Mathematical and Computer Modelling. On the characterization of Aloha in underwater wireless networks

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1 Mathematical and Comuter Modelling 53 (2011) Contents lists available at ScienceDirect Mathematical and Comuter Modelling journal homeage: On the characterization of Aloha in underwater wireless networks Swades De a,, Priyatosh Mandal b, Shyam S. Chakraborty c a Electrical Engineering Deartment, Indian Institute of Technology Delhi, New Delhi, India b Centre for Develoment of Telematics, New Delhi, India c Intelligent Systems esearch Centre, Faculty of Engineering, University of Ulster, Ireland, UK a r t i c l e i n f o a b s t r a c t Article history: eceived 19 December 2009 eceived in revised form 4 June 2010 Acceted 29 June 2010 Keywords: Underwater wireless ad hoc network Acoustic sensor network Short-range underwater communication andom access erformance modeling Aloha Modified slotted-aloha Standard analyses of wireless random access rotocols that are available in the literature assume negligible roagation delay between any two nodes. This assumtion holds good in reasonably short-range terrestrial F (radio frequency) wireless networks. On the contrary, in wireless communications involving acoustic wave roagation, as in underwater wireless networks, even short distance roagation has areciably large roagation delay. This observation has led to several recent simulation and exerimental studies on underwater Aloha and slotted-aloha (S-Aloha) rotocols and also a few new roosals on random access rotocols for underwater wireless ad hoc networks (UWN). To study the efficiency of more advanced multiaccess communication rotocols for UWN, it is imortant to benchmark their erformances with resect to the two basic random access rotocols, Aloha and S-Aloha. This aer rovides an analytic framework to cature the erformance of Aloha and S-Aloha rotocols in an underwater environment with high and random internodal signal roagation delay. The erformance of underwater Aloha and S-Aloha are contrasted with those in short-range terrestrial F wireless networks. The analysis shows that random internodal roagation delay has no effect on the underwater Aloha erformance. It also sheds light on the throughut degradation of underwater S-Aloha with a slotting concet that achieves F S-Aloha equivalent one-slot vulnerability. Additionally, a modified slotting concet is introduced where the slot size is judiciously reduced such that even by allowing some collisions the overall system throughut can be increased. Our calculations show that, with the modified slotting aroach u to 17% throughut erformance gain can be achieved over the naive (F S-Aloha equivalent) slotting aroach in UWN. Our analytic results are suorted by discrete event simulations Elsevier Ltd. All rights reserved. 1. Introduction Short-range underwater wireless ad hoc networks (UWN) are aimed at remotely monitoring various aquatic activities, such as marine biological and zoological lives, geological changes, and underwater human activities. There are some similarities in UWN and terrestrial radio frequency (F) wireless sensor networks, such as, limited channel bandwidth, high bit error rate caused by the wireless channel, and limited battery ower of sensor nodes. Therefore, both tyes of networks have common erformance measures, such as, throughut, delay, and battery life. Yet, UWN and terrestrial wireless networks differ in many asects; roagation delay is the most sensitive arameter of them all. F networks universally use electromagnetic frequency (EM) waves at various frequency bands. However, due to high attenuation, underwater wireless (UW) communication systems cannot use EM waves. Instead, UW systems use acoustic waves. The atmosheric roagation seed of F carrier is close to m/s, that is, seed of light in free sace. On the other hand, roagation seed Corresonding author. Tel.: ; fax: addresses: swadesd@ee.iitd.ac.in (S. De), riyatos@cdotd.ernet.in (P. Mandal), ss.chakraborty@ulster.ac.uk (S.S. Chakraborty) /$ see front matter 2010 Elsevier Ltd. All rights reserved. doi: /j.mcm

