Stochastic Properties of Mobility Models in
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Mobile Ad Hoc Networks Seema Bandyopadhyay, Member, IEEE, Edwrad J Coyle, Fellow, IEEE, and Tillmann Falck, Student Member, IEEE
S. Bandyopadhyay is with the Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611. Email:
[email protected]. E. J. Coyle is with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907. Email:
[email protected]. T. Falck is with the Department of Electrical Engineering, Ruhr-Universität Bochum, Germany. Email:
[email protected].
Abstract The stochastic model assumed to govern the mobility of nodes in a mobile ad hoc network have been shown to significantly affect the network’s coverage, maximum throughput, and achievable throughputdelay tradeoffs. In this paper, we compare several mobility models, including the random walk, random waypoint and Manhattan models, on the basis of the number of states visited in a fixed time, the time to visit every state in a region, and the effect of the number of wandering nodes on the time to first entrance to a set of states. These metrics for a mobility model are useful for assessing the achievable event detection rates in surveillance applications where wireless-sensor-equipped vehicles are used to detect events of interest in a city. We also consider mobility models based on Correlated Random Walks, which can account for time dependency, geographical restrictions, and nonzero drift. We demonstrate that these models are analytically tractable by using a matrix-analytic approach to derive new, closedform results in both the time- and transform-domains for the probability that a node is at any location at any time for both semi-infinite and finite one-dimensional lattices. We also derive first entrance time distributions for these walks. We find that a correlated random walk (i) covers more ground in a given amount of time and takes a smaller amount of time to cover an area completely than a random walk with the same average transition rate; (ii) has a smaller first entrance time to small sets of states than the random waypoint and random walk models and (iii) leads to a uniform distribution of nodes (except at the boundaries) in steady state.
Index Terms MANET, mobility models, correlated random walk, random walk, random waypoint model.
I. I NTRODUCTION A mobile ad-hoc network (MANET) consists of mobile nodes that rely only on wireless connections to communicate with each other. These networks are still being developed, so few of 2
them have actually been deployed. Researchers have thus had to rely on analysis and simulations to predict how these networks will perform. From this preliminary work, it is clear that the mobility patterns of the nodes will have a very significant impact on the network coverage, maximum throughput and achievable throughput-delay tradeoffs. The accuracy of the predictions made about network performance in real-life scenarios will depend on how well the mobility models used in the simulations captured the real-life behavior of nodes. It is therefore important to have realistic mobility models. Hence, many mobility models have been proposed in the literature. A survey of mobility models that have been proposed for MANETs can be found in [1], [2]. These models vary widely in terms of their complexity, analytical tractability, correlation structure and restrictions on movements. No single one, however, is detailed enough to be appropriate in all situations. Several past studies [3]–[5] have explored the impact of mobility models on network performance and have found that they significantly affect the protocol performance and delay-capacity tradeoffs. In this paper, we compare widely used mobility models like the random walk, random waypoint and manhattan mobility models in terms of the number of states visited in a fixed time, the time to visit every state in a region, and the effect of the number of wandering nodes on the time to first entrance to a set of states. The full coverage time and first entry times of a mobility model are useful for assessing the achievable event detection rates in surveillance applications where wireless-sensor-equipped vehicles are used to detect events of interest in a city. These results provide guidelines for the density with which the nodes should be distributed in an area to achieve event detection within a specified amount of time, given the information about their mobility behavior.
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Fig. 1.
Model for streets in a city. The horizontal and vertical motions of the node combine to create diagonal movements
along the solid lines that are shown. These diagonal solid lines form the grid of city streets.
We also propose and investigate a Correlated Random Walk (CRW) based mobility model that is designed to characterize the movements of nodes, such as wireless-equipped vehicles, on the streets of a city. The streets are modeled as a finite, two-dimensional square grid, and each point on the grid corresponds to an intersection of streets, as shown in Fig. 1. A node’s motion is controlled by the following rules. Two-dimensional CRW Mobility Model: •
a node takes a step in the same direction as its previous step with probability p.
