Int J Wireless Inf Networks DOI 10.1007/s10776-011-0163-z
The Temporal Effect of Mobility on Path Length in MANET Jyoti Prakash Singh • Paramartha Dutta
Received: 24 February 2011 / Accepted: 13 July 2011 Ó Springer Science+Business Media, LLC 2011
Abstract Ad hoc network consists of a set of identical nodes that move freely and independently and communicate among themselves via wireless links. The most interesting feature of this network is that they do not require any existing infrastructure of central administration and hence is very suitable for temporary communication links in an emergency situation. This flexibility, however, is achieved at a price of communication uncertainty induced due to frequent topology changes. In this article, we have tried to identify the system dynamics using the proven concepts of time series modeling. Here, we have analyzed variation of path length between a particular source destination pair nodes over a fixed area for different mobility patterns under different routing algorithm. We have considered four different mobility models—(i) GaussMarkov mobility model, (ii) Manhattan Grid mobility model and (iii) Random Way Point mobility model and (iv) Reference Point Group mobility model. The routing protocols under which, we carried out our experiments are (i) Ad hoc On demand Distance Vector routing (AODV), (ii) Destination Sequenced Distance Vector routing (DSDV) and (iii) Dynamic Source Routing (DSR). The path length between two particular nodes behaves as a random variable for all mobility models for all routing algorithms. The pattern of path length for every combination of mobility model and for every routing protocol can J. P. Singh (&) Department of Information Technology, National Institute of Technology, Patna, Bihar, India e-mail:
[email protected] P. Dutta Department of Computer and System Sciences, Visva-Bharati University, Santiniketan, West Bengal, India e-mail:
[email protected]
be well modeled as an autoregressive model of order p i.e. AR(p). The order p is estimated and it is found that most of them are of order unity only. We also calculate the average path length for all mobility models and for all routing algorithms. Keywords Ad hoc network Mobility modeling Path length Time series analysis Autoregressive modeling Autocorrelation
1 Introduction Multi-hop wireless networks [1, 2], commonly referred to as ad hoc wireless networks do not require a fixed infrastructure because a mobile node can relay packets to another node without using base stations. The nodes are mobile and changing locations regularly. The transmission range of a wireless node is the maximum distance within which the signals received from the node can be used to extract meaningful information. Two nodes within their transmission range can communicate directly and we say a link is up between them. If two nodes are not within each others transmission range, they have to communicate using a number of links involving one or more intermediate nodes. The ordered list of links between a source and destination is called a path. The path length between a source destination pair is the total number of links between that said source destination pair. Since mobile nodes operate with limited battery power, they are often configured to work on a smaller transmission range. So, most of the time the length of path is more than one. A node in mobile ad-hoc network (MANET) is moving and its neighbourhood varies constantly with time and place. Due to change of this neighbourhood, the link between two nodes may be up and down. Due to variation in
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link condition, the path length varies constantly. The mobility of nodes makes fixed multi-hop routing very difficult in MANETs. In a dynamically changing environment, the position of the nodes changes over a period of time. As a result, a route determined between any pair of nodes breaks frequently and has to be re-discovered. The length of the rediscovered routes is normally different from the earlier one. In this article, we have modelled the variation of path length for a particular path between a source destination pair over time. If we look at the previous path lengths of a particular path, we may find that variation of path length shows a specific pattern and this pattern may be modelled in terms of time series. This time series modelling may be used for the purpose of prediction of future path lengths. The predicted values of path lengths may help us in designing QoS routing algorithms. Let N = {Ni | 1 B i B n} represent the collection of nodes in the system. A duplex link Lij is formed between two nodes say Ni and Nj if and only if Ni is neighbour of Nj and Nj is neighbour of Ni. The length Pijt of a path between node Ni and Nj is changing constantly with time t. At every instant some new nodes are coming into the transmission range of Ni and Nj whereas some old neighbour nodes are leaving the transmission range of Ni and Nj thereby leading to change in the neighbourhood of the nodes Ni and Nj. Currently we have considered four mobility models—(i) Gauss Markov mobility [3, 4] , (ii) Manhattan Grid mobility, (iii) Random Way Point mobility [5, 6] and (iv) Reference Point Group mobility [7] models for our experiment. A survey of the most frequently used mobility models is presented in [8]. The main motivation of the present work is to find a relationship among the path lengths of a particular path Pij between two nodes Ni and Nj in an ad hoc network. This information can be utilized to other areas such as augmenting the performance of existing routing protocols to include QoS parameters. However, these issues are kept out of the purview of present endeavour in order to avoid shift in the focus of the present article. The rest of this paper is organized as follows. Section 2 contains a brief survey of the relevant works. Section 3 covers our proposal for using autoregressive model AR(p) of order p. We also discuss the techniques to find out the coefficients of AR(p) model as well as how to determine the order p of the AR(p) model in this section. In Section 4, we provide the simulation results to supplement our proposal of modelling the path length using AR(p) model. Section 5 describes the forecasted values using the fitted AR model. In Section 6, we conclude and suggest future directions.
