Tamkang Journal of Science and Engineering, Vol. 13, No. 4, pp. 443-454 (2010)
443
Analysis of Internet Connectivity and Number of
Isolated Clients in Finite One-Dimensional Multihop
Wireless Access Networks
Chun-Yen Hsu1*, Shun-Te Wang2 and Jean-Lien C. Wu3
1
Networks and Multimedia Institute, Institute for Information Industry,
Taipei, Taiwan 105, R.O.C.
2
Department of Electronic Engineering, Hwa Hsia Institute of Technology,
Taipei, Taiwan 235, R.O.C.
3
Department of Computer and Communication Engineering, St. John’s University,
Tamsui, Taiwan 251, R.O.C.
Abstract
Multihop wireless access networks, where clients can access the Internet through Internet
gateways by direct or multihop transmissions, are potential for future wireless data services and
vehicular network applications. In the context of multihop transmissions, less number of gateways is
required for Internet access while a certain level of Internet connectivity still holds. In this paper, we
investigate the Internet connectivity, which is the probability that all clients are Internet-reachable, in
one-dimensional multihop wireless access networks. We show the relationship between the Internet
connectivity and the deployment of Internet gateways. We also investigate the mean number of
isolated clients, who are not Internet-reachable, in a multihop wireless access network. These results
are useful for network operators to deploy Internet gateways, manage clients and design protocols for
future vehicular network applications. The theoretical results are corroborated through extensive
simulations.
Key Words: Internet Gateway, Multihop Wireless Access Network, Vehicular Ad Hoc Network,
Internet Connectivity, Isolated Client
1. Introduction
The remarkable advances of wireless network technologies and protocols start a new era where vehicles can
communicate with the world, with each other, and with
the electronic devices within them. Vehicular networking is critical to the intelligent transportation systems
(ITS) and automotive electronics applications [1,2]. The
integration of sensors, actuators, computer processors,
wireless communication devices and human interfaces
helps the realization of intelligent vehicles which provide drivers with required information under a variety of
driving conditions. Although research on vehicular net*Corresponding author. E-mail: hcy@iii.org.tw
works is very system and application dependent, there
are still some common issues that must be dealt with. To
enable quick and cost-efficient distribution of data in a
vehicular network, advanced network technologies and
reliable communication protocols therefore need to be
developed.
Among the applications of mobile ad hoc networks
(MANETs), a vehicular ad hoc network (VANET) provides communications among nearby vehicles and roadside equipments [3-6]. Because most vehicles have
ample power and tend to move in an organized manner,
e.g., along a highway, limitations commonly assumed in
MANETs, such as computational and power resources,
and node mobility patterns, are mitigated in VANET research. Hence, high-performance, highly scalable and
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secure communication networks can then be investigated more effectively.
From the application point of view, future wireless
systems are being built to deliver novel and various types
of high-speed multimedia services, so network service
providers can extend their offerings besides preliminary
services. For example, the Internet Protocol Multimedia
Subsystem (IMS) [7] initiated by the third generation partnership project (3GPP) is the platform for providing a unified session control on top of various access network
technologies for realizing real-time interactive multimedia services. The IMS equipment that provides an interface between the radio network (i.e., access network) and
the IP-based network is the access gateway. Before configuring the gateways for an access network, a network
operator must consider the number of access gateways
needed, and their location. However, in the determination
of the number of gateways, there is a tradeoff between
the deployment cost and the network performance.
Providing Internet access services ubiquitously is
one of the most promised goals in the research domain of
networking, which presents an extraordinary challenge
to the VANET research community. In this paper, we
investigate the Internet connectivity and the mean number of isolated clients in one-dimensional multihop wireless access networks. Our results are helpful to the deployment of Internet gateways, client management and
future protocol design. Note that the Internet connectivity is defined as the probability that all clients in a network are Internet-reachable, and an isolated client is defined as a client that is not Internet-reachable. Furthermore, Internet gateways are naturally Internet-reachable.
The rest of this paper is organized as follows. In section 2 we describe the model of the system under consideration. Then in section 3, the Internet connectivity
is analyzed given the type of one-dimensional network
topology. The mean number of isolated clients is discussed and derived in section 4. In section 5, a number of
simulation experiments are carried out to verify against
theoretical results. Finally, section 6 summarizes the
study of this paper.
