On Flow-Induced Diffusive Mobile Molecular Communication: First Hitting Time and Performance Analysis

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arXiv:1806.04784v1 [cs.IT] 12 Jun 2018 oeua omncto,nano-network. communication, molecular Terms Index th conditions. into presente mobility insights yield different are to for results and performance results presence Simulation theoretical fro the derived. the noise verify in are and capacity sources errors, decisi the other counting based and interference, (LRT) error, inter-symbol test of ratio probability Likelihood rule, optimal part with detectio In alarm of investigated. probabilities the is for expressions channels performance closed-form modulati diffusive the particle (OOK) flow-induced results, keying through in on-off these with validated on communication are molecular Based in- expressions simulations. dr and positive PDF based with nanomachines derived derived motion the hitti Brownian The are of first using scenarios movement molecules the formation mobile the of fixed, aforementioned characterizing expression (PDF) is Closed-form by the function (RX) RX mobile. under density is receiver and RX time probability mobile and and the fixed is (TX) is for TX TX transmitter (ii) (iii) both and mobile, (i) are as nanomachines mobility such various under ditions communication molecular diffusive Abstract igo eal [email protected]). 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Simi- 2) This work also provides a comprehensive performance larly, the work in [12] investigates different coding strategies analysis for molecular communication between mobile for mitigating transposition errors for flow-induced diffusion nanomachines in flow-induced diffusion channels, consid- mobile molecular communication. In the works [10]–[12] the ering multiple-source interference (MSI), ISI and count- mobility of the nanomachines is modeled through a time- ing errors at the RX. In particular, an optimal decision varying distance (e.g., distance increases with equal difference rule and an adaptive decision threshold are derived based [10]). Mobile bacteria networks are proposed in [13], applying on the optimal Likelihood ratio test (LRT). Unlike the an independently and identically distributed mobility model. decision threshold presented in [20, (42)], the decision That is, in each time slot a network node chooses its new threshold in this work does not require knowledge on position independently and identically distributed over the the previously transmitted bits, which is not known in entire network. Mobile ad-hoc nano-networks are considered practice. Based on the derived optimal decision rule in [14] and [15], where the mobile nanomachines freely diffuse the detection performance in terms of receiver operating in a three-dimensional (3D) environment via Brownian motion. characteristic, error probability, and the channel capacity In [14] the information transfer is accomplished through are investigated. Both analytical expression and computer neurospike communication after collision and adhesion of the simulations for the aforementioned metrics are presented nodes and in [15] it based on Förster resonance energy transfer to develop several important and interesting insights into if the nodes are in close proximity of each other. A mathemat- the system performance under various mobile scenarios. ical model of non-diffusive mobile molecular communication networks is presented in [16], describing the two-dimensional C. Organization (2D) mobility of the nanomachines through the Langevin equa- tions. In [17] the channel impulse response (CIR) for diffusive The rest of the paper is organized as follows. Section II mobile molecular communication in a 3D environment is presents the system model for diffusive molecular commu- presented, assuming that only the receiver (RX) is mobile and nication with mobile TX and RX under drift channel. The a fully absorbing receiver. The CIR is obtained by modifying derivations of first hitting time distribution under various mo- the diffusion coefficient of the information molecules. In [18] bile scenarios, along with validation using the particle-based the first hitting time distribution for a one-dimensional (1D) simulations, are presented in Section III. This is followed diffusion channel without drift is derived, taking the mobility by the comprehensive performance analysis in Section IV of the transmitter (TX) and RX into account. Further, using the where the analytical results for the optimal decision rule at first hitting time distribution [18], the performance of diffusive the RX, probabilities of detection and false alarm, the total mobile molecular communication system without drift has error probability and the channel capacity are derived. Section been recently analyzed in [19]. In [20] the CIR for a mobile V demonstrates the performance analysis through computer molecular communication system is presented, considering the simulations under various mobile conditions. Finally, Section mobility of both TX and RX as well as the impact of flow. VI provides concluding remarks. However, in contrast to [17]–[19] TX and RX are assumed to be as passive. It is important to note that the information II. SYSTEM MODEL FOR DIFFUSIVE MOBILE MOLECULAR transfer in [17]–[20] is accomplished through freely diffusing COMMUNICATION particles, whereas in [14]–[16] other mechanisms (e.g., Förster Consider a diffusive molecular communication system with resonance energy transfer) are exploited. TX and RX in motion while communicating in a semi-infinite 1D fluid medium with drift. Moreover, this work assumes constant temperature and viscosity. TX and RX are placed B. Contributions on a straight line at a certain distance from each other and the In this work, we derive the first hitting time distributions for movement of both nanomachines is modeled as a 1D Gaussian mobile molecular communication in flow-induced diffusion random walk [18]. Similar to [22]–[24] and the references channels and provide a comprehensive performance analysis therein, both nanomachines are assumed to be synchronized for such a system. In particular this work makes the following in time. The channel is divided into time slots of duration T contributions: as shown in Fig. 2, where the jth slot is defined as the time 1) Closed-form expressions for the first hitting time distri- period [(j 1)T,jT ] with j 1, 2, . At the beginning bution in flow-induced diffusion channels are derived, of each time− slot, the TX either∈ { emits···} the same type of considering various mobility conditions such as (i) both molecules in the propagation medium with prior probability TX and RX are mobile, (ii) TX is mobile and RX is β for transmission of information symbol 1 or remains silent fixed, and (iii) TX is fixed and RX is mobile. The RX is for transmission of information symbol 0. Let [j] denote modeled as fully absorbing RX [21], which is in contrast the number of molecules released by the TX forQ information to [20] where only a passive RX is investigated. The symbol x[j]=1 at the beginning of the jth slot. The molecular derived expressions are validated through particle-based propagation from the TX to RX occurs via Brownian Motion simulations. To the best of our knowledge, this is the first with diffusion coefficient denoted by Dm. Similar to [23], time that the analysis for the first hitting time distribution [24] and the references therein, it is also assumed that the for flow-induced diffusion channels with mobile TX and transmitted molecules do not interfere or collide with each mobile active RX is reported. other. Once the molecules reach the RX, they are immediately 3