2 Related Works MANETs are usually deployed in myriads of scenarios having complex node mobility and connectivity dynamics.
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For example, in a MANET on a battlefield, the movement of the soldiers will be influenced by the commander. In a city-wide MANET, the node movement is restricted by obstacles or maps. The node mobility characteristics are very application specific. Widely varying mobility characteristics are expected to have a significant impact on the performance of the routing protocols like DSR [9], DSDV [10] and AODV [11]. Bai et al. [12] proposed a framework for analyzing the Impact of Mobility on the Performance Of RouTing protocols in Adhoc NeTworks abbreviated as (IMPORTANT) framework. Through this framework they illustrated the importance of mobility modelling in routing performance and understanding the mechanism of ad hoc routing protocols. They presented a reasonable explanation as to why some routing protocols behave differently under different mobility patterns. To show this difference and supplement their argument, they used a number of useful metrics such as degree of spatial dependence, degree of temporal dependence, connectivity graph, link duration etc. They also provided a reasonable explanation as to why mobility affects routing performance. However they failed to give a satisfactory explanation for why routing protocols behave differently even for same mobility pattern. Helmy et al. [13, 14] tried to answer the question ‘‘why routing protocols behave differently even for same mobility pattern?’’ They tried to answer this question by decomposing a protocol into its protocol mechanistic building blocks. They used reactive protocols like DSR [9] and AODV [11] to explain this. In their work, they decomposed a protocol into its mechanistic building blocks where each mechanistic building block is used to implement a specific function. Then the effect of different mobility patterns on each building block is evaluated. Their approach provides a methodology to justify the contribution of each mechanistic building block on overall protocol performance under various mobility patterns. Time series [15, 16] modelling is drawing a lot of attention in the modelling of internet traffic, wireless sensor and ad hoc network traffic. Basu et al. [17] model the internet traffic using the Autoregressive Moving Average Model of order p and q (ARMA(p,q)) model. Using this model, they forecast the traffic which was generated by a TCP source using FDDI protocol. They also develop a system to generate synthetic traffic which could be useful for simulation studies of internet traffic and in resource management algorithms. Singh and Dutta [18, 19] pointed out the importance of neighbour count of a mobile node and modelled that information using autoregressive model. They showed through extensive simulation that the distribution of neighbour count of a node Ni under a threshold value of speed, range and sampling time for different mobility models are well correlated and can be represented by AR(p) model for suitable choice of p. However, beyond
Int J Wireless Inf Networks
that threshold value, the autocorrelation of node distribution does not remain significant enough. They also calculated the order of autoregressive model and used that model to predict the neighbour count values.
