Noname manuscript No.(will be inserted by the editor)

Performance Analysis of Unslotted CSMA/CA in WirelessNetworks

Ben Lauwens · Bart Scheers · Antoine Van de Capelle

the date of receipt and acceptance should be inserted later

Abstract In this paper a novel analytical model forthe saturation throughput of unslotted Carrier SensingMultiple Access with Collision Avoidance (CSMA/CA)in wireless networks is proposed. A fixed point proce-dure is developed based on the interaction of the Phys-ical layer (PHY) and the Medium Access Control sub-layer (MAC). The output of the Clear Channel Assess-ment (CCA), i.e. idle or busy medium in the neigh-borhood of a node, serves as a feedback mechanism forthe dynamical scheduling rate controlled by the back-offprocedure. The PHY is described by a renewal processbetween successful transmissions with failed attemptsand collided packets in between. A semi-Markov pro-cess of the internal states of a node is used as a modelfor the MAC. An event-driven simulator for the non-beacon enabled IEEE Std 802.15.4TMMAC is devel-oped to verify the numerical results of the analyticalmethod. A detailed analysis of the idle period after atransmission is carried out based on the proposed ana-lytical approach. The probability that the CCA sensesthe channel idle depends clearly on the actual back-offstage and the first back-off expiration after a transmis-sion cannot be modeled by a exponential distributionwhen a finite number of nodes are in contention. Theoutput of the event-driven simulations confirms both

Ben Lauwens, Bart ScheersRoyal Military Academy, CISS, 30 Renaissancelaan, B-1000 Brus-sels, BelgiumE-mail: {ben.lauwens,bart.scheers}@rma.ac.bePhone: +32 2 7426626Fax: +32 2 7426622

Antoine Van de CapelleKULeuven, ESAT-TELEMIC, 10 bus 2444 Kasteelpark Aren-berg, B-3001 Heverlee, BelgiumE-mail: antoine.vandecapelle@esat.kuleuven.bePhone: +32 16 321063Fax: +32 16 321986

statements in great detail and the saturated through-put evaluated with the analytical procedure is verifiedby event-driven simulations.

Keywords saturation throughput · wireless · unslot-ted CSMA/CA · IEEE Std 802.15.4TM · analyticalmodeling · semi-Markov process · event-driven simula-tion

1 Introduction

The use of wireless transmissions has known an enor-mous growth since the start of mobile telephony. Largescale wireless networks transporting data and voice arenowadays available. One of the disadvantages of theexisting single-hop wireless networks is the overhead ininfrastructure. Only transmissions between a terminaland a kind of base station or coordinator are allowed. Ina multi-hop network a terminal, called a node, can alsoplay the role of relay. Less infrastructure is needed butthe scheduling of transmissions can no longer be coordi-nated in a centralized way. A main characteristic of thedifferent types of wireless networks is the way how theycontrol the access to the medium. Several access mech-anisms are known, e.g. Time Division Multiple Access(TDMA), Frequency Division Multiple Access (FDMA)and Code Division Multiple Access (CDMA). In caseof a shared medium for which each node can trans-mit to every node in its transmission range, FDMA isdifficult to implement and CDMA needs a scheme todistributed the codes. TDMA is not possible withoutexternal synchronization, e.g. GPS signals or beacons.A new kind of medium access was born in 1970 withthe ALOHA protocol. The novel idea is twofold. Thedata is split into packets and the nodes self make theirown decision whether to transmit or not. However if

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two or more nodes transmit at the same time, a col-lision can occur and the payload of the packet getslost. These transmission failures can be remedied bya positive acknowledgment scheme (ACK), i.e. retrans-mitting the packet if no acknowledgment from the re-ceiver is heard by the transmitter. The effective datarate decreases due to the retransmission of the datapackets and the transmission of the ACK packets. Thenode can make a more intelligent decision by first sens-ing the shared medium before starting a transmission.With Carrier Sensing Multiple Access (CSMA) colli-sions are avoided by not transmitting if the medium issensed busy; collisions due to hidden nodes are howeverstill possible. The best known sibling of CSMA is Car-rier Sensing Multiple Access with Collision Detection(CSMA/CD) used by the IEEE Std 802.3TM-2005 [1].In a wireless medium this collision detection is not fea-sible. In order to further reduce the number of collisionsin a wireless channel a random back-off scheme can beimplemented to limit the probability that two or morenodes start transmitting at almost the same time. Thisis called Carrier Sensing Multiple Access with CollisionAvoidance (CSMA/CA). The CSMA family of proto-cols, including ALOHA, comes in two variants: slottedand unslotted. In slotted CSMA the time is dividedinto time intervals, called slots, and a transmission canonly start at a slot boundary. Timing information is in-cluded in the transmissions or beacons can be used tosynchronize the different nodes in the network. It canbe shown that the performance is better with slottedCSMA because the vulnerable period, i.e. the time in-terval in which two or more nodes can start to transmitsimultaneously, is reduced due to the alignment of thetime slots. In this paper however the throughput perfor-mance of unslotted CSMA/CA is discussed. The disad-vantage of the unslotted version is also its forte: the lackof synchronization. The overhead is minimized and themedium access control is completely decentralized. Asecond reason to model the unslotted CSMA/CA is theimpact of collisions due to hidden nodes in a multi-hopwireless network. Two transmitters that are hidden foreach other but having a mutual receiver will no longertransmit in a synchronized way and the vulnerable pe-riod will be the same for the slotted and the unslottedversion. The unslotted CSMA/CA is also easier to im-plement and the throughput can be higher due to lesssynchronization overhead.

The throughput of a node in a wireless network de-pends on the interaction between the Physical layer(PHY) and the Medium Access Control sub-layer (MAC)[3]. On the one hand, the Carrier Sensing Multiple Ac-cess / Collision Avoidance (CSMA/CA) functionalityof the MAC controls the attempt scheduling discipline

of the nodes to access the shared channel through aback-off procedure. On the other hand, the PHY sensesthe state of the channel and encodes the data to en-sure the correct reception of frames. The likelihood ofa successful reception depends on how well the channelcoding and the modulation defends against noise andinterference. The Clear Channel Assessment (CCA) ofthe PHY can be regarded as a feedback for the dynamicscheduling rate governed by the MAC. The connectiv-ity of the network is determined by the topology of thenetwork, i.e. the relative position of the nodes, and thesensitivity of the receiver. The idle probability of themedium as seen by the sensing mechanism of a nodedepends on this connectivity and the scheduling ratesof the adjoining nodes. With a contention based MACthe relationship between PHY and MAC is not triv-ial. Besides the collisions due to hidden nodes, theycan be caused by the finite propagation delay or bythe timing constraints to switch from receive mode totransmit mode and vice-versa. In a real-world imple-mentation the CCA is not a point process and takesa finite amount of time. The simulations and calcula-tions hereafter are modeled after the non-beacon en-abled IEEE Std 802.15.4TM-2006 [2] MAC, which usesunslotted non-persistent CSMA with a truncated ex-ponential back-off procedure. The different algorithmsand timings are respected as specified in the standard.The used techniques however are fairly general and caneasily be adapted to include other unslotted CSMA im-plementations.