•
it takes a step in the opposite direction with probability q.
•
it takes a step in the two orthogonal directions with probability r where p + q + 2r = 1.
•
on reaching the boundary of the square region it is reflected back with probability one.
•
The time instant at which the node takes its steps is governed by a Poisson process of intensity λ. This means that a node (vehicle) moves from one intersection to a neighboring intersection with a random velocity.
Existing mobility models like the random walk mobility model and the Manhattan mobility model [3] are special cases of the above mobility model. The random walk mobility model is
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a two-dimensional CRW with p = q = r = .25 and the Manhattan mobility model is the above model with p = .5, q = 0, and r = .25. In the Manhattan model, there are additional constraints: the speed of a node in a step is correlated with its speed in the previous step and is restricted by the speed of any node preceding it on the same street. We derive analytical results on both the transient and steady state behavior of this CRW mobility model. Such an analysis is important to assessments of the validity of simulation results on performance of protocols that are almost always obtained by taking an average over a time period. The main contributions of this paper are: •
Proposal of a mobility model called the Correlated Random Walk model that accounts for temporal dependence, captures such geographic restrictions as the motion of nodes on streets in a city, and allows non-zero drift.
•
Use of a new technique for transient and steady state analysis of CRW mobility models on finite and semi-infinite lattices in both one and two dimensions.
•
Derivation in the transform domain of the first-entry time distributions for 1-D CRW mobility models.
•
Simulation of the proposed CRW mobility model and comparisons with the random walk, the random waypoint and the Manhattan mobility models in terms of first-entry times and time to full coverage.
An initial version of these results can be found in [6].
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II. R ELATED W ORK A. Background in Mobility Models Many mobility models have been proposed in the literature. A survey of these models can be found in [1], [2] where the authors categorized the existing mobility models into four classes: random models, models with temporal dependency, models with spatial dependency, and models with geographic restrictions. Temporal dependency means that a node’s movements are correlated in time. Spatial dependence refers to correlations between the movements of nodes that are close to each other. By mobility models with geographic restrictions, they mean that the movements of nodes are bounded by streets and obstacles in the region. Models such as random walks and the random waypoint model and its variations fall under the category of random models; they do not consider geographical restrictions, temporal correlation, or spatial correlation. Gauss-Markov mobility models and Smooth Random models consider temporal dependence but do not consider spatial dependence or geographical restrictions. Group mobility models [1], [7] capture spatial dependence but do not account for temporal dependence or geographical restrictions. Pathway mobility models and Obstacle mobility models restrict the nodes’ movements to pathways in the simulation field but do not consider temporal or spatial dependence. In [3], two new mobility models, the Freeway and the Manhattan mobility models, were proposed as models of the mobility patterns on highways and city streets. In [8], the authors proposed a modified version of the random waypoint model in which there is a weight (that depends on both the current location and time) and a pause-time distribution associated with destinations in order to model the popularity of certain destinations. [9] proposed the random trip model which contains the random waypoint (RWP) and random walk models as special cases. In [10], the authors have
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proposed a mobility model in which the nodes move in an area with obstacles and the destination and pathways of the nodes are determined by the location of these obstacles. Recent research has also focussed on studying how realistic these models are [3]. Researchers have also explored the possibility of using WLAN traces to develop more realistic mobility models [11], [12]. A significant effort is being made to study how mobility models affect the performance of different MANET protocols [3], [4], [10], [13]–[15]. In [3], the authors proposed a framework to evaluate the performance of different routing protocols when the nodes move according to different mobility models. They observed that the protocol performance is highly influenced by the mobility models. In [4], [13]–[15], the authors studied how the distribution of path duration in a network depends on the mobility model and which in