A. Distance Variation between TX and RX In case of mobile TX and/or RX the diffusion coefficients of the mobile TX and RX are denoted by Dtx and Drx, respectively. Each of them is either fixed or moves with the flow v, i.e., the velocities of TX and RX are given by vtx, vrx 0, v . This work assumes that the movement of TX and RX∈{ does} not disrupt the propagation of the information Fig. 2. Diffusion based molecular communication over time slotted molecules and that TX and RX can pass each other. The channel, where [(j − 1)T,jT ] is the current slot and q0 is the probability that a molecule reaches RX within the current slot. Euclidean distance between TX and RX at the beginning of the kth time slot, i.e., at time τ = kT , is given by

2 2 r = (x rx x tx) = d , (5) absorbed, followed by detection of the transmitted information k k, − k, k based on the number of molecules received. q q with dk = xk,rx xk,tx and xk,u, u tx, rx , denotes the Due to the stochastic nature of the diffusive channel, the times x-coordinate position− of TX and RX∈ at { time} τ = kT . It of arrival at the RX, of the molecules emitted by the TX, is important to note that the Euclidean distance used in (2) are random in nature and can span multiple time slots. Let corresponds to the Euclidean distance in (5) at time τ = 0. qj k denote the probability that a molecule transmitted in slot The positions of TX and RX at time τ = kT considering their k − 1,2, ,j arrives in time slot j and can be obtained as ∈{ ··· } mobility in drift channels can be defined as follows [1] (j k+1)T − qj k = fTh (t; k)dt, (1) xk,u = xk 1,u + ∆xu − − (j k)T k Z − where f (t; k) denotes the probability density function (PDF) = x0,u + ∆xu, u tx, rx , (6) Th ∈{ } i=1 of first hitting time at the RX. The following section derives X the closed-form expression for the PDF of first hitting time where x , u tx, rx , denote the initial position of TX and 0,u ∈{ } considering the various mobility conditions of TX and RX in RX and ∆xu = vuT + ∆˜ xu. The random displacement ∆˜ xu flow-induced diffusive channel with drift. follows a Gaussian distribution with zero mean and variance 2 ˜ σu =2DuT , i.e., ∆xu (0, 2DuT ). Thus, the position and III. FIRST HITTING TIME the distance of TX and∼ RX N at time τ = kT follows a Gaussian This section begins by considering the first hitting time dis- distribution tribution for fixed TX and RX under flow-induced diffusive x (¯x , kσ2 ), (7) channel. Then, the variation of the Euclidean distance when k,u ∼ N k,u u ¯ 2 TX and RX are moving is characterized, which is subsequently dk (dk, σk), (8) used to derive the first hitting time distribution for various mo- ∼ N with , ¯ , and bility scenarios. Finally, the derived expressions are validated x¯k,u = x0,u + vukT dk = d0 + kT (vrx vtx) σ2 = k(σ2 + σ2 )=2kTD , with D = D +− D . The through particle-based simulations. k tx rx tot tot tx rx distribution of the Euclidean distance defined in (5) is The first hitting time for flow-induced diffusion channels with rk provided in the following theorem. fixed TX and RX follows an inverse Gaussian (IG) distribution with its PDF given by [25] Theorem 1. Since the distance dk follows a Gaussian dis- ⋆ 2 tribution, the Euclidean distance r = d2 follows a scaled r0 (r0 v t) k k f h (t)= exp − , t> 0, (2) T 3 noncentral chi distribution with one degree of freedom. The √4πDmt − 4Dmt p PDF is given by where the Euclidean distance between TX and RX is given 2 2 by r0 = (x0,rx x0,tx) , and x0,tx and x0,rx denote the rk 2 − 2 + λ x-coordinate position of TX and RX, respectively. The velocity 1 √σk rk rk 1 ⋆ p frk (rk)= exp   λ I λ  , v is given by 2 − 2 2 − 2 2 σk s σk σk ! ⋆     v = sgn(x rx x tx)v = sgn(d )v, (3)     0, − 0, 0 p    p p     (9)  where sgn( ) denotes the sign function, d0 = x0,rx x0,tx and v corresponds· to the positive flow velocity from TX− to RX. The with noncentrality parameter λ = d¯2 /σ2 and the modified definition in (3) allows to model molecular communication k k 1 Bessel function of the first kind I q1 (y) [26]. with both positive and negative flow . − 2 In case of no flow, i.e., v =0, the IG distribution in (2) turns Proof. The Euclidean distance rk (cf. (5)) can be written as into a Lévy distribution given by 2 2 σ r0 r k 2 2 0 rk = 2 dk = σkr˜k, (10) fTh (t)= exp , t> 0. (4) σ √4πD t3 −4Dmt s k m q q 2 2 1Please note that the first hitting time in case of a negative flow can be where r˜k = dk/σk denotes the scaled Euclidean distance. modeled through an IG distribution with negative flow velocity v [25]. Since d follows a Gaussian distribution (cf. (8)), the scaled k p 4