In this paper two relations are proposed to modelthe basic interaction between the PHY and the MAC.Firstly, the channel occupation as seen by a node can beregarded as a renewal process between successful trans-missions with failed attempts in between. The successrate of an individual node is given as a function of themean back-off delay after a transmission and the prob-ability that this node breaks the idle period and ini-tiates the transmission. Secondly, a semi-Markov chainanalysis of the back-off algorithm gives the steady-stateback-off probabilities as a function of the probabilitythat the medium is sensed idle by the CCA. This idleprobability depends however on the back-off stage. Afixed point procedure can be applied to the completeset of non-linear equations, including the mean back-off delay and the idle probabilities. After convergence,the saturation throughput, i.e. each node has always apacket ready to transmit, can be calculated.

The following hypotheses are made in this paper tomake the analytical calculation tractable:

– the physical sensing range of each node equals thetransmission range;

– there are no power capture effects;

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– the communication channel is error-free;– the propagation delay is zero;– packets have a fixed length;– acknowledgments always arrive reliable and at not

cost;– the internal states of a node are ergodic and– the system is decoupled, i.e. the carrier sensing prob-

abilities of the nodes are mutually independent.

The first three restrictions make the resulting equa-tions tractable. A method similar to [3] can be usedto model a more realistic PHY with noise and fadingbut the collision probabilities are linearized with a lossin accuracy as a consequence. For moderate values ofthe fading parameters and a relatively low noise floorthe results of the event-simulations are not significantlydifferent compared to the simplified model. In this pa-per the analysis of the MAC is more important thanpossible PHY effects. The vulnerable period due to anon-zero propagation delay is much smaller than thevulnerable period due to the time interval needed toswitch between receive and transmit mode. The num-ber of collisions in the former case is negligible com-pared with the latter case and event-driven simulationsconfirm this hypothesis. The next assumption simplifiesthe MAC protocol and eliminates collisions with ACKs.The state space of both the renewal process and thesemi-Markov chain increases enormously. The ergodichypothesis is needed to use the steady-state probabil-ities of the internal states of a node as time averagefor the N nodes in contention. The decoupling featureis needed to model the internal state of a node inde-pendently of the internal states of the others nodes. In[4] a mean field analysis provides a theoretical justifi-cation in the case of a slotted CSMA with an explicitexponential back-off algorithm.

The main contributions in this papers are twofold.Firstly, a novel method is proposed to study the inter-action between the PHY and the MAC for unslottedCSMA/CA. The PHY is modeled as a renewal processbetween successful transmission and the MAC is de-scribed by a semi-Markov chain. The saturation through-put of the nodes in the network can be calculated. Themodel is also valid in a multi-hop scenario where thesaturation throughput of a node is highly dependenton its connectivity. Secondly, a detailed analysis of theback-off delay for the IEEE Std 802.15.4TM-2006 [2]unslotted MAC is provided, based on both simulationsand analytical expressions. The main results are thatthe output of the CCA depends on the back-off stageand that the use of an exponential distribution to modelthe idle period after a transmission is not appropriate.

The main differences with existing models are dueto the continuous time analysis needed to model the

non-persistent unslotted CSMA/CA used in the non-beacon enabled IEEE Std 802.15.4TM-2006 [2] whereasthe IEEE Std 802.11TM-2007 [15] uses a persistent slot-ted CSMA/CA:

– the MAC is described by a semi-Markov chain hav-ing states with variable duration instead of an ordi-nary Markov chain for which the states have all thesame duration;

– the conditional collision probability is replaced bythe conditional idle probability of a CCA to modelthe transitions between the states of the semi-Markovprocess;

– the state transition probability depends on the back-off stage and can be non-homogeneous between dif-ferent nodes whereas the existing models assumethat the state transition probability is identical forall back-off stages and for all nodes and

– the use of an exponential distribution to model theidle period after a transmission is not appropriateand the calculcation scheme uses a realistic descrip-tion of the MAC to evaluate the idle period after atransmission.

The non-homogeneousness of the state transition prob-ability allows a straightforward generalization to bothnodes with different back-off settings and to multi-hopnetworks for which the topology introduces a hetero-geneity between nodes. This is the first paper present-ing a realistic saturation throughput model for the non-beacon enabled IEEE Std 802.15.4TM-2006 [2] so thatthe results can only be verified by simulations.

This paper is organized as follows. In section 2 therelated work is discussed and the differences with theapproach outlined in this paper are commented. Thenon-beacon enabled version of the IEEE Std 802.15.4TM-2006 [2] is detailed by calculating the maximum date-rate in the case of one transmitter-receiver pair in sec-tion 3. To validate the novel analytical method an event-driven simulator is developed. Section 4 describes brieflythe implementation of the events and some graphs arepresented which give the maximum data-rate as a func-tion of the payload length in the case of one transmitter-receiver pair. Section 5 starts with a description of thechannel state as seen by a node. A renewal processbetween successful transmission attempts is proposed.Then the back-off algorithm based on a semi-Markovmodel is analyzed and the density function of the mini-mum back-off delay is established. The probability thatthe CCA sense the medium idle is evaluated for eachback-off stage. The section ends with a description ofthe fixed point procedure to calculate the saturationthroughput. Some graphs of the saturation through-put as a function of the number of competing nodes

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are shown for different settings of the back-off parame-ters. The analytical results are verified with the event-simulator. In section 6 a conclusion is formulated andfuture work is discussed.