turn affects the network protocol performance. [10] proposed a mobility model in which the nodes move in an area with obstacles and studied the effect of this model on network performance taking the signal fading in presence of obstacles into consideration. In [5], [16], the authors have focused on how mobility models affect achievable capacity-delay √ tradeoffs . Gupta and Kumar [17] found that per-node capacity decreases as O(1/ n log n) as the number of nodes n increases in static wireless multi-hop networks. Grossglauser and Tse [18] showed that node mobility can improve the per-node capacity in multi-hop wireless networks, but at the cost of increased delay. In [5], [16], the author has found that the delay-capacity tradeoffs are radically different for different (Brownian motion, i.i.d., and the random way-point) mobility models. Recently, researchers have also studied the effect of mobility on various other performance measures in MANETs. In [19], [20], the authors have studied the impact of mobility on connec-
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tivity and coverage in MANETs. In [21], the authors have studied the effect of planned mobility on coverage in sensor networks in the presence of noise and obstacles. [22] studied the latency in target detection by sensors moving according to an unccordinated radom walk (sensors choose a random location within radius r around its current location in each measurement period). In [23], the authors have analyzed the distribution of time to detect an object by sensors moving according to a Brownian Motion mobility model. In the last few years, researchers have also paid attention to the study of stochastic properties of mobility models that are widely used in simulation studies. Getting a better understanding of how the models behave as time evolves in simulation studies is very important in order to avoid misinterpretations of simulation results. Examples of such research are [24]–[31], in which the authors have looked at the steady-state characteristics of the random waypoint and random direction model. In [32], the authors showed that mobility models like the RWP have a transient period which, if not taken into account, can impair the accuracy of simulations results. They then developed stationary mobility models that can eliminate these transient periods and hence lead to reliable simulation results. [9] proposed the random trip model, which contains the random waypoint (RWP) and random walk models as special cases, and found conditions for a stationary regime to exist. [33] analyzed the relation between the mobility model parameters and the duration of the transient period and proposed a generalized random direction model to achieve the desired steady state node distribution. To best of our knowledge, none of the existing studies have compared mobility models in terms of their time to first entry to a given region and their time to full coverage of an area, which is the focus of the study in this paper. Also, no earlier paper has considered and analyzed
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the stochastic properties of a correlated random walk based mobility model as general as the one proposed in this paper.
B. Background in Correlated Random Walks The earliest available work on the subject of correlated random walks appears to be that of Goldstein [34], who gave limiting distributions under various conditions for the discretetime, one-dimensional CRW on (−∞, +∞). In a one-dimensional correlated random walk, the walker takes a step in the same (opposite) direction as its previous step with probability p1 or p2 (q1 = 1 − p1 or q2 = 1 − p2 ) depending on whether its previous step was in the positive or the negative direction. He also derived formulae for the moments and obtained an asymptotic estimate of the distribution function in terms of hyper-geometric functions. Researchers [35]– [38] have obtained generating functions or exact combinatorial expressions for the probability of being at any lattice point at the nth step for a discrete-time CRW on a one-dimensional lattice that is unrestricted1 or has one or two absorbing boundaries. Zhang [39] obtained explicit expressions for the absorbing probability and expected duration of the discrete-time asymmetric2 CRW (ACRW) in the presence of elastic or absorbing barriers. In [40], [41], the authors obtained generating functions of the probabilities for the time to the first and j th passage to the origin or any other lattice point in a discrete-time, unrestricted, ACRW. In [42], the authors proposed the use of transition probability matrices associated with discrete-time CRWs to obtain various probability distributions for CRWs, even when the steps are of unequal lengths. The authors in [38] used the matrix-geometric results [43] to obtain the steady state probabilities for a 1
the term "unrestricted" is used to mean random walks on (−∞, +∞).
2
A CRW is called asymmetric if p1 6= p2 and symmetric otherwise.
