Wireless Personal Communications

, Volume 97, Issue 2, pp 2037–2052 | Cite as

Performance Analysis of Collision of Exponential Jitter Mechanism in Wireless Networks

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Abstract

Jittering, a small delay imposed before forwarding a packet, has been used in wireless communication for many purposes. For instance, AODV routing protocol uses jitter mechanism to prevent simultaneous transmission of nodes in route discovery stage, which reduces collisions. Recently, many works have studied possibility of using different random variables with different parameters for jittering rather than a simple uniform random variable. It has been shown that other random variables including Exponential and Pareto distributions can also be beneficial. In this paper, we first propose a discrete time Markov model to capture the behavior of nodes in route discovery stage when they use exponential distribution for their jitter mechanism. With this model, we obtain the number of collisions and route discovery time mathematically, which is proven to be accurate by simulation. We also use our model to find the optimum value of \(\lambda\), exponential distribution parameter, which somehow minimize the probability of collision and route discovery time. We further obtain some equations that give us the relation between parameters of different jitter mechanisms such that their route discovery stage takes almost equal time, which is used for fair comparison. Finally, we show that the exponential jitter mechanism using our optimum \(\lambda\) outperform other jitter mechanisms.

Keywords

Markov model Jitter mechanism Wireless routing protocol 

1 Introduction

Broadcast or flooding has many applications in wireless networks such as Wireless Sensor Networks (WSNs), Mobile Ad Hoc Networks (MANETs), and Vehicular Ad Hoc Networks (VANETs) [1]. In many routing protocols, flooding is an important part of route discovery phase in which a source tries to find a path to a destination using flooding, like AODV [2].

In simple flooding algorithm, all nodes that receive RREQ packets forward the packet only once. However, that all adjacent nodes receive the RREQ packet and schedule forwarding almost at the same time leads to packet collisions. To ameliorate this situation, each node schedules a random back-off time so that the chance of simultaneous transmission reduces [3]. This mechanism, standardized in [4], is called jitter mechanism.

The effect of jitter has been studied in some wired and wireless IP networks [5, 6]. The influence of broadcast mechanism for different applications in wireless ad hoc networks has been also studied. For instance, the effect of broadcast traffic on CSMA, which uses back-off mechanism, has been studied in [7]. One of the most useful application of Jitter mechanisms is in route discovery stage. This mechanism has been already recommended for many routing protocols such as AODV [2] and LOADng [8].

The effect of uniform jitter mechanism on route discovery stage has been widely studied in [9, 10, 11]. In uniform jitter mechanism, each node selects the jitter value randomly from a uniform distribution, \(U\sim [0,C]\), where C is the maximum value of jitter [12]. Other jitter mechanisms based on uniform distribution, in which the parameters of uniform distribution are only different, have been also studied. For instance, window jitter mechanism uses \(U\sim [\alpha C,C]\) where \(\alpha \in (0,1)\) [3, 13]. In [14], the usefulness of jitter mechanisms based on other distributions such as Exponential and Pareto has been studied.

In broadcast and jitter mechanisms, collision and time required to finish the whole stage is highly important. Some works have already proposed analytical models for flooding in MANET [15] and collision in AODV routing protocol [16]. However, the effect of behavior of other mechanisms, such as exponential jitter mechanism, has not been studied yet. In this paper, we analytically model the route discovery stage of exponential jitter mechanism and use our model to find the optimum value of Exponential distribution parameter.

In Sect. 2, we propose two discrete time Markov chains (DTMC) to capture the behavior of the networks in route discovery stage. We also obtain the optimum value of Exponential distribution parameter which leads to fewer collisions and low discovery time in Sect. 3. In Sect. 4, we obtain some formulas that allow us to select parameter of uniform and window jitter mechanisms such that their comparison with exponential jitter mechanism becomes fair. In Sect. 5, not only our models are validated by simulations, but also our analysis toward finding optimum parameter for exponential jitter mechanism is shown to be beneficial. Finally, the conclusion is given in Sect. 6.