suggests that the past p values rt-i ði ¼ 1; . . .; pÞ jointly determine the conditional expectation of the past data. The associated polynomial equation of the AR model, called characteristic equation, is given by xp /1 xp1 /2 xp2 /p ¼ 0
3 Proposed Model As we pointed out in Section 2, that the path length Pij between node Nj and Ni is changing with time and is a function of the path lengths Pij at previous instances. Our first concern is to study whether the path length under a specific mobility scenario varies for different routing protocols. If they do so, we try to find out the amount of variation and correlation for the path length under a specific mobility model using different routing protocol. We also attempt to answer as to why and how they are varying. We shall try to identify the system dynamics using the proven concepts of time series in this section. A time series [15, 16] is a sequence of observations that are arranged according to the time of their outcome. By recording and analysing the data, we can avail a better understanding of the data generating mechanism and make a prediction of future values. The main characteristic of a time series is that the data are often governed by a trend and they have inherent periodic components. An important part of the analysis of a time series [15, 16] is the selection of a suitable model (or class of models) fitting that data. Naturally, the more appropriate the model selection is, the better is the expected prediction. The path length Pij between nodes Ni and Nj is a parameter varying with time that appears to be a suitable candidate for such modelling. The relation of path lengths with the previous values is also supported with the autocorrelation function or the autocorrelation of nodes. The experimental data confirms that the autocorrelation is very high at initial lags and is constantly decreasing with higher lags. This implies that the current path length can be modelled as a function of values of Pij on previous instances with a white noise with mean zero and known variance r2. This behaviour of data can be well represented by an AR(p) model [15, 16]. An autoregressive AR(p) model is a type of random process which is often used to model and predict various types of natural phenomena. The autoregressive model is one of a group of linear prediction formulas that attempt to predict an output of a system based on the previous outputs. The AR(p) model can be defined as rt ¼ /0 þ /1 rt1 þ /2 rt2 þ þ /p rtp þ at
ð1Þ
where p is a nonnegative integer and /i 2 < are parameters of the AR(p), 0 B i B p model. at is a white noise sequence with mean zero and variance r2 and is independent of ri V i, t - 1 B i B t - p. This model
ð2Þ
The characteristic equation indicates that the plot of the autocorrelation, known as autocorrelation function, (ACF) of AR(p) model shows a mixture of damping sine and cosine patterns and exponential decays depending on the nature of its characteristic roots. This is in strong conformity with our experimental data. One major hurdle in representing data using AR(p) model is solving the AR coefficients and finding the appropriate value of p. A number of techniques exist for computing AR coefficients. The most common least squares method is based upon the Yule-Walker equations. We have used the Yule-Walker Equations to determine the AR coefficient for the arbitrary p. The Yule-Walker equation in matrix form is given as 3 2 3 2 3 2 r1 1 x1 a1 xN1 7 6 a2 7 6 r2 7 6 x 1 x x 2 N2 7 6 7 6 7 6 6 r3 7 6 x 2 6 7 x3 xN3 7 ð3Þ 7 6 a3 7 6 7¼6 6 .. 7 6 .. .. 7 6 .. 7 .. 4 . 5 4 . . 5 4 . 5 . xN1 xN2 rN aN 1 where xd represents the autocorrelation coefficient at lag d. We can find the xd values by the following method. Let us consider the general AR(p) model once again as given in Eq. 1. rt ¼ /0 þ /1 rt1 þ /2 rt2 þ þ /p rtp þ at
ð4Þ
Multiplying both side of the equation by rt-1 gives rt rt1 ¼
p X ð/j rt rtjþ1 Þ þ rt1 at þ rt1 /0
ð5Þ
t¼1
where j and t are term and time indices respectively. Considering expectations on both sides E½rt rt1 ¼
p X ð/j E½rt rtjþ1 Þ þ E½rt1 at þ E½rt1 /0
ð6Þ
t¼1
where the /j values are kept outside the purview of the expectation because they are parameters rather than random variables. It is to be noted that E[rt-1at] = 0 and E[rt-1]/0 = 0 because the random perturbation a of the current time is statistically uncorrelated with the previous values of the process. Therefore, we get E½rt rt1 ¼
p X ð/j E½rt rtjþ1 Þ
ð7Þ
t¼1
Next dividing by (N - 1) throughout and using the evenness of the auto-covariance i.e. c1 = c-1, we get
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c1 ¼
p X
/j cj1
ð8Þ
t¼1
Finally dividing throughout by c0 we get x1 ¼
p X
/j rj1
ð9Þ
t¼1