Euclidean distance r˜k follows a noncentral chi distribution 3) Mobile TX and RX: In this case, TX and RX are mov- with one degree of freedom [27]. Its PDF is given by ing along with the information molecules with flow v, i.e., vtx = vrx = v, Dtx = 0, and Drx = 0. The PDF of the first 2 2 6 6 ⋆ r˜k + λ hitting time is given by (15), with D , D and v = sgn(d¯ )v f (˜r )=exp λr˜ I 1 (λr˜ ) , (11) m tx k r˜k k k 2 k ⋆ − 2 − substituted by Deff, Dtot and v = sgn(d¯ )veff = sgn(d¯ )(v k k − p vrx)=0, respectively. It can be easily verified that (15) for ¯2 2 mobile TX and RX in drift channel reduces to with the noncentrality parameter λ = dk/σk. The Euclidean 2 2 distance rk is defined by σkr˜k and,q thus, the corresponding √kTDtotDeff d0 fTh (t; k)= exp − PDF can be expressed as π(kTD + D t)√t 4kTDtot p tot eff kTDtot 1 rk + fTh t + f (r )= f , (12) Deff rk k 2 r˜k 2 σ σ ! k k d D t erf 0 eff , (16) p p 2 kTD (D t + kTD ) where the PDF fr˜k ( ) is defined in (11). × s tot eff tot ! · which is equivalent to the first hitting time PDF [18, Eq. (6)] for diffusion channels without flow and mobile TX and RX. B. First Hitting Time Distribution This is because TX, RX and the information molecules are all Similar to [18], [20] this work distinguishes between three moving with the same flow v. cases, based on the mobility of TX and RX, for deriving the Further, it can also be easily verified that in case of fixed TX first hitting time distribution of molecules released at time and mobile RX, (15) reduces to a Lévy distribution as defined τ = kT . in (4) with the parameters d¯k and Deff. 1) Fixed TX and mobile RX: In this case, TX is fixed and only RX is moving along with the information molecules with C. Evaluation of the First Hitting Time Distribution flow v, i.e., vtx = Dtx = 0, Drx = 0, and vrx = v. The first hitting time distribution can be obtained6 through considering For the evaluation of the first hitting time PDF, this the relative motion between the information molecules and work adopts the parameters used in [18], [20]: Without the RX. This can be accomplished by an effective diffusion loss of generality this work assumes that TX and RX coefficient [28] and an effective flow veloc- are located at the x-coordinates x0,tx = 0 and x0,rx = Deff = Dm + Drx 6 6 ity . Since the RX is moving with flow the 10− m, i.e., r0 = 10− m. The diffusion coefficients veff = v vrx v 9 2 10 2 effective velocity− is zero. Thus, the first hitting time for are given by Dm =0.5 10− m /s, Dtx =1 10− m /s, veff 12 ×2 × fixed TX and mobile RX follows a Lévy distribution as defined and Drx =0.5 10− m /s. The drift velocity of the fluid × 3 medium is considered as v = 10− m/s. Figs. 3-5 demonstrate in (4), with r0 and Dm substituted by d¯k = d0 + kT v and the impact of releasing molecules at different time slots k Deff = Dm + Drx, respectively. on the first hitting time PDF under various mobility con- 2) Mobile TX and fixed RX: In this case, TX is moving along ditions. Moreover, Fig. 6 compares the arrival probabilities with the information molecules with flow v and RX is fixed, q = T f (t; k)dt with T = 0.3 ms for various flow i.e., v = D = 0, D = 0, and v = v. The PDF of the 0 0 Th rx rx tx tx velocities 3 3 m/s and mobility scenarios. first hitting time can be computed6 as follows v 10− , 0.5 10− Fig. 3 showsR ∈{ the PDF of the× first} hitting time for fixed TX and

∞ mobile RX. In this case the first hitting time follows a Lévy distribution with the parameters d¯ and D (cf. Sec. III-B1). fTh (t; k)= fTh rk (t rk)frk (rk)drk, (13) k eff | | Since only the RX is moving, the distance between TX and RX rkZ=0 becomes larger as the time slot k increases (cf. (8) and (9)) and, as a result, the probability for a small first hitting time where frk (rk) is defined in (9) and fTh rk (t rk) is given by | | decreases compared to the scenario of fixed TX and RX. This ⋆ 2 r (r v t) observation is confirmed through the arrival probability q0 f (t r )= k exp k − , t> 0, Th rk k 3 shown in Fig. 6, which is almost zero from and | | √4πDmt − 4Dmt k =2 k =4 3 3 (14) for v = 10− m/s and v =0.5 10− m/s, respectively. Fig. 4 shows the PDF of the× first hitting time for mobile ⋆ with v = sgn(d¯k)v. The variable k denotes the time slot TX and fixed RX. In this case, first the TX moves towards in which the molecules are released; in the kth time slot the the fixed RX, and, thus, the distance between them reduces. molecules are released at time τ = kT . The variable t denotes After a certain time, the TX passes the RX and the distance the relative hitting time at the RX for a fixed time slot number between them becomes larger. Once the TX has passed the k. The closed-form solution for (13) is given in (15) and the RX, molecular communication with negative flow occurs [25]. corresponding proof can be found in Appendix A. In (15), the This behavior can be observed in Fig. 6, where the arrival

PDF fTh (t) corresponds to the first hitting time PDF for fixed probability q0 first increases (TX moves towards RX) and then TX and RX in (2) and the error function erf(x) is defined by decreases (TX has passed RX). In particular, an increase in the erf(x)= 2 x exp( t2)dt [26, Eq. (8.250.1)]. arrival probability can be observed as long as sgn(d¯ ) > 0, √π 0 − k R 5

¯2 ⋆ 2 √kTDtxDm Dmdk kTDtx(v ) t fTh (t; k)= exp − − π(kTD + D t)√t 4kTDtxDm tx m ⋆ ⋆ 2 ⋆ ¯ 2 ¯ 1 kTDtx v (v (kTDtx) +2v kTDtxDmt 2dkDmt 2dkkTDtxDm) + f h t + exp − − 2 T D 4D2 (D t + kTD ) m m m tx 2 ⋆ ⋆ ⋆ l v kTDtx v d¯ t v kTDtx + ( 1) d¯ Dm t + ( 1)l exp ( 1)l k 1+ erf − k . × Dmd¯ − − 2(Dmt + kTDtx) √4kTD D Dmt + kTDtx l=1 k tx m r X (15)

500 1200 Fixed TX and RX (Analytical) Fixed TX and RX (Analytical) Fixed TX and RX (Simulated) Fixed TX and RX (Simulated) Fixed TX and mobile RX, k=1 (Analytical) Mobile TX and RX, k=1 (Analytical) 400 Fixed TX and mobile RX, k=1 (Simulated) Mobile TX and RX, k=1 (Simulated) Fixed TX and mobile RX, k=10 (Analytical) Mobile TX and RX, k=10 (Analytical) Fixed TX and mobile RX, k=10 (Simulated) 800 Mobile TX and RX, k=10 (Simulated) 300 (t;k) (t;k) h h T T f 200 f 400

100

0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 t [ms] t [ms]

Fig. 3. First hitting time PDF fTh (t; k) for molecules released at different Fig. 5. First hitting time PDF fTh (t; k) for molecules released at different time slots k, with ﬁxed TX and mobile RX. time slots k, with mobile TX and mobile RX.