2 Markov Models

In this section, we aim to model the route discovery process of reactive routing protocols by an absorbing DTMC. The transmission range of all nodes are assumed to be equal. Whenever a node sends a packet, all nodes in its transmission range will receive the packet. We called the area whose nodes receive this packet a receiving area which is a circle of radius R, defined as the tranmission range. We assume that if one observes the behavior of any arbitrary receiving area, almost same behavior will be measured regarding route discovery process. Although it might not be completely true in reality, this assumption makes the analytical modeling feasible. Hence, we aim to model one of these receiving area.

We have provided two models. Our first model is simpler, but not very accurate. However, since it contains fewer number of states and has a 2-dimensional structure, it can be shown and understand easier. Our second model is built upon our first model and is more accurate. Since it has a 3-dimensional structure, it is not possible to show the corresponding Markov chain clearly, but its similarity to our first model helps to comprehend the model.

2.1 First Model

We assume a source node sends RREQ packet and n neighboring nodes receiving this RREQ packet choose random delay from an exponential distribution and forward the RREQ packet after the delay. This is called exponential jitter mechanism [14]. We assume all receiving nodes are contributing in route discovery process and there is no other traffic in the network. Given that the time needed to send and propagate a packet is s, we divide the time into slots of size s and assume that nodes only send RREQ packets at the beginning of slots. This assumption will be elminated in the second model. The first slot, at time \(t=0\), indicates the reception of RREQ packet by all neighboring nodes. We assume all nodes forward the RREQ packet only once whether the transmission is successful or unsuccessful. The probability that a node forward the RREQ packet at the first slot, denoted by P(1), is as follows
$$\begin{aligned} P(1)= \int _{0}^{s} \lambda e^{-\lambda x} dx = 1-e^{-\lambda s}. \end{aligned}$$
(1)
Given that the node has not forwarded the RREQ packet yet, the probability of forwarding at slot i, that is P(i), is equal to P(1) owing to the memoryless property of exponential distribution. Thus, the probability of forwarding is hereafter represented by P without any time index. It is assumed that all nodes use the same parameter for exponential distribution, \(\lambda\). Hence, The probability that k nodes forward in the slot t, knowing that i nodes have already forwarded in the previous slots, has a binomial distribution, denoted by \(N_{(i,k)}(t)\). This can be expressed as follows (\(0\le i\le n , 0\le k\le n-i\))
$$\begin{aligned} N_{(i,k)}(t)= \left( {\begin{array}{c}n-i\\ k\end{array}}\right) P^{k} (1-P)^{(n-(k+i))} . \end{aligned}$$
(2)
Since the probability of forwarding at slot i, P, is independent of the time, \(N_{(i,k)}(t)\) is also independent of the time. As a result, we drop the t index and write \(N_{(i,k)}\). Now, an absorbing discrete time Markov chain (DTMC) is built upon these slots. Let \(\pi _{(i,k)}\) (\(0\le i\le n ,\ 0\le k\le n-i\)) denotes the states of the DTMC in which i represents the number of nodes that have already forwarded the RREQ packet in the previous slots and k represents the number of nodes that are forwarding the packet at this slot. The corresponding DTMC is shown in Fig. 1 for the case when there are only 3 nodes.
Fig. 1

First Markov model when there are only 3 nodes

At the begining of the forwarding process, when n neighboring nodes receive the RREQ packet for the first time, no node has forwarded the RREQ yet and DTMC is in state \(\pi _{(0,0)}\). With probability N(0, 0), DTMC stays at this state and with probability N(0, i) it moves to state \(\pi _{(0,i)}\). The only absorbing state is \(\pi _{(n,0)}\) in which all nodes had already forwarded their RREQ packet. Thus, transition probabilities of the DTMC is given by
$$\begin{aligned} p_{(i,j),(i+j,k)} = N_{(i+j,k)} \qquad 0 <i+j+k \le n \end{aligned}$$
(3)
Now we prove some lemmas and define some functions to solve our absorbing DTMC.