2. System Model
domly deployed and that radio links can be established if
and only if any two nodes are located within a transmission range of r, are usually modeled by geometric random graphs [8-10]. Given the number of nodes n, Desai
and Manjunath [11] show the probability that all the n
randomly deployed nodes are connected in an interval
[0, z]. Based on the result of [11], A. D. Gore [12,13], investigates the network connectivity of a sensor network
with a sink node located at the boundary. Foh et al. [14]
investigate the network connectivity of one-dimensional
MANETs where nodes move according to the random
waypoint model [15].
Based on the result of [11], we study here the Internet connectivity and the number of isolated clients in
one-dimensional multihop wireless access networks. In
the rest of this paper, we follow the notation used in [11],
where the authors have shown the probability that an ad
hoc network where n clients are uniformly deployed in
[0, z] is connected is
(1)
where Un (z) is the volume of the set of all realizable
networks in [0, z],
(2)
and Uc (n, z, r) is the volume of the set of all connected
networks in [0, z],
(3)
Note that here u(z - kr) is the unit step function.
2.2 Network Types and Definitions
Consider a one-dimensional finite multihop wireless
access network with gw Internet gateways located in [0,
a], a > 0. These gateways partition the network into gw 1 segments with respective length z1, z2, …, z gw -1 , where
gw - 1
åz
i
= a, as shown in Figure 1. Each segment contains a
i =1
2.1 Related Work
Multihop wireless networks, where nodes are ran-
number of clients and two Internet gateways located at
the left and right boundary. Clients located outside the
Analysis of Internet Connectivity and Number of Isolated Clients in Finite One-Dimensional Multihop Wireless Access Networks 445
interval [0, a] are not considered. Assume that the number of clients in a segment is statistically independent of
each other. Clients in a segment are uniformly distributed. All nodes, including Internet gateways and clients, have the same transmission range of r. Multihop
transmissions can occur in segment i only if zi is greater
than 2r. A client is Internet-reachable either it is located
within the transmission range of an Internet gateway or it
has any multihop relay path to an Internet gateway.
In a segment i containing ni clients, ICi (ni, zi, r) and
niso ,i ( ni , z i , r ) stand for its Internet connectivity and mean
number of isolated clients, respectively. Internet connectivity, IC, and mean number of isolated clients, N iso , of
the whole network can be obtained as follows,
(4)
Internet connectivity and mean number of isolated clients of a generalized segment i that accommodates ni
clients in [0, zi].
Assume all clients in segment i are Internet-reachable. An access network must be one of the four C-Types
as shown in Figure 2. In a C-Type-1 network, all nodes,
including the two Internet gateways and clients, are connected. In a C-Type-2 network, all clients are connected,
and only the left Internet gateway connects with the clients. In a C-Type-3 network, all clients are connected,
and only the right Internet gateway connects with the clients. In a C-Type-4 network, a gap that partitions clients
into two groups exists, and each Internet gateway serves
only one group of clients. Obviously, a C-Type-4 network is possible if and only if ni is greater than one. The
Internet connectivity can be obtained as
(5)
(6)
In the rest of this paper, we focus on the analysis of the
Figure 1. One-dimensional wireless multihop access network.
Figure 2. Internet connection types.
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where Pri {C - Type - k} is the probability that the network in segment i is of C-Type-k. Let UIC, i, k (ni, zi, r) be
the volume of the set formed by all C-Type-k networks
in segment i. Eq. (6) can then be rewritten as
(7)
Let x1, x2, …, x ni be the ordered positions of the ni
clients. Define yj = xj + 1 - xj, x0 = 0, thus y$ = [y0, y1, ...,
yni -1 ] represents the network. Convex polytope (8) describes all realizable networks in segment i.
A network may contain isolated clients. In a segment, the network containing isolated clients must be
any of the four I-Types as shown in Figure 3. In an IType-1 network, all clients are isolated. In an I-Type-2
network, only the left Internet gateway connects with
certain clients. In an I-Type-3 network, only the right
Internet gateway connects with certain clients. Apparently, an I-Type-2 or I-Type-3 network is possible if and
only if ni is greater than one. In an I-Type-4 network, two
or more gaps partition clients into three groups, i.e., two
groups of Internet-reachable clients and one group of
isolated clients. An I-Type-4 network is possible if and
only if ni is greater than two.
The mean number of isolated clients in segment i can
be obtained as follows.
(8)
Although y$ is sufficient to describe all realizable networks in [0, zi], for the sake of easy interpretation, define x ni + 1 = z i , thus
(9)
So, the following convex polytope describes all realizable networks in segment i
(10)
(11)
where niso ,i ,k ( ni , z i , r ) is the mean number of isolated clients of I-Type-k networks in segment i.
A notation table is provided in Appendix A for reference.
