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Deterministic Formulization of Bandwidth
Efficiency for Multicast Systems
Syed S. Rizvi1 and Mustafa A. Khan
Computer Science and Engineering Department
University of Bridgeport, Bridgeport, CT 06601
{srizvi, mustafak}@bridgeport.edu
Tel: (757) 576-0928
Fax: (203) 576-4766
Aasia Riasat
Department of Computer Science
Institute of Business Management, Pakistan 78100
Aasia.riasat@iobm.edu.pk
Tel:92 (111) 002-004
Fax:92 (021) 509-0968
Abstract
End-System multicasting (ESM) is a promising application-layer scheme that has been recently proposed for implementing
multicast routing in the application layer as a practical alternative to the IP multicasting. Moreover, ESM is an efficient
application layer solution where all the multicast functionality is shifted to the end users. However, the limitation in bandwidth
and the fact that the message needs to be forwarded from host-to-host using unicast connection, and consequently incrementing
the end-to-end delay of the transmission process, contribute to the price to pay for this new approach. Therefore, supporting
high-speed real-time applications such as live streaming multimedia, videoconferencing, distributed simulations, and multiparty
games require a sound understanding of these multicasting schemes such as IP multicast and ESM and the factors that might
affect the end-user requirements. In this paper, we present both the analytical and the mathematical models for formalizing the
bandwidth efficiency of both IP and ESM multicast system. Specifically, our proposed formalization of the bandwidth efficiency
is based on the end-to-end delays proposed by [11] for both IP and ESM multicast systems. For the sake of the experimental
verifications of the proposed models, several numerical and simulation results are presented in this paper. Finally, the proposed
formulization can be used to design and implement a more robust and efficient multicast systems for the future networks.
Keywords – Unicast, multiple unicast, IP multicast, End-System multicasting, and overlay networks.
1Contact Author : srizvi@bridgeport.edu
978-1-4244-3314-8/09/$25.00 ©2009 IEEE
This full text paper was peer reviewed at the direction of IEEE Communic ations Society subject matter experts for publication in the IEEE IC4- 2009 proceeding.
Authorized licensed use limited to: University of Bridgeport. Downloaded on March 24, 2009 from IEEE Xplore, Restrictio ns apply.
Abstract— End-System multicasting (ESM) is a promising
application-layer scheme that has been recently proposed for
implementing multicast routing in the application layer as a
practical alternative to the IP multicasting. Moreover, ESM is an
efficient application layer solution where all the multicast
functionality is shifted to the end users. However, the limitation
in bandwidth and the fact that the message needs to be
forwarded from host-to-host using unicast connection, and
consequently incrementing the end-to-end delay of the
transmission process, contribute to the price to pay for this new
approach. Therefore, supporting high-speed real-time
applications such as live streaming multimedia,
videoconferencing, distributed simulations, and multiparty
games require a sound understanding of these multicasting
schemes such as IP multicast and ESM and the factors that might
affect the end-user requirements. In this paper, we present both
the analytical and the mathematical models for formalizing the
bandwidth efficiency of both IP and ESM multicast system.
Specifically, our proposed formalization of the bandwidth
efficiency is based on the end-to-end delays proposed by [11] for
both IP and ESM multicast systems. For the sake of the
experimental verifications of the proposed models, several
numerical and simulation results are presented in this paper.
Finally, the proposed formulization can be used to design and
implement a more robust and efficient multicast systems for the
future networks.
I. INTRODUCTION
There is an emerging class of Internet and Intranet multicast
applications that are designed to facilitate the simultaneous
delivery of information from a single or multiple senders to
multiple receivers. Different approaches of multicasting have
been suggested to improve the overall performance of
networks especially the Internet. These approaches are:
multiple unicast, IP multicast, and end-system multicast
(ESM). All of these methods have some advantages and
disadvantages but the last two approaches (IP multicast, and
ESM) mentioned above have had more research effort in terms
of performance evaluation of networks. ESM uses an overlay
structure, which is established on top of the traditional unicast
services. In this way, every pair of edges (source-destination)
2Contact Author : srizvi@bridgeport.edu
is a unicast connection. The overlay has its meaning from the
fact that the same link can have multiple unicast connections
for multiple pair of edges. Although, ESM seems to have
many advantages (no further changes to the network are
required, user has mo re control of the application layer, no
need of special multicast router capability, etc), there is a
penalty to pay. In the overlay structure, hosts are able to
multicast information and consequently use the same link to
redirect packets increasing the end-to-end delay of the entire
transmission process. Another problem is the number of
receivers that a potential “multicast” host can support. End
users have a limited bandwidth and suffer the last mile
problem.
