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An analytical model of delay in multi-hop wireless ad hoc networks

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Abstract

Several analytical models of different wireless networking schemes such as wireless LANs and meshes have been reported in the literature. To the best of our knowledge, all these models fail to address the accurate end-to-end delay analysis of multi-hop wireless networks under unsaturated traffic condition considering the hidden and exposed terminal situation. In an effort to gain deep understanding of delay, this paper firstly proposes a new analytical model to predict accurate media access delay by obtaining its distribution function in a single wireless node. The interesting point of having the media access delay distribution is its generality that not only enables us to derive the average delay which has been reported in almost most of the previous studies as a special case but also facilitates obtaining higher moments of delay such as variance and skewness to capture the QoS parameters such as jitters in recently popular multimedia applications. Secondly, using the obtained single node media access delay distribution, we extend our modeling approach to investigate the delay in multi-hop networks. Moreover, probabilities of collisions in both hidden and exposed terminal conditions have been calculated. The validity of the model is demonstrated by comparing results predicted by the analytical model against those obtained through simulation experiments.

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Correspondence to A. Diyanat.

Appendices

Appendix 1: Table of notations

Notation

Description

A p

Average duration of payload

A s

Average service time

A w

Average waiting time

A d

Average End-to-end delay

s b

Backoff service time

T w

Average time that a packet waits for another one to be served

m

Maximum number of retransmission

m'

Maximum contention window

c

Probability of collision occurrence

q

Probability of empty queue after processing current packet

T c

Collision slot duration

T s

Successful slot duration

P s

Probability of busy slot being a successful transmission

P tr

Probability of sensing a busy slot

P nk

Probability of generating no packet in a given period

U

Normalized channel utilization factor

τ

Probability of transmission in an Idle slot

W

Minimum contention window

W i

ith contention window

σ

Slot duration

\(\bar{\sigma} \)

Average slot duration

p i,j

Probability of being in the (i,j) state at the State transition diagram.

p FirstTR

State of receiving new packet when both node and channel are idle

p IDLE

State of the node when it has no packet in queue to serve

λ g

Average packet arrival rate

r(N)

Communication range of node n

\(\bar{r}_{i,j}\)

Average distance between each two nodes in the network

n Hid

Average number of nodes located in hidden area of a random node

n c

Average number of node located in common area of two random adjacent nodes

t v

Vulnerable period

\(\bar{d} \)

Mean message distance

p i

Probability of a message crossing i channels to reach the destination

D

Diameter of the network

α

Indicating the degree of communication locality

θ (α, n)

Normalizing constant

N

Number of nodes in the networks (Network size)

z

Size of the network(side of a square operational area)

λ n

Mean traffic of the network

b s

Indicating backoff stages(the number of tries before the successful transmission of the packet)

Appendix 2: Deriving second moment of E[T w 2]

In Sect. 3.1.2, the second moment of T w can be further simplified. The detailed computations are depicted here.