Lemma 1

The multiplication of \(N_{(i,j)}\) and \(N_{(0,i)}\) can be expressed as a multiplication of \(N_{(i,0)}\) and \(N_{(0,i+j)}\) as follows
$$\begin{aligned} N_{(i,j)} N_{(0,i)} = \left( {\begin{array}{c}i+j\\ i\end{array}}\right) N_{(i,0)} N_{(0,i+j)}. \end{aligned}$$
(4)

Proof

It can be easily proven by substituting N from its definition in Eq. (2). \(\square\)

Lets define a function \(\psi _{(m)}\) as follows
$$\begin{aligned} \psi _{(m)} \triangleq \left\{ \begin{array}{ll} \frac{\sum _{i=0}^{m-1} \left( {\begin{array}{l}m\\ i\end{array}}\right) N_{(i,0)} \psi _{(i)}}{1-N_{(m,0)}} & \quad \text{ if } 1\le m < n \\ \frac{1}{N_{(0,0)}} & \quad \text{ if } m = 0 \end{array} \right. \end{aligned}$$
(5)

Lemma 2

The function \(\psi _{(m)}\) can also be expressed as follows
$$\begin{aligned} \psi _{(m)} = \sum _{i=0}^{m} \left( {\begin{array}{c}m\\ i\end{array}}\right) N_{(i,0)} \psi _{(i)} \qquad 1\le i < n \end{aligned}$$
(6)

Proof

From definition, we have
$$\begin{aligned} \begin{aligned} \psi _{(m)}&= \frac{\sum _{i=0}^{m-1} \left( {\begin{array}{c}m\\ i\end{array}}\right) N_{(i,0)} \psi _{(i)}}{1-N_{(m,0)}} \\ \psi _{(m)} - N_{(m,0)} \psi _{(m)}&= \sum _{i=0}^{m-1} \left( {\begin{array}{c}m\\ i\end{array}}\right) N_{(i,0)} \psi _{(i)} \\ \psi _{(m)}&= \sum _{i=0}^{m-1} \left( {\begin{array}{c}m\\ i\end{array}}\right) N_{(i,0)} \psi _{(i)} + N_{(m,0)} \psi _{(m)} \\ \psi _{(m)}&= \sum _{i=0}^{m-1} \left( {\begin{array}{c}m\\ i\end{array}}\right) N_{(i,0)} \psi _{(i)} + \left( {\begin{array}{c}m\\ m\end{array}}\right) N_{(m,0)} \psi _{(m)} \\ \psi _{(m)}&= \sum _{i=0}^{m} \left( {\begin{array}{c}m\\ i\end{array}}\right) N_{(i,0)} \psi _{(i)}\\ \end{aligned} \end{aligned}$$
\(\square\)
Fig. 2

Our first Markov model

The Markov chain of the case in which there are n nodes is shown in Fig. 2. Since, the absorbing DTMC always starts at state \(\pi _{(0,0)}\), we only need to compute the first row of the fundamental matrix. The components of the first row can be obtained by solving the following system of equations [17]
$$\begin{aligned} \pi _{(0,0)}= & {} 1+N_{(0,0)} \pi _{(0,0)} \end{aligned}$$
(7)
$$\begin{aligned} \pi _{(i,j)}= & {} N_{(i,j)} \sum _{k=0}^{i} \pi _{(k,i-k)} \qquad 1\le i < n ,\ 0\le j\le n-i. \end{aligned}$$
(8)
The value of \(\pi _{(0,0)}\) can be obtained by moving \(N_{(0,0)} \pi _{(0,0)}\) to the left side of the equation (7) and factoring \(\pi _{(0,0)}\). Thus,
$$\begin{aligned} \pi _{(0,0)} = \frac{1}{1-N_{(0,0)}}. \end{aligned}$$
(9)
Now, Theorm 1 gives a formula to compute all \(\pi _{(i,j)}\) in terms of \(\pi _{(0,0)}\).

Theorem 1

The number of visit of the state \(\pi _{(i,j)}\), that is the components of the first row of the fundamental matrix of the DTMC of exponential jitter mechanism, is as follows
$$\begin{aligned} \pi _{(i,j)} = N_{(i,j)} N_{(0,i)} \psi _{(i)} \pi _{(0,0)} \qquad 1\le i < n ,\ 0\le j\le n-i. \end{aligned}$$
(10)