While these different multicast approaches can displace
some of the costs of face-to-face communications, their true
potential business benefit lies in improving the accessibility
and timeliness of information, vastly increasing its value to
both the organization and individual employees. Although
research on multicast dates back to the early days of the
Internet, it has yet to produce a multicast service that is
ubiquitously and economically available. In spite of the
performance advantages, commercial deployment of multicast
has not yet been fully realized. One of factors that prevent the
wide-range deployment of multicast is the difficulty in
providing reliable multicast transport.
II. THEORETICAL ANALYSIS
In this section, we will theoretically analyze the problems of
different level of multicasting, which hinder their performance
with respect to the bandwidth utilization and latency.
A. IP Multicast
The IP-multicast capable version of the network shown in
Fig. 1 consists of network with native multicast support. IP
multicast capable routers are consider along the path. The
traditional process includes the construction of a source-rooted
tree together with the members of the multicast group. Since
only one copy of the message is required, we can say that a
minimum bandwidth effort is being used for the transmission
of the message to all group members connected in the
network. The problem for IP multicast is that there is no
commercial support for multicast routers. Investors still think
Deterministic Formulization of Bandwidth
Efficiency for Multicast Systems
Syed S. Rizvi
2
and Mustafa A. Khan
Computer Science and Engineering Department
University of Bridgeport, Bridgeport, CT 06601
{srizvi, mustafak}@bridgeport.edu
Tel: (757) 576-0928
Fax: (203) 576-4766
Aasia Riasat
Department of Computer Science
Institute of Business Management, Pakistan 78100
Aasia.riasat@iobm.edu.pk
Tel:92 (111) 002-004
Fax:92 (021) 509-0968
that there is not enough multicast application demand and that
multicast traffic could take their routers down due to
congestion problems. The IP-multicast transmission takes the
same bandwidth on source host's network as a single copy,
regardless of how many clients are members of the destination
host group in the Internet.
Besides the advantages of IP multicast, there are also certain
drawbacks of this approach. One of them is the deployment
problem of IP multicast, which imposes dependency on
routers. The main disadvantage of IP multicast is the need of
commercial routers supporting multicast protocol. Existing IP
multicast proposals [1] [2] embed an assumption of universal
deployment, as all routers are assumed to be multicast capable.
The lack of ubiquitous multicast support limits the deployment
of multicast applications, which in turn reduces the incentive
for network operators to enable multicast. Therefore, from the
above discussion one can expect that we need another
multicast alternative in which network routers have not to do
all of the work; instead each of the host will equally contribute
in the overall multicast process of the messages.
B. End-system Multicast (ESM)
Because of the limitations in IP multicast, researchers have
explored an alternative architecture named ESM, which built a
system on top of the unicast services with multicast
functionalities. ESM is a very promising application layer
solution where all the multicast functionality is shifted to the
end users as shown in Fig. 2. However, doing multicasting at
end-hosts incurs in some performance penalties. The structure
of the ESM is an overlay in a sense that each of the paths
between the end systems corresponds to a unicast path. Here
the membership and replication functionality is performed by
the end receivers, which connect together over unicast
channels to form a multicast tree, rooted at one data source.
The end receivers could play the role of parent or children
nodes. The parent nodes perform the membership and
Fig.2. Example of ESM, solid and dotted lines represent
two ways and one way packet transmission, respectively.
R-3
R
-
4
Source 3
Sender 4
1
2
3
4
5
1
2
3
4
5
6
7
8
9
10
D10
D9
Source 1
Router 1
Source 2
Router 2
1
2
3
1 2
1 2
1 2 3 4
1
2
3
4
5
6
7
8
9
1
2
3
1
D8
D7
D6
D5
D4
D3
D2
D1
Fig.1. Example of IP Multicast with four sources and routers along with 10 destination systems
replication process. The children nodes are receivers who are
getting data directly from the parent nodes. There is one
central control server and one central data server residing in
the same root source. Any receiver can play the role of parent
to forward data to its children. Each client has two
connections: a control connection and a data connection.