2 Related work

The seminal work of Kleinrock and Tobagi, publishedin several parts [5,6,7,8], gives the basis for the per-formance analysis of contention based MAC protocols.Both the slotted and the unslotted version of ALOHAand the CSMA variants1 are detailed and analyzed.They assume that the traffic is generated by an infinitenumber of nodes who collectively form an independentPoisson source. All packets are of constant length andare transmitted over a noiseless channel. Another as-sumption states that the interarrival times of the pointprocess defined by the start time of all the packets plusretransmissions are independent and exponentially dis-tributed. Collisions are caused by a small, but non-zeropropagation delay which is identical for all transmitter-receiver pairs. Their analysis is based on renewal theoryand probabilistic arguments requiring the independenceof random variables. The performance is measured asthe maximum achievable throughput, called the capac-ity. The effect of acknowledgment traffic is analyzed in[9] by the same authors. The delay due to the acknowl-edgment scheme is studied but the turnaround time toswitch from a transmission mode to a reception modeand vice versa is considered to be negligible. Collisionswith an ACK packet are not modeled. In [10] Takagiand Kleinrock extend the previous works to include afinite number of nodes in the case of non-acknowledgedtraffic. Not only the mean transmission rate is calcu-lated but also the variance. This allows to determinethe coefficients in the diffusion process approximationof a node’s queue length distribution. The renewal pro-cess between successful transmissions in our paper isbased on the previous work. The vulnerable period isno longer the propagation delay but the much larger2

turnaround time needed to switch from a transmissionmode to a reception mode. The duration of the delaybetween two successive transmission is not modeled asan exponential distribution but evaluated by a semi-Markov process of the internal states of the nodes.

CSMA techniques applied to ground-based packetradio systems showed that there are many situationsin which some users cannot hear transmissions from

1 The different CSMA variants are known as 1-persistent, non-persistent and p-persistent. The protocols differ in their way ofhandling a busy channel.

2 In this paper the vulnerable period of 64 µs is large comparedto a typical propagation delay of 1 µs.

certain other users. Such a situation is called a hidden-terminal problem. Besides the basic introduction of thisphenomenon in [6], Takagi and Kleinrock give an ap-proximate output process in the case of hidden nodesin [11]. Not only collisions due to a non-zero propaga-tion delay are taken into account but also the probabil-ity that hidden nodes cause collisions is approximatelyevaluated. The main problem is the dependent behav-ior of nodes once any transmission in their neighbor-hood has started, i.e. a node stops initiations of trans-missions when a transmission is heard. A solution isproposed by Boorstyn, et al, in [12]. They consider aCSMA with perfect capture and use a Markov modelto develop a product form solution to efficiently analyzethe throughput of arbitrary topology multi-hop packetradio networks. In [13] the validity of the method isanalyzed. A product form solution is only accurate ifthe corresponding Markov chain is reversible and theCSMA protocol may not introduce a bias in the lengthstatistics of packets rescheduled because of collisions.The first assumption excludes an asymmetric channeland the second states that only fixed length packets canbe modeled. In the case of the IEEE Std 802.11TM-2003[15] the latter condition is always met using Request ToSend (RTS) and Clear To Send (CTS) messages withthe Distributed Coordination Function (DCF). The ef-fect of acknowledgment traffic is analyzed by Boorstyn,et al, in [14] and includes the possibility of collisionswith ACK messages. Our analysis of the scheduling rateis based on the view of a specific node, i.e. the methodcan easily be combined with the product form solutionof the Markov model for a multi-hop network with anarbitrary topology. Due to paper size limits this is notdiscussed in detail.

The analysis of real-world wireless systems has knowna breakthrough with the work of Bianchi [16]. He calcu-lated the saturation throughput of a single-hop IEEEStd 802.11TM[15] network when N nodes compete. Adecoupling strategy is applied and a single tagged nodeis analyzed. The state of a node is modeled as a Markovprocess based on the assumption that the conditionalcollision probability is the same regardless the numberof previous attempts. In [17] the saturation throughputis evaluated in non-saturated conditions for which thenumber of idle states is described by a geometrical dis-tribution. A linearization procedure is introduced in [3]to model a non-error free channel with fading effects.An overview of the fixed point analysis is presented in[18] which uses the decoupling approximation to evalu-ate the attempt rate per node as the long run averageback-off rate per slot. A complete analysis of the impactof hidden nodes on the DCF is given in [19] with an em-phasis on the starvation effects. In [20] the concept of a

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transmission link, a specific transmitter-receiver pair, isintroduced to evaluate the saturation throughput whichis highly directional in the case of a network with anarbitrary topology.

Several papers, e.g. [21,22,23], analyze the perfor-mance of the IEEE Std 802.15.4TM-2006 [2] with thebeacon enabled access mode, i.e. the slotted CSMA/CA.In most cases a similar Markov model as introduced byBianchi is proposed. However the conditional collisionprobability is replaced by the conditional idle proba-bility of a CCA and this probability is also the sameregardless the back-off stage. An extension to the multi-hop case has not yet been found. The excellent resultsof the previous papers, inspired several authors to ap-ply the same methodology to the non-beacon enabledaccess mode. In [24] an overview of the standard is pre-sented and the data rate of a single transmitter-receiverpair is calculated. In section 3 of our paper this datarate is reevaluated to include a real-world CCA whichis not a point process and similar results are obtained.The authors of [25] present a model for the non-beaconenabled mode in a star topology. Two successive CCAs,as for the beacon-enabled mode, are considered whichis not compliant to the IEEE Std 802.15.4TM-2006. Theback-off expiration after a transmission is given by anexponential distribution which is shown to be faulty inour paper. The same principal author proposes a math-ematical model based on the busy cycle of a M/G/1queueing system to analyze the throughput and the en-ergy consumption. The turnaround time is consideredto be smaller than the CCA so that collisions with ACKmessages are avoided which in reality do happen. Anon-stationary Markov model is used for which the at-tempt probabilities are time-dependent in [27]. The idleprobability of the CCA is considered to be independentof the back-off stage and the CCA is modeled as a pointprocess. In this paper a stationary semi-Markov processis proposed for which the idle probabilities of the CCAdepends on the back-off stage. The non-zero durationof the CCA is analyzed by a renewal process with anon-exponential distribution for the first back-off expi-ration after a transmission. Each node can be modeledindependently so that the analysis can be extended toa product form solution in the case of a multi-hop wire-less network.

3 Non-beacon enabled IEEE802.15.4

The IEEE Std 802.15.4TM-2006 [2] specifies the MACsub-layer and the PHY layer for Low-Rate Wireless Per-sonal Area Networks (LR-WPANs).

The PHY layer is responsible for the data transmis-sion and the reception using a certain radio channel and

Table 1 Frequency bands, data rates and modulation types ofIEEE802.15.4TM-2006 [2]

Frequency Spreading Parameters Data ParametersChip Rate Modulation Bit Rate Symbols

868 MHz 300 kchip/s BPSK 20 kb/s Binary915 MHz 600 kchip/s BPSK 40 kb/s Binary2400 MHz 2000 kchip/s O-QPSK 250 kb/s 16-ary

according to a specific modulation and spreading tech-nique. The IEEE802.15.4 offers three operational fre-quency bands and corresponding data rates (see Table1 ). A Direct Sequence Spread Spectrum (DSSS) tech-nique is used for all frequency bands. The PHY layeris also in charge of the activation and deactivation ofthe radio transceiver. The turnaround time from trans-mitting to receiving and vice versa should be no morethan 12 symbol periods according to the standard. ThePHY layer also performs the Clear Channel Assessment(CCA). The CCA reports the medium activity state:busy or idle. In this paper it is assumed without lossof generality that the medium is busy when the meanpower received during the CCA is above the Energy De-tection threshold (ED). This is one of the three meth-ods allowed by the IEEE Std 802.15.4TM-2006 [2] toperform the CCA.