2 2094 S. De et al. / Mathematical and Comuter Modelling 53 (2011) of acoustic waves in normal water is about m/s. Thus, the roagation delay in UW networks is several orders of magnitude higher than that in F networks. Another imortant issue is that, the carrier frequency of UW acoustic signals are tyically in the range of khz [1], while that of the F carrier is tyically in the range of GHz. Therefore, the bandwidth of UW networks is also a few orders of magnitude lower than that of the terrestrial wireless networks. This clearly means that the rotocols designed for F networks are unlikely to be directly alicable in UWN [2 4], necessitating that the network rotocols be re-looked for UWN. Similar to the terrestrial wireless networks, MAC (medium access control) schemes lay a very imortant role in shortrange UW wireless networks where the acoustic channel is used as a shared medium by many nodes. It is well known that, in a large and random deloyment setting with distributed control and bursty data, contention-free access rotocols, such as TDMA (time-division multile access), FDMA (frequency-division multile access), are not efficient. Moreover, energy-constrained sensor nodes could save energy by hysical event-driven communications, which can be a random henomenon. Therefore, for internodal communications random (contention-based) access rotocols are more aroriate. Many basic random access rotocols have been develoed for conventional multile access environments. The basic contention schemes are Aloha and slotted-aloha (S-Aloha). For longer bursts of data, CSMA and its two imortant variants CSMA with collision detection (CSMA/CD) and CSMA with collision avoidance (CSMA/CA) are used, which rovide a combination of random access and reservation. Naturally, these schemes would also be considered for UW multile access systems. Several recent works on UW multiaccess schemes underline that slight difference of internodal distance has areciable roagation delay difference, which in turn affects the erformance of a random access rotocol, and these findings are logical. A few variants of UW network random access rotocols have been roosed to mitigate the effects of long roagation delay. We rovide here a brief survey of rior works that are ertinent to our current study Prior work There have been some recent works on UWN multiaccess networks (e.g., [5 11]). Based on simulation studies of UWN it was suggested in [6] that the imum erformance of S-Aloha is the same as that of Aloha. The effects of internodal roagation delay on many-to-one Aloha and S-Aloha throughut erformance was studied via simulations in [7]. The Aloha erformance was shown to be unaffected by satial uncertainty. With a slot size equal to a (fixed) frame transmission time, their simulation results on S-Aloha showed the throughut degrades to that of Aloha at any roagation delay. Further, to enhance the S-Aloha erformance, the authors roosed to increase the slot size by some fractional amount. An analytic study of the many-to-one rotocols roosed in [7] was erformed in [9]. In [12], two Aloha based variants namely, Aloha with collision avoidance and Aloha with advance notification were roosed, where, a node uon overhearing the neighboring nodes communication, takes aroriate backoff measure so as to minimize the collision robability. It was qualitatively observed in the aer that simle Aloha as an UW random access rotocol could be inefficient, but it did not rovide any guideline as to how the basic Aloha and S-Aloha rotocols would erform under different oerating arameters. To counter the effect of UW roagation delay, TS/CTS (request-to-send/clear-to-send) based reservation rotocol was roosed in [8], where based on the roagation delay of TS frame and the data length information in it, the receiving node decides a receive window for a collision-free data frame recetion. In another work [10], communication between a master (gateway) node and the slave (non-gateway) nodes was considered, where searate channels for control (reservation) and data were suggested in TS/CTS handshake based reservation rotocol. The TS frames from the non-gateway nodes are sent using the Aloha rotocol, and until a desired CTS frame is received at a non-gateway node, it does not transmit its data frame. Note that, such schemes are efficient with relatively longer frames and infrequent transmissions. This rocess also ensures collision-free data transmission in a single-cell scenario. However, when smaller frames comarable to the size of TS CTS frames are transmitted frequently, such exlicit reservation mechanisms are clearly not efficient. This is also reflected in the rovisions of direct (without TS/CTS mechanism) data transmission in the IEEE standards. Further, the erformance of such a scheme may deteriorate in a multi-cell scenario, where a gateway node may be reachable from the nodes outside its cell boundary. Thus, while reservation based multiaccess rotocols, such as CSMA/CA with TS/CTS, may offer a higher throughut, basic Aloha rotocols would be of interest in situations where the return channel for reservation is unavailable or infeasible to use. In other words, basic Aloha rotocols are exected to be used in UW communications for short frame transmissions or as a reservation rotocol for suorting longer sessions (as in [10], similar to the contention-based channel access in wireless LANs and for aging in the GSM cellular systems) Contribution In this aer, we rovide a detailed theoretical basis for the erformance evaluation of two basic random access rotocols, namely Aloha and S-Aloha for one-to-one communications in underwater environment. Our secific contributions are as follows: (a) We derive generalized throughut erformance exressions for Aloha with a random internodal delay setting, and with fixed as well as variable frame size, which can be used in UWN as well as F networks. Our analysis and simulations show that ure Aloha erformance is indeed indeendent of signal roagation seed. Note that, while the Aloha-uw

3 S. De et al. / Mathematical and Comuter Modelling 53 (2011) erformance with a fixed internodal roagation delay is rather aarent, the outcome in a random delay is not so obvious. (b) When roagation delay is non-negligible, we suggest that the slot size in a randomly deloyed network be dictated by the imum roagation delay within a nodal coverage range, which will achieve an equivalence of one-slot vulnerability as in traditional S-Aloha-rf. This also imly that, the condition for a higher throughut erformance of S-Aloha-uw with resect to Aloha is governed by the nodal coverage range. (c) To imrove the S-Aloha-uw erformance we further roose a modified slotting concet, where for a given communication range, slot size can be aroriately chosen as a function of the frame size. Via a closed form analysis suorted by simulations we demonstrate that, an otimal choice of slot size can lead u to 17% throughut erformance gain with resect to the naive slotting decision. Note that, although the concet of modified S-Aloha resented in the aer is intuitive, the exact analytic roof of throughut erformance gain is rather involved. The objectives in this aer match closely with that of [6,7,9]. However, in contrast with these studies, we rovide analyses of one-to-one Aloha-uw erformance under a random node deloyment in an ad hoc network setting, and S-Alohauw for any value of internodal roagation delay. Our analysis aroach is different from that rovided in [9] for many-toone Aloha rotocols. The slotting aroach in S-Aloha-uw roosed in this aer is different in that, instead of one frame transmission time as the slot size, we roose to have a slot size which is the sum of and the imum internodal roagation delay T, where T can be of any value such that T < or, = or, >. To increase the throughut efficiency of S-Aloha-uw, we roose and analyze an otimum slot size reduction factor k. A reliminary version of the work was resented in [13] Paer organization The remainder of the aer is organized as follows. General assumtions, definitions, and a list of major notations used throughout the aer are rovided in Section 2. The throughut erformance analysis of Aloha in UW environment is resented in Section 3. Section 4 contains S-Aloha-uw slotting concet that achieves S-Aloha-rf equivalent vulnerability duration, and the throughut analysis. Our roosed modified S-Aloha-uw is resented and analyzed in Section 5. Numerical and simulation results and remarks are rovided in Section 6. Section 7 concludes the aer. 2. Assumtions, definitions, and notations The following assumtions and definitions are used in our subsequent discussion. 1. The network consists of homogeneous nodes, with all nodes having equal communication range. That is, irresective of the underwater nodes temoral and satial locations, nodal coverage range and signal roagation seed are considered fixed. 2. Nodes in the network are uniformly randomly distributed. Besides ensuring that the internodal roagation delay is a random number, uniform random distribution of node locations simlifies the comutation of collision robability in modified S-Aloha-uw. 3. Internodal communications are event-driven, which is considered random. This random traffic arrival rocess, including the backlog retries, is aroximated as Poisson distributed with a rate indeendent of the state of the network. Poisson (memoryless) arrival rocess with state-indeendent rate hels simlify the erformance evaluation of Aloha rotocols. 4. A node outside the communication range is unreachable. Physical channel related frame errors are discounted. A frame can be corruted and lost due to MAC level collisions only. 5. Temoral variability of internodal roagation delay due to underwater current is not accounted. 6. Throughut erformance is measured in terms of normalized system throughut, defined as the average number of successful frames in the network er average frame transmission time. Note that, location and time deendent variability of nodal coverage range and signal roagation seed could be more ractical considerations. However, there has not been any suitable model available to characterize them. Also, some other assumtions, namely, 2, 3, and 4 can be relaxed, but the rimary claims with these relaxations will remain unchanged, although the analysis will be more comlicated. Major notations used in the aer are listed in Table Aloha in UWN In this section, we analyze the ure Aloha rotocol erformance in UWN considering fixed as well as exonentially distributed frame size Aloha-uw with fixed frame size In Fig. 1, the collision vulnerability windows in short-range Aloha-rf and Aloha-uw multiaccess schemes, resectively, are shown. Note from Fig. 1(a) that, in short-range F communications, such as in mobile ad hoc networks, F wireless sensor