III. PROPOSED FORMULIZATION OF BANDWIDTH
EFFICIENCY FOR MULTICAST SYSTEMS
This section presents the formulization of the bandwidth
efficiency for all multicasting schemes. For the ease of
simplicity, we divide our proposed formulization for each type
of multicasting approach such as unicast, multiple unicast, IP
multicast, and ESM.
A. Model and Assumptions
Let G is an irregular graph that represents a network with a
set of N vertices and M edges such as:
{
}
,
G N M
=. Let L is
a direct communication link between a single pair of source
(s) and destination (d) where both source and destination
belong to N such as:
{
}
,
s d N
∈
. In addition, each packet
transmitted between source (s) and destination (d) must
traverse one or more communication links in order to reach
the final destination.
Let the value of D(L) denotes packet-delay that is associated
with each direct communication link. Therefore, each
transmitted packet will typically experience a delay of D(L) on
a particular link. The delay includes transmission, processing,
and propagation delays such as:
( )
Link Delay D L
− = =
Transmission Delay + Propagation Delay + Processing Delay
where L
∈
M. In connection less communication such as IP
network, there might be multiple routes exist between a pair of
source and destination. As a result, each packet might follow a
different route in order to reach the final destination where
each route requires traversing of one or more communication
links (L). A single route between a pair of source and
destination can be defined as:
{
}
{
}
, ,
R s d where s d N
∈
In order to approximate the bandwidth for Unicast,
multicast, and ESM, we use the classical definition of
computing the transmission time. Based on this definition, the
transmission time (TT) can be defined as a product of the
packet size (Ps) which we transmit between a pair of source (s)
and destination (d) and the inverse of the bandwidth (BW).
Mathematically, this can be expressed as
follows:
( )
1
T s
W
T P
B
=
Let the value of (
(s d )
D→ ) denotes the total packet-delay
which is associated with each direct communication link.
Therefore, each transmitted packet will typically experience a
delay of
(s d )
D→ on a particular link (L) between a pair of
source (s) and destination (d). This delay is the sum of the
transmission time, the queuing and the propagation delays
such as:
(s d )
D→ = Transmission Time (
T
T
) + Propagation
Delay (
τ
) + Queuing Delay (
Q
D
). Also, it can be expressed
as: (s d)
T Q
D T c D
→ = + + . Changing the above
expression for the transmission delay, we got
(
)
(
)
(s d )
( )
T Q
T D D
τ
→
= − − (1)
Recall our classical definition for the transmission time,
equation (1) can be rewritten as:
( )
( )
(s d )
1 ( )
s Q
W
P D D
B
τ
→
= − −
(2)
Solving equation (2) for approximating the bandwidth, we got
( )
( )
(s d )
( )
s
W
Q
P
B
D D
τ
→
=− − (3)
It should be noted that the propagation delay (∏) is a ration
between the distance for a communication link (LD) and the
speed of light (SoL). This allows us to further extend equation
(3).
(s d )
( )
s
W
D
Q
P
BL
D D
SoL
→
=
− −
(4)
Simplifying the above equation (4), we got
(
)
(
)
( )
(s d )
( ) – ( )( )
s
W
D Q
P SoL
B
SoL D L SoL D
→
=
−
(5)
For the sake of simplicity, we can ignore the queuing delay.
Equation (5) can now be written as:
(
)
(
)
( )
(s d )
( ) –
s
W
D
P SoL
B
SoL D L
→
= (6)
B. Bandwidth Efficiency Formulization for a Unicast System
In unicast, a packet is sent from one point (source) to
another point (destination). As mentioned earlier, when packet
transmit from one source (s) to a specified destination (d),
there exist multiple routes where each route can have multiple
links. This implies that the packet-delay for unicast is entirely
dependent on the number of links a packet needs to traverse in
order to reach the final destination system. Based on the above
argument, one can define the packet delay such as:
1 2
( ) ( ) ( ) ......... ( )
n
D R D L D L D L
= + + + where n is the
maximum number of links that need to be traversed on route R
between s and d.
We generalize the delay for one particular route (R) that
exists between source (s) and destination (d) as:
1
( ) ( )
n
i
i
D R D L
=
=∑where
1 2
1
( ) ......
n
i n
L L L L M
= + + + ∈
∑. This expression is
further extended as:
( ) ( )
( )
s d L R s d
Delay D D L
−∈ −
= =
∑
where
(
)
L R s d
∈ −
represents the value of the total delay
associated with the route R between source s and destination d.