The MAC sub-layer provides an interface betweenthe PHY layer and higher layer protocols. The MACsub-layer of IEEE802.15.4 uses Carrier Sense MultipleAccess / Contention Avoidance (CSMA/CA) as channelaccess protocol. The MAC protocol supports two oper-ational modes: beacon-enabled and non beacon-enabledmode. The latter is modeled in this paper. In the non-beacon enabled mode, the nodes can simply send theirdata by using unslotted CSMA/CA. All messages whichhave to be transmitted, with the exception of acknowl-edgment (ACK) frames, must be dispatched accordingto this mechanism. The use of ACK frames is optional.

The IEEE Std 802.15.4TM-2006 [2] supports a max-imum over-the-air data rate of 250 kbps in the 2.4 GHzband, the reference implementation used in this paper.In practice, the effective data rate is lower due to theframe structure and the timings given by the MAC andthe PHY.

3.1 Channel Access Timing

The unslotted CSMA/CA mechanism of the IEEE Std802.15.4TM-2006 [2] uses two variables to schedule theaccess to the medium:

NB is the number of times the CSMA/CA algorithmwas required to back-off while attempting to access

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Idle

NB > macMaxCSMABackoffs

NB = 0BE = macMinBE

Delay for random[0, 2BE − 1

]back-off unit periods

CCA

NB = NB + 1BE = min(BE + 1, aMaxBE)

Failure

CSMA/CA

Transmit

y

y

n

n

Fig. 1 The unslotted CSMA/CA channel access mechanism ofthe IEEE Std 802.15.4TM-2006 [2]

the current channel. This value is initialized to zerobefore each new transmission attempt.

BE is the back-off exponent, which is related to howmany back-off unit periods a node must wait beforeattempting to perform a CCA.

Figure 1 depicts a flowchart describing the unslottedversion of the CSMA/CA access protocol. It can besummarized in five steps:

1. Initialization of NB and BE: NB is initialized to 0and BE is set to macMinBE, which is equal to 3by default.

2. Random delay for collision avoidance: the algorithmattempts to avoid collisions by waiting during a givendelay randomly generated in the range

[0, 2BE − 1

]back-off unit periods. One back-off unit period equals20Ts with Ts = 16µs.

3. Clear channel assessment: the CCA starts immedi-ately after the expiration of the random delay. If thechannel is detected to be in a busy state, the algo-rithm goes to step 4, otherwise, i.e. the channel isidle, the algorithm goes to step 5.

4. Busy channel: the values of NB and BE are in-creased by one. However, BE cannot exceed mac-MaxBE, which has a default value of 5. If the num-ber of retries is less thanmacMaxCSMABackoffs,with a default value of 4, the algorithm returns tostep 2, otherwise the algorithm terminates with achannel access failure status.

5. Idle channel: the MAC sub-layer starts immediatelytransmitting its current frame just after the channelis assessed to be idle by the CCA procedure.

Initially the back-off exponentBE is set tomacMinBE.Using the default value of 3 formacMinBE and assum-

ing the channel is found to be free, the mean access timeof the first back-off stage can be calculated as:TACCESS = TB1 + TCCA + TRx→Tx = 1312µs (1)

with TB1 =0+(23−1)

2 20Ts and TCCA = 8Ts. TRx→Tx

is the turnaround time needed to switch from receivemode to transmit mode. The clear channel assessment isperformed during the turnaround time in the CC2420and CC2520 chipsets of Texas Instruments3. The useof the STXONCCA command strobe is advised in thenon-beacon enabled mode with the unslotted CSMA/CAchannel access mechanism and takes aTurnaroundT ime(= 12Ts) as specified by the IEEE Std 802.15.4TM-2006[2]. This behavior is simulated and modeled in this pa-per.

3.2 Frame Structure

The IEEE Std 802.15.4TM-2006 [2] specifies the maxi-mum number of bytes that can be transmitted in theMAC data payload (MSDU) as 114 byte using shortaddressing (16-bit instead of 64-bit addresses). Fig. 2gives a schematic view of a data frame. The PPDU iscomposed of the MPDU (127 byte), the PHR (1 byte)and the SHR (5 byte).

The maximal frame transfer time can be calculatedas:

TDATA =(LMPDU + LPHR + LSHR) 8

RPHY= 4256µs (2)

with RPHY = 250 kbps.

3.3 Acknowledged Transmission Timing

The use of acknowledgment frames is optional in theIEEE Std 802.15.4TM-2006 [2]. The ACK transmissiontime can be evaluated as:TACK =

LACK8RPHY

= 352µs (3)

where the ACK frame size equals 11 byte. The trans-mission of an ACK does not use CSMA/CA and startsaTurnaroundT ime after the reception of the last byteof the data frame.

To allow the received data to be processed, an in-ter frame space (IFS) must follow the reception of theACK frame by the initial transmitter respectively thetransmission of the DATA frame in case of an unac-knowledged transmission. For MPDUs with a length ofup to 18 byte, a short IFS (SIFS) of at least 12Ts iswaited, for larger MPDUs a long IFS (LIFS) of at least40Ts is introduced before starting a new transmission.When the back-off delay is in the range [0, 1] back-off

3 Former Chipcon.

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FCSDataPayload

AdressingFields

SequenceNumber

FrameControl

MPDUFrameLength

Start ofFrame

PreampleSequence

PPDU

MHR MFR

PSDUPHRSHR

2 1 4 to 20 LDATA 2MAC

Sublayer

PHYLayer

4 1 1 5+(4 to 20)+LDATA ≤127MSDU

Fig. 2 Schematic view of a data frame

unit periods a delay of 448µs respectively 128µs is in-troduced before the back-off interval. This increases themean access time after a successful transmission with72µs in the case of a maximal data payload:TACCESS = 1384µs . (4)

It is unlikely that a 0 % packet error rate (PER)will be achieved. The transmitting node will wait amacAckWaitDuration (=54Ts) for an acknowledgmentbefore it attempts a retry. The number of retries, mac-MaxFrameRetries (default value of 3), can be specified.

