Worst case delay analysis for a wireless point-to-point transmission

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1 Worst case deay anaysis for a wireess point-to-point transmission Katia Jaffrès-Runser University of Tououse IRIT - INPT ENSEEIHT Tououse, France Emai: katia.jaffres-runser@irit.fr Abstract Wireess technoogies are currenty being intensivey investigated for rea-time appications because of their appeaing ease of depoyment and scaabiity. Dimensioning a wireess network for safety-critica appications is sti an open probem mainy because of the intrinsic non-deterministic nature of the wireess medium. This paper discusses the derivation of a worst case deay (WCD) measure for a point-to-point wireess transmission. A WCD performance measure is centra to the performance evauation of wireess networks subject to hard rea-time constraints. To capture the non-deterministic nature of the wireess channe, our measure reies on a probabiistic ink mode where transmissions are guaranteed using an acknowedgement mechanism. The deay is expressed by the number of emissions necessary for a packet to arrive at its destination. The WCD is expressed as the P d -percentie of this number of emissions. The proposed WCD metric is computed for an interference-free scenario considering AWGN and Rayeigh fading channes. Interference-imited scenarios are discussed as we to highight the perspectives of this work. Keywords-Wireess transmission, unreiabe ink mode, performance evauation, worst case deay anaysis I. INTRODUCTION The depoyment of wireess technoogies for rea-time appications is rapidy gaining momentum because of their appeaing ease of depoyment and scaabiity. First anaysis of egacy wireess protocos [1] (e.g. IEEE802.11, Buetooth or IEEE ) in the factory automation context caed for the design of nove soutions meeting the needs of rea-time systems. New protocos have been specified for industria process contro such as WireessHART [2] [3] or ISA100.11a [4]. Both soutions provide a pure time division mutipe access to its rea time users to prevent unbounded channe access deays. Channe hopping techniques with backisting is impemented at the physica ayer to be more robust to interference. In the context of nucear pant or warship monitoring, dedicated wireess sensor network protocos such as OCARI and MACARI [5] have been deveopped. Tempora behavior of such protocos have to be thoroughy assessed for such critica appications. As such, a comprehensive performance evauation of transmission deay is needed. Together with controing the variance of transmission deays, it is of foremost importance to derive a safe bound on the worst case transmission deay. This safe bound can be accounted for to check that transmission deays meet their tempora requirements in the protoco integration process. Worst case deay anaysis in wired networks has been performed using two types of derivations: deterministic (network cacuus [6], trajectory approach [7]) and probabiistic (stochastic network cacuus [8]). In this paper we propose a probabiistic derivation of the worst case deay (WCD) bound for a point-to-point wireess communication. This choice is ceary motivated by the non-deterministic nature of the wireess channe whose most vaid modes are stochastic. Thus, our WCD bound reies on a probabiistic ink mode where transmissions are guaranteed using an acknowedgement mechanism. The overa transmission deay is measured as a function of the number of emissions necessary for a packet to arrive and be decoded at its destination. The WCD deay is defined as the P d -percentie of the overa transmission deay. As such, there is a probabiity of = (1 P d )/100 for the deay to be arger than the WCD, providing a confidence eve on the cacuated WCD bound. The proposed WCD metric is competey derived and cacuated for an interference-free scenario considering AWGN and Rayeigh fading channes. Interference-imited scenarios are discussed at the end of the paper to show the perspectives of this work. This paper is organized as foows. Section II presents our WCD anaysis for a point-to-point interference-free transmission. Next, Section III discusses the main issues reated to the WCD anaysis in interference-imited scenarios. Section IV concudes the paper. II. WORST CASE DELAY FOR INTERFERENCE-FREE TRANSMISSIONS This section detais firsty the unreiabe wireess ink mode, then it briefy presents the average transmission deay computation before introducing the derivation of the stochastic WCD bound. A. Unreiabe wireess ink mode The unreiabe ink mode captures the wireess ink avaiabiity between two nodes i and j. It is defined as the probabiity p of a successfu transmission over the ink = (i,j). Characterization of the ink probabiity is