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discrete-time, one-dimensional asymmetric CRW with one or two reflecting boundaries. They further obtained the probability of first passage to the boundaries at the nth step in a discretetime ACRW on a one-dimensional lattice with two absorbing boundaries. Gillis [44] studied the discrete-time CRW on a d-dimensional lattice in which the walker takes a step in the same direction as its previous step with probability p, in the direction opposite to its previous step with probability q, and in one of the directions orthogonal to his previous step with probability r and p + q + 2(d − 1)r = 1. He obtained a generating function for the probability that the walker is at a particular lattice point at the nth step and the probability of return to the origin at the nth step for the case p = r. To the best of our knowledge, none of the existing studies provide transient probabilities and first passage times to any lattice point starting from 0 for a continuous-time CRW on a finite lattice in one or two dimensions; we derive these quantities in this paper.
III. S TOCHASTIC P ROPERTIES OF CRW M OBILITY M ODELS Before comparing the stochastic properties of various mobility models, we derive the transient and steady state probabilities for the two-dimensional (2-D) CRW mobility model described in Section I.
A. Transient Probability Distributions for CRW on one-dimensional lattice In this section, we first derive the transient probability distributions of a continuous-time one-dimensional CRW on [0, ∞) with a reflecting boundary at 0. In a one-dimensional continuous-time CRW, the walker moves according to the following rules: 10
•
the walker takes a step in the same direction as its previous step with probability p1 or p2 or depending on whether its previous step was in the positive or the negative direction
•
it takes a step in a direction opposite to its previous step with probability q1 = 1 − p1 or q2 = 1 − p2 depending on whether its previous step was in the positive or the negative direction
•
on reaching a reflecting boundary, it takes a step in the opposite direction with probability one
•
the time at which it takes its next step is governed by a Poisson process of intensity λ.
Then, we propose a mapping of a continuous-time CRW on [0, ∞) to a continuous-time CRW on [0, N ] with reflecting boundaries at 0 and N . This mapping allows us to obtain the Laplace transform of the transient probability distributions in closed form for the CRW on [0, N ] with two reflecting boundaries. 1) Transient Analysis of a CRW on [0, ∞): An asymmetric CRW on [0, ∞) with a reflecting boundary at 0 can be modeled as a quasi-birth-death (QBD) process with state transition diagram shown in Fig. 2. The state n− (or n+ ) at any level n is reached when the walker reaches the location n from the location n + 1 (or n − 1). Note that the boundary location 0 has only one state 0− . If Q denotes the generator for this QBD process, the Laplace transform (Π(s)) of the transient probabilities π(t) satisfy the equations [43], [45], [46]: Π(s)(Q − sI) = −π(0), π(0) = π(t)|t=0 .
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(1)
Fig. 2.
State Transition Diagram of a correlated walk on [0, ∞).
For this QBD process,
−λ Q=
q1 λ p2 λ 0 0 0 0
.. .
λ −λ 0 0 0 0 0
0 0 −λ q1 λ p2 λ 0 0
0 p1 λ q2 λ −λ 0 0 0
.. .
.. .
.. .
0 0 0 0 −λ q1 λ p2 λ
0 0 0 p1 λ q2 λ −λ 0
.. .
0 0 0 0 0 0 0 0 0 0 0 p1 λ −λ q2 λ
... ... ... ... ... . ... ...
.. . . . .