Proof

We can prove the theorem by mathemathical induction. For \(i=0\) and \(j=1\), \(\pi _{(0,1)}\) is obtained as follows
$$\begin{aligned} \begin{aligned} \pi _{(0,1)}&= N_{(0,1)} \pi _{(0,0)} \qquad \qquad \text {(by\, Eq.\, 8)} \\&= N_{(0,1)} \pi _{(0,0)} \times \frac{N_{(0,1)}}{N_{(0,1)}} \\&= N_{(0,1)} \pi _{(0,0)} N_{(0,1)} \psi _{(0)} \qquad \text {(by\, Definition\, 5)}. \\ \end{aligned} \end{aligned}$$
For \(i=1\) and \(j=0\), \(\pi _{(1,0)}\) is obtained as follows
$$\begin{aligned} \begin{aligned} \pi _{(1,0)}&= N_{(1,0)} (\pi _{(0,1)}+\pi _{(1,0)})\ \ \text {(by\, Eq.\, 8)} \\ \pi _{(1,0)} - N_{(1,0)} \pi _{(1,0)}&= N_{(1,0)} \pi _{(0,1)} \\ \pi _{(1,0)}&= \frac{N_{(1,0)} \pi _{(0,1)}}{1-N_{(1,0)}} \\ \pi _{(1,0)}&= \frac{N_{(1,0)} N_{(0,1)} \pi _{(0,0)}}{1-N_{(1,0)}}\ \ \text {(by\, Lemma\, 1)} \\ \pi _{(1,0)}&= N_{(1,0)} N_{(0,1)} \psi _{(1)} \pi _{(0,0)} \qquad \text {(by\, Lemma\, 2)} \\ \end{aligned} \end{aligned}$$
For any \(1\le i < n\) and \(0\le j\le n-i\), we have
$$\begin{aligned} \begin{aligned} \pi _{(i,j)}&= N_{(i,j)} \sum _{k=0}^{i} \pi _{(k,i-k)} \qquad \text {(by\, Eq.\, 8)} \\&a= N_{(i,j)} \sum _{k=0}^{i} N_{(k,i-k)} N_{(0,k)} \psi _{(k)} \pi _{(0,0)} \\&= N_{(i,j)} \sum _{k=0}^{i} \left( {\begin{array}{c}i\\ k\end{array}}\right) N_{(k,0)} N_{(0,i)} \psi _{(k)} \pi _{(0,0)} \ \ \text {(by\, Lemma\, 1)} \\&= N_{(i,j)} N_{(0,i)} \pi _{(0,0)} \sum _{k=0}^{i} \left( {\begin{array}{c}i\\ k\end{array}}\right) N_{(k,0)} \psi _{(k)} \\&= N_{(i,j)} N_{(0,i)} \psi _{(i)} \pi _{(0,0)} \qquad \text {(by\, Lemma\, 2)} \\ \end{aligned} \end{aligned}$$
\(\square\)
Knowing the number of visit to each state of our absorbing DTMC, it is possible to find the average number of successful tranmissions. A node can forward the RREQ packet successfully only if it is the only sender in a slot. Hence, the number of successful tranmissions is as follows
$$\begin{aligned} E[N_s]= \sum _{i=0}^{n-1} \pi _{(i,1)}. \end{aligned}$$
(11)
Note that, in an absorbing DTMC, the elements of fundamental matrix give the average number of visit to that state, which can be even more than one. In our DTMC, however, states in which transmission occur, that is \(\pi _{(i,j)}\) where \(j>0\), cannot be visited twice. Therefore, \(\sum _{i=0}^{n-1} \pi _{(i,1)}\) represents the number successful transmissions. Since each node forwards RREQ packets only once, total number of tranmissions would be n. Thus, the number of collisions is as follows
$$\begin{aligned} E[N_c]= n-E[N_s]. \end{aligned}$$
(12)
We are also interested in finding the average time needed for all nodes to forward their RREQ packet. This can be obtained by finding the expected number of steps (slots) before being absorbed. Since we have only one absorbing state, this can be obtained by summing all states except the absorbing one as follows
$$\begin{aligned} E[T]= \sum _{i=0}^{n-1} \sum _{j=0}^{n-i} \pi _{(i,j)} \end{aligned}$$
(13)

2.2 Second Model

In our first Markov model, we assumed that nodes start their transmission at the beginning of each slot. In reality, however, no common routing protocol such as AODV and DSR imposes such a restriction. In other words, packets may be transmitted at any time during each slot. As a result, a packet sent in the middle of a slot can collide with a packet sent at the beginning of the next slot. This kind of collision that happens between two adjacent slots cannot be captured by our first model. In this subsection, we extend our model to capture these kinds of collisions as well.