3. Internet Connectivity
In this section, we in turn interpret the calculation of
the Internet connectivity of the four C-Types.
Figure 3. Types of network with isolated clients.
Analysis of Internet Connectivity and Number of Isolated Clients in Finite One-Dimensional Multihop Wireless Access Networks 447
3.1 C-Type-1 Network
Convex polytope (12) describes all networks that are
of C-Type-1 in segment i.
(12)
3.4 C-Type-4 Network
In a network of C-Type-4, clients are partitioned by a
gap into two groups. The length of the gap is at least r.
Convex polytope (18) describes all networks that are of
C-Type-4 in segment i.
Note that a C-Type-1 network is a network that all clients are connected in [0, zi] while in both intervals [0, r)
and [zi - r, zi] there is at least one client. Thus
(13)
3.2 C-Type-2 Network
Convex polytope (14) describes all networks that are
of C-Type-2 in segment i.
(14)
Note that a C-Type-2 network is a network that all clients are connected in [0, zi] while in [0, r) there is at
least one client and in (zi - r, zi] there is no clients. In
other words, a C-Type-2 network is a network that all
clients are connected in [0, zi - r] while in [0, r) there is
at least one client. So
(15)
3.3 C-Type-3 Network
Convex polytope (16) describes all networks that
are of C-Type-3 in the segment i.
(18)
In a C-Type-4 network, { y0 , y1 , ..., yh , ..., yni }C-Type- 4 ,
since yh is the only gap in the netwrk, by interchanging
y h and yni , we can obtain a C-Type-2 network, { y0 ,
y1 ,..., yh- 1 , yni , yh+ 1 , ..., yni - 1 , yh }C-Type- 2 . Here the mapping between C-Type-4 and C-Type-2 networks is not
one-to-one correspondence. In fact, transforming a CType-4 network into a C-Type-2 network, by interchanging yh and yni , results in a (ni - 1)-to-one and onto
mapping. In other words, each C-Type-2 network represents exactly (ni - 1) realizable C-Type-4 networks.
Hence, we have
(19)
Thus, Internet connectivity of the segment can be
obtained using Eq. (7).
4. Isolated Clients
In this section, we in turn interpret the calculation of
mean number of isolated clients in a segment.
4.1 I-Type-1 Network
Convex polytope (20) describes all networks that are
of I-Type-1 in segment i.
(16)
(20)
Note that there is a one-to-one correspondence mapping between C-Type-2 and C-Type-3 networks because each C-Type-3 network can be uniquely obtained
by inverting the yk sequence of a C-Type-2 network,
and vice versa. Therefore
(17)
An I-Type-1 network is present if and only if all clients
are located in the interval [r, zi - r]. Thus, the probability that an I-Type-1 network is present is
(21)
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Therefore, the mean number of isolated clients is
(22)
4.2 I-Type-2 and I-Type-3 Networks
Convex polytope (23) describes all I-Type-2 networks in segment i.
(24)
Then, by shifting all mk to the right of the network, convex polytope (25), which representing C-Type-2 networks, is obtained
(23)
An I-Type-2 network contains at least two gaps, i.e., yh
and yni . Clients are partitioned into one group of Internet-reachable clients and one group of isolated clients. Note that clients in the Internet-reachable group
are surely pairwise-connected. This feature may not
hold for isolated clients because gap(s) may appear between them. In case the number of gaps is greater than
two, isolated clients are further partitioned into isolated
subgroups, and clients in different isolated subgroups
are mutually unreachable.
Derivation of the mean number of isolated clients in
any I-Type-2 network is shown as follows. Define yk¢ =
yk - mkr, 0 £ yk¢ < r, mk = 0, 1, …, thus yk¢ = yk and mk = 0
if and only if yk < r. Convex polytope (23) can be rewritten as
(25)
Figure 4 shows an example of transforming a threegap I-Type-2 network to a C-Type-2 network.
The mapping of I-Type-2 networks to C-Type-2 networks is not one-to-one correspondence. In an I-Type-2
network, given the number of gaps, Ng, the isolated clients are partitioned into (Ng - 1) isolated subgroups.
Given Ng and the number of isolated clients, Niso, there
æ N iso - 1 ö
÷ ways to distribute Niso clients into (Ng - 1)
are ç
ç Ng -2 ÷
ø
è
isolated subgroups such that each isolated subgroup
contains at least one isolated client. In other words,
Figure 4. Example of transforming an I-Type-2 network into a C-Type-2 network.