We assume that the initial position of the walker is 0; i.e., π(0) = [1, 0, 0, . . . ]. Let Πi (s) denote the vector of Laplace transforms of the probability of being in the states at level i; i.e., Π0 (s) = [Π0− (s)] and Πi (s) = [Πi+ (s), Πi− (s)] for i > 0. Also, Π(s) = [Π0 (s), Π1 (s), . . . ]. Then, from (1), we get,
q1 λ −(λ + s)Π0 (s) + Π1 (s) p2 λ
= −1,
(2)
−(λ + s) = 0, λΠ0 (s) + Π1 (s) 0 Πn (s)B(s) + Πn+1 (s)C(s) = 0,
where B(s) =
(3) n ≥ 1,
(4)
p1 λ q1 λ −(λ + s) . and C(s) = p2 λ 0 −(λ + s) q2 λ 0
From (4) we get, Πn+1 (s) = Πn (s)W (s), 12
(5)
where
W (s) = −B(s) [C(s)]−1
p1 λ λ+s = q2 λ λ+s
p 1 q1 λ − p2 (λ + s) . λ+s q1 q2 λ − p2 λ p2 (λ + s)
Observe that to obtain the complete transient probability distribution we need to obtain the three boundary variables Π0− (s), Π1− (s) and Π1+ (s); the rest of the entries in the vector Π(s) can then be easily obtained by using the recursive relation in (5). But also observe that we have only two equations, (2) and (3), in these three variables. Hence, we need another equation to be able to solve for the boundary variables. We resort to the results in [46], which prove that the boundary variables also satisfy the following equation: Π1 (s)p1 (s) = 0,
(6)
where p1 (s) denotes the right eigenvector corresponding to the eigenvalue of W (s) whose magnitude is greater than or equal to one for all possible values of s. We find that W (s) is diagonizable and its eigenvalues are given by: p (p1 + p2 )λ2 + 2sλ + s2 − (−1)k ((p1 + p2 )λ2 + 2sλ + s2 )2 − 4p1 p2 λ2 (λ + s)2 , λk (s) = 2p2 λ(λ + s) for k = 1, 2. The right eigenvectors pk (s), k = 1, 2 corresponding to the eigenvalues λ1 (s) and λ2 (s) are given by:
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p (s) def k1 pk (s) = pk2 (s) p (p1 + p2 − 2p1 p2 )λ2 + 2sλ + s2 + (−1)k ((p1 + p2 )λ2 + 2sλ + s2 )2 − 4p1 p2 λ2 (λ + s)2 2p2 (p2 − 1)λ2 = 1 We find numerically that for all possible values of p1 , p2 , λ and s, the magnitude of λ1 (s) is greater than or equal to 1 . Hence, (6) implies that Π1− (s) = −Π1+ (s)p11 (s). Using this equation with (2) and (3), we find, Π1+ (s) = (p1 + p2 −
1)λ2
+ (1 − 2p2 )(s +
λ)2
+
p
2λ(1 − p2 ) (s + λ)4 − [(1 − 2p1 )(1 − 2p2 ) + 1]λ2 (s + λ)2 + (p1 + p2 − 1)2 λ4
Without loss of generality, we assume λ = 1 from this point. It can be proved that the inverse transform of Π1+ (s) is given by: µ −t
π1+ (t) = 2q2 e
1 − 2p2 p1 + p2 − 1 t ∗ sinh(t) + sinh(t) 4p2 (p2 − 1) 4p2 (p2 − 1) ¡ −t√p1 p2 ¡ √ ¢¢ ¡ t√p1 p2 ¡ √ ¢¢ 1 e − J0 t −c ∗ e J0 t −c 4p2 (p2 − 1) ¡ √ ¡ √ ¢¢ ¡ √ ¡ √ ¢¢ (1 − 2p1 )(1 − 2p2 ) + sinh(t) ∗ e−t p1 p2 J0 t −c ∗ et p1 p2 J0 t −c 4p2 (p2 − 1) ¶ ¡ −t√p1 p2 ¡ √ ¢¢ ¡ t√p1 p2 ¡ √ ¢¢ (p1 + p2 − 1)2 − J0 t −c ∗ e J0 t −c t ∗ sinh(t) ∗ e 4p2 (p2 − 1)
where q2 = 1 − p2 , c = (1 − p1 )(1 − p2 ) and J0 (t) is the zero’th order Bessel function of first kind. Note that the arguments of the Bessel functions in the above equation are complex-valued.
