal Analysis of Throughput forSingle-Hop CSMA NetworksMehmet Koseoglu and Ezhan Karasan, Member, IEEEAbstract—Throughput model for non-persistent CSMA net-works which was proposed by Kleinrock and Tobagi has beenwidely used, although it provides a loose lower bound when nodesare distributed in a large area because the analysis assumesthat the propagation delay between each pair of users equalsto the largest propagation delay in the network. We present athroughput analysis which considers the spatial distribution ofnodes. We obtain a simple throughput expression which predictsthroughput with an 8% maximum error whereas the earliermodel results in a 44% error when the maximum propagationdelay equals to the packet transmission time.Index Terms—Spatio-temporal analysis, single-hop CSMA.I. INTRODUCTIONA CSMA network with non-negligible propagation delay issensitive to spatio-temporal effects because the locationsof the nodes affect their time of exposure to the transmissionsof other nodes in the network. In the seminal analysis of non-persistent CSMA [1], these effects are ignored by assumingthat the propagation delay between each node in the networkequals to the largest propagation delay in the network. Thisassumption results in a loose lower bound for the networkthroughput [2]. We here propose a novel analysis of non-persistent CSMA without ignoring the spatial distribution ofthe nodes, which predicts the throughput much more accu-rately.The model proposed in [1] is still frequently referred in thestudies on the performance analysis of terrestrial networks [3]–[7] and on the performance analysis of underwater networkswhere the propagation delay is much higher than the terrestrialnetworks [8]. Despite the model’s widespread usage, it is notaccurate for scenarios where the nodes are distributed overa large area. We propose an exact model of non-persistentCSMA and, then, an approximate simpler model which resultsin an 8% estimation error whereas the model in [1] resultsin a 44% error when the maximum propagation delay in thenetwork equals to the packet transmission time.The rest of the letter is organized as follows: The nextsection describes the system model that we study, we explainthe main idea behind our analysis in Sec. III. Derivation of thethroughput expression is presented in Sec. IV and numericalresults are given in Sec. V.II. SCENARIO DESCRIPTIONWe consider uniformly distributed nodes over a circulararea which are transmitting to the base station located at theManuscript received October 11, 2013. The associate editor coordinatingthe review of this letter and approving it for publication was P. Serrano.The authors are with the Department of Electrical and Electronics En-gineering, Bilkent University, TR-06800, Bilkent, Ankara, Turkey (e-mail:{kmehmet, ezhan}@ee.bilkent.edu.tr).Digital Object Identifier 10.1109/LCOMM.2014.020414.132292center. We assume that a large number of nodes are spatiallydistributed according to a homogeneous Poisson point process.We only consider the traffic from the nodes to the base station,and we assume that the nodes always have a packet to send.We do not consider retransmissions; that is, each node hasalways a single packet to send. Nodes attempt to transmitaccording to the Poisson distribution where λ denotes the totalattempt rate in the network. We also assume that consecutivetransmissions are independent, i.e., we analyze a transmissionin isolation from a previous transmission.The largest propagation delay normalized to the packettransmission time is denoted as T ; hence a node can havea maximum normalized propagation delay of T/2 to the basestation. A node ni is described by its polar coordinates (ri, θi)where ri ≤ T2 is the propagation delay between ni and thebase station, such that all distances are expressed in terms ofnormalized propagation delay.III. MAIN IDEAThe main difference between the proposed analysis ofCSMA and the traditional analysis of CSMA [1] is that weconsider the spatial distribution of nodes in our analysis. Dueto the spatial distribution of nodes, some nodes may stayexposed to a finished transmission for a longer time than othernodes or may be exposed to a starting transmission earlierthan other nodes. This results in a time-varying packet arrivalrate at the base station different than the time-homogeneousarrival rate assumption used in [1]. In this part, we explainthis phenomenon.A. Decreasing arrival rate after the start of a transmissionIn the CSMA protocol, at the beginning of a transmission,the transmission undergoes a vulnerable period