4 2096 S. De et al. / Mathematical and Comuter Modelling 53 (2011) Table 1 Summary of notations. Nodal communication (transmit/receive) range v Acoustic signal roagation seed c F signal roagation seed λ Frame arrival rate in the system er unit time η Normalized system (network) throughut F Frame length c Channel rate Frame transmission time; = F c T Signal roagation delay, a function of transmitter receiver distance r;t = r v ;r T Maximum signal roagation delay; T = v T rf s Slot size in S-Aloha-rf; T rf s = T uw s, T s1 Slot size in S-Aloha-uw, or slot size in modified S-Aloha-uw with k = 1; T uw s = + T = T s1 T sk Slot size in ms-aloha-uw (modified S-Aloha-uw) with 0 k 1; T sk = + kt k Slot size reduction factor in ms-aloha-uw; 0 k 1 r Distance of the receiver from a neighboring transmitter, that has a frame in revious slot r n Distance of the receiver from a neighboring transmitter, that has a frame in next slot n Number of frames scheduled in revious slot n n Number of frames scheduled in next slot a b Fig. 1. Collision vulnerability window in Aloha rotocols. (a) Terrestrial short-range F wireless network; (b) UW networks. networks, and urban cellular wireless networks, where internodal roagation delay is negligible, irresective of the nodal coverage range, the vulnerability window is 2. In such F wireless networks, since the internodal roagation delay of the F signal is insignificant comared to, the collision robability of a frame is simly the robability of a frame arrival in the window of size 2. Accordingly, with Poisson distributed traffic arrival rocess in the network at a total rate λ er unit time, the normalized system throughut in Aloha-rf scheme with fixed frame size is: η (fixed) Aloha-rf = λ Pr[no collision with any other frame] = λ e 2λ. In UW networks, on the other hand, due to areciable signal roagation delay T comared to, the collision vulnerability window is larger than 2. Particularly, it can be observed from Fig. 1(b) that, an additional frame generated at a neighboring transmitter at time t, where t + T T t t + T +, may lead to collision with a frame that is being received at the receiver in question from time t +T. That is, irresective of the receiver s distance from its transmitter, traffic generated from the neighboring transmitters within a time window 2 + T can cause collision at the receiver, where T = is the imum roagation delay u to a node s communication range, and v is the underwater acoustic v signal roagation seed. However, unlike in short-range F networks, only some of the neighborhood generated frames in the interval 2 + T will lead to collision with a frame that is currently being received. Secifically, referring to Fig. 1(b), a collision with the frame currently being received at time t + T occurs if the frame generation instant t at a neighboring transmitter and the associated roagation delay T u to the receiver in question satisfy either of the two conditions in (1). t + T < t + T < t + T + (1a) or, t + T < t + T + < t + T +. (1b) Looking from the receiver s ersective, as long as its frame recetion duration does not overla with any other frame arrivals from its neighbors, the frame will be successful. Thus, a frame of size, whose recetion starts at time t + T, will