which gives x1 in terms of 1; x1 ; . . .; xN1 : Repeating the same process we get x2 in terms of x1 ; x2 ; x3 ; . . .; xN2 and so on. Finally writing all this together gives the YuleWalker equation given in Eq. 5 which can be solved to get the coefficients of AR(p) model. We can also get partial autocorrelation function using Yule-Walker equation. Partial autocorrelation plots are a commonly used tool for identifying the order of an autoregressive model. The partial autocorrelation of an AR(p) process becomes nearly zero on lag k where p\k 1: If the sample autocorrelation plot indicates that an AR model may be appropriate, then the sample partial autocorrelation plot is examined to help identify the order. To find out the order p of AR process, we look for the point p on the plot where the partial autocorrelations for all higher lags are essentially zero. Once that point p is found, we use Akaike Information Criterion (AIC) to confirm that point. The AIC is a way of selecting a model from among a set of models. The chosen model is one that minimizes the KullbackLibler distance metric between the models and the truth. AIC not only rewards goodness of fit, but also imposes a penalty that is an increasing function in respect of the number of estimable parameters. This penalty discourages over fitting. To apply AIC in practice, we start with a set of candidate models, and then find the models corresponding to AIC values. We identify the model with the minimum AIC value. Models having their AIC values in the range of 1 and 2 of the minimum have substantial support and should be considered in making inferences. This is how we fix order p of our model. The least square method finds the best estimate of unknown parameters, given a specific model. Here, the question is different. Here our problem is of fixing the most suitable model based on available data. Least square regression does not serve the purpose as that offered by AIC. The PACF of the data set representing path length Pij of link between nodes i and j moving under Gauss Markov mobility model and using AODV as routing protocol shown in Fig. 3. The AIC values of data set representing path length Pij between nodes i and j moving under Gauss Markov mobility model and using AODV as routing protocol are tabulated in Table 1. The PACF shown in Fig. 3 tells us that this data can be modelled using an AR(3) process and the AIC values also confirms that. Thus it is justified to take the value of p to be 3.
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Table 1 Finding the order of Autoregressive Model using AIC for path length using AODV routing under Gauss Markov Mobility pattern AIC value
Order of AR model
478.7405
1
475.9975
2
473.9003 475.6748
3 4
476.9659
5
4 Simulation Results Our first concern is to study whether the path length under a specific mobility scenario varies under different routing protocol. If they do so, we try to find out the amount of variation and correlation for the path length under a specific mobility model using different routing protocol. We also attempted to determine why and how they are varying. We use Bonn-Motion [20] for generating mobility scenarios. We generate four mobility patterns with 70 nodes moving in an area of 1200 m 9 1200 m for a period of 1000 s after ignoring the first 3600 s of node movement for each mobility pattern. It has been observed that with the Random way point model, nodes have a higher probability of being near the center of the simulation area, while they are initially uniformly distributed over the simulation area initially. Similarly, in Manhattan Grid model, all nodes start at (0,0) and then they are distributed over the simulation area. So, we skip 3600 s at the beginning to mitigate the boundary effects of node movement during simulation. The maximum speed Vmax of a node is set to 10 m/s. The minimum speed vmin of a node is always set to 0.5 m/s. The vmin is set to a positive value because Yoon and Liu [21] proved mathematically that the average speed of the nodes using Random way point mobility model decreases constantly and would eventually reach zero. One of their suggestion for getting rid of this problem was to use non zero minimum speed. This is what we followed here. The cbrgen tool which is a part of ns-2 [22] distribution is used to generate Constant Bit Rate traffic for 1000 s with 1 packet/s per source. The number of sources and destinations were chosen randomly by cbrgen tool. We have used ns-2 [22] for network simulation and traces are generated in new trace format. The path length values for each time period is calculated from those trace files using some AWK scripts. The computed values of path length are then taken to Minitab for further analysis. We test the pattern of path length for stationarity and then determined the autocorrelation function (ACF) and partial autocorrelation function (PACF). We analyze the path length for four mobility models (i) Gauss Markov, (ii) Manhattan Grid, (iii)