2 Tabe I TRANSMISSION PARAMETERS [12] Symbo Description Vaue N b Number of bits per packet 2560 R Transmission bit rate 1 Mbps N 0 Noise eve 154dBm/Hz f c Carrier frequency 2.4GHz G T Transmitter antenna gain 1 G R Receiver antenna gain 1 α Path-oss exponent 3 L Circuitry osses 1 impacted by enhancements and impairments at the physica ayer: transmission power, moduation type, channe fading, etc. Such a reaistic ink mode captures the nondeterministic nature of a wireess transmission and has been used in recent performance studies [9] [10] [11] focusing on various metrics such as energy consumption, average deay or reiabiity. It is derived for the transmission of a packet of N b bits. Formay, p (γ ) = (1 BER(γ )) N b (1) where BER(γ ) is the bit error rate (BER) corresponding to the signa to noise ratio (SNR) γ on ink. Note that consequenty p (γ ) = 1 PER(γ ), with PER(γ ) the corresponding packet error rate. The BER depends on the transmission chain technoogy (moduation, coding, etc.) and channe type (AWGN, Rayeigh, Rician). It is defined as the average probabiity to decode one bit. Thus it is a function of the SNR γ experienced by the destination cacuated by [12]: with γ = K 1 d α, (2) N 0 B K 1 = G T G R λ 2 (4π) 2 L, (3) where d is the transmission distance between nodes i and j, α 2 is the path oss exponent, is the transmission power, N 0 the noise power density in mw/hz, G T and G R are the antenna gains for the emitter and receiver respectivey, B is the bandwidth of the channe and is set to the emission rate (B = R), λ is the waveength and L 1 summarizes osses through the transmitter and receiver circuitry. For a given transmission technoogy, K 1 is constant and p (γ ) is a function of d and : p (d, ). Vaues considered herein are isted in Tabe I. In the foowing, a transmission scenario using Binary Phase Shift Keying (BPSK) moduation and coherent detection is assumed. Cosed form expressions of BER(γ ) for AWGN (Additive White Gaussian Noise) and Rayeigh fat fading channes as foows: AWGN CHANNEL: Derivation of BER(γ ) for BPSK and coherent detection foows the derivation in [13]: BER(γ ) = α m Q( β m γ ) (4) with the Q function, Q(x) = 1 x π e u2 /2 du and α m, β m the moduation type and order, respectivey. For BPSK, α m = 1 and β m = 2. RAYLEIGH FLAT FADING: The genera expression for the BER in Rayeigh fat fading channe for γ 5 [13] is assumed: BER f (γ ) α m (5) 2β m γ Link success probabiity Link success probabiity =100mW =300mW =600mW =1000mW AWGN channe Rayeigh channe =100mW =300mW =600mW =1000mW Figure 1. Link probabiity as a function of distance at different transmission power vaues, for AWGN and Rayeigh fat fading channes. Link probabiity vaues for both channe types and different vaues of are represented in Figure 1. The curves can be divided into three parts: reiabe transmission, unreiabe transmission and impossibe transmission. For instance, in an AWGN channe at a transmission power of 100mW, transmission is aways successfu unti about 150 meters. Transmission is impossibe beyond 180 meters and in between, the transmission is unreiabe. In the Rayeigh fat fading environment, the perfecty reiabe transmission is

3 neary inexistent and most of the inks are unreiabe inks. Rayeigh fading characterizes harsher propagation environments where nodes are usuay not in ine of sight and transmission is deepy affected by muti-path such as it is the case in heaviy buit up city centers. There is no main ine of sight transmission component. In this case, the enveope of the received SNR is Rayeigh distributed. Other channe modes can be considered, depending on the environment the wireess network is depoyed in. For instance, Rician fading is appropriate when communication with a direct ine of sight is possibe in a harsh propagation environment with ots of scatterers. The probabiistic mode contrasts to previous modes such as the switched ink mode where a transmission between nodes i and j is successfu if and ony if the SNR is above a minima threshod vaue. With the switched ink mode, there are either competey reiabe inks or no communication is possibe. Unreiabe inks have been everaged to propery evauate connectivity [14], derive muti-objective performance trade-offs [10] [11] and design optima routing and resource aocation strategies [11]. We show in this paper that the unreiabe ink mode is particuary suited to become the buiding bock of a worst case deay anaysis of wireess networks. B. Average deay metric To combat packet osses on an unreiabe radio ink, we assume here a genera acknowedgement procedure where the compete packet is retransmitted if no acknowedgement is received before T NACK miiseconds have eapsed. A maximum number of retransmissions NR max can be set. If transmission is successfu, the acknowedgement packet is received within T ACK miiseconds. For simpicity, we assume T ACK = T NACK but