1600 0.8 Fixed TX and RX (Analytical) Fixed TX and RX Fixed TX and RX (Simulated) Fixed TX and RX

Mobile TX and fixed RX, k=1 (Analytical) ) Mobile TX and RX Mobile TX and fixed RX, k=1 (Simulated) 0 1200 Mobile TX and RX Mobile TX and fixed RX, k=10 (Analytical) 0.6 Fixed TX and mobile RX Mobile TX and fixed RX, k=10 (Simulated) Fixed TX and mobile RX Mobile TX and fixed RX 800 Mobile TX and fixed RX

(t;k) 0.4 T f

400 0.2 Arrival Probability (q

0 0 0 1 2 3 4 5 6 0 2 4 6 8 10 12 14 t [ms] kth Time Slot

T Fig. 4. First hitting time PDF fTh (t; k) for molecules released at different Fig. 6. Arrival probability q0 = R0 fTh (t; k)dt for molecules released at time slots k, with mobile TX and ﬁxed RX. time t = kT for v = 10−3 m/s (solid line) and v = 0.5 × 10−3 m/s (dashed line).

3 which is the case for k 3 and k 6 for v = 10− m/s and 3 ≤ ≤ v =0.5 10− m/s, respectively. increases, but ultimately it becomes zero for k as shown × → ∞ Fig. 5 shows the PDF of the first hitting time for mobile TX in [18, Fig. 3]. and RX in a diffusion channel with flow v. This scenario is The analytical expression of the first hitting time PDF is identical to the case of mobile TX and RX in a diffusion verified through a particle-based simulation. For this purpose channel without flow (cf. Sec. III-B3). As the time slot k we modeled the the movement of TX and RX as well as the increases the variation of the distance between TX and RX the motion of the information molecules as random walk [1] becomes larger (cf. (8) and (9)), resulting in a non-zero prob- and determined the relative frequency of the first molecule ability that TX and RX are in close proximity. Thus, a non- arrivals. As can be seen from Figs. 3-5, the analytical PDF zero probability for very small hitting times can be observed, expressions in Section III-B match exactly with the particle- as shown in Fig. 5 for k = 10. The arrival probability q0 first based simulations under aforementioned mobility conditions, 6 thereby validating the derived analytical results. RX corresponding to the different hypotheses are distributed as 2 IV. PEFORMANCE ANALYSIS 0 : R[j] (µ0[j], σ0[j]) H ∼ N (20) : R[j] (µ [j], σ2[j]), This section comprehensively analyzes the performance of H1 ∼ N 1 1 flow-induced diffusive mobile molecular communication sys- where the mean µ [j] and variance σ2[j] under hypothesis , tem in terms of detection, bit error probability, and capacity 0 0 H0 considering the impact of ISI, counting errors, and noise from as derived in Appendix B, are given as the other sources at the RX. The number of molecules received µ [j]=µ [1] + µ [2] + + µ [j 1]+ µ at the RX during time slot [(j 1)T,jT ] can be expressed as 0 I I ··· I − o j 1 − − R[j]= S[j]+ [j]+ N[j]+ C[j]. (17) =β [j k]qk + µo, (21) I Q − k=1 The quantity S[j] denotes the number of molecules received X σ2[j]=σ2[1] + σ2[2] + + σ2[j 1]+ σ2 + σ2[j] in the current slot [(j 1)T,jT ] and is binomially distributed 0 I I ··· I − o c − j 1 with parameters [j]x[j] and q0, where x[j] 0, 1 is the − Q ∈ { } = [β [j k]qk(1 qk)+ β(1 β) symbol transmitted by the TX in the jth time slot. The quantity Q − − − k=1 N[j] denotes the MSI, i.e., noise arising due to molecules X ( [j k]q )2]+ σ2 + µ [j], (22) received from the other sources, which can be modeled as × Q − k o 0 a Gaussian random variable with mean µ and variance σ2 o o and the mean µ [j] and variance σ2[j] under hypothesis , assuming a sufficiently large number of interfering sources 1 1 1 as derived in Appendix C, are given as H [29]. Further, the noise N[j] is independent of the number of molecules S[j] and [j] received from the intended TX [23]. µ [j]=µ[j]+ µ [1] + µ [2] + + µ [j 1]+ µ I 1 I I ··· I − o The term C[j] denotes counting error at the RX and can be j 1 modeled as a Gaussian distributed random variable with zero − 2 = [j]q0 + β [j k]qk + µo, (23) mean and variance σc [j] that depends on the average number Q Q − 2 E k=1 of molecules received, i.e., σc [j] = R[j] [23], [30]. The 2 2 2 X 2 2 2 2 { } σ1 [j]=σ [j]+ σI [1] + σI [2] + + σI [j 1]+ σo + σc [j] quantity [j] models the ISI, arising from transmission in the ··· − I j 1 previous j 1 slots, and is determined as − − = [j]q (1 q )+ [β [j k]q (1 q ) Q 0 − 0 Q − k − k [j]= I[1] + I[2] + + I[j 1], (18) k=1 I ··· − X 2 2 + β(1 β)( [j k]qk) ]+ σ + µ1[j]. (24) where I[k] Binomial( [j k]x[j k], q ), 1 k j 1, − Q − o ∼ Q − − k ≤ ≤ − denotes the number of stray molecules received from the previ- The optimal decision rule at the RX for symbol detection is ous (j k)th slot. Assuming the number of molecules released presented next. by the− TX to be sufficiently large, the binomial distribution for S[j] can be approximated by a Gaussian distribution2 with Theorem 2. The optimal decision rule at the RX correspond- 2 mean µ[j]= [j]x[j]q0 and variance σ [j]= [j]x[j]q0(1 ing to the jth time slot is obtained as q ), i.e., S[jQ] ( [j]x[j]q , [j]x[j]q (1Q q )) [31].− 0 0 0 0 1 ∼ N Q Q − H Similarly, the binomial distribution for I[k], 1 k j 1 T (R[j]) = R[j] ≷ γ′[j], (25) ≤ ≤ − 0 can be approximated as I[k] (µI [k] = [j k]x[j H k]q , σ2[k] = [j k]x[j ∼k]q N(1 q )).Q Further,− it can− k I k k where the optimal decision threshold γ [j] is given as be noted that SQ[j] and− I[k],− k =1, 2, − , j 1 are mutually ′ independent since the molecules transmitted··· − in different time γ′[j]= γ[j] α[j]. (26) slots do not interfere with each other [23], [24]. − The quantities γ[j] and α[j]pare defined as A. Optimal Decision Rule at the RX µ [j]σ2[j] µ [j]σ2[j] α[j]= 1 0 − 0 1 , (27) The symbol detection problem at the RX can be formulated σ2[j] σ2[j] as the binary hypothesis testing problem 1 − 0 2 2 2 2σ1 [j]σ0[j] (1 β) σ1[j] 2 0 : R[j]=I[1] + I[2] + + I[j 1]+ N[j]+ C[j] γ[j]= 2 2 ln − 2 + (α[j]) H ··· − σ1 [j] σ0[j] " β s σ0[j]# 1 : R[j]=S[j]+ I[1] + I[2] + + I[j 1] − H ··· − µ2[j]σ2[j] µ2[j]σ2[j] + N[j]+ C[j], + 1 0 − 0 1 . (28) σ2[j] σ2[j] (19) 1 − 0 where the null and alternative hypotheses , correspond H0 H1 Proof. The optimal log likelihood ratio test (LLRT) at the RX to the transmission of binary symbols 0, 1 respectively during is given as the jth time slot. The number of molecules R[j] received at the 1 p(R[j] 1) H 1 β 2This approximation is reasonable when Q[j]q0 > 5 and Q[j](1−q0) > 5 Λ(R[j]) = ln |H ≷ ln − . (29) p(R[j] 0) 0 β [23]. |H H 7