5 S. De et al. / Mathematical and Comuter Modelling 53 (2011) Fig. 2. Pictorial reresentation of collision vulnerability concet. be successful if no additional arrival at the receiver occurs during the interval 2 (from t + T to t + T + ), even though the ossible arrivals during this time could be caused by the generation rocess over a larger time duration (which is 2 + T in case of UWN). This concet is further ictorially deicted in Fig. 2, where the duration t 1 t 3 is the frame generation window causing ossible collision vulnerability, and t 2 t 3 is the vulnerability window with resect to the recetion rocess. Consider the number of frames that arrive in window [t 2, t 3 ] = m, and the generated ones during [t 1, t 3 ] = n. Irresective of the signal roagation delay, we have [14, Ch. 3] n Pr m out of n frames arrive during [t 2, t 3 ] = P n (m) = m (1 ) n m (2) m where = t 2 t 3 t 1 t 3 = 2 t 1 t 3. Since the frame generation rocess in the system is Poisson, the arrivals in the window [t 2, t 3 ] can also be aroximated as Poisson distributed, as follows. The frame arrival rate in the system is λ = n. In case of a t 1 t 3 homogeneous frame generation rocess, the window t 1 t 3 can be increased arbitrarily, leading to n and 0, keeing the roduct n = 2λ a constant. Hence, (2) can be aroximated as: n (n)m P n (m) e m! = e 2λT (2λ t) m. m! The frame success robability is, P n (0) = e 2λ. Therefore, the normalized system throughut of Aloha-uw with fixed frame size is given by: (3) η (fixed) Aloha-uw = λe 2λ which is the same as the Aloha-rf throughut, and is valid for any roagation delay. (4) 3.2. Aloha-uw with variable frame size Normalized system throughut in Aloha-rf with Poisson distributed arrival rocess and variable (exonentially distributed) frame size can be found as [15, Ch. 3]: η (ex) = Aloha-F λ Pr[system idle at the frame arrival instant] Pr[next interarrival time τ > current frame duration ] = λ e λ = λ e λ 0 0 Pr[τ > t = t] Pr[ = t] e λt 1 e t Tt dt = λ 1 + λ e λ. (5) Following a similar logic as in the case of Aloha-uw with fixed frame size, irresective of the signal roagation delay, the normalized system throughut η (ex) Aloha-uw is also given by (5). 4. Slotted-Aloha in UWN First, irresective of the nodal coverage range in a short-range F network, we have the throughut exression for S- Aloha-rf as [15, Ch. 4]: η S-Aloha-rf = λ e λ. In a UWN with randomly located nodes, roagation delay T of a frame to the receiver varies between 0 and T (see Fig. 3(b) and (c)). Since the synchronization in a slotted access rotocol is done at the transmitter nodes, to resemble the one-slot S-Aloha vulnerability concet as in short-range F communications, i.e., to ensure that a frame collision robability in S-Aloha-uw is only due to non-zero additional arrivals in one slot, a buffer time T is needed to accommodate the arrival uncertainty due to roagation delay. Thus, unlike in S-Aloha-rf, where the slot size is T rf s = (see Fig. 3(a)), the slot size in S-Aloha-uw should be T uw s = + T = T s1 (see Fig. 3(c)). Note from Fig. 3(b) that, a slot size T s1 = + T ensures that the frames generated in a slot do not collide with the ones generated in another slot. Also, S-Aloha-rf like frame success robability is achieved as long as T <. However, if T, more than one frame generated in a slot do (6)

6 2098 S. De et al. / Mathematical and Comuter Modelling 53 (2011) a b c Fig. 3. Slotting concets. (a) Slot size in S-Aloha-rf; (b) effect of signal roagation delay on the time lag between a frame transmission and its recetion rocess; (c) slot size in S-Aloha-uw. not necessarily cause a frame collision at the receiver. So, the S-Aloha-uw frame throughut for ad hoc networks has to be comuted differently for the two regimes of roagation delay. Case 1: T < The throughut comutation in this regime is done similarly as in S-Aloha-rf. Thus, the normalized system throughut in S-Aloha-uw for T < is given by: η S-Aloha-uw (T < ) = λ Pr[no additional arrival in a slot] = λ e λ( +T ). (7) Case 2: T Let the receiver s distance from its intended transmitter be r = r. The frame success robability P S of S-Aloha-uw is obtained from the conditional success robabilities as: P S = Pr[success r = r] (r) r=0 (8) where (r) = Pr[r = r] = Pr[intended transmitter s distance to the receiver, r = r]. In a network with uniformly random distributed nodes, if a transmitter receiver air is chosen indeendent of the distance between them, considering the receiver is at the centre of its circular communication range, a transmitter can be located at any oint in the circular region. Then, the density function (df) of the distance r between a transmitter and the receiver is: 2r f r (r) =, 0 r 2 (9) 0, elsewhere. Hence, (r) Pr[r r r + dr] = f r (r)dr = 2rdr The regime of T is further divided into two: (a) T 2 2. (10) 2, and (b) T > 2. Case 2-a: T With denoting the nodal communication range and v denoting the acoustic signal roagation seed, we have the following three sub-regions of r for which the conditional frame success robabilities are calculated searately as follows: In sub-region 1, where 0 r v, if n additional frames are generated in the same slot along with the intended frame, the frame will still be successful as long as all n other frames have roagation delay T r v +. Since T = r v, where r is a random variable (V) reresenting the distance of a neighboring transmitter from the intended receiver, the above condition reduces to r r + v. Accordingly, the conditional frame success robability is given by P S1a = Pr[r r + v] n Pr[n additional arrivals in a slot] = n=0 r + Tt v 2 n (λts1 ) n 1 e λt s1 = e λt s1 n! n=0 where, from (9), (r + v) 2, Pr[r 0 r T r + v] = 2 t v 1, v r 0, elsewhere. r+ v 2 (11) (12)

7 S. De et al. / Mathematical and Comuter Modelling 53 (2011) In sub-region 2, where v < r < v, the frame will be successful if there are no additional frames generated from any neighboring transmitters in the same slot. Accordingly, P S2a = e λt s1. Similarly as in sub-region 1, in sub-region 3, where v r, the intended frame will be successful if there are n additional frames in the same slot generated at a distance r such that 0 r r v. The conditional frame success robability is given by: P S3a = Pr[0 r r v] n (λt s1 ) n e λt s1 n! = n=0 r Tt v 2n (λt s1 ) n e λt s1 = e λt s1 n! n=0 [ ] r T 1 t v 2 where (r v) 2, Pr[0 r T r v] = 2 t v r 0, 0 r v 1, elsewhere. Using (8), the net frame success robability P Sa ( T 2 ) is obtained as: Tt v Tt v P Sa = P S1a (r) + P S2a (r) + P S3a (r). (16) r=0 v v Hence, the normalized system throughut is: η S-Aloha-uw ( T 2 ) = λ P Sa. Case 2-b: T > 2 In this case, sub-region 1 is 0 r v, where the success robability P S1b is given by (11). In sub-region 2, v < r < v. If n additional frames from neighboring transmitters are generated, of which n are from a distance r such that r r + v and n n are from a distance r such that 0 r r v, the intended frame to the receiver will still be successful. Thus, the frame success robability P S2b is given by P S2b = = n=0 n =0 n=0 n =0 n Pr[ r r + v] n Pr[0 r r v] n n (λt s1 ) n n! e λt s1 n r + Tt v 2 n r Tt v 2(n n ) (λt s1 ) n 1 e λt s1. (18) n! The sub-region 3 is v r, where the success robability P S3b is given by (14). Combining, the unconditional success robability is given by Tt v Tt v P Sb = P S1b (r) + P S2b (r) + r=0 r= v Hence, the normalized system throughut is obtained as: (13) (14) (15) (17) P S3b (r). (19) r= v η S-Aloha-uw T > 2 = λtt P Sb. (20) 5. A modified S-Aloha for UWN From the analysis in Section 4 it can be noted that, with the naive slotting concet in S-Aloha-uw, the slot size has to be larger than that in S-Aloha-rf by T =, in anticiation that a transmitter receiver air can be u to distance aart. v However, as deicted in Fig. 3(b) and (c), in most cases transmitter receiver airs are less than distance aart, and so a recetion rocess is comleted before the S-Aloha-uw slot ends. Note that, in one-to-one communication, after the frame recetion at a node is comleted, the system remains idle for the duration T T, thereby causing reduction in system throughut. It is also clear from (7) that, with T < and for a given λ, the higher the ratio T, the lesser the system throughut η S-Aloha-uw comared to η S-Aloha-rf in (6). Similar trends are exected at T (see (17) and (20)), which