Int J Wireless Inf Networks
4.1 Analysis Under Gauss Markov Mobility Model Our first experiment deals with finding the average path length Pavg for mobile nodes moving under Gauss Markov mobility model for different routing algorithm. Gauss Markov mobility model is characterized by the twin parameters viz. speed s and direction d. The value of speed and direction at nth instance is calculated based upon the value of speed and direction at the (n - 1)st instance using Eqs. 10 and 11. sn ¼ asn1 þ ð1 aÞs þ qð1 a2 Þsxn1
ð10Þ
dn ¼ adn1 þ ð1 aÞd þ qð1 a2 Þdxn1
ð11Þ
where sn and dn are the new speed and direction at time interval n and a : 0 B a B 1 is the tuning parameter used to vary the randomness. s and d are constants representing the mean value of speed and direction as n ! 1. sxn1 and dxn1 are random variables following Gaussian distribution. For our experiment, we have taken a = 0.75 whereas sxn1 and dxn1 are chosen from N(0,1) distribution. Since the AR(p) model depends on the previous p instances and on a white noise that is normally distributed. The nodes following Gauss Markov mobility are also normally distributed, they are the suitable candidate for AR(p) model as supported by ACF shown in Fig. 3. As can be seen from Fig. 1 that the average path length is lower for nodes following DSDV routing compared to AODV and DSR routing moving under Gauss Markov mobility pattern. DSDV is table driven routing protocol and hence routes are subject to periodic updates and hence, they maintain the
Fig. 1 Variation of path length for different routing protocol using Gauss Markov Mobility pattern
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Random way point [5] and (iv) Reference Point Group Mobility models. We study the path length under three different routing protocols (i) AODV, (ii) DSDV and (iii) DSR.
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Fig. 2 Time series plot of path length for different routing protocol using Gauss Markov Mobility pattern
ACF for aodv_gm 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
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Fig. 3 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for AODV routing protocol using Gauss Markov Mobility pattern
shortest path. On the other hand DSR and AODV are reactive protocol and search for routes as and when packets are ready to be sent. Figure 2 illustrates the time series plot of path lengths for nodes using AODV [11], DSDV [10] and DSR [9] routing protocols. Figure 3 shows the autocorrelation function and partial autocorrelation functions of the path length values for nodes moving with Gauss Markov mobility model and using AODV routing protocol. The autocorrelation function and partial autocorrelation functions of the path length values for nodes moving with Gauss Markov mobility model and using DSDV and DSR routing protocols are depicted in Figs. 4 and 5. It is evident from Figs. 2, 3 and 4 that the path length values Pij between node i and j show a strong autocorrelation with high peaks at initial lags but decreases at higher lags. The characteristics of the autocorrelation functions shown in Figs. 2, 3, and 4 confirm that the Pij value between node i
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Int J Wireless Inf Networks ACF for dsdv_gm 1
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Fig. 6 Variation of path length for different routing protocol using Manhattan Grid Mobility pattern
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Fig. 4 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for DSDV routing protocol using Gauss Markov Mobility pattern
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ACF for dsr_gm 0.4 0.2 0 -0.2 -0.4
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aodv_man dsdv_man dsr_man
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Fig. 7 Time series plot of path length for different routing protocol using Manhattan Grid Mobility pattern
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Fig. 5 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for DSR routing protocol using Gauss Markov Mobility pattern
and j moving under Gauss Markov Mobility model can be fitted to AR(p) model with suitable value of p. 4.2 Analysis Under Manhattan Grid Mobility Model The second experiment deals with finding comparison of average path lengths Pij for Manhattan Grid mobility model for different routing algorithm. As can be seen from Figs. 6 and 7 that the average path length is higher for AODV compared to DSR and DSDV as was the case with Gauss Markov mobility model. But this time the differences in the path length values across routing protocols are not so prominent. This is because, in Manhattan Grid mobility model, the nodes move in a restrictive lane and hence the overhead of link break is comparatively low. The autocorrelation function of path length values for nodes using