Analysis of Internet Connectivity and Number of Isolated Clients in Finite One-Dimensional Multihop Wireless Access Networks 449
æ N iso - 1 ö
÷ I-Type-2 networks are possible. Define Lg =
ç
ç Ng -2 ÷
ø
è
ni
æ Lg - 1 ö
÷ possible commk . Given Ng and Lg, we have ç
å
ç N g -1÷
k= 0
è
ø
binations such that each gap has at least a length of r.
Thus, given Lg, Ng and Niso, the mapping of I-Type-2 netæ N iso - 1 ö æ Lg - 1 ö
÷÷ç
works to C-Type-2 networks is ç
ç N g - 2 ÷ ç N -1÷
øè g
è
ø
to-one and onto mapping. Note that in an I-Type-2 network, Ng follows
(26)
(31)
Finally, we can obtain the mean number of isolated clients in the network of I-Type-2,
(32)
For I-Type-3 networks, convex polytope (33) describes all I-Type-3 networks in segment i.
Lg follows
(33)
(27)
Since I-Type-2 and -3 are dual, the mean number of isolated clients in the network of I-Type-3 is
and Niso follows
(34)
(28)
Thus, the probability that an I-Type-2 network is present in segment i is
4.3 I-Type-4 Network
In a network of I-Type-4, isolated clients are present
in the middle of the segment. Thus we have three groups
of clients: two groups of Internet-reachable clients and
one group of isolated clients. An I-Type-4 network contains at least two gaps. Convex polytope (35) describes
all I-Type-4 networks in segment i.
(29)
where U iso ,i ,2 ( ni , z i , r, L$g ) is the volume of the convex
polytope formed by all I-Type-2 networks whose Lg
value is equal to L$g . Convex polytope (30) describes all
such networks.
(35)
Similar to the transformation of convex polytopes (23)
into (24) described in Section 4.2, convex polytope (35)
can be transformed as
(30)
The volume of convex polytope (30), i.e., Uiso, i, 2 (ni, zi,
r, L$g ), can be calculated as
(36)
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Then, by shifting all mk to the right of the network, convex polytope (37) that represents all C-Type-2 networks is obtained
And we can obtain the mean number of isolated clients
in the network of I-Type-4
(41)
(37)
The mapping of I-Type-4 networks to C-Type-2
networks is not one-to-one correspondence. Note that,
given Niso, we have (ni - Niso - 1) different ways to distribute the Internet-reachable clients over the two Internet-reachable groups such that each group contains at
least one client. Furthermore, given Lg, Ng and Niso, as
æ N iso - 1 ö æ Lg - 1 ö
÷
֍
described in section 4.2, we have ç
ç N g - 2 ÷ ç N -1÷
g
øè
è
ø
ways to distribute Niso isolated clients into (Ng - 1) isolated subgroups such that each isolated subgroup contains at least one isolated client while each gap has at
least a length of r. Thus, given Lg, Ng and Niso, the mapping of I-Type-4 networks to C-Type-2 networks is
æ N iso - 1 ö æ Lg - 1 ö
÷ -to-one and onto map÷ç
( ni - N iso - 1) ç
ç N g - 2 ÷ ç N -1÷
øè g
è
ø
ping. Note that in an I-Type-4 network, Ng follows
(38)
Lg follows Eq. (27) and Niso follows
(39)
Therefore, the probability that I-Type-4 network is present is
(40)
Now, the mean number of isolated clients in a segment
and in the whole network can be obtained using Eqs.
(11) and (5), respectively.
5. Verification
In this section, we verify our analysis on Internet
connectivity and mean number of isolated clients against
simulation using C++ program. In the simulations, the
number of clients, ni, varies from 5 to 50. Clients are randomly distributed over the segment i. The ratio of transmission range to segment length, r/zi, varies from 0.001
to 0.45. Figure 5 depicts respectively the occurrence
probabilities of C-Type-k network. Curves show theoretical results while marks are simulation results. Simulation and theoretical results match within a reasonable
error tolerance (say, 0.1%). Each simulation result comes
from the statistic of 108 different random generated networks. The Internet connectivity in a segment, which is
the aggregation of occurrence probabilities of C-Type-1
to -4, is depicted in Figure 6. Given a desired Internet
connectivity and expected number of clients in a segment, we can obtain the required value of the ratio of
transmission range to segment length, i.e., r/zi. Note that,
in many cases, r is predefined and fixed, so the maximum
allowable distance between two neighboring Internet
gateways, i.e., zi, can be calculated from Figure 6.