8 2100 S. De et al. / Mathematical and Comuter Modelling 53 (2011) a b Fig. 4. Modified slotting concet in UWN. T sk = + kt, where 0 k 1 and T <. (a) A frame from a r 1 distance away transmitter scheduled in the revious slot may cause collision with a frame in the current slot if k < r 1 <. (b) A frame in the current slot from a r 2 distance away transmitter may encounter collision with a frame scheduled in the next slot if k < r 2 <. (c) A frame scheduled from a r 3 distance away transmitter, 0 < r 1 < k, does not cause collision with the frames in other slots. c are resented in Section 6. In ad hoc network multiaccess communication, other than having reduced system throughut, no additional intuition is derived from the cases of T. Therefore, we restrict our further studies on S-Aloha-uw to T <. Since it is likely that almost in all cases r <, it may be wise to reduce the slot size aroriately, such that the frames in most cases are successful, while in some cases they may collide with the receding or/and subsequent frames. An otimally chosen slot size would minimize the system idling time without increasing the collision vulnerability, so as to increase the overall system throughut. We call this modified slotted-aloha rotocol as ms-aloha-uw. The modified slotting concet with T < is shown in Fig. 4. In this aroach, the buffer time to accommodate the transmitter-to-receiver roagation delay is reduced to kt, where k is termed as the slot size reduction factor. Since 0 k 1, we have kt T, and hence the modified total slot size T sk = + kt T s1. Note that, k = 0 corresonds to the slot size in S-Aloha-rf, but it will introduce frame vulnerability in S-Aloha-uw from the revious slot as well as the next slot; whereas k = 1 corresonds to the naive S-Aloha-uw, in which case there would not be any collision with frames from any other slots. The throughut of ms-aloha-uw can be comuted using the general exression for the success robability P S given in (8), where the V r (now identified as r i ) is the distance of the transmitter that has a frame scheduled in the current slot (slot i) to an intended receiver. However, in addition to the collision robability due to more than one frame scheduled in slot i (i.e., more than one arrivals in slot i 1), deending on the value of k, two conditions for a frame collision exist. For k 0.5, a frame transmitted in slot i can be vulnerable simultaneously due to the neighboring nodes transmissions in the two adjacent slots i 1 and i+1; whereas, for k 0.5, vulnerability of a frame can be caused by a transmission in either the revious slot i 1 or the next slot i + 1. Accordingly, the successful recetion robability of a frame is comuted differently for 0 k 0.5 and 0.5 k 1.0. In each of these two cases, the frame success robability varies at different windows of r i = r. For examle, with k 0.5, at r i = r, 0 r k, a frame recetion beginning in slot i is successful if there is only one frame scheduled in slot i, and there are ossibly n neighboring frame transmissions scheduled in slot i 1 but all of them have roagation delay T ( j) = T ( j) to the receiver in question such that T ( j) r+k, j n v. Since T ( j) = r v, the above condition reduces to r r + k, where r is an i.i.d. V reresenting the distance of the receiver in question from a neighboring transmitter that has a frame scheduled in slot i 1. So, the robability that the current frame does not collide with the one scheduled in slot i 1 is, Pr r r + k, given by (12) with v relaced by k. Likewise, the condition for no collision with a frame in the next slot (slot i + 1) is: T (n j) r k, j n v n, where T (n j) is the roagation delay u to the receiver from the j-th neighboring transmitter with a scheduled frame in slot