For a unicast system, taking the above expressions into
account, the available estimated bandwidth (BW) for a
communication link (L) that exists between a pair of source (s)
and destination (d) can be approximated in the following
equation:
(
)
(
)
( )
( )
( ) –
s
W
D
L R s d
P SoL
B
SoL D L L
∈ →
=∑ (7)
The D(L) in (7) represents the link delay where as the
(
)
L R s d
∈ −
represents the value of the total delay
associated with the route R between source s and destination d.
The above equation (7) represents the approximated
bandwidth which can be used by the transmitted packet for
each individual communication link between the source and
destination. It should be noted that the above equation is not
representing the bandwidth approximation for one particulate
route between the source and destination. Instead, it represents
the bandwidth approximation for n number of links that need
to be traversed on route R between source (s) and destination
(d).
Based on the above derivation, one can also simply derive a
mathematical expression for an average-bandwidth, denoted
by ABW. The average bandwidth represents the available
bandwidth that each transmitted packet may utilize if it
traverses one of the available routes. Equation (6) can be
modified for the average delay between a pair of source (s)
and destination (d), denoted by as follows:
(
)
(
)
( )
( )
{ }
SoL
–
s
W
D
s d
P
AB
SoL AD L
→
= (8)
The mathematical expression for an estimated ABW can be
derived as follows:
(
)
(
)
( )
( )
1
–
s
Wy
i
i
D
P SoL
AB
D R
SoL L
y
=
=
∑ (9)
Where
1
( ) ( )
n
i
i
D R D L
=
=∑
1 2
1
( ) ......
n
i n
L L L L M
= + + + ∈
∑and y represents the
maximum number of possible routes between source s and
destination d.
In addition to the average bandwidth, one can also choose
the optimal route with respect to the minimum bandwidth that
each packet may require when traverses from one particular
source (s) to a destination (d). In order to derive an expression
for the optimal bandwidth, we may need to modify equation
(6) for the optimal delay. This is due to the fact that we
assume that for each link that offers minimum bandwidth must
have an optimal delay. This leads us to the following
modification of (6), such as:
(
)
(
)
( )
(s d )
( ) –
s
W
D
P SoL
OB
SoL OD L
→
= (10)
where
(s d )
OD → represents the optimal delay with respect
to the minimum delay that each packet may experience when
traverses from one particular source to a destination. Based on
(10), we can derive an expression for the optimal bandwidth,
denoted by
W
OB
, between a pair of source (s) and destination
(d) such as:
(
)
(
)
( ) ( )
1
–
s
Wy
i D
i
P SoL
OB
SoL Min D R L
=
=
∑
(11)
where 1 2
( ) ( ) ( ) ......... ( )
n
D R D L D L D L
= + + + and n
is the maximum number of links that need to be traversed on
route R between s and d.
C. Bandwidth Efficiency Formulization for a Multiple
Unicast
In addition to unicast systems, we can derive the similar
mathematical expressions for the multiple unicast system
where a single source (s) can transfer a packet simultaneously
to multiple destinations. In other words, in a multiple unicast
system, there exist a unicast route between a source (s) and
one of the destinations. This hypothesis leads us to the
following argument: multiple routes can be established
between the source (s) and each destination (d1, d1, d1,…….,dy)
where y represents the maximum number of unicast routes
established in multiple unicast. Based on this hypothesis, we
can modify (6) that account the total delay such as:
( )
( ) ( )
( )
( )
1 2
1 2
, ,......,
, ,......,
–
multiple
y
multiple
y
W s d d d
s
D
s d d d
B
P SoL
SoL D L
→
→
= (12)
The following mathematical expression can be used to
estimate the total bandwidth that the entire packet
transmission utilizes in a multiple unicast system:
( )
(
)
(
)
( ) ( )
, ,......,
1 2
1
–
multiple
W s d d d y
s
WB y
i
i
D
P SoL
B
SoL D R
L
→
=
=
∑
(13)
Although, in multiple unicast system, a single packet can be
transmitted from one source to multiple destinations, the
transmitted packet may follow a different route in order to
reach the appropriate destination. In particular, a bandwidth is
always associated with links rather than the complete routes
between the source and destinations.