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AODV [11] routing protocol and moving with Manhattan Grid mobility pattern is shown in Fig. 8. Figure 9 illustrates the autocorrelation function of path length values for nodes using DSDV [10] routing protocol and moving with Manhattan Grid mobility pattern. The autocorrelation function of path length values for nodes using DSR [9] routing protocol and moving with Manhattan Grid mobility pattern is shown in Fig. 10. In case of Manhattan Grid mobility model also the path length values Pij between node i and j show a strong autocorrelation with peaks at initial lag as evident from Figs. 8, 9 and 10. Hence, the path length values Pij between node i and j moving under this mobility models can also be modelled using AR(p) model with suitable p. 4.3 Analysis Under Reference Point Group Mobility Model The third experiment deals with finding the comparison of average path length Pij for Reference Point Group mobility model for different routing algorithm. As can be seen from Figs. 11 and 12 that the average path length is lower for
Int J Wireless Inf Networks
0.6 0.4 0.2 0 -0.2 -0.4 -0.6
ACF for aodv_man
ACF for dsr_man 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
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PACF for aodv_man
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Fig. 8 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for AODV routing protocol using Manhattan Grid Mobility pattern
Fig. 10 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for DSR routing protocol using Manhattan Grid Mobility pattern
ACF for dsdv_man 1
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Fig. 11 Variation of path length for different routing protocol using RPGM Mobility pattern
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Fig. 9 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for DSDV routing protocol using Manhattan Grid Mobility pattern
DSDV compared to AODV and DSR. In case of DSDV, the path length was almost constant and similar was the case for AODV routing but with higher value. The path length shows a strong fluctuation in case of DSR routing. The autocorrelation function of path length values for nodes using AODV [11] routing protocol and moving with Reference Point Group mobility pattern is shown in Fig. 13. Figure 14 plots the autocorrelation function and partial autocorrelation function of path lengths for nodes moving under RPGM mobility models and using DSR routing protocol. The ACF of path length for AODV and DSR routing shown in Figs. 13 and 14 respectively shows a constant decrease with increasing lags. We can not plot ACF of path length for DSDV routing as the path length values are almost constant.
4.4 Analysis Under Random Way Point Mobility Model The fourth experiment deals with finding the relationship between the average path length Pij for Random Way Point mobility model for different routing algorithm. As can be seen from Figs. 15 and 16 that the average path length is lower for DSDV compared to AODV and DSR. The autocorrelation function of path length values for nodes using AODV [11] routing protocol and moving with Random Way Point mobility pattern is shown in Fig. 17. Figure 18 and 19 plots the autocorrelation function and partial autocorrelation function of path lengths for nodes moving under Random way point mobility models and using DSDV and DSR routing protocols respectively. The autocorrelation function of path length values for nodes using DSR [9] routing protocol and moving with Random way point mobility pattern is shown in Fig. 16. As evident from Figs. 17, 18 and 19 that the path length values Pij between node i and j moving with Random way point
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aodv_rpgm (left) dsdv_rpgm (left) dsr_rpgm (right)
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Fig. 15 Variation of path length for different routing protocol using Random Way Point Mobility pattern
Fig. 12 Time series plot of path length for different routing protocol using RPGM Mobility pattern
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Fig. 16 Variation of path length for different routing protocol using Random Way Point Mobility pattern
lag
Fig. 13 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for AODV routing protocol using RPGM Mobility pattern ACF for dsr_rpgm 1
mobility model show a strong autocorrelation with peaks at initial lag and decreases with higher lags for. Hence, the path length values Pij between node i and j moving under this mobility model can also be modelled using AR(p) model with suitable p.