3.4 Data Rate of a Single Transmitter-Receiver Pair

The maximum effective data rate between a single trans-mitter receiver pair without acknowledgment can becalculated based on the following assumptions:

– CSMA/CA algorithm never finds that the channelis busy and

– there are no collision between packets,

i.e. the transmitter senses the channel always idle dur-ing the first back-off stage:

RDATA =LMSDU8

TACCESS + TDATA

= 161.7 kbps (5)

where LMSDU = 114 byte.With acknowledgment the previous assumptions are

extended:

– No retries are required.

RDATA,ACK =LMSDU8

TACCESS + TDATA + TRx→Tx + TACK

= 147.5 kbps (6)where LMSDU = 114 byte.

4 Simulator

4.1 Main Simulation Loop

A MAC event-driven simulator has been developed tovalidate the analytic results. It mimics with a high de-gree of precision the behavior of the IEEE Std 802.15.4TM-2006 [2] MAC in non-beacon enabled mode with theunslotted CSMA/CA procedure.

The different steps during the transmission and re-ception of a frame are implemented as events contain-ing a timestamp. An efficient binary heap is used as aqueue for future events. In the main simulation loop,the following sequence is repeated until the end of thesimulation: the event with the smallest timestamp ispopped out of the heap, the current simulation time,tcurrent, is updated and the event is executed. Duringthe execution of the event, new events can be pushedinto the heap.

The simulator is written in fortran 2003 [30] andthe object oriented features of this recent ISO standardare used to ease the implementation, e.g. events are im-plemented as fortran types. Both data and procedurescan be linked to a type and inheritance can be used toeliminate redundancy in the code.

4.2 Channel matrix

The channel is modeled by a matrix where the rowscorrespond to transmitting nodes and the columns toreceiving nodes. In each cell of this matrix the state ofthe transmitting node, the last modification time andthe received power are stored.

The propagation model used in section 5 is based ona fixed transmission range, which also equals the sens-ing range. The code is however pluggable and a modulewith a Rayleigh fading model with additive Gaussiannoise is also implemented. Distances are calculated from2D coordinates and the sensing range can be differentfrom the transmission range. For moderate values of thefading parameters and a relative low noise floor the re-sults are not significantly different than with the basicpropagation model.

4.3 Events

The unslotted version of the CSMA/CA MAC withACK can be implemented in only 5 events, which are al-ways linked to the node transmitting the DATA frame.Fig. 3 represents the event generation flow.

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CCA Event

DATA Transmit Event

DATA Receive Event

ACK Transmit Event

ACK Receive Event

Fig. 3 Schema depicting the event generation flow in the simu-lator.

4.3.1 CCA Event

The channel matrix is checked for an ongoing transmis-sion between [tcurrent − TCCA, tcurrent] by comparingthe mean received power to the ED. If the channel isidle a DATA Transmit Event is pushed into the heapto be scheduled at t = tcurrent + TRx→Tx, else BE andNB are updated, a new random back-off period is cho-sen and a CCA event is pushed into the heap with thetimestamp set to t = tcurrent + TB + TCCA.

4.3.2 DATA Transmit Event

The mean received power is calculated by the pluggablepropagation model. The state of the cells in the channelmatrix with the rows corresponding to the transmittingnode is updated and the modification time and receivedpower are changed. A DATA Receive Event is scheduledat time t = tcurrent + TDATA.

4.3.3 DATA Receive Event

The channel matrix is checked for simultaneous trans-missions heard by the receiver between [tcurrent−TCCA,

tcurrent]. The received power indicated in the cells ofthe channel matrix corresponding to the transmittingnode, is set to the noise level and the state is tog-gled. BE and NB are reset and a new random back-off period is chosen. Without the use of acknowledg-ment frames, a CCA Event with a new packet is pushedinto the heap with the timestamp set to t = tcurrent +max (TB + TCCA, tIFS). The packet is lost if a collisionhappens. When the ACK procedure is used, an ACK

Transmit Event is scheduled at time t = tcurrent +aTurnaroundT ime if no other transmission collides atthe receiver, else a CCA Event is pushed into the heapto be scheduled at t = tcurrent+macAckWaitDuration+TB + TCCA and the retry count is incremented. If theretry limit is exceeded, the packet is lost and a newpacket is created.

4.3.4 ACK Transmit Event

The mean received power is calculated by the pluggablepropagation model. The state of the cells in the channelmatrix with the rows corresponding to the node, whichis sending the ACK, is updated and the modificationtime and received power are changed. An ACK ReceiveEvent is scheduled at time t = tcurrent + TACK .

4.3.5 ACK Receive Event

The channel matrix is checked for simultaneous trans-missions heard by the receiver between [tcurrent−TCCA,

tcurrent]. The received power indicated in the cells of thechannel matrix corresponding to the transmitting node,is set to the noise level and the state is toggled. BE andNB are reset and a new random back-off period is cho-sen. If the ACK is successful received, a CCA Eventwith a new packet is pushed into the heap with thetimestamp set to t = tcurrent +max (TB + TCCA, tIFS),else a CCA Event is scheduled at time t = tcurrent −aTurnaroundT ime−TACK +macAckWaitDuration+TB + TCCA and the retry count is incremented. In thelatter case if the retry limit is exceeded, the packet islost and a new packet is created.

4.4 Data Rate of a Single Transmitter-Receiver Pair

To validate the simulator, the saturation data-rate asa function of the payload length is simulated and theresults are shown in figure 4. Each simulation run sim-ulates the behavior of the saturated node during 1000seconds. Both the scheme with acknowledgment andwithout acknowledgment are evaluated by equations5 and 6 for different settings of the back-off param-eters and the results are compared with the simula-tions. No anomalies can be observed. The bump atLDATA = 18byte is due to the IFS which changes fromSIFS to LIFS.

One run of the simulator takes less than 0.3 s CPUtime on a recent 3 GHz Intel processor and during thattime almost one million events are executed.

9

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120

S(LDATA

)[kbp

s]

LDATA [byte]

Without ACKWith ACK

Fig. 4 The saturation throughput of a single transmitter-receiver pair as a function of the length of the payload LDATA

5 Analytical Model of the Interaction betweenPHY and MAC

5.1 Non-persistent CSMA Renewal Process

Let{X(k); k ∈ N0

}be a sequence of packet interdepar-

ture times (from the entire system) beginning at the endof an arbitrarily chosen successful transmission (t = 0).These interdeparture times are independent and identi-cally distributed. If the CSMA protocol is memorylessthen

Z(n) =n∑

k=1

X(k) n ∈ N0 (7)

defines the time at which the nth successful transmis-sion completes. By definition

{Z(n);n ∈ N0

}is a re-

newal process; see for example [31]. For time t > 0, letD(t) be the number of successful transmissions com-pleted during an interval [0, t]:D(t) = max

Z(n)≤t(n) . (8)

Renewal theory states that

limt→∞

D(t)t

=1X

= S (9)

where S is the channel throughput expressed in packetsper second.