Let \(\pi _{(k,j,i)}\) (\(i+j+k\le n\) and \(0 \le k,j,i\)) denotes the states of the DTMC in which k represents the number of nodes that had already forwarded the RREQ packet before the last slot, j represents the number of nodes that have already forwarded the RREQ packet in the last slot and i represents the number of nodes that are forwarding the packet at this slot. Since this DTMC has 3-dimensional structure, it is not possible to draw its figure clearly.

Transition probabilities of the corresponding DTMC is as follows:
$$\begin{aligned} \pi _{(0,0,0)}= & {} 1+N_{(0,0)} \pi _{(0,0,0)} \end{aligned}$$
(14)
$$\begin{aligned} \pi _{(0,0,i)}= & {} N_{(0,i)} \pi _{(0,0,0)} \qquad 0< i \le n. \end{aligned}$$
(15)
$$\begin{aligned} \pi _{(0,j,i)}= & {} N_{(j,i)} \pi _{(0,0,j)} \qquad i+j\le n. \end{aligned}$$
(16)
$$\begin{aligned} \pi _{(k,j,i)}= & {} N_{(k+j,i)} \sum _{l=0}^{k} \pi _{(l,k-l,j)} \qquad i+j+k\le n. \end{aligned}$$
(17)
Since the DTMC starts at \(\pi _{(0,0,0)}\) where no node has yet sent its packet, Eq. (14) is obviously true. Note that it is easy to solve Eq. (14) for \(\pi _{(0,0,0)}\), as follows
$$\begin{aligned} \pi _{(0,0,0)} = \frac{1}{1-N_{(0,0)}} \end{aligned}$$
(18)
Equation (15) indicates that in order to have i packets in the current slot while considering that no RREQ packet has been sent in previous slots, i nodes out of n nodes (\(N_{(0,i)}\)) must choose this slot. To have i packets in the current slot while considering that j RREQ packets have been sent in last slot and no RREQ has been sent before the last slot, i nodes out of remaining \(n-j\) nodes (\(N_{(j,i)}\)) must choose this slot, which is shown by Eq. (16). Finally, in order to have i packets in the current slot while considering that j and k RREQ packets have been sent in last slot and before the last slot, respectively, i nodes out of remaining \(n-j-k\) nodes (\(N_{(j+k,i)}\)) must choose this slot. Note that this can happen in k different ways which is shown by Eq. (17).

Theorem 2

The number of visit of the state \(\pi _{(k,j,i)}\) of our second model is as follows
$$\begin{aligned} \pi _{(k,j,i)} = N_{(k+j,i)} N_{(k,j)} N_{(0,k)} \psi _{(k)} \pi _{(0,0,0)} \qquad i+j+k\le n\;\text {and}\;0 < k+j \end{aligned}$$
(19)

Proof

Like our 2-dimensional model, we prove the theorem by mathemathical induction. For \(k=0\), \(\pi _{(0,j,i)}\) is obtained as follows
$$\begin{aligned} \begin{aligned} \pi _{(0,j,i)}&= N_{(j,i)} \pi _{(0,0,j)} \\&= N_{(j,i) }N_{(0,j)}\pi _{(0,0,0)} \\&= N_{(j,i)} N_{(0,j)} \pi _{(0,0,0)} \times \frac{N_{(0,0)}}{N_{(0,0)}} \\&= N_{(j,i)} N_{(0,j)} {N_{(0,0)}} \psi _{(0)} \pi _{(0,0,0)}\\ \end{aligned} \end{aligned}$$
Now, for any \(0\le i,j \le n\) and \(1\le k\le n-i-j\), we have
$$\begin{aligned} \begin{aligned} \pi _{(k,j,i)}&= N_{(k+j,i)} \sum _{l=0}^{k} \pi _{(l,k-l,j)} \\&= N_{(k+j,i)} \sum _{l=0}^{k} N_{(k,j)} N_{(l,k-l)} N_{(0,l)} \psi _{(l)} \pi _{(0,0,0)} \\&= N_{(k+j,i)} N_{(k,j)} \pi _{(0,0,0)} \sum _{l=0}^{k} N_{(l,k-l)} N_{(0,l)} \psi _{(l)} \\&= N_{(k+j,i)} N_{(k,j)} \pi _{(0,0,0)} \sum _{l=0}^{k} \left( {\begin{array}{c}k\\ l\end{array}}\right) N_{(l,0)} N_{(0,k)} \psi _{(l)} \qquad \text {(by\, Lemma\, 1)} \\&= N_{(k+j,i)} N_{(k,j)} \pi _{(0,0,0)} N_{(0,k)} \sum _{l=0}^{k} \left( {\begin{array}{c}k\\ l\end{array}}\right) N_{(l,0)} \psi _{(l)} \\&= N_{(k+j,i)} N_{(k,j)} N_{(0,k)} \psi _{(k)} \pi _{(0,0,0)} \qquad \text {(by\, Lemma\, 2)} \\ \end{aligned} \end{aligned}$$
\(\square\)
Fig. 3