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Also, it can be proved that, π0− (t) = δ 0 (t) ∗ π1+ (t), π(n+1)+ (t) = p1 πn+ (t) + q2 πn− (t), π(n+1)− (t) = −(p1 q1 /p2 )πn+ (t) +
1 0 q1 q2 δ (t) ∗ πn− (t) − πn− (t). p2 p2
Special Case: When p1 = p2 = p µ
¶ 2p − 1 1 (2p − 1)2 π1+ (t) = e + I0 (t) ∗ I0 ((2p − 1)t) − t ∗ I0 (t) ∗ I0 ((2p − 1)t) 2p 2p 2p 2p − 1 1 1 + I1 (t) ∗ I0 ((2p − 1)t) + I0 ((2p − 1)t) 2p 2p 2p −t π0− (t) = e 2 (2p − 1) I0 (t) ∗ I0 ((2p − 1)t) − 2p · 2 ¸ 8p − 10p + 3 (2p − 1)(p − 1) t− I0 (t) ∗ I0 ((2p − 1)t) 2p2 2p2 2 (2p − 1) (p − 1) π1− (t) = e−t + t ∗ I (t) ∗ I ((2p − 1)t) 0 0 2 2p 1 2p − 1 + 2 I2 (t) ∗ I0 ((2p − 1)t) + I1 ((2p − 1)t) 4p 2p −t
q2 1 π(n+1)+ (t) = pπn+ (t) + qπn− (t), π(n+1)− (t) = −qπn+ (t) + δ 0 (t) ∗ πn− (t) − πn− (t), p p
where Iv (t) is the v th -order modified Bessel function of first kind. 2) Transient Analysis of a CRW on [0, N ]: The CRW on [0, N ] with reflecting boundaries can be modeled as a finite QBD process with the state transition diagram shown in Fig. 3. The transient probability distribution for such a walk can be obtained by solving the equation: π(t) = π(0)eQf in t , where Qf in is the generator of the finite QBD process. But as the number of states in the process increases, it becomes increasingly difficult to solve this equation. Hence, in this paper, we take a different approach that allows transient analysis even when the number of 15
Fig. 3.
State Transition Diagram of a correlated walk on [0, N ).
states is large. We map a CRW on [0, ∞) to a CRW on a finite lattice to and use the transient probabilities obtained for CRW on [0, ∞) to obtain the transient probability distributions for continuous-time CRW on a finite lattice with reflecting boundaries at the two ends. The correlated random walk on [0, N ] with reflecting boundaries at 0 and N can then be obtained by mapping the locations (2lN ± i), l = 1, 2, . . . in the correlated random walk on [0, ∞) with a reflecting boundary at 0 to the location i, i = 1, 2, . . . , N in the correlated random walk on [0, N ]. By this we mean that the walker of CRW on finite lattice is at location i, i = 1, 2, . . . , N when the walker of the CRW on [0, ∞) is at any one of the locations in the set {j : j = 2lN ± i, l = 1, 2, . . . }. This mapping is shown in Fig. 4 for N = 6. In Fig. 4, the black dots represent the lattice points and the number adjacent to these dots denote the location of the walker of the CRW on [0, ∞). The walker of the CRW on [0, 6] is at location 2 when the walker of the CRW on [0, ∞) is at any one of the locations in the set {2, 10, 14, . . . }. Hence, the probability that the walker of the CRW on [0, N ] is at location i, i = 1, 2, . . . , N at any time t is the sum of the probabilities that the walker of the CRW on [0, ∞) is at any ¯ i (s) denote the Laplace one of the locations in the set {j : j = 2lN ± i, l = 1, 2, . . . }. Let Π transform of transient probability distributions of being at ith location for the correlated random
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Fig. 4.
Mapping of a correlated random walk on [0, ∞) to a correlated random walk on [0, 6].
walk on [0, N ] with reflecting boundaries at both ends. Then, using (5) recursively, we get, ¯ i (s) = Π =
∞ X l=0 ∞ X
Π2lN +i (s) +
∞ X
Π2lN −i (s)
l=1
Π1 (s)[W (s)]2lN +i−1 +
l=0
¯ 0 (s) = Π
∞ X
Π1 (s)[W (s)]2lN −i−1 (s),
(7)
l=1 ∞ X
Π2lN (s) = Π0 (s) +
∞ X
∞ X
Π1 (s)[W (s)]2N l−1 ,
(8)
l=1
l=0
¯ N (s) = Π
0