Substituting the Gaussian PDFs p(R[j] ) and p(R[j] ) Proof. The probabilities of detection (P [j]) and false alarm |H1 |H0 D from (20), the test statistic Λ(R[j]) can be obtained as (PFA[j]) at the RX in the jth time slot for the decision rule in (25) are obtained as 2 σ0 [j] 1 Λ(R[j]) = ln 2 + 2 2 PD[j]=Pr(T (R[j]) >γ′[j] 1) "sσ1 [j]# 2σ0[j]σ1 [j] |H =Pr(R[j] >γ′[j] 1), (35) 2 2 2 2 |H (R[j] µ0[j]) σ1[j] (R[j] µ1[j]) σ0 [j] . (30) × − − − PFA[j]=Pr(T (R[j]) >γ′[j] 0) , |H f(R[j]) =Pr(R[j] >γ′[j] ), (36) |H0 | {z } The expression for f(R[j]) given above can be further sim- where the number of received molecules R[j] is Gaussian plified as distributed (20) under hypotheses 0 and 1 respectively. Subtracting their respective meansH followedH by division by 2 2 2 2 R[j] µ1[j] R[j] µ0[j] f(R[j]) =R [j](σ [j] σ [j]) + 2R[j](µ1[j]σ [j] the standard deviations, i.e., − and − yields 1 − 0 0 σ1[j] σ0[j] µ [j]σ2[j]) + (µ2[j]σ2[j] µ2[j]σ2[j]) standard normal random variables for hypotheses 1 and − 0 1 0 1 − 1 0 H 2 2 2 0 respectively. Subsequently, the expressions for PD[j] and =(σ1 [j] σ0[j])(R[j]+ α[j]) H − PFA[j], given in (33) and (34) respectively, can be obtained (µ [j]σ2[j] µ [j]σ2[j])2 1 0 − 0 1 employing the definition of the Q( ) function. − σ2[j] σ2[j] · 1 − 0 + (µ2[j]σ2[j] µ2[j]σ2[j]), (31) 0 1 − 1 0 C. Bit Error Probability where α[j] is defined in (27). Substituting the above equation The end-to-end probability of error for communication be- for f(R[j]) in (30) and subsequently merging the terms tween TX and RX follows as described in the result below. independent of the received molecules R[j] with the detection threshold, the test can be equivalently expressed as Lemma 1. The average probability of error (Pe) for slots 1 to i at the RX in the diffusion based molecular nano-network

1 with mobile TX and RX is (R[j]+ α[j])2 H≷ γ[j], (32) 0 i 1 γ′[j] µ [j] H P = β 1 Q − 1 e i − σ [j] where γ[j] is deﬁned in (28). Further, taking the square root j=1 1 X of both sides where γ[j] 0, Equation (32) can be simpliﬁed γ′[j] µ [j] ≥ +(1 β)Q − 0 . (37) to yield the optimal test in (25). − σ [j] 0

Proof. The probability of error Pe[j] in the jth time slot is B. Probabilities of Detection and False Alarm deﬁned as [32]

The detection performance for the optimal test derived in (25) Pe[j]=Pr(decide 0, 1 true)+ Pr(decide 1, 0 true) H H H H at the RX considering also the mobility of the TX and RX, is =(1 PD[j])P ( 1)+ PFA[j]P ( 0), (38) obtained next. − H H where the prior probabilities of the hypotheses P ( 1) and Theorem 3. The average probabilities of detection (P ) and H D P ( 0) are β and 1 β respectively. The quantities PD[j] and false alarm (P ) at the RX in the diffusion based mobile H − FA PFA[j] denote the probabilities of detection and false alarm at molecular communication nano-network, corresponding to the the RX during the jth time slot as obtained in (33) and (34) transmission by the TX over i slots, are given as respectively. The average probability of error for slots 1 to i follows as stated in (37). 1 i P = P [j] D i D j=1 X D. Capacity i 1 γ′[j] µ [j] = Q − 1 , (33) Let X[j] and Y [j] be two discrete random variables that i σ [j] j=1 1 represent the transmitted and received symbols, respectively, X in the jth slot. The mutual information I(X[j], Y [j]) between 1 i P = P [j] X[j] and Y [j] with marginal probabilities Pr(x[j] = 0) = FA i FA j=1 1 β, Pr(x[j]=1)= β is given by X − i 1 γ′[j] µ0[j] I(X[j], Y [j]) = Pr(y[j] x[j])Pr(x[j]) = Q − , (34) | i σ0[j] x[j] 0,1 y[j] 0,1 j=1 X∈{ } X∈{ } X Pr(y[j] x[j]) log2 | , (39) where γ′[j] is deﬁned in (26) and Q( ) denotes the tail Pr(y[j] x[j])Pr(x[j]) · × probability of the standard normal random variable [32]. x[j] 0,1 | ∈{P } 8

1 with and without ﬂow, where each point on the curve cor- Analytical, Number of molecules Q[j]=30 for x[j]=1 ) 3 D 0.9 Analytical, Number of molecules Q[j]=90 for x[j]=1 responds to a value of (PFA, PD) for a given threshold Mobile TX and RX with D =10−10,D =0.5×10−12 (Simulated) 0.8 tx rx γ′[j] = γ′, j [32]. For these simulations, the range of γ′ is Mobile TX and RX with D =10−11,D =0.5×10−13 (Simulated) ∀ 0.7 tx rx considered from 1 to 80. Fig. 7 demonstrates the detection