Effect of shifting real slots such that the current packet fits into one slot

With our 3-dimensional model, we can obtain collision probability more accurately. First we obtain the number of successful transmission. If there is no transmission in the previous slot and there is only one transmission in the current slot, the transmission would be successful. In the case where there are transmissions in both adjacent slots, there is still a chance for transmissions to be successful. Analyzing this case in our model would be extremely difficult since a collision may happen when the next slot also contains transmission. Therefore, instead of looking at our real slot, we shift all slots so that the transmission in this slide occurs at the beginning of the slot, as it is shown in Fig. 3. Note that due to the memoryless property of exponential distribution, changing position of slots does not change the result. By looking at the imaginary shifted slots, it easy to observe that if the last slot contains a transmission, collision happens. Moreover, transmission in the next slot has no impact on this slot’s transmission. Consequently, the number of successful transmissions is as follows
$$\begin{aligned} E[N_s]= \sum _{k=0}^{n-1} \pi _{(k,0,1)} \end{aligned}$$
(20)
Therefore, the number of collisions is again obtained by Eq. (12). As it is shown with our 2-dimensional model, time required for all nodes to send their RREQ packet is equal to sum of all states except the absorbing one as follows
$$\begin{aligned} E[T]= \sum _{i=0}^{n-1} \sum _{j=0}^{n-i} \sum _{k=0}^{n-i-j} \pi _{(i,k,j)} \end{aligned}$$
(21)
Fig. 4

Value of V parameter for different number of nodes

3 Optimizing Exponential Jitter Mechanism

In the exponential jitter mechanism, the distribution parameter (\(\lambda\)) has a crucial role. By increasing the value of \(\lambda\), number of collisions increases, but, at the same time, the discovery time (or the time required to finish the route discovery stage) decreases. Therefore, it is not clear what would be the best value for \(\lambda\).

For jitter mechanisms based on Uniform distribution, some characteristics and formulas were proposed which help to choose the distribution parameters [10, 11, 13]. However, the only paper studied other jitter mechanisms [14], such as Exponential and Pareto, did not investigate how one should set the parameters of such jitter mechanisms, which is in fact non-trivial. In this section, using the performance metrics we obtained from our Markov models, we propose a formula to help set the Exponential parameter of jitter mechanism which tries to minimize probability of collision and discovery time simultaneously.

In many routing protocols such as AODV and DSR, the first packet that reaches the destination indicates the route, unless another RREQ packet coming from a shorter path arrives. In such cases, routing protocol does not wait until the route discovery stage finishes. Therefore, it is reasonable to choose \(\lambda\) such that it favors the first RREQ packet. To do so, we define a new parameter, denoted by V, as follows
$$\begin{aligned} V(\lambda ) = \frac{\pi _{(0,0,1)}}{\pi _{(0,0,0)}} = N_{(0,1)} = n (1-e^{-\lambda s}) e^{-\lambda s(n-1)} \qquad \text {(by\, Eq.\, 15)} \end{aligned}$$
(22)
Here, we want to maximize the successful transmission probability of first RREQ (\(\pi _{(0,0,1)}\)) and minimize the time until the transmission of the first RREQ (\(\pi _{(0,0,0)}\)), which will be called the first RREQ delay, hereafter. That is the main reason for definition of V which will be shown to be useful in comparison with other jitter mechanism.1 As the number of nodes increases, the probability of collision also increases if all nodes use a fixed value of \(\lambda\). Hence, the maximum value of V must occurs at lower \(\lambda\) when the number of nodes increases, as it is shown in Fig. 4. To find the value of \(\lambda\) which maximizes the preceding equation, we solve the derivative of the equation when it is equal to zero as follows
$$\begin{aligned} \frac{dV}{d\lambda } = -s(n-1)ne^{-\lambda s(n-1)} + sn^2 e^{-\lambda sn} =0 \end{aligned}$$
(23)
Solving this equality gives the optimum value of \(\lambda\), denoted by \(\lambda _{opt}\), as follows
$$\begin{aligned} \lambda _{opt} = \frac{ ln(\frac{n}{n-1}) }{s} \end{aligned}$$
(24)