Figures 7 (a)-(d) depict the mean number of isolated
clients in an I-Type-k network, respectively. The mean
number of isolated clients in a segment, which is the aggregation of mean number of isolated clients of I-Type-1
to -4 networks, is provided in Figure 8.
Our theoretical results are valid for networks where
clients are uniformly deployed. We have also tested two
simulation scenarios with mobile clients to see whether
or not clients’ mobility influences our results. In the first
Analysis of Internet Connectivity and Number of Isolated Clients in Finite One-Dimensional Multihop Wireless Access Networks 451
Figure 5. The occurrence probabilities of C-Type-1 to -4 networks.
Figure 6. The Internet connectivity in a segment.
scenario, 25 mobile clients are deployed randomly in a
segment. Clients’ velocity is uniformly distributed in
[10] m/s with a randomly chosen direction. Each client’s
velocity is randomly assigned at the beginning of a simulation run and is fixed during that simulation. Boundary
wrap-around is used in the simulation, i.e., when a client
crosses the boundary of the segment, a new client immediately enters the segment from the other boundary with
the same velocity as the one leaves. In this way, the segment always accommodates 25 clients. In the second
scenario, the number of clients in the segment varies.
Clients enter the segment according to a Poisson process.
Clients are classified into three groups. Table 1 summarizes arrival rate and velocity of clients. Each simulation
run lasts for 3600 seconds. Simulation results shown in
Figures 9 and 10 are the average of 1000 runs. Note that,
in the second scenario, no client exists in the segment at
the beginning of simulation. Therefore, a 1000-sec presimulation time is used in each run in the simulation of
the second scenario.
Figures 9 and 10 depict the Internet connectivity and
mean number of isolated nodes in a segment, respectively. Simulation results of the first scenario are comparable with those shown in Figure 6 (ni = 25). In the
second scenario, the number of clients in the segment
varies with time. Therefore, the Internet connectivity is
(42)
where Pr{ni = ns} is the probability that there are exactly ns clients in the segment. Since ns clients may con-
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6. Conclusion
sist of the clients belonging to a separate group or a mix
of the three groups, we have
(43)
A multihop wireless access network can be used as
the network architecture for VANETs and certain sensor
networks, which may be the most promising architecture
for future access networks. In a multihop wireless access
And the mean number of isolated nodes in segment i in
the second scenario is
(44)
Figures 9 and 10 illustrate that clients’ mobility makes
no difference to the theoretical results derived from
sections 3 and 4.
Figure 8. The mean number of isolated clients in a segment.
Figure 7. The mean number of isolated clients in I-Type-1 to -4 networks.
Table 1. Arrival rate and velocity of clients in the second scenario
Arrival rate (1/sec)
Moving speed (zi/sec)
Moving direction
Group 1
Group 2
Group 3
l1 = 0.1
v1 = 0.01
Randomly chosen
l2 = 0.05
v2 = 0.02
Randomly chosen
l3 = 0.04
v3 = 0.025
Randomly chosen
Analysis of Internet Connectivity and Number of Isolated Clients in Finite One-Dimensional Multihop Wireless Access Networks 453
Figure 9. Internet connectivity (mobile clients).
network, the arrangement of Internet gateways is crucial
for the network performance and cost. In this paper, we
investigate the Internet connectivity and mean number of
isolated clients in finite one-dimensional multihop wireless access networks. Based on the knowledge of the number of clients and radio transmission range, we can obtain
the maximum allowable distance between two neighboring Internet gateways to meet the required Internet connectivity. With further knowledge of the distance between
two neighboring Internet gateways, we can also obtain the
mean number of isolated clients in the network. We verify
the theoretical results against simulation results and show
that they agree with each other. These results will be useful in the design, deployment and management of future
multihop wireless access networks.
Appendix A
Notation
gw
Description
The number of Internet gateways in the
network
IC
The Internet connectivity of the network
ICi(ni, zi, r) The Internet connectivity of segment i
ni
The number of vehicles in segment i
The mean number of isolated clients in the
N iso
network
niso ,i ( ni , z i , r ) The mean number of isolated clients in
segment i
r
Transmission range
Uc(ni, zi, r)
The volume of the set of all connected
networks in segment i
UIC,i,k(ni, zi, r) The volume of the set formed by all CType-k networks segment i
The volume of the set of all realizable netU ni ( z i )
Figure 10. Mean number of isolated clients (mobile clients).
xk
yk
zi
works in segment i
The position of the kth vehicle
yk = xk+1 - xk
The length of segment i
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Manuscript Received: Oct. 27, 2008
Accepted: Sep. 9, 2009