different, reaistic vaues of both durations can be accounted for if needed. The deay for a packet to be emitted once by i and acknowedged by j over, d 1, is the sum of three deay components. The first component is the queuing deay during which a packet waits at i for being transmitted. The focus of this paper is on the deay introduced by the transmission and thus, queuing deay is out of the scope of this anaysis. The second component is the transmission deay equa to N b /R and the third component is T ACK. Propagation deay is negected because transmission distances in current technoogies emitting in the 2.4GHz band are usuay short ( 100m). T ACK and N b /R being constant, d 1 is set to be 1 unit. Due to ink unreiabiity, packets suffer from the deay introduced by their possibe retransmissions. As such, we introduce the random variabe N R which represents the number of retransmissions needed before receiving a positive acknowedgement. For a given vaue of N R, the compete transmission deay D over is thus given by: D = (N R +1) d 1 (6) since N R unsuccessfu and one successfu transmissions are needed. From (6), D is a random variabe giving the time before a positive acknowedgement is received in j. Having d 1 constant, the expectation of random variabe D is derived from the average number of retransmissions N R [15] using D = (N R + 1) d 1. Assuming a maximum number of retransmissions NR max, N R foows: N R = NR max r=0 r P[N R = r] (7) with P[N R = r] = p (1 p ) r the probabiity for a packet to necessitate r retransmissions. For a perfect transmission, N max R = and N R = 1/p (d, ). C. Worst case deay metric The distribution of N R knowing the ink probabiity p is given by P[N R = x] = p (1 p ) x. The distribution of the deay D is derived according to (6): P[D = x] = P[N R = x d 1 1] (8) Definition The worst case deay is defined in this paper by the vaue D w of D beow which P d percent of the observations fa, with P d = (1 ) 100. Formay: max D D w D w s.t. P[D D w ] (9) The worst case deay D w is the P d -percentie of the transmission deay D on ink. D w is a function of the random variabe N R. It is thus a probabiistic bound that can be exceeded with probabiity. Cosed form expression of D w is: D w d1 n( ) = (10) n(1 p ) Proof: We have P[D D w] = p D w d 1 1 x=0 (1 p ) x from (8). This is a geometric serie of rate (1 p ) and thus P[D D w ] = p (1 p ) Dw /d1 1 = 1 (1 p ) Dw /d1 p From (9), 1 (1 p ) Dw /d1 1, eading to D w d 1 n( ) n(1 p ). Since we are ooking for the argest integer vaue of D satisfying (9), we have: D w d1 n( ) = n(1 p ) Average deay D and worst case bounds D w expressed for different vaues are represented in Figures 2 and 3, for both AWGN and Rayeigh fading channes. Figure 2

4 Deay (number of transmissions) 10 3 Worst case deay AWGN Channe d=80m Deay (number of transmissions) 10 3Worst case deay AWGN Channe Pt=1000mW Deay (number of transmissions) Transmission power (mw) 10 3 Worst case deay Rayeigh Channe d=30m Deay (number of transmissions) Worst case deay Rayeigh Channe Pt=1000mW Transmission power (mw) Figure 2. and worst-case deay as a function of the transmission power for different percentie vaues, for AWGN and Rayeigh fat fading channes. Figure 3. and worst-case deay as a function of the distance for different percentie vaues, for AWGN and Rayeigh fat fading channes. focuses on the impact of the transmission power for a fixed inter-node distance whie Figure 3 concentrates on the impact of the inter-node distance for a fixed transmission power. In Figure 2, deay decreases with the increase in power. Indeed, as power is increased for a fixed inter-node distance, the ink becomes more and more reiabe, reducing the number of retransmissions needed to transmit a packet. For the AWGN channe, no communication is possibe for a power beow 20 mw: average and WC deay are infinite. Practicay, infinite deays are not toerabe in a transmission and a maximum number of retransmissions is introduced (which is not represented in this figure). WCD bounds are presented for vaues as sma as , providing a reay tight probabiistic bound on the worst case deay in this context. Impact of inter-node distance at fixed power is represented in Figure 3. As expected, deay (and thus ink reiabiity) NR max increases with distance. Simiar concusions to Figure 2 can be drawn here: a tight bound is obtained, at the cost of itte computation since a cosed form expression exists in (10). III. ACCOUNTING FOR INTERFERENCE IN WCD ANALYSIS This section introduces the main issues in accounting for interference created by mutipe concurrent transmissions in our WCD anaysis. Firsty, we concentrate on the interference-imited unreiabe ink mode. Next, we discuss the main steps and probems to integrate eaborated channe access protocos if interference-free medium access is not achievabe. A. Interference-imited ink mode In this section, we sti discuss a point-to-point wireess communication on ink between two nodes i and j. We assume here that this communication is interference-imited due to the other active inks in the network as represented in