+- 1.96/T^0.5
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5 Forecasting Path Length Using AR Model
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Fig. 14 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for DSR routing protocol using RPGM Grid Mobility pattern
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We have used our model to forecast the next path length values. We have selected the path length data set moving under AODV routing protocol and following Gauss Markov Mobility model for forecasting. We have modeled the said data set using autoregressive model of order 3 as determined in Section 3 using PACF shown in Fig. 3 and AIC values given in Table 1. We have used the said model to forecast for next 10 values of the data set. The comparison of forecast and original data obtained is shown in Fig. 18. The actual data and the forecasted data values are very close to each other as evident from Fig. 20. We have formulated the following two hypothesis based on the
Int J Wireless Inf Networks ACF for dsr_rwp
ACF for aodv_rwp 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4
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Fig. 17 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for AODV routing protocol using Random Way Point Mobility pattern
Fig. 19 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for DSR routing protocol using Random Way Point Mobility pattern 9
forecasted actual
ACF for dsdv_rwp 1
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Fig. 18 Autocorrelation function (ACF) and Partial autocorrelation functions (PACF) of path length values for DSDV routing protocol using Random Way Point Mobility pattern
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Fig. 20 The forecasted data vs actual path length for AODV routing protocol using Gauss Markov Mobility pattern
6 Conclusion observed and expected data to validate our point statistically that the forecasted values are indeed close to actual value. H0: The expected values and predicted values match. H1: The expected values and predicted values do not match. We have used v2 test to test the hypothesis. The test 2 P iÞ : Here Oi refers to criterion for v2 test is v20 ¼ ni¼1 ðOi E Ei the observed values and Ei refers to the expected values. For the data shown in Fig. 18, the computed v20 value is 4.77 which is much smaller than the critical value 16.92 of v20:05;9 with 9 degree of freedom at 5 % significance level. Thus the hypothesis that the forecasted value is similar to actual value may be accepted with 95% confidence.
In this article, we compared the routing protocols with respect to their path lengths across different mobility models and find that DSDV achieves the shortest path length across all mobility models considered here. We also tried to model the path length distribution between two nodes using a Auto-regressive AR(p) model used in stationary time series analysis. We found through our experiments that path length distribution between two nodes for all four mobility models considered in this article are well correlated and can be represented by AR(p) model for suitable value of p. We have also predicated the path length between two nodes in future time frames and found that the prediction is close enough to the real values. These predicated values of path length may be used for routing with
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quality of service parameters. Currently, the authors are engaged in using the predicted path length to design a QoS based routing algorithm. We are also trying to supplement the existing routing protocols with more information to the finding of the present work.
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Author Biographies Jyoti Prakash Singh did his B.Tech. in Computer Science and Technology from Kalyani Government Engineering College, West Bengal, India in the year 2000. He completed his M.Tech. in Information Technology in the year 2005 from Sikkim Manipal Institute of Technology, Sikkim, India. He is currently an Assistant Professor in the Department of Information Technology in National Institute of Technology, Patna, Bihar, India. He has been visiting/guest faculty in Kalyani Government Engineering College, West Bengal, India. He has co-authored five books in the area of C programming, Data Structures and Operating systems. He has more than 25 research publications in various national and international journals and conference proceedings. His research interests include sensor ad hoc network, network security and data mining. He is Member of IEEE Computer Society, Computer Society of India, International Association of Engineers, Hong Kong, International Association of Computer and Information Technology Singapore. Paramartha Dutta did his Bachelors and Masters in Statistics from Indian Statistical Institute, Kolkata, India in 1988 and 1990 respectively. Subsequently, he did his Master of Technology in Computer Science in 1993 from Indian Statistical Institute, Kolkata, India. He completed his Ph.D. in Engineering in 2005 from Bengal Engineering and Science University, Shibpur, India while he was in service. He is currently a Professor in the Department of Computer and System Sciences, Visva-Bharati University, Santiniketan, West Bengal, India. He has been visiting/guest professor in (i) University of Kalyani, Nadia, West Bengal, India, (ii) Bengal Engineering and Science University, Shibpur, West Bengal, India, (iii) University of Tripura, India. He has co-authored four books and has one edited book to his credit. Apart
Int J Wireless Inf Networks from this he has more than 100 research publications in various national and international journals and conference proceedings. His research interests include evolutionary computation, soft and intelligent computing, pattern recognition and mobile computing. Dr. Dutta is a Fellow of Optical Society of India and Institution of Electronics and Telecommunication Engineering. He is Life Member of Indian Science Congress
Association, Computer Society of India, Indian affiliate of International Association of Pattern Recognition, Indian Society for Technical Education, Advanced Computing and Communication Society and International Association of Engineers, Hong Kong. He is also a Member of Association for Computing Machinery, USA and IEEE Computers Society USA.
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