4 A Step Toward Fair Comparison

In the last section, we obtained \(\lambda _{opt}\) that maximizes the probability of successful transmission and minimizes delay to some extent. Comparison of exponential jitter with other jitter mechanisms such as uniform jitter would be inconclusive if one mechanism is better at one performance parameter and the other is better at another. Therefore, we aim to set the distribution parameters of these mechanisms such that their route discovery time becomes almost equal. Therefore, comparing the number of collisions would be fair. Note that in all common routing protocols the first RREQ packet that reaches the destination indicates the final route. Since we assume that the wireless network is homogeneous and the behavior of the network regarding RREQ transmission are similar in all parts of the network, we only need to study the first RREQ delay in one part of the networks instead of evaluating the whole route discovery time. We will later show by simulation that by setting distribution parameters of different jitter mechanisms in a way that their first RREQ delay becomes equal, their route discovery time becomes also equal.

For the exponential distribution jitter, the probability that the first RREQ delay is greater t, which is the complementary cumulative distribution function (CCDF), is equal to the probability that all nodes get greater value than t from exponential distribution, as follows
$$\begin{aligned} P_{T_{exp}}(T>t) = \prod _{i=1}^{n} \left( 1-\int _0^t \lambda e^{-\lambda t} dt\right) = e^{-n \lambda t} \end{aligned}$$
(25)
It is easy to obtain the probability density function from CCDF.
$$\begin{aligned} f_{T_{exp}}(t) = n \lambda e^{n \lambda t} \end{aligned}$$
(26)
Hence, the expected value of the first RREQ delay is as follows
$$\begin{aligned} E[T_{exp}] = \int _0^\infty n \lambda e^{n \lambda t} t \ dt = \frac{1}{n \lambda } \end{aligned}$$
(27)
In uniform jitter mechanism, delay is selected from \(U \sim [0, C_{uni}]\). Using the same method as exponential distribution, the probability density function of the first RREQ delay is as follows
$$\begin{aligned} f_{T_{uni}}(t) = \frac{n(C_{uni}-t)^{n-1}}{C_{uni}^n} \end{aligned}$$
(28)
Consequently, one can obtain the expected value of the first RREQ delay.
$$\begin{aligned} E[T_{uni}] = \int _0^{C_{uni}} \frac{n(C_{uni}-t)^{n-1}}{C_{uni}^n} t \ dt = \frac{C_{uni}}{n+1} \end{aligned}$$
(29)
Finally, window jitter mechanism selects delay from \(U \sim [\alpha C_{w}, C_{w}]\), where \(0<\alpha <1\). Using the same approach as before, the probability density function of the first RREQ delay is as follows
$$\begin{aligned} f_{T_{w}}(t) = \frac{n(C_w-t)^{n-1}}{(C_w - \alpha C_w)^n} \end{aligned}$$
(30)
Hence, the expected value of the first RREQ delay for window jitter mechanism is as follows
$$\begin{aligned} E[T_{w}] = \int _{\alpha C_w}^{C_w} \frac{n(C_w-t)^{n-1}}{(C_w - \alpha C_w)^n} t \ dt = C_w \left( \frac{1+\alpha n}{n+1}\right) \end{aligned}$$
(31)
In order for these mechanisms to have an equal route discovery time, one should equate \(E[T_{exp}]\) with \(E[T_{uni}]\) and \(E[T_{w}]\). This gives us the following equations which will be used in the next section for simulation.
$$\begin{aligned} C_{uni}= & {} \frac{n+1}{n\lambda } \end{aligned}$$
(32)
$$\begin{aligned} C_w= & {} \frac{n+1}{n \lambda (1+\alpha n)} = \frac{C_{uni}}{1+\alpha n} \end{aligned}$$
(33)