0.6 performance at the RX considering the mobile nature of 3 0.5 the TX and RX under drift channel with v = 10− m/s. It can be clearly seen that an increase in the number of 0.4 molecules emitted by the TX results in a higher probability 0.3 of detection at the RX for a fixed value of the probability 0.2 10 2 of false alarm. For example with Dtx = 10− m /s and 0.1 12 2 Probability of Detection (P Drx =0.5 10− m /s, the probability of detection PD can 0 × −7 −6 −5 −4 −3 −2 −1 0 be increased from 0.1 to 0.52 as the number of molecules [j] 10 10 10 10 10 10 10 10 Q Probability of False Alarm (P ) released for symbol x[j]=1 increases from 30 to 90 for a FA 4 fixed probability of false alarm PFA = 10− . On the other hand, the detection performance significantly deteriorates as Fig. 7. Probability of detection versus probability of false alarm considering −10 −11 2 the diffusion coefficients D and D increase due to higher mobile TX and RX scenario with Dtx ∈ {10 , 10 } m /s, Drx ∈ tx rx −12 −13 2 −3 {0.5 × 10 , 0.5 × 10 } m /s, vtx = vrx = v = 10 m/s, and mobility. T = 2 ms. In Figs. 8(a)-(c), one can observe that the detection performance at the RX, considering [j] = 30 molecules released Q where the conditional probabilities Pr(y[j] 0, 1 x[j] by the TX for information symbol x[j] = 1, significantly ∈ { }| ∈ improves as slot duration (T ) increases from 1 ms to 10 ms. 0, 1 ) can be written in terms of PD[j] and PFA[j] as { } Further, the detection performance at the RX considering the 3 Pr(y[j]=0 x[j]=0)=1 PFA[j] mobility of the TX and RX with drift v = 10 m/s and | − − γ′[j] µ0[j] without drift v = 0 are identical. This is because of the =1 Q − , fact that the molecule arrival probabilities (cf. (1) and − σ0[j] qj k (16)) are equivalent under both the scenarios.− Furthermore, Pr(y[j]=1 x[j]=0)=PFA[j] 3 | the system with fixed TX and RX under drift, i.e., v = 10− γ′[j] µ [j] =Q − 0 , m/s outperforms the other scenarios. However, without drift, σ [j] 0 i.e., v = 0 as shown in Fig. 8(a), fixed TX and RX scenario Pr(y[j]=0 x[j]=1)=1 PD[j] performs poor for T =1 ms in comparison to the mobile TX | − γ′[j] µ1[j] and RX case. This is owing to the fact that the probability =1 Q − , of a molecule reaching the RX within current slot, i.e., is − σ1[j] q0 lower and the probabilities of a molecules received from the Pr(y[j]=1 x[j]=1)=PD[j] | previous slots, i.e., qj k, j = k, are higher under the fixed γ′[j] µ [j] − 6 =Q − 1 . TX and RX scenario in comparison to the one obtained for σ [j] 1 mobile TX and RX scenario. On contrary, as T increases from The mutual information between the TX and RX can be 1 ms to 2 ms and 10 ms shown in Figs. 8(b)-(c), the system maximized as where a fixed TX communicates with a fixed RX under no i drift outperforms the mobile scenario. 1 C[i] = max I(X[j], Y [j]) bits/slot, (40) Figs. 9(a) and 9(b) demonstrate the detection performance β i j=1 at the RX under fixed TX and mobile RX, and mobile X TX and fixed RX scenarios respectively, where the follow- which equals the channel capacity as the number of slots i ing interesting observations are obtained. First, the detection [22]. → ∞ performance considering fixed TX and mobile RX scenario significantly degrades under drift channel and the performance V. SIMULATION RESULTS further deteriorates as the drift velocity (v) increases from 5 4 This section presents simulation results to demonstrate the 5 10− m/s to 2 10− m/s. This is due to the fact that performance of the diffusive mobile molecular communication the× mobility of RX× under drift channel increases the distance system under various mobile conditions and also to validate between TX and RX that in turn decreases the probability the derived analytical results. For simulation purposes, the of a molecule arriving in same slot and also increase the various system parameters are set as, diffusion coefficient ISI with higher arrival probabilities from previous slots. This 10 2 Dm = 5 10− m /s, slot duration T 1, 2, 10 ms, can also be clearly seen in Fig. 3 with k = 10 slots. Under × ∈ { } distance r0 = 1 µm, prior probability β = 0.5, and i = 10 the scenario with mobile TX and fixed RX as shown in Fig. slots. The MSI at the RX is modeled as a Gaussian distributed 9(b), the detection performance at the RX with drift velocity 2 4 random variable with mean µo = 10 and variance σo = 10 v = 10− m/s improves in comparison to the scenario with no unless otherwise stated. drift, i.e., v = 0. This arises due to the fact that the mobile Figs. 7-9 demonstrate the detection performance of optimal LRT based detector in (25) under various mobile scenarios 3It is important to note that this is not an optimal threshold derived in (26). 9

1 1 Analytical 0.9 ) 0.9 Mobile TX and RX with drift (Simulated) D 0.8 Mobile TX and RX without drift (Simulated) 0.8 Fixed TX and RX with drift (Simulated) 0.7 Fixed TX and RX without drift (Simulated) 0.7

0.6 0.6

0.5 0.5 −5 −4 −4 0.4 0.4 v = [5× 10 , 10 , 2× 10 ] m/s

0.3 0.3 Fixed TX and mobile RX with drift (Analytical) 0.2 0.2 Fixed TX and mobile RX with drift (Simulated) Fixed TX and mobile RX without drift (Analytical) 0.1 0.1 Probability of Detection (P Fixed TX and mobile RX without drift (Simulated) 0 −7 −6 −5 −4 −3 −2 −1 0 0 10 10 10 10 10 10 10 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability of False Alarm (P ) FA (a) (a) 1 1 v = 10−4 m/s ) )

D 0.9 0.9 D

0.8 0.8

0.7 0.7 Analytical 0.6 0.6 Mobile TX and RX with drift (Simulated) −4 −3 0.5 Mobile TX and RX without drift (Simulated) 0.5 v = 3× 10 m/s v = 10 m/s Fixed TX and RX with drift (Simulated) 0.4 Fixed TX and RX without drift (Simulated) 0.4