of T duringwhich a colliding transmission may arrive. In the traditionalanalyses of CSMA, it is assumed that colliding transmissionsarrive at a fixed rate during this vulnerable period. However, inpractice, as time progresses within the vulnerable period, thearrival rate of colliding transmissions reduces as more nodesare exposed to the started transmission.A sample network where node n0 has started a transmissionis depicted in Fig. 1 and the timeline of the transmission isdepicted in Fig. 2. Assume n0 has started a transmission att = t0. Node n1 becomes exposed to this transmission att = t0 + d(n0, n1), where d(n0, n1) is the temporal distancebetween nodes n0 and n1. After t = t0 + d(n0, n1), node n1cannot start a transmission since it now senses the channelbusy. A transmission which is started by n1 just before itbecomes exposed, will arrive to the base station at time t =t0 + d(n0, n1) + r1. Since the transmission of n0 will arrive1089-7798/14$31.00 c© 2014 IEEEKOSEOGLU and KARASAN: SPATIO-TEMPORAL ANALYSIS OF THROUGHPUT FOR SINGLE-HOP CSMA NETWORKS 565n1:(r1, θ1)r0-d(n0,n1)r1d(n0,n1)n0:(r0, θ0)d(n0,n1)APFig. 1. A sample single-hop network scenario with marked distances. Grayedcircle denotes the exposure region for the transmission by node n0.t0+d(n0,n1) t0+r0 t0+r1+d(n0,n1)n0n1APr1+d(n0,n1)-r0t0r1+d(n0,n1)-r0Fig. 2. Timeline for a transmission by n0. Time difference between thearrival of n0’s transmission at the base station and the arrival of the latestpossible transmission from n1 is r1+d(n0, n1)−r0. Similarly, transmissionof n1 cannot arrive at the base station for a duration of r1+ d(n0, n1)− r0after the transmission of n0 is fully received at the base station.to the base station at time t = t0 + r0, n0’s transmissionstays vulnerable to n1’s transmission for a duration of (r1 +d(n0, n1)− r0) at the beginning.We define λstn(r1,θ1)→n0(t) as the rate of colliding transmis-sions from nodes located at (r1, θ1) to the transmission of n0where t denotes the time passed after the transmission of n0arrives to the base station. The superscript st is an abbreviationof start indicating the arrival rate after the transmission startsat the base station. Since nodes are uniformly distributedwithin the circular region, the attempt rate of nodes located at(r1, θ1) is λπT 2/4dA where dA = r1dr1dθ1. The latest timethat n0’s transmission can experience a collision from a nodelocated at (r1, θ1) is (r1 + d(n0, n1) − r0), so, the rate ofcolliding arrivals can be written as:(1)λstn(r1,θ1) →n0(t)={4λπT 2 r1dr1dθ1 if t < (r1 + d(n0, n1)− r0)0 if t > (r1 + d(n0, n1)− r0)whered(n0, n1) =√r20 + r21 − 2r0r1 cos(θ0 − θ1). (2)Total rate of colliding transmissions from all nodes after atransmission by node n0 is obtained by integrating over theentire circle:λst∪in(ri,θi)→n0(t) =∫ 2π0∫ T20λstn(r1,θ1)→n0(t). (3)To obtain the average arrival rate of colliding transmissionsafter a transmission by an arbitrary node in the network,λst(t), we take the average over all nodes, n0. Since eachnode has an equal probability of starting a new transmission,we obtainλst(t) =∫ 2π0∫ T204πT 2λst∪in(ri,θi)→n0(t)r0dr0dθ0. (4)B. Increasing arrival rate after the end of a transmissionA similar phenomenon occurs after the end of a trans-mission. A transmission from n1(r1, θ1) cannot arrive for aduration of (r1 + d(n0, n1)− r0) after the transmission of n0is fully received at the base station as depicted in Fig. 2.We define λendn(r1,θ1)→n0(t) as the rate of arrivals from nodeslocated at (r1, θ1) after the transmission of n0 finishes wheret denotes the time elapsed after transmission of n0 finishes atthe base station. This arrival rate is given by:(5)λendn(r1,θ1) →n0(t)={0 if t < (r1 + d(n0, n1)− r0)4λπT 2 r1dr1dθ1 if t > (r1 + d(n0, n1)− r0)The arrival rate of starting transmissions from all nodes in thenetwork is given by:λend∪in(ri,θi)→n0(t) =∫ 2π0∫ T20λendn(r1,θ1)→n0(t). (6)and the average arrival rate distribution can be found as:λend(t) =∫ 2π0∫ T204πT 2λend∪in(ri,θi)→n0(t)r0dr0dθ0. (7)IV. THROUGHPUT COMPUTATIONA. Linear approximation of the arrival rate function.Closed form expressions for the arrival rate functions givenby (4) and (7) cannot be obtained. However, simulation resultsand numerical computation of these arrival rate functionsdemonstrate that these functions can be approximated by thefollowing linear functions:λst(t) ={λT−tT if t ≤ T ,0 if t > T .(8)λend(t) ={λ tT if t ≤ T ,λ if t > T .