5 d kj Node k i is using. Formay, I D is defined as: I D = k D K 2 k d α kj (12) Figure 4. Node i Link Node j Interference-imited ink mode. Figure 4. The compete network is static. More specificay, this scenario iustrates the study case where i is transmitting data to an access point j. Other nodes may interfere this communication because they have ad hoc communications with other nodes and can not detect the ongoing transmission between i and j for some reason (hidden termina probem for instance). Interference originates from concurrent transmissions in the wireess channe ink. Medium access contro prevents nodes from the same network to interfere with each other. Interference can be competey mitigated using Time Division Mutipe Access (TDMA) or Frequency Division Mutipe Access (FDMA). In this case, each user is assigned its own resource (time sot or frequency) and no other node is aowed to transmit in this resource. Worst case deay anaysis resumes in this case to the previousy defined pointto-point interference-free mode of Section II. TDMA or FDMA medium access may suffer from both under-utiization of the network bandwidth and additiona overhead for resource aocation. This is mosty the case when the network is ighty oaded. In this case, Carrier Sense Medium Access (CSMA) is an aternative that reduces resource aocation overhead and provides a faster access to the wireess channe. The drawback of CSMA is that interference can not be competey mitigated anymore, mosty because of the hidden termina probem. As for the interference-free case, the formuas for the BER hod, but this time they depend on the Signa to Noise and Interference ratio γ I (SINR) instead of the SNR γ. Interference I experienced at receiver j is added to the therma noise in equation (2) to derive the SINR: γ D = K 1 Pi t d α N 0 B +I D (11) with I D defined as the sum of the power at j received from a other emitters transmitting at the same time. In this notation, D represents the set of interfering nodes. In this formuation, nodes can use different transmission power vaues. Thus, Pi t represents the transmission power a node where K 2 = G T (λ/4π) 2 and d kj the distance between interferer k and destination node j. This computation of interference power captures the geometry of the network. As such, if the ocation of a nodes in the network is known, I D can be cacuated using (12) and its corresponding ink probabiity using (11) and (1). Simiary, if the node distribution foows a given aw (e.g. a Poisson point process or a power aw distribution for scae-free networks), interference distribution may be derived as we. B. Worst case deay metric and medium access contro The set of interferers D affecting the communication on ink depends on the decisions made by the medium access contro (MAC) ayer. For idea TDMA (one user is assigned to one time sot at any time), the set D is empty. For a CSMA-oriented MAC protoco, we are interested in deriving the distribution of the bit error rate vaues over a possibe interfering sets. A set of interferers D beongs to the power set P(N) of N, with N the set of a nodes of the network different from i and j. For each set of interferers D P(N), a SINR vaue can be computed with (11) and its corresponding BER using (4) or (5). The distribution of BER(γ D ) is given by the distribution of the set of interferers: P[BER(γ D) = x] = P[D active], with D the set producing BER(γ D). A node is said to be active if it can emit in the same channe than i. The activity of a node is captured by the probabiity it is emitting on the channe as proposed in [11]. Two types of such emission probabiities may be considered: 1. Independent emission probabiity: It is captured by τ i, the probabiity node i is emitting. Using τ i vaues, it is shown in [11] that it is possibe to derive the probabiity of any set D of interferers to be active using: P[D active] = i Dτ i (1 τ j ) j N\D The average ink probabiity is deduced from the distribution of BER vaues using the aw of tota probabiities: p = p,d P[D active] (13) D P(N) where p,d = [ 1 BER(γ D )] N b is the ink probabiity experienced for the set D of interferers. 2. Conditiona emission probabiity: The independent channe access mode is a simpified mode where transmission decisions are independent from eachother, which is usuay not the case in a MAC protoco. Interaction between nodes coud be for instance captured by τ i/j, the probabiity the channe is occupied by a transmission of node i knowing j is not transmitting. This conditiona channe