$$ \begin{aligned} E[T_{w} ^{2}]&=\frac{d^{2} G_{b_{s} } (z)}{dz^{2} } |_{z=1} +E(T_{w})\\ E[T_{w} ^{2}]&=T_{s} ^{2} (1-c)q(1-p_{tr}(n-1))\\ &\quad+\sum _{i=1}^{m}\left( T_{s} ^{2} +T_{c} ^{2} i^{2}+\frac{\bar{\sigma }^{2} }{4} \sum _{j=0}^{i-1}(W_{j} -1)^{2} + 2T_{c} T_{s} i+2T_{s} \frac{\bar{\sigma }}{2} \sum _{j=1}^{i-1}(W_{j} -1)+ iT_{c} \bar{\sigma }\sum _{j=0}^{i-1}(W_{j} -1 )\right) c^{i} (1-c)\\ &\quad+(m+1)^{2} T_{c} ^{2} c^{m+1}+c^{m+1} \frac{\bar{\sigma }^{2} }{4} \sum _{j=0}^{m-1}(W_{j} -1)^{2}+c^{m+1} (m+1)T_{c} \bar{\sigma }\sum _{j=0}^{m-1}W_{j} -1 \\ \end{aligned} $$
$$ \begin{aligned} E[T_{w} ^{2}]&=T_{s} ^{2}(1-c)q(1-p_{tr} (n-1))+ T_{s}^{2} (c-c^{m+1} )+2T_{s} T_{c}\left(-mc^{m+1}-\frac{c(1-c^{m} )}{1-c}\right)\\ &\quad+T_{c}^{2} \left(\frac{2mc^{m+1} }{1-c} -m^{2} c^{m+1} +\frac{c(c+1)(1-c^{m} )}{(1-c)^{2} } \right)\\ &\quad+\frac{\bar{\sigma }^{2} }{4} \left(c\sum _{j=0}^{m-1}(W_{j} -1)^{2} c^{j} -c^{m+1} \sum _{j=0}^{m-1}(W_{j} -1)^{2} \right)\\ &\quad+T_{s} \bar{\sigma }\left(c\sum _{j=0}^{m-1}\frac{W_{j} -1}{2} c^{j} -c^{m+1} \sum _{j=0}^{m-1}W_{j} -1 \right)\\ &\quad+\bar{\sigma }T_{c} \left(\sum _{j=0}^{m-1}(W_{j} -1)(j+1)c^{j+1} +\sum _{i=j+2}^{m}\sum _{j=0}^{m-2}(W_{j} -1 )c^{j}-mc^{m+1} \sum _{j=0}^{m-1}(W_{j} -1 )\right)\\ &\quad+(m+1)^{2} T_{c}^{2} c^{m+1}+c^{m+1} \frac{\bar{\sigma }^{2} }{4} \sum _{j=0}^{m-1}(W_{j} -1)^{2} +c^{m+1} (m+1)T_{c} \bar{\sigma }\sum _{j=0}^{m-1}(W_{j} -1) \\ \end{aligned} $$
$$ \begin{aligned} E[T_{w} ^{2}]&=T_{s} ^{2} (1-c)q(1-p_{tr} (n-1))+T_{s} ^{2} (c-c^{m+1} )+2T_{s} T_{c} \left(-mc^{m+1} -\frac{c(1-c^{m} )}{1-c} \right)\\ &\quad+T_{c} ^{2} \left( \frac{2mc^{m+1}}{1-c} +(2m+1)c^{m+1} +\frac{c(1+c)(1-c^{m} )}{(1-c)^{2}}\right)\\ &\quad+\frac{-\bar{\sigma }}{4} \sum _{j=0}^{m-1}(W_{j} -1)^{2} c^{j+1} +T_{s}\bar{\sigma }\sum _{j=0}^{m-1}(W_{j} -1)c^{j+1} +(T_{c} -T_{s} )\bar{\sigma }c^{m} \sum _{j=0}^{m-1}W_{j} -1\\ &\quad+{T_{c} \bar{\sigma }\sum _{j=0}^{m-1}(W_{j} -1)(j+1)c^{j+1} +T_{c} \bar{\sigma }\sum _{i=j+2}^{m}\sum _{j=0}^{m-2}(W_{j} -1) c^{j}} \\ \end{aligned} $$

We derive the average distance between two nodes inside a cell with radius r n . If r represents the distance from the center of the cell, then the average distance using infinitesimal rings could be formulated as:

$$ \bar{r}_{i,j} =\frac{\int _{0}^{r_{n} }2\pi r(rdr) }{\pi r_{n} ^{2} } =\frac{2}{3} r_{n} .$$

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Ghadimi, E., Khonsari, A., Diyanat, A. et al. An analytical model of delay in multi-hop wireless ad hoc networks. Wireless Netw 17, 1679–1697 (2011). https://doi.org/10.1007/s11276-011-0372-5

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