(9)Fig. 3 depicts the numerical computations along with thesimulation results for a 1000-node network and the linearapproximation for λ = 0.2 and λ = 1 at T = 0.9.Numerical computations approximate the arrival rate functionsvery closely and the linear approximation provides a closeestimation while allowing mathematical tractability.Reduction in the accuracy at λ = 1 is due to increaseddependency between consecutive transmissions at a higher λ.566 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 4, APRIL 20140 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91λst for λ=0.2λend for λ=0.2λst for λ=1 λend for λ=1TimeArrival rate  Linear approx.Numerical computationSimulationFig. 3. Comparison of the linear approximation against the numericalsolutions of (4) and (7) and simulation results λ = 0.2 and λ = 1 at T = 0.9.If a new transmission starts immediately after a previous trans-mission, some of the nodes in the network are already exposedto the previous transmission which results in a reduced arrivalrate of colliding transmissions for the new transmission. Thiseffect is not captured in our model which assumes consecutivetransmissions are independent.B. ThroughputThroughput of a CSMA network can be analyzed byinspecting the packet receptions at the base station. Fig. 4demonstrates a series of packet receptions at the base station.The timeline consists of renewal intervals where each renewalinterval corresponds to the time between the start of consec-utive transmissions. The busy period of the renewal intervalincludes either a single successful transmission or a number ofcollided transmissions. If the busy period consists of a singletransmission, its length is one. If a colliding transmissionarrives after the start of the busy period, the length of thebusy interval gets longer depending on the arrival time of thecolliding transmission. The idle period in the renewal intervalis the time required for a new transmission to arrive after theend of a busy period. Hence, the throughput of the CSMAnetwork can be written asσTCSMA =PsE[tbusy ] + E[tidle](10)where tbusy and tidle denote the lengths of the busy andidle periods, respectively and Ps represents the probability ofsuccess for the busy period, i.e., the probability that a singlepacket arrives in the busy period.To derive the probability distribution of lengths of theseperiods, we refer to a branch of statistics called survival orreliability analysis [9]. The varying arrival rate of a collidingtransmission is analogous to the hazard rate in this literatureand probability of success is analogous to the survival proba-bility up to the duration of a packet.1) Probability of success: For a given hazard rate function,the probability of survival up to time t is analogous to theprobability of no collisions up to t, which is written as [9]:Sst(t) = exp{−∫ t0λst(t)dt}. (11)Busy period Idle periodRenewal intervalBusyperiod Idle periodRenewal interval1 xxFig. 4. Illustration of renewal intervals and busy periods. The length ofthe busy period is 1 + X where X is the arrival time of the last collidingtransmission.Using (8), the approximate probability of no collision duringthe vulnerable period can be obtained as follows:Ps = Sst(T ) = exp{−∫ T0λst(t)dt} = e−T2 λ. (12)2) Expected duration of the idle period: For a givensurvival function, the expectation of life can be obtained as[9]μ =∫ ∞0S(t)dt. (13)which is analogous to the expected time till the arrival of anew transmission. Using (9), the survival function is given by(14)Send(t) = exp{−∫ t0λend(τ)dτ}={exp{−λt22T } if t ≤ T ,exp{−(t− T2 )λ} if t > T .Then, the mean duration of the idle period can be found as(15)E[tidle] =∫ ∞0Send(t)dt=√πT2λerf(√λT2)+e−λT2λ.3) Expected duration of the busy period: Duration of thebusy period is one for successful transmissions which occurswith probability Ps. If one or more colliding transmissionsarrives in the vulnerable period, then the length of the busyperiod is 1+X where X is the arrival time of the last collidingtransmission as it can be seen from Fig. 4.To obtain E[X ], we first find the probability of no arrivalsoccurring between t and T .FX(t) = exp{−∫ Ttλst(τ)dτ} = e−λt2−2λtT+λT22T (16)Then the expected value of X can be found as:E[X ] =∫ T0(1− FX(t))dt (17)The expected length of the busy period is given by:(18)E[tbusy] = 1 + E[X ]= 1 + T −√πT2λerf(√λT2)KOSEOGLU and KARASAN: SPATIO-TEMPORAL ANALYSIS OF THROUGHPUT FOR SINGLE-HOP CSMA NETWORKS 5670 0.5 1 1.5 200.050.10.150.20.250.30.35Total attempt rate, λThroughput  SimulationNumerical IntegrationLinear ApproximationKleinrock−Tobagi ModelFig. 