6 probabiity can be everaged to derive the probabiity of a set of interferer K to be active. Two types of worst case deays can be computed. The first one can be derived from (10) using the average ink probabiity p derived in (13). A safer estimation but more pessimistic probabiistic bound can be computed from the worst case ink probabiity which is experienced as the channe between i and j is the most interfered. Knowing the BER distribution and simiary to the definition of the worst case deay D w, we can define the worst case ink probabiity. Definition The worst case ink probabiity is defined as the vaue p w of p above which P d percent of the observations fa, with P d = (1 ) 100. Formay: min p w p w p s.t. P[p p w ] (14) The safer deay bound is then computed from (10) using the worst case ink probabiity of (14). Worst case deay bounds are straightforward to cacuate if channe activity of each node is known (i.e. node emission probabiities). Different medium access protocos can be characterized using such node emission probabiities. Future studies wi study the impact of these node emission probabiities on the worst case deay, and work on modeing medium access decisions either as independent or conditiona emission probabiities, possiby accounting for incoming traffic modes, memory size or node distribution. IV. CONCLUSION This paper discusses the derivation of the worst case deay (WCD) bound for a point-to-point wireess communication. This bound is guaranteed not to be exceeded with a probabiity of (1 ), with arbitrariy sma. The proposed WCD metric is computed for an interferencefree scenario considering AWGN and Rayeigh fading channes. Interference-imited scenarios are discussed as we to highight the perspectives of this work. The next step is to fuy characterize the WCD for the interference-imited case and concentrate on mapping MAC protoco decisions to emission probabiities. Therefore, protoco performance evauation modes derived from the one proposed by Bianchi in [16] for IEEE DCF can be everaged. REFERENCES [1] A. Wiig, K. Matheus, and A. Woisz, Wireess technoogy in industria networks, Proceedings of the IEEE, vo. 93, no. 6, pp , june [2] A. N. Kim and F. Hekand and S. Petersen and P. Doye, When HART goes wireess: Understanding and impementing the wireesshart standard, in Proceedings of IEEE ETFA, 2008, pp [3] J. Song, S. Han, A. Mok, D. Chen, M. Lucas, and M. Nixon, Wireesshart: Appying wireess technoogy in rea-time industria process contro, in Rea-Time and Embedded Technoogy and Appications Symposium, RTAS 08. IEEE, apri 2008, pp [4] S. Petersen and S. Carsen, WireessHART versus ISA100.11a: The format war hits the factory foor, Industria Eectronics Magazine, IEEE, vo. 5, pp , Dec [5] A Agha, K. and Bertin, M.-H. and Dang, T. and Guitton, A. and Minet, P. and Va, T. and Vioet, J.-B., Which wireess technoogy for industria wireess sensor networks? The deveopment of OCARI technoogy, Industria Eectronics, IEEE Transactions on, vo. 56, no. 10, pp , October [6] J.-Y. Le Boudec and P. Thiran, Network cacuus: a theory of deterministic queuing systems for the internet. Berin, Heideberg: Springer-Verag, [7] H. Bauer, J.-L. Scharbarg, and C. Frabou, Appying trajectory approach with static priority queuing for improving the use of avaiabe afdx resources, Rea-Time Systems, vo. 48, no. 1, pp , [8] J.-L. Scharbarg, F. Ridouard, and C. Frabou, A probabiistic anaysis of end-to-end deays on an avionics switched ethernet, IEEE Trans. Industria Informatics, vo. 5, no. 1, pp , [9] Ruifeng Zhang and Jean-Marie Gorce, Optima transmission range for minimum energy consumption in wireess sensor networks, in Proceedings of IEEE WCNC, 2008, pp [10] R. Zhang, J.-M. Gorce, R. Dong, and K. Jaffrès-Runser, Energy efficiency of opportunistic routing with unreiabe inks, in Wireess Communications and Networking Conference, WCNC IEEE, Apri 2009, pp [11] K. Jaffrès-Runser, M. R. Schurgot, C. Comaniciu, and J.-M. Gorce, A mutiobjective performance evauation framework for routing in wireess ad hoc networks, in Proceedings of IEEE WiOpt 2010, May 2010, pp [12] H. Kar and A. Wiig, Protocos and Architectures for Wireess Sensor Networks. John Wiey and Sons, [13] A. Godsmith, Wireess Communications. New York, NY, USA: Cambridge university press, September [14] J.-M. Gorce, R. Zhang, and H. Parvery, Impact of radio ink unreiabiity on the connectivity of wireess sensor networks, EURASIP Journa on Wireess Communications and Networking, vo. 2007, no. 1, [15] R. Zhang, J.-M. Gorce, and K. Jaffrès-Runser, Energydeay bounds anaysis in wireess muti-hop networks with unreiabe radio inks, INRIA, Rapport de recherche RR- 6598, [16] G. Bianchi, Performance anaysis of the ieee distributed coordination function, Seected Areas in Communications, IEEE Journa on, vo. 18, no. 3, pp , mar 2000.