9 S. De et al. / Mathematical and Comuter Modelling 53 (2011) i + 1. For an areciable (non-zero) value of T (n j), the robability of no collision with a frame scheduled in slot i + 1 becomes: Pr r n r k, where r n is an i.i.d. V reresenting the receiver s distance from the neighboring transmitter. Pr r n r k = 1 Pr r n r k is obtained from (15) with v relaced by k. Considering all values of r in (0, ), the frame success robability for the two regimes of k is obtained below. Case 1: 0 k 0.5 In this range, the frame success robability is given by (21), k P S (0 k 0.5) = (0 (i) ) (n (i 1) ) Pr[r r + k] n (r) r=0 k + + k k n =0 (0 (i) ) (0 (i) ) n =0 n n =0 (n (i 1) n ) Pr[r r + k] n n =0 nn(r) (n (i+1) n ) Pr[r n r k] nn(r) (n (i+1) n ) Pr[r n r k] = P S 1 + P S 2 + P S 3 (21) where (f (j) ) = Pr[f frames scheduled in slot j], and (r) is defined in (10). Note that, in addition to accounting the ossibility of more than one arrival in the current slot, P S 1 catures the frame vulnerability due to arrivals in the revious slot, P S 2 absorbs the vulnerability due to arrivals in the revious slot as well as the next slot, whereas P S 3 accommodates the vulnerability due to arrivals in the next slot. With the assumtion of Poisson distributed traffic arrival rocess at a rate λ, and using (10) and (12), the exression for P S 1 in (21) is obtained as: P S 1 = k e λt sk r=0 n =0 = e 2λT sk λt sk (λt sk ) n e λt sk r + k n! 2n 2rdr 2 e 4λT skk 2 e 2 λt skk 2ke 2λT sk e 4λT skk 2 D + 2k λtsk e λt skk 2 D + k λtsk λtsk (22a) (22b) where D + (x) = e x2 x 0 et2 dt is the Dawson s integral [16, Ch. 7]. Using (10), (12) and (15), P S 2 is obtained as: P S 2 = k e λt sk k n =0 (λt sk ) n e λt sk r + 2n k (λt sk ) n n ] e λt nn sk (r k)2 2rdr [1 n! n n! 2 2 n n =0 = 2e 2λT sk (4λT sk k) 2 (4λT sk k(1 k) 1) e 4λT skk(1 k) 4λT sk k 2 1 e 4λT skk 2 Similarly, the exression for P S 3 is given by: P S 3 = e λt sk r= k n n =0 = e λt sk λt sk (λt sk ) n n ] e λt nn sk (r k)2 2rdr [1 n n! 2 2 e λt sk(1 2k) 2 e λt sk(1 k) 2 + πke λt sk erf λtsk (1 k) erf λtsk (1 2k) λtsk (23a) (23b) (24a) (24b) where erf(x) = 2 π x 0 e t2 dt. Using (22b), (23b) and (24b), the normalized system throughut for 0 k 0.5 is found as: η ms-aloha-uw (0 k 0.5) = λ (P S 1 + P S 2 + P S 3 ). (25) At a limiting case k 0, the integrations in (22a) and (24a) vanish, and thus, from (23a) and (25) the normalized system throughut is reduced to: lim η ms-aloha-uw = η 0 = ms-aloha-uw λe 2λ (26) k 0 which is the same as the Aloha throughut with fixed frame transmission time, given in (4).

10 2102 S. De et al. / Mathematical and Comuter Modelling 53 (2011) Case 2: 0.5 k 1.0 The frame success robability in this case is given by (27). P S (0.5 k 1.0) = k r=0 k + (0 (i) ) r= k n =0 (0 (i) )(r) + = P s 1 + P s 2 + P s 3. (n (i 1) ) Pr[r r + n k] (r) r=k (0 (i) ) n n =0 (n (i+1) n ) (Pr[r n r k]) n n (r) Note that, unlike in (21), P s 2 in (27) is reresents the vulnerability due to additional arrivals in the current slot (i.e., slot i) only. Using (12) and (15), similarly as in (22) (24), we have the exressions for P s 1, P s 2, and P s 3 : (27) P s 1 = k r=0 = e 2λT sk λt sk e 2λT sk e λt sk r+k 2 2rdr 2 e λt sk e 2 λt skk 2k e λt sk D + λtsk e λt sk(2 k 2 ) D+ k λtsk λtsk (28a) (28b) k P s 2 = e λt sk 2rdr = e λt sk (2k 1) (29) r= k 2 2 P s 3 = e λt sk e λt r k sk 2rdr r=k 2 = e λt 1 e λt sk(1 k) 2 sk π + k erf λtsk(1 k). λt sk λt sk (30a) (30b) The corresonding normalized system throughut is: η ms-aloha-uw (0.5 k 1.0) = λ (P s 1 + P s 2 + P s 3 ). (31) Again, in the limit k 1, the integrations in (28a) and (30a) reduce to 0, and hence, from (29) and (31) the normalized system throughut becomes: lim η ms-aloha-uw = η 1 = ms-aloha-uw λe λ( +T ) (32) k 1 which is the same as the naive S-Aloha-uw throughut erformance given in (7) Validity of the analysis in short distance F wireless environment Let us now check how the ms-aloha-uw throughut analysis alies to the S-Aloha-rf case. Since in F wireless communication the signal roagation seed c underwater acoustic signal roagation seed v, the slot size is T sk = + k T ( j) c, for any value of k. Also, the roagation delay associated with a transmitter receiver distance r is r c r+k v in a short distance F wireless communication becomes T ( j) 0. Accordingly, the condition 0, j n, and hence, Pr[T r+k c ] 1, which imlies, Pr[r r + k] can be relaced by 1. Likewise, since T (n j) 0, j n n, for any value of k and r, Pr[r n r k] can be relaced by 1. With the above reduced exressions, irresective of the value of k, from (8), the robability of a frame success in any slot i is given by, P S = r=0 Pr[no other frame scheduled in slot i] Pr[r i = r] = which leads to the same normalized throughut exression as in (6). In the following Section, relative throughut erformance results are discussed. e λt 2rdr t = e λ (33) r=0 2