Let ∀i ∈ {1, . . . N} : Di(t) be the number of success-ful transmissions completed by node i and let di be theprobability that a successful transmission is achievedby node i;

∑Ni=1 di = 1. In the case where all the nodes

have the same back-off settings, di equals 1N . In the

general case the interdeparture times between success-ful transmission of the same node are not identicallydistributed although they are independent. In [10] it isshown that forDi(t) = max

Z(n)i ≤t

(n) (10)

the asymptotes still exist

limt→∞

Di(t)t

=1Xi

(11)

withDi(t) = diD(t) (12)and the throughput of an individual node is thenSi = diS (13)

5.2 Modeling the IEEE Std 802.15.4TM-2006CSMA/CA

Figure 5 illustrates how X is composed of j − 1 unsuc-cessful transmissions and one successful transmissionY (j). The channel throughput can then be evaluatedby

S =γ

TACCESS + TDATA + (1− γ)TCOL

(14)

where γ is the success probability of a transmission andTCOL is the mean transmission start delay of the lastcolliding packet in reference to the start of the transmis-sion breaking the channel idle period. The mean accessperiod TACCESS equals the sum of the delay betweenthe previous transmission and the first back-off timerexpiration, the time to evaluate a CCA and the time toswitch from receive mode to transmit mode:TACCESS = min

i(ti) + TCCA + TRx→Tx . (15)

To calculate the first term, the classic CSMA litera-ture [5] assumes that the interarrival times of the pointprocess defined by the CCA attempts are independentamong the different nodes and that they are all expo-nentially distributed. Simulations have shown howeverthat the back-off expiration times after a transmissionare mutually independent but not exponentially dis-tributed. In subsection 5.4 the exact density functionfi(t) of the point process is analytically established andcompared with the simulation results.

After a transmission, successful or failure, all nodeshave a packet ready to transmit due to the saturationscenario and are in a back-off state. Since the first back-off expiration after a transmission is the minimum of allnodes back-off delays, its distribution is given by

F (t) = = Pr(min

i(ti) ≤ t

)= 1−

N∏i=1

Pr (ti > t)

= 1−N∏

i=1

(1−ˆ t

0

fi (ti) dti

). (16)

The mean value of the minimum access delay TACCESS

becomes

TACCESS =ˆ +∞

0

tdF (t) + TCCA + TRx→Tx . (17)

10

Y (1)X(k)

Y (2) Y (j)

t

TACCESS TDAT A TCOL TCCA TRx→T x

t = 0

Fig. 5 Packet interarrival times X for the IEEE Std 802.15.4TM-2006 CSMA; CCA attempts are indicated by arrows and thevulnerable period is shaded, i.e. more than one CCA attempt during this interval causes a collision

The probability that node i among others breaksthe idle period can be calculated

νi = Pr(ti ≤ minj 6=i

tj) =ˆ +∞

0

fi(ti)ˆ +∞

ti

dF ∗(t)dti (18)

where F ∗(t) is the minimum back-off delay after a trans-mission of all nodes except the node i

F ∗(t) = 1−N∏

j=1;j 6=i

(1−ˆ t

0

fj (tj) dtj

). (19)

Collisions can occur if one or more nodes start aCCA after the CCA attempt of a first node and if theirCCAs ends before the start of the transmission of thatfirst node. The vulnerable period in which simultaneousCCAs can happen has a duration TRx→Tx after thestart of the first CCA. The probability of success in acycle where node i initiates the transmission period isgiven by

γi =ˆ +∞

0

fi(ti)ˆ +∞

ti+TRX→T x

dF ∗(t)dti (20)

and the probability that a transmission is successful, γ,can be evaluated

γ =N∑

i=1

νiγi . (21)

The time between the start of the first and thelast colliding transmission, TCOL, has a value in therange [0;TRx→Tx] which is small compared to the cy-cle length. An upper bound and a lower bound of theeffective data rates are obtained by setting respectivelyTCOL = 0 and TCOL = TRx→Tx.

5.3 Internal Semi-Markov Model

The internal states of a node can be described by asemi-Markov model. In [31] a semi-Markov process isdefined as a process with a finite number of states. Eachtime the process enters a state i it remains there for arandom amount of time with mean µi and then makesa transition into state j with probability Pij . Fig. 6depicts the semi-Markov chain for a node which doesnot use the acknowledgment scheme. Two logical statescan be distinguished:

– Back-off and Clear Channel Assessment state, notedby CBi where i indicates the back-off stage, and

– Transmitting and Inter Frame Space state, noted byT .

CB1

CB2

CBm

T

p1

p2

q1

q2

pm

qm

qm−1

Fig. 6 Internal Semi-Markov Chain of a node without ACK

Table 2 The idle probabilities of the different back-off stages inthe internal semi-Markov model without the use of acknowledg-ments

macMinBE 3 3 3 4 3 4

macMaxBe 5 5 5 5 3 4

m 5 5 5 5 5 5

N 3 5 10 5 5 5

LMSDU 114 114 114 114 114 114

p1 0.47 0.27 0.12 0.23 0.09 0.15

p2 0.19 0.14 0.10 0.20 0.06 0.14

p3 0.20 0.15 0.10 0.20 0.09 0.14

p4 0.20 0.15 0.10 0.20 0.10 0.14

p5 0.20 0.15 0.10 0.20 0.09 0.14

The transition probabilities are expressed as a functionof pi and qi = 1− pi, where i is the index of the previ-ous back-off stage. The probability pi is called the idleprobability of the ith back-off stage. Table 2 shows thecorresponding idle probabilities from the event simula-tion for different values of macMinBE, macMaxBE,macMaxCSMABackoffs (= m−1) and N , the num-ber of transmitting nodes. One of the main assump-tions in the literature of CSMA modeling states thatp1 = p2 = . . . = pm and that this value equals thesteady-state idle probability of the medium, PI . In thispaper the idle probabilities pi can be different for eachback-off stage. Figure 7 illustrates this behavior. For alarge number of nodes in contention the idle probabilityno longer depends on the back-off stage. The idle prob-

11

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2 4 6 8 10 12 14 16 18 20

pidle

N

p1p2p3p4p5

Fig. 7 Event-driven simulation of the idle probabilities for eachback-off stage with macMinBE = 3, macMaxBE = 5 andmacMaxCSMABackoffs = 4 as a function of the number ofnodes N

abilities for each back-off stage are analytically modeledin subsection 5.5. The figure shows clearly that the idleprobability of the first back-off stage differs significantlyfrom the other idle probabilities for a small number ofnodes N .