5. Throughput as a function of attempt rate in the network for T = 1.4) Throughput of CSMA: Using (12), (15) and (18), we canobtain the throughput expression given by (10) as follows:σTCSMA =λe−λT2λ(T + 1) + e−λT2(19)It should be noted that the proposed throughput function hasthe same form with the expression in [1] but the maximumdelay is replaced by half of the maximum delay.V. NUMERICAL RESULTSIn this section, we evaluate the proposed throughput analy-sis against simulation results. Fig. 5 depicts the throughput ver-sus the total attempt rate in the network for T = 1 and Fig. 6plots the change in the maximum throughput (both in absolutevalue and percentage error with respect to simulations) asT increases. Both figures include the throughput obtainedby exact computation of λst and λend by using numericalintegration in (4) and (7), throughput expression obtained bythe linear approximation as given by (19) and the throughputexpression given in [1]. The discrepancy between the proposedmodels and the simulated throughput increases as the totalattempt rate increases since this increase results in an increaseddependency between consecutive transmissions as discussedin Sec. IV-A. The estimation errors for all three models alsoincrease as T increases. The numerical integration provides theclosest result to the simulation as it estimates the maximumthroughput with a 3% error. The linear approximation ofthe arrival rate function results in a 8% underestimationwhereas the model given in [1] underestimates the maximumthroughput with a 44% error.VI. CONCLUSIONWe proposed an analysis of the throughput for a non-persistent CSMA network in which we incorporate the spatio-temporal effects caused by large propagation delay by mod-eling the arrival rate of colliding transmissions after the startof a transmission and after the end of a transmission. Wecomputed the throughput using an exact computation and withan approximate model which results in the simple throughput0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.10.20.30.40.50.6Maximum ThrougputPropagation Delay, T  01020304050Percentage ErrorSimulationNumerical IntegrationLinear ApproximationKleinrock−Tobagi ModelFig. 6. The maximum throughput and the percentage error as T increases.Dashed lines indicate the percentage error with respect to simulations.expression (19). In estimating the maximum throughput, themodel with numerical integration deviates from simulationresults with 3% error and the linear approximation modelresults in an 8% error whereas the Kleinrock-Tobagi modelresults in an error of 44% when the maximum propagationdelay equals to the packet transmission time. The simpleexpression given by (19) can be used for accurate estimationof throughput for CSMA networks in emerging applicationswith large propagation delays.ACKNOWLEDGMENTThis work was supported by the European Commission inthe framework of the FP7 Network of Excellence in WirelessCommunications NEWCOM#.REFERENCES[1] L. Kleinrock and F. Tobagi, “Packet switching in radio channels—part I: carrier sense multiple-access modes and their throughput-delaycharacteristics,” IEEE Trans. Commun., vol. 23, no. 12, pp. 1400–1416,Dec. 1975.[2] R. Rom and M. Sidi, “Multiple access protocols: performance andanalysis,” 1990.[3] D. Chan, T. Berger, and L. Tong, “Carrier sense multiple access commu-nications on multipacket reception channels: theory and applications toIEEE 802.11 wireless networks,” IEEE Trans. Commun., vol. 61, no. 1,pp. 266–278, 2013.[4] B. Nardelli and E. Knightly, “Closed-form throughput expressions forCSMA networks with collisions and hidden terminals,” in Proc. 2012IEEE INFOCOM, pp. 2309–2317.[5] L. Dai, “Toward a coherent theory of CSMA and Aloha,” IEEE Trans.Wireless Commun., vol. 12, no. 7, pp. 3428–3444, 2013.[6] M. Koseoglu and E. Karasan, “Throughput modeling of single hop CSMAnetworks with non-negligible propagation delay,” IEEE Trans. Commun.,vol. 61, no. 7, pp. 2911–2923, 2013.[7] R.-H. Gau, “Modeling the slotted nonpersistent CSMA protocol forwireless access networks with multiple packet reception,” IEEE Commun.Lett., vol. 13, no. 10, pp. 797–799, 2009.[8] Y. Zhu, Z. Jiang, Z. Peng, M. Zuba, J.-H. Cui, and H. Chen, “Towardpractical MAC design for underwater acoustic networks,” in Proc. 2013IEEE INFOCOM, pp. 683–691.[9] G. Rodrı´guez, “Lecture notes on generalized linear models,” 2007.Available: http://data.princeton.edu/wws509/notes/.

Spatio-Temporal Analysis of Throughput for Single-Hop CSMA Networks

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Spatio-Temporal Analysis of Throughput for Single-Hop CSMA Networks

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