11 S. De et al. / Mathematical and Comuter Modelling 53 (2011) Fig. 5. Performance comarison of Aloha and S-Aloha with fixed frame size. F = 40 Bytes, = 20 m. 6. esults and discussion System throughut erformance of the Aloha variants in UW as well as F wireless networks have been studied in MATLAB using the analytic exressions develoed in Sections 3 5, and via C based discrete event simulations of a random network. We have not used a standard network simulator in this study for the following reasons: (a) Our current study has been rather focussed on MAC layer only; it does not involve multile layers or the system as a whole. Also, as it is aarent from the analytic roofs, the isolated underwater MAC layer roblem itself is quite involved. (b) While the basic underwater MAC characterization remains the same, we anticiate that, besides the roagation seed, there might be some imact of channel characteristics on the multiaccess erformance, which could be tested using a standard network simulator. However, the standard simulators do not have a ractical underwater channel model built in. In fact, to the best of our knowledge, a thorough characterization of (variability of) underwater wireless channel for network alication is yet to be available. Therefore, our underwater network simulation studies focussed on the effect of random roagation delay in addition to the random acket arrival rocess on the system erformance. In the numerical comutations and network simulation studies, following the underwater modem secifications [1], the channel rate was considered c = 16 kbs. Acoustic signal seed is v = 1500 m/s. The nodes were assumed to have homogeneous circular communication range, and they can have any-to-any communication. Default value of communication range was = 20 m. Since the TS frame size is 36 Bytes (as in b/g standard without interoerability) or 44 Bytes (as in b/g standard with interoerability), the default frame size was taken as an average, F = 40 Bytes. Also, as allowed in standard sensor motes (e.g., Crossbow MICA2 motes), the largest frame size taken was F = 240 Bytes. For S-Aloha, the value of was chosen aroriately to ensure if the imum internodal signal roagation delay T = is less than, v or equal to, or greater than the frame transmission time = F. c In the simulation, to study the effect of frame collisions at a receiving node, N = 200 randomly located nodes were taken around the receiving node s communication range. In each iteration, a randomly located transmitter was chosen, and the other neighboring transmitters activities were controlled by varying the (Poisson distributed) frame arrival rate λ 0 at a node, where λ 0 and the system-wide arrival rate λ are related as λ = Nλ 0. These neighboring transmitters may send data to their chosen resective receivers. To comute the throughut erformance, for every desired frame recetion, we checked for any ossible time overla with the frames that may have been generated from the neighboring transmitters. For each set of arameters, average erformance was comuted over 5000 iterations to obtain sufficiently high confidence over the simulated data. In Fig. 5, throughut erformances of Aloha and S-Aloha are comared when alied in short-range F networks and UW networks, resectively, with constant sized frames. The analytic observations match very well with the simulation results. Matched results of Aloha-uw and Aloha-rf confirm that the signal roagation seed does not have an effect on the Aloha throughut erformance. However, the sensitivity of roagation delay in S-Aloha quite aarent, as S-Aloha-uw erforms oorer comared to the S-Aloha-rf. The erformance degradation is more rominent because of the chosen high T (13.3 ms), which is comarable to the value of (20 ms). Note, the analysis indicates that, under the condition > T the S-Aloha-uw throughut erformance would be in between S-Aloha-rf and Aloha-rf. The Aloha-uw erformance with variable (exonentially distributed) frame size is shown in Fig. 6, which further verifies the lack of sensitivity of signal roagation delay on ure Aloha erformance. The deendence of the ratio T on S-Aloha-uw erformance is aarent from the simulated imum throughut results in Fig. 7, where the communication range is ket fixed, but the frame size is varied. While the Aloha and S- Aloha-rf erformances are fairly constant (nearly and 0.368, resectively), S-Aloha-uw erformance imroves as,

12 2104 S. De et al. / Mathematical and Comuter Modelling 53 (2011) Fig. 6. Performance comarison of Aloha-uw and Aloha-rf with exonentially distributed frame size. Average frame size F = 128 Bytes. Fig. 7. Variation of imum throughut with frame size. = 20 m. the transmission time increasingly dominates over the roagation time. This is because, relatively less roagation delay imlies lesser system idling time in S-Aloha-uw. Fig. 8 further shows the nature of variation of imum system throughut for different values of T, in which our > 1. First, note that the imum throughut is monotonically decreasing articular interest is the region where T increases. This observation romted us to restrict our ms-aloha-uw studies to T as T 1, beyond which the erformance of simle Aloha will always be better. Second, the rate of decrease in imum throughut is not shar after T = 1, which is because, beyond this value there is a finite robability of receiving a frame correctly even though there could be more than one transmissions within the coverage range of a receiver. Throughut erformance of ms-aloha-uw at different values of slot reduction factor k are shown in Fig. 9. The lots indicate that, by choosing roerly reduced slot size (via controlling k) the underwater S-Aloha erformance can be significantly imroved. Note that k = 0 imlies the slot size T sk =, and it gives the same throughut erformance as in Aloha. This is because, having no buffer time, a frame recetion vulnerability duration becomes 2 (as in Aloha), and it can collide with a frame in the receding slot and/or the next slot. At the other extreme, with very high buffer time the sill over duration of an arriving frame beyond the slot boundary is minimized. But most of the time the frame arrivals to the receivers are comleted well within the slot time, thus leaving much room to system idling. The S-Aloha-rf throughut lot on the same grah also indicates that, due to added randomness in frame arrival rocess in acoustic wireless networks, ms-aloha-uw erformance is quite oorer, and the arrival rate corresonding to the eak erformance of ms-aloha-uw tends to that of Aloha. A good match of the analytically obtained lots with the simulated results also verify correctness of the analysis. In the subsequent discussion, we resent some analytic lots to show the conditions for imum system throughut.