The transmission scheduling rate of a node is calcu-lated by first expressing the steady-state probabilitiesof the embedded Markov-Chain:πCBi+1 = qiπCBi

πT =m∑

i=1

piπCBi

1 =m∑

i=1

πCBi + πT (22)

with m = macMaxCSMABackoffs+ 1. The steady-state probabilities of the different states are then givenby

PCBi=

πCBiTCBi∑mi=1 πCBi

TCBi+ πTTT

PT =πTTT∑m

i=1 πCBiTCBi

+ πTTT(23)

where TCBi= TBi

+TCCA and TT = TRx→Tx+TDATA+TIFS . The scheduling rate can be evaluated consideringthe steady-state probability that a node enters a stateg =

πT∑mi=1 πCBi

TCBi+ πTTT

. (24)

5.4 Distribution of the Back-off Delays

After a successful transmission or a failure, all nodeshave a packet ready to transmit due to the saturationassumptions. The nodes involved in the previous trans-mission are in the inter frame space (IFS) state while

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 1000 2000 3000 4000 5000

fB

(t)[

1 µs]

t [µs]

BE = 3

BE = 4

BE = 5

Fig. 8 The density function fB (t) for BE = 3, BE = 4 andBE = 5 with TACCESS + TDATA = 5055

the others are in one of the back-off states. The steady-state probabilities of the internal node state can be ex-pressed as a function of the probabilities of the embed-ded Markov-Chain:PT =

πTTT∑mi=1 πCBi

TCBi+ πTTT

PB1 =(πCB1 − πT )TB1∑m

i=1 πCBiTCBi

+ πTTT

PBi=

πCBiTBi∑mi=1 πCBi

TCBi+ πTTT

; i 6= 1 . (25)

In case of back-off stage i, the back-off period has adiscrete uniform distribution with values in the range of[0, 2BEi − 1

]back-off unit periods. Considering the fact

that the node is in the specific back-off state, the densityfunction of the back-off delay after a transmission andbefore performing a CCA is then given by

fBi (t) =1

2BEi

2BEi−1∑j=0

χ (0 ≤ t ≤ j 20TS)(

1j 20TS

)(26)

where χ is the indicator function. If TACCESS+TDATA >

j 20TS the density function is truncated at TACCESS +TDATA and the remainder is added at t = 0. The partof the density function after the truncation attributesto the delay of the next CCA. Figure 8 shows the den-sity function of the back-off delay after a transmissionfor different values of BE when the node is in a back-offstate.

In case the node has just finished a transmission thedensity function is discretefT (t) =

n

2macMinBE − 1(27)

where n = 1 for values of t = 2i 20TS > TIFS−TCCA−TRx→Tx and n equals the number of times 2i 20TS ≤TIFS−TCCA−TRx→Tx for t = TIFS−TCCA−TRx→Tx.Figure 9 shows the density function of the back-off delay

12

0

0.05

0.1

0.15

0.2

0.25

0 1000 2000 3000 4000 5000

fT

(t)[

1 µs]

t [µs]

Fig. 9 The density function fT (t) for macMinBE = 3 withTACCESS + TDATA = 5055

1e-05

0.0001

0.001

0.01

0.1

0 1000 2000 3000 4000 5000

f(t

)[

1 µs]

t [µs]

Fig. 10 The weighted density function f (t) withmacMinBE =

3, macMaxBE = 5, macMaxCSMABackoffs = 4

after a transmission for macMinBE when the node isin an IFS state after the previous transmission.

By taking the weighted sum of all these functionsthe density function of the back-off delay of one nodeafter a transmission can be expressed as

f (t) = P ∗T fT (t) +m∑

i=1

P ∗BifBi (t) (28)

with

P ∗T =PT

PT +∑m

i=1 PBi

P ∗BI=

PBi

PT +∑m

i=1 PBi

.

Figure 10 shows the weighted density function which isa superposition of an almost exponential function witha discrete spectrum. Figure 11 gives the same graphobtained by simulation. The discrete part of the den-

1e-05

0.0001

0.001

0.01

0.1

0 1000 2000 3000 4000 5000

f(t

)[

1 µs]

t [µs]

Fig. 11 The simulated weighted density functionf (t) with macMinBE = 3, macMaxBE = 5,macMaxCSMABackoffs = 4

sity function is due to the first back-off stage after atransmission and is the main reason for the high valueof p1.

5.5 Calculation of the Idle Probabilities

Figure 7 illustrates the variance of the idle probabilityof each back-off stage pi as a function of the numberof nodes N . For small values of N the idle probabilityof the first back-off stage is significantly higher thanthe others. In the previous subsection it is shown bysimulation and analytical modeling that this behavioris caused by the discrete part of the weighted densityfunction.

The probability that a CCA attempt of a node whichhas not participated in the previous transmission, isidle, is found to be independent of the back-off stage.This idle probability can be approximated by the steadystate channel idle probability PI

PI =TACCESS − TCCA

TACCESS + TDATA + (1− γ)TCOL

(29)

where TCOL can be approximated by TRx→T x

2 .The probability that a CCA attempt of a node which

has just finished a transmission, is idle, needs a moreinvolved calculation. Let ν∗i be the probability that thisnode, indicated by i, among all others breaks the idleperiod

ν∗i = Pr(ti ≤ minj 6=i

tj) =ˆ +∞

0

fT (ti)ˆ +∞

ti

dF ∗(t)dti(30)

where F ∗(t) is the minimum back-off delay after a trans-mission of all nodes except the node i

F ∗(t) = 1−N∏

j=1;j 6=i

(1−ˆ t

0

fj (tj) dtj

). (31)

13

pi

πCBi, πT

PBi, PT

fBi, fT , f

fBi, fT , f

other nodes

PI , PC

ν, ν∗, γ

F, F ∗

Fig. 12 Illustration of the iterative method for one node

and fT (ti) is the discrete density function calculated inthe previous section.

If this nodes does not break the idle period, the nodecan sense the medium as idle when the CCA attempthappens in the vulnerable period of another transmis-sion attempt. This probability PC can be evaluated by

PC =ˆ T I+TRx→T x

T I

fT (t)dt (32)

with

T I =ˆ +∞

0

tdF ∗(t) . (33)

The idle probability of the first back-off stage canthen be calculated byp1 = P ∗T (ν∗i + (1− ν∗i )PC) + P ∗B1

PI (34)withP ∗T =

PT

PT + PB1

P ∗BI=

PB1

PT + PB1

.