0.3 0.3 Mobile TX and fixed RX with drift (Analytical) 0.2 0.2 Mobile TX and fixed RX with drift (Simulated) 0.1 Mobile TX and fixed RX without drift (Analytical)

Probability of Detection (P 0.1 Probability of Detection (P Mobile TX and fixed RX without drift (Simulated) 0 −7 −6 −5 −4 −3 −2 −1 0 0 10 10 10 10 10 10 10 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability of False Alarm (P ) Probability of False Alarm (P ) FA FA (b) (b) 1 Fig. 9. Probability of detection versus probability of false alarm considering )

D 0.9 Q[j] = 30 molecules released by the TX for information symbol x[j] = 1, slot duration T = 2 ms, and (a) ﬁxed TX and mobile RX scenario with 0.8 −12 2 Dtx = 0, Drx = 0.5 × 10 m /s, vtx = 0, vrx = v m/s and (b) mobile 0.7 −10 2 TX and ﬁxed RX with Dtx = 10 m /s, Drx = 0, vtx = v m/s, vrx = 0. 0.6

0.5

0.4 scenario when mobile TX communicates with ﬁxed RX over

0.3 Analytical diffusive channel without drift. Mobile TX and RX with drift (Simulated) 0.2 Mobile TX and RX without drift (Simulated) Fig. 10 presents the error rate versus MSI noise variance Fixed TX and RX with drift (Simulated) 2 0.1 (σo ) performance under various mobile scenarios, where the Probability of Detection (P Fixed TX and RX without drift (Simulated) optimal decision rule (25) is employed at the RX. First, it can 0 −7 −6 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 10 be observed from Fig. 10 that the analytical Pe values obtained Probability of False Alarm (P ) FA using (37) coincide with those obtained from simulations, thus (c) validating the derived analytical results. One can also observe 2 that an increase in the MSI noise variance (σo ) results in a Fig. 8. Probability of detection versus probability of false alarm considering −10 2 higher probability of error at the RX. The error rate further mobile TX and RX scenario with Dtx = 10 m /s, Drx = 0.5 × −12 2 10 m /s, and fixed TX and RX scenario with Dtx = Drx = vtx = increases as the mobility increases as shown in Fig. 10(a). vrx = 0 over slot duration (a) T = 1 ms (b) T = 2 ms, and (c) T = 10 ms. Moreover, it can also be seen therein that the system with T = 10 ms experiences lower probability of error for the fixed TX and RX scenario with and without drift in comparison to 4 TX with v = 10− m/s under drift channel approaches the the mobile scenario. Further, similar to detection performance, RX that in turn increases the probability q0 as slot number the system considering mobile TX and fixed RX with flow 4 j progressively increases. However, as drift velocity increases v = 10− m/s experiences lower probability of error in 4 4 3 from 10− m/s to 3 10− m/s and 10− m/s, the TX crosses comparison to the scenario when both mobile TX and fixed the RX and the distance× between them further increases as RX communicate with no flow v = 0. On the other hand, slot number j increases. Therefore, in comparison to a scenario for fixed TX and mobile RX as shown in Fig. 10(b), the 4 3 with mobile TX and fixed RX with drift v 3 10− , 10− system performance significantly degrades under drift channel m/s, a significant performance improvement∈{ can× be seen for} a and the performance further deteriorates as the drift velocity 10

0.9

) −1 e 10 0.8

T = 10 ms 0.7

0.6 −2 10 0.5

0.4

Analytical −3 0.3 Analytical, T=2 ms 10 −10 −13 Mobile TX and RX with D =10 , D =5× 10 (Simulated) −10 −12 tx rx Mobile TX and RX with D =10 , D =0.5× 10 , i=10 (Simulated) −11 −14 tx rx Mobile TX and RX with D =10 , D =5× 10 (Simulated) 0.2 −10 −12 tx rx × Capacity C[i], bits/slot Mobile TX and RX with D =10 , D =0.5 10 , i=20 (Simulated) Probability of Error (P tx rx Fixed TX and RX without drift (Simulated) Fixed TX and RX without drift (Simulated) −3 0.1 Fixed TX and RX with drift v = 10 (Simulated) −3 −4 Fixed TX and RX with drift v=10 (Simulated) 10 0 1 2 3 0 10 10 10 10 0 20 40 60 80 100 120 140 160 180 200 2 MSI Noise Variance (σ ) MSI Noise Variance (σ2) o o (a) (a)

0 10 1 With drift (Analytical) Mobile TX and fixed RX with T=2 ms With drift (Simulated) −4 0.9 ) v=3×10 m/s Without drift (Analytical) e Without drift (Simulated) 0.8 v=5×10−6 m/s −4 v=10−4 m/s 0.7 v=10 m/s −1 10 −6 0.6 v=8×10 m/s Fixed TX and mobile RX Fixed TX and v=8×10−6 m/s 0.5 with T=10 ms mobile RX with T=10 ms −4 0.4 v=3×10 m/s

−2 −6 With drift (Analytical) 10 × 0.3 v=5 10 m/s With drift (Simulated) Capacity C[i], bits/slot Probability of Error (P Without drift (Analytical) 0.2 Without drift (Simulated) Mobile TX and fixed RX with T=2 ms

0 1 2 3 0.1 10 10 10 10 0 20 40 60 80 100 120 140 160 180 200 MSI Noise Variance (σ2) MSI Noise Variance (σ2) o o (b) (b)

2 2 Fig. 10. Probability of error versus MSI noise variance (σo ) with µo = 0 and Fig. 12. Capacity versus MSI noise variance (σo ) under various mobile Q[j] = 30 molecules released by the TX for information symbol x[j] = 1. conditions with µo = 10 and Q[j] = 60 molecules released by the TX for information symbol x[j] = 1.