5 Simulation

In this section, we first validate our models with simulation and show that our second model perfectly fits simulation results. Then, we compare the exponential jitter mechanism with uniform and window jitter mechanisms. To do so, we use \(\lambda _{opt}\) as the exponential jitter parameter and use Eqs. (32, 33) to obtain the value of uniform and window jitter parameters. Finally, we will show that our analysis leading to \(\lambda _{opt}\) establish a jitter mechanism which outperforms other jitter mechanisms in terms of collision.

We conducted our simulations with ns2 in which 200 nodes were randomly deployed in region of 1000 \(\times\) 3000 m. Transmission range of all nodes was 250 m and 5 pairs of connection established during each simulation. Note that different nodes, even if they are in vicinity of each others, may experience different number of collisions. Moreover, nodes that are busy sending a packet cannot detect any collision. To solve these problems, we deployed 10 listening nodes that only observe the network and capture the number of collisions. These listening nodes were not allowed to contribute to any packet transmission.

In Fig. 5, our models are compared with simulation. n represents the number of nodes in each forwarding region, or the number of nodes that each listening node can hear. As it is mentioned before, our first model underestimates the number of collisions since it cannot capture the collision between two adjacent slots. Our second model, however, predicts the number of collisions accurately, as it is shown in Fig. 5a. Interestingly, both models almost predict the number of slots to finish all transmission in a region accurately, as it is shown in Fig. 5b.
Fig. 5

Comparison of our models with simulation

Fig. 6

Effect of distance on routing performance

In Fig. 6, the number of collisions and route discovery time of 3 jitter mechanisms are shown. We used \(\lambda _{opt}\) as the exponential jitter parameter. Equations (32, 33) were also used to obtain distribution parameters of uniform and window jitter mechanisms. Note that n in these equations represents the number of nodes in each forwarding region, not the total number of nodes in the network, denoted by N. Needless to say, n can be obtained using \(n=\frac{N \pi R^2 }{S}\), in which R is the transmission range and S is the area of the network. As it is shown in Fig. 6a, exponential jitter mechanism with \(\lambda _{opt}\) always outperforms the other mechanisms. Fig. 6b shows that our analysis in Sect. 4 is valid and almost all jitter mechanisms have equal route discovery time.

As it is shown in Fig. 7a, increasing the number of nodes in the wireless network can dramatically increase the number of collisions. As the number of nodes increases, more and more nodes contending for channel to forward RREQ packets. Window jitter has the most number of collisions since nodes select the delay from a bounded uniform distribution with smaller range than uniform distribution. In exponential jitter mechanism, since the distribution is unbounded, it even possible that some nodes select the delay so much big that they forward the RREQ packet when the contention in the region has gone. That is why exponential jitter outperforms the other mechanisms. In Fig. 7b, the route discovery time is illustrated which shows that regardless of the number of nodes, the route discovery time of these mechanisms are almost equal.
Fig. 7

Effect of node density on routing performance

6 Conclusion

In this paper, we formally analyzed the performance metric of exponential jitter mechanism. By modeling the route discovery stage of wireless routing protocols with absorbing discrete time Markov chain, we captured the behavior of this stage regarding number of collisions and route discovery time. This allowed us to see the effect of exponential distribution parameter on the number of collision and route discovery time. Having defined a new performance parameter, we found a formula to obtain the optimum value exponential parameter. We have also obtained two formulas that relate the parameter of uniform and window jitter mechanisms such that their route discovery time becomes almost equal. Finally, we conducted a set of simulations to validate our model and show that our analysis of the optimum value of exponential jitter parameter contributes to fewer collisions in comparison with other jitter mechanisms.

Footnotes

  1. 1.

    This formula is picked arbitrarily for convenient. In fact, any equation that allows us to maximize \(\pi _{(0,0,1)}\) and minimize \(\pi _{(0,0,0)}\) at the same time can be used here.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Computer EngineeringSharif University of TechnologyTehranIran

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