5.6 Fixed Point Procedure

The calculation detailed in this section cannot be donein one step. The resulting equations are non-linear andthe variables are not separable. A fixed point procedureis proposed to calculate the saturation throughput. Fig-ure 12 illustrates the iterative method for one node. Thecalculation loops for all nodes are done in parallel. If all

300400500600700800900

10001100120013001400

2 4 6 8 10 12 14 16 18 20

TACCESS[µ

s]

N

Fig. 13 The mean time between two successive trans-mission, TACCESS , as a function of the number ofnodes N with macMinBE = 3, macMaxBE = 5,macMaxCSMABackoffs = 4

0.7

0.75

0.8

0.85

0.9

0.95

1

2 4 6 8 10 12 14 16 18 20

γ

N

Fig. 14 The probability that a transmission is successful,γ, asa function of the number of nodes N with macMinBE = 3,macMaxBE = 5, macMaxCSMABackoffs = 4

nodes are coordinated, i.e. they have the same view ofthe channel, the method can be simplified because thefunction evaluations for all nodes are identical. In thecase of a multi-hop network however, this is no longertrue and all the equations have to be evaluated for eachnode separately.

Once the fixed point procedure has converged, TACCESS

and γ are used to calculate the saturation throughput.Figure 13 shows the analytical evaluation of TACCESS

as a function of the number of contending nodes N forthe default values specified in the IEEE Std 802.15.4TM

and figure 14 the probability that a transmission is suc-cessful, γ. As a final result the saturation throughputas a function of the number of nodes is plotted in figure

14

140

145

150

155

160

165

170

2 4 6 8 10 12 14 16 18 20

S(N

)[kbp

s]

N

SimulationLower BoundUpper Bound

Fig. 15 Saturation throughput S(t) with macMinBE = 3,macMaxBE = 5, macMaxCSMABackoffs = 4 as a functionof the number of nodes N

15. This graph shows also the 95% confidence intervalof 25 simulation runs. The upper and lower bound fitclosely the simulated results. For a large value of Nboth bounds coincide.

6 Conclusion

In this paper a analytical model for the saturation through-put of N contending nodes in a wireless network is pro-posed. The unslotted CSMA/CA scheme is analyzedwith a non-zero duration of the CCA and the parame-ters of the non-beacon enabled access mode of the IEEEStd 802.15.4TMMAC are used. To evaluate the impactof the contention based MAC a renewal process betweensuccessful transmission is defined. The internal statesof the nodes are assumed to be independent. A nodestate is modeled as a semi-Markov chain. An event-simulator is also developed to analyze the unslottedCSMA/CA and to verify the results of the analyticalcalculations. The simulations show that the idle prob-ability of a CCA is clearly dependent on the back-offstage and that the first back-off expiration after a trans-mission cannot be modeled as an exponential distribu-tion. These results are used in the analytical model. Thesaturation throughput of each node can be indepen-dently calculated so that an extension to a multi-hopwireless network is straightforward. A good fit betweenthe analytical model and the simulations is obtained.

This methodology will be used to analyze the queue-ing behavior in a multi-hop wireless network. The anal-ysis in [32] can be adapted to the analytical procedurein this paper so that the ideal CSMA protocol can be re-placed by a realistic unslotted CSMA/CA scheme with

truncated exponential back-offs. Topics for further re-search also include:

Implementation of the ACK scheme. The semi-Markovmodel has to be adapted to include two states: asuccessful DATA and ACK message reception stateand a waiting state in which the DATA message orthe ACK message has collided and is lost. This willhave an impact on the density function of the firstexpiration of a back-off after a transmission and willalso alter the idle probability analysis. The collisionprobability will be extended to include the case ofa collided ACK message. Simulations have alreadyshown that this collision probability is mildly de-pendent on the specific back-off stage.

Extension to the multi-hop scenario. A Markov chainof simultaneous transmissions can be formed to ana-lyze the impact of hidden nodes [12,14]. The outputof the product form analysis serves as a feed-back forthe calculation of the back-off delay before a trans-mission. The saturation throughput of the nodes ina network with an arbitrary topology is highly de-pend on the connectivity of each node.

More realistic physical layer. The effect of an error-pronePHY can be evaluated by linking the Signal / Noiseratio in a Rayleigh of Ricean fading channel with ad-ditive Gaussian noise to the Bit Error Rate (BER)[3]. The analysis of the probability that a transmis-sion is successful will be extended to include the sup-plementary packet loss which depends on the spe-cific modulation type and the receiver sensitivity.

The developed event-driven simulator is ready to tacklethese problems: the propagation model is pluggable, thechannel is modeled as a matrix which contains specificinformation for each transmitter-receiver pair and theACK scheme is completely implemented. Without thedevelopment of this event-driven simulator the analyt-ical results of this paper would be very difficult to vali-date and the simulator provides the information to ab-stract the complex interaction of the contending nodesbased on an unslotted CSMA/CA scheme in order toobtain a tractable but still valid model.

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Ben Lauwens (M.Eng.’00, M.Sc.’05) is born in Leuven,Belgium, in January 1977. He received a Master in Engineer-ing degree, option telecommunications, from the Royal MilitaryAcademy in 2000. After his studies he served as an officer in theterritorial CIS unit of the Belgian Armed Forces. In 2005 he ob-tained a Master in Science degree, option ICT, at the KatholiekeUniversiteit Leuven. Since 2006 he is occupied as a research andteaching assistant in the Electrical and Computer Engineeringdepartment of the Royal Military Academy and he is working to-wards a PhD in Engineering Science in collaboration with theElectrical Engineering Departement of the Katholieke Univer-siteit Leuven. His current research interests include the fast sim-ulation of large queueing networks and the performance analysisof ad-hoc wireless networks.

Bart Scheers, Dr. Ir. is born in Rumst, Belgium, in Novem-ber 1966. He obtained his degree of engineer, with a specializa-tion in telecommunications, at the Royal Military Academy in1991. After his studies he served as an officer in the territorialsignal unit of the Belgian Army. In 1994 he became assistant atthe Royal Military Academy in the field of signal processing. In2001 he presented his PhD thesis on the use of ground penetrat-ing radars in the field of humanitarian demining. From 2000 heworks as a lecturer in the telecommunication department. Hiscurrent domains of interest are ad-hoc networks, Wireless SensorNetworks and Software Defined Radio.

Antoine Van de Capelle, Dr. Ir., full professor, head ofTelemic, a research laboratory of the Department of ElectricalEngineering (ESAT) of the Katholieke Universiteit Leuven, Leu-ven, Belgium.