0 10 able to achieve the minimum probability of error at the RX. )

e The optimal value of the threshold (cf. (26)) as shown in Fig. 11 increases from 40.30 to 80.4 as the MSI increases from 2 2 µo = σo =1 to µo = σo = 40. Fig. 12 shows the capacity performance where the maximum −1 10 mutual information is achieved for equiprobable information symbols, i.e., β =0.5. Similar to the error rate performance, Optimal Threshold=80.4 with µ =σ2=40 the maximum mutual information C[i] obtained for slots 1 o o 2 Optimal Threshold=60 to i also decreases as the MSI noise variance (σo ) increases. Probability of Error (P 2 Optimal Threshold=40.30 with µ =σ =20 2 o o Analytical with µ =σ =1 Interestingly, it is also worth noting that the maximum mu- o o Simulated −2 10 tual information C[i] in bits per channel use progressively 0 20 40 60 80 100 120 Detection Threshold at RX decreases as the number of slots (i) increases due to the mobile nature of TX and RX. This is owing to the fact that the probability of a molecule reaching the RN within the Fig. 11. Error rate of diffusion based molecular systems at j = 10 time slot versus detection threshold for mobile TX and RX scenario with Q[j] = 120 current slot, i.e., q0 progressively decreases while the ISI from molecules released by the TX for information symbol x[j] = 1. previous slots increases as the value of i increases. Further, the value of C[i] considering mobile TX and fixed RX with flow 4 v = 10− m/s as shown in Fig. 12(b) is higher compared to v increases. the scenario when both mobile TX and fixed RX communicate Fig. 11 shows the error rate performance at (j = 10)th time over a diffusive channel without drift. slot versus detection threshold at the RX considering mobile 10 2 VI. CONCLUSION TX and RX with diffusion coefficients Dtx = 10− m /s, 12 2 Drx = 0.5 10− m /s and slot duration T = 10 ms. It Closed-form expressions for the PDF of the first hitting can be seen× that the optimal threshold obtained using (26) is time in flow-induced diffusive molecular communication under 11 various mobile scenarios were derived. Based on these results Further, using the identity [26, Eq. (3.462.5)] the performance for molecular communication in flow-induced ∞ diffusion channels in the presence of ISI, MSI, and counting x exp( ux2 2wx)dx errors was investigated. Closed-form analytical expressions − − Z were derived for the optimal decision test statistics and optimal 0 1 w π w2 w decision threshold, together with the resulting probabilities of = exp 1 erf , (44) 2u − 2u u u − √u detection, false alarm as well as the probability of error. In r addition, an analytical expression for the capacity was derived. leads to Simulation results were conducted to verify the theoretical Z[k] results and to gain insights into the performance under various 2 mobility conditions. The results reveal that molecular commu- 1 w π (w )2 w = l exp l 1 erf l , nication of mobile TX and RX in a drift channel performs C 2u − 2u u u − √u l=1 r identical to mobile TX and RX in a non-drift environment. X 2 2 For fixed TX and moving RX, the error performance degrades 1 1 π (wl) wl = wl exp 1 erf , for an increasing drift velocity, since the distance between TX C u − 2u u u − √u " r l=1 # and RX increases faster. In case of mobile TX and fixed RX, X (45) the error performance improves as long as TX moves towards with = exp( (ab)2)/√2πd, w = (2ab + ( 1)ld) and RX, but significantly deteriorates when TX has passed RX C − l − because molecular communication with negative flow occurs u = a + c. Re-substituting wl and u into (45) and Z[k] into in this case. Finally, future studies can focus on the first hitting (41) results, after some straightforward mathematical manip- time distribution in 3D environment. However, this task is not ulations, in (15). straightforward since, to the best of our knowledge, no closed- APPENDIX B form expression for the first hitting time is known even for MEAN µ [j] AND VARIANCE σ2[j] UNDER HYPOTHESIS fixed TX and RX and a fully absorbing RX. 0 0 H0 Using (19), the mean µ [j] under can be calculated as 0 H0 APPENDIX A j 1 PROOF OF (15) − µ0[j]=E I[k]+ N[j]+ C[j] Substituting (9) and (14) into (13) results in (k=1 ) j 1X 1 1 λ λ2 − f (t; k)= exp = E I[k] + µo, (46) Th 3 2 2 √4πDmt σ s σ − 2 { } k k kX=1 ∞ p⋆ 2p 2 where E I[k] is given as 3/2 (v t rk) rk λrk r exp exp I 1 dr . { } k − 2 2 k 4D t 2σ − 2 E E × − m − k σk! I[k] =Pr(x[j k]=1) I[k] x[j k]=1 rkZ=0 { } − { | − } (41) + Pr(x[j k]=0)E I[k] x[j k]=0 p − { | − } =βE I[k] x[j k]=1 +(1 β)E I[k] x[j k]=0 The integral in (41) can be written as { | − } − { | − } =β [j k]q . (47) ∞ ⋆ 2 2 k 3/2 (v t rk) rk λrk Q − Z[k]= r exp exp I 1 dr 2 k − 2 2 Thek variance σ [j] under can be derived as 4D t 2σ − 2 0 0 − m − k σk ! H rkZ=0 j 1 − ∞ p σ2[j]= σ2[k]+ σ2 + σ2[j] 3/2 0 I o c 2 2 1 = rk exp( a(b rk) ) exp( crk)I (drk) drk k=1 − − − − 2 Xj 1 rkZ=0 − 2 2 = exp( (ab)2) = σI [k]+ σo + µ0[j], (48) − k=1 ∞ X 3/2 2 2 2 1 where σ [j] = µ0[j] and the variance σ [k] of the ISI term rk exp( (a+c)rk) exp(2abrk)I (drk) drk, c I × − − 2 can be obtained as rkZ=0 (42) σ2[k]=E (I[k])2 E2 I[k] I { }− { } 1 ⋆ 1 λ 2 2 2 =E (I[k]) (β [j k]q ) , (49) with a = 4D t , b = v t, c = 2σ and d = 2 . Applying k m k √σk { }− Q − 1 E 2 the relation I 1 (y)= [exp(y) + exp( y)] gives where (I[k]) is given as 2 2πy { } − − 2 2 q E (I[k]) =Pr(x[j k]=1)E (I[k]) x[j k]=1 2 2 ∞ exp( (ab) ) { } − { | 2 − } Z[k]= − r exp( (a + c)r2) + Pr(x[j k]=0)E (I[k]) x[j k]=0 √ k − k − { | − } 2πd l=1 E 2 XrkZ=0 =β (I[k]) x[j k]=1 l { | − } 2 exp((2ab + ( 1) d)rk)drk. =β [j k]qk(1 qk) + ( [j k]qk) . × − (43) Q − − Q − (50) 12

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