13 S. De et al. / Mathematical and Comuter Modelling 53 (2011) Fig. 8. Variation of imum throughut with internodal distance. F = 40 Bytes, λ = 0.4. Fig. 9. ms-aloha-uw erformance at different values of k, and comarison with S-Aloha-rf rotocol. T Tt = 2 3. The deendence of the imum throughut erformance on slot size reduction factor k is shown in Fig. 10, where T (obtained by choosing suitable ) is taken as the arameter. Observe that, for a given T (i.e., for a given communication =, η ms-aloha-uw = (achieved = (when k = 1). Hence, ms-aloha-uw offers a 17.3% gain in imum throughut range ), there is an otimum k that offers the imum system throughut. At T at k = 0.52), whereas η S-Aloha-uw at T = by otimally choosing k, where the ercentage throughut gain is defined as: Gain = η ms-aloha-uw η S-Aloha-uw 100. η S-Aloha-uw (34) Gain at a smaller values of T is less, which is because of a smaller system idling ossibility with lesser T, and hence the room for imroved erformance at an otimal k is also less. The variation of imum system throughut as a function of T (by controlling the nodal communication range ), with k as arameter, is shown in Fig. 11. The lots clearly indicate the imortance of choosing right k for a given ratio T, because no articular value of k offers the highest throughut erformance as the roagation delay factor is increased. In Fig. 12, on the Y1 axis the otimum slot size reduction factor k that achieves η ms-aloha-uw is lotted with resect to T, which can be controlled either by varying or T. In conjunction, the ercentage throughut gain with resect to the t naive S-Aloha-uw (defined in (34)) at the k ot values is lotted on the Y2 axis. The lots further demonstrate that, while naive S-Aloha-uw does not offer a system throughut as good as in S-Aloha-rf, an otimal choice of slot size can offer an areciable increase in throughut, esecially for a large nodal coverage range.

14 2106 S. De et al. / Mathematical and Comuter Modelling 53 (2011) Fig. 10. Throughut imization via controlling k, with T as the arameter. Fig. 11. Variation of imum system throughut as a function of T Tt. Fig. 12. Otimum slot size reduction factor k ot for imum achievable throughut η ms-aloha-uw, and the corresonding imum throughut gain with resect to naive S-Aloha-uw, as a function of roagation delay to transmission delay ratio.

15 S. De et al. / Mathematical and Comuter Modelling 53 (2011) Conclusion In this aer, we have resented a theoretical framework for throughut erformance comutation of the basic random access rotocols, namely Aloha and S-Aloha, in underwater wireless networks with a random internodal signal roagation delay. We have shown that, ure Aloha throughut erformance does not have any imact, while S-Aloha does have a strong imact, of signal roagation seed. Further, we have roosed a new aggressive slotting concet, wherein the slot size can be otimally chosen such that, even by allowing some collisions due to overshooting the slot boundary, the overall system throughut can be significantly increased. The validity of our general analysis has been roven to hold for the secial cases of conventional underwater slotted-aloha as well as in short-range F roagation environments. Our analytic conclusions have been verified by discrete event simulations. The develoed framework in the current study could be useful to benchmark the erformance of advanced multiaccess rotocols in roagation delay intensive ad hoc networks. Acknowledgements This research was suorted by the Det. of Science and Technology (DST) under grant no. S/S3/EECE/054/2007 and the Council of Scientific and Industrial esearch (CSI) under grant no. 22/448/07/EM-II. eferences [1] LinkQuest Underwater Acoustic Modems Data Sheet (2009). htt:// [2] I.F. Akyildiz, D. Pomili, T. Melodia, Underwater acoustic sensor networks: research challenges, Elsevier Ad Hoc Networks 3 (2005) [3] J. Partan, J. Kurose, B. Levine, A survey of ractical issues in underwater networks, in: Proc. ACM Intl. Wks. Under-Water Networks, WUWNET, Los Angeles, CA, USA, 2006, [4] N. Parrish, L. Tracy, S. oy, P. Arabshahi, W.L.J. Fox, System design considerations for undersea networks: link and multile access rotocols, IEEE Journal on Selected Areas in Communications 26 (9) (2008) [5] G. Xie, J. Gibson, A networking rotocol for underwater acoustic networks, Tech. re. (Dec. 2000). [6] L. Vieira, J. Kong, U. Lee, M. Gerla, Analysis of Aloha rotocols for underwater acoustic sensor networks, in: Poster Abstract, Proc., ACM WUWNET, Los Angeles, CA, USA, [7] A. Syed, W. Ye, B. Krishnamachari, J. Heidemann, Understanding satio-temoral uncertainty in medium access with ALOHA rotocols, in: Proc. ACM Intl. wks. Under-Water Networks, WUWNET, Montreal, Quebec, Canada, 2007, [8] X. Guo, M.. Frater, M.J. yan, An adative roagation-delay-tolerant MAC rotocol for underwater acoustic sensor networks, in: Proc. IEEE OCEANS 2007 Euroe, Aberdeen, Scotland, 2007, [9] J. Ahn, B. Krishnamachari, Performance of roagation delay tolerant aloha rotocol for underwater wireless networks, in: Proc. Intl. Conf. Distributed Comuting in Sensor Systems, DCOSS, Santorini Island, Greece, [10] L.T. Tracy, S. oy, A reservation mac rotocol for ad-hoc underwater acoustic sensor networks, in: Proc. ACM Intl. Wks. Wireless Network Testbeds, Exerimental Evaluation and Characterization, WuWNeT 08, San Francisco, CA, USA, 2008, [11] N. Chirdchoo, W.-S. Soh, K.C. Chua, IPT: A receiver-initiated reservation-based rotocol for underwater acoustic networks, IEEE Journal on Selected Areas in Communications 26 (9) (2008) [12] N. Chirdchoo, W.-S. Soh, K.C. Chua, Aloha-based MAC rotocols with collision avoidance for underwater acoustic networks, in: Proc. IEEE INFOCOM Minisymosium, Anchorage, AK, USA, [13] P. Mandal, S. De, S.S. Chakraborty, Characterization of aloha in underwater wireless networks, in: Proc. Nat. Conf. Commun., Chennai, India, [14] A. Paoulis, Probability, andom Variables, and Stochastic Processes, 3rd ed., McGraw-Hill International Editions, [15] D. Bertsekas,. Gallager, Data Networks, 2nd ed., Prentice Hall, [16] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1970.