As a result, each transmitted packet may use a different
amount of bandwidth with respect to the number of links that
the packet needs to traverse on the chosen unicast route. This
implies that, in order to estimate an average bandwidth that
each packer might utilize, one should consider the number of
maximum links a unicast route has. This leads us to the
following mathematical expression for an average available
bandwidth for the multiple unicast system:
(
)
(
)
( )
( )
{ }
SoL
–
s
W
D
s d
P
AB
SoL AD L
→
=
(14)
Further, solving (14) for average bandwidth approximation
results (15) such as:
( )
(
)
(
)
( )
( )
( ) ( )
1
1
–
s
yW s d
i
i
D
n
i i
i
P SoL
AB
D R
SoL L
y
where D R D L
→
=
=
=
∑
∑
(15)
where y represents the maximum number of unicast routes
between a source (s) and multiple destinations and n
represents the maximum number of links that a unicast route
has.
D. Bandwidth Efficiency Formulization for a IP Multicast
In IP multicast system, a single source (s) sends a packet to
a group that consists of multiple destination systems. In
addition, a packet is sent only once by the source system
where as the intermediate routers along the route perform
replications with respect to the number of destinations a group
has. Lets MG denotes a multicast group that consists of one or
more destination systems whereas Z represents the size of the
group such as Z = | MG |. In an IP multicast system, in order to
efficiently transmit a packet from a specific multicast source
to all multicast destinations, all multicast groups (MG) can be
typically organized in a spanning tree (T). For the ease of
mathematical expression, we only consider a spanning tree
rooted at the multicast source (s) consisting of one of the
multicast groups (MG) that has a size of Z.
Based on the above discussion, we describe the spanning
tree such as: T = (NT, MT) rooted as multicast source (s) where
the numbers of destinations in one multicast group (MG)
belong to the total number of nodes present in the network
such as: MG
∈
M where M represents the total edges that the
network has. The terms NT and MT represent the vertices and
the edges of the spanning tree (T), respectively. It should be
noted that we consider a spanning tree (T) that includes only
the multicast destinations of a multicast group (MG) with the
exception of intermediate routers. In other words, we assume
that NT of the spanning tree (T) only consists of one or more
destination nodes. The reason for this assumption is to
simplify the process of estimating the total available
bandwidth involves with the packet transmission in an IP
multicast system.
Based on the above proposed model, we can give the
following hypothesis: The total available bandwidth (TBW)
utilizes by multicast packets when transmitted from a root
node (s) to a multicast group (MG) can be defined as a ration
of packet size and the available bandwidth on each link of a
spanning tree (T) from the root nodes (s) to all destinations (d
∈
MG) with the bandwidth available to one or more
intermediate routers of each link.
The above hypothesis leads us to the following expression
for total available bandwidth (TBW) utilizes by multicast
packets transmitted from root node (s) to a destination node
(d):
(
)
(
)
( )
(s )
( ) –
G
s
W
M D
P SoL
TB
SoL D L
→
= (16)
Computing the total delay for the IP multicast and using its
resulting values in (16), we provide an expression for the total
available bandwidth such as:
( ) ( ) ( )
1 1
–
s
WZ n
D
i i
i i
P
TB
L
D L D L
SoL
= =
=
+
∑ ∑
(17)
where Z in the denominator of (17) represents the number of
destination systems in one multicast group of a spanning tree
(T) where n represents the total number of links a route has.
The first term of the denominator of (17)
(
( )
1
Z
i
i
D L
=
∑)yields the total delay associated with the number
of links within a spanning tree when a packet is transmitted
from a root node (source) to all the leaf and non-leaf nodes
(i.e., the multiple receivers with in T excluding the source
node (s)). In other words, according to spanning tree, if a
message is transmitted from a source node (s) to all the
destination nodes, then the packet must traverse Z (i.e., the
number of destinations in one multicast group) number of
links which consequently experience a delay on each link.
The second main term of the denominator (
( )
1
n
i
i
D L
=
∑) in
(17) provides a total delay that a packet may experience when
transmitted along a certain route (i.e., from a source node (s)
to multicast group (MG) via one or more routers along the
route). The last term of the denominator can not be considered
as a design parameter and therefore its value completely
depends on the type of network.
The above equation can be further generalized for one of the
specific destinations (d) within a multicast group such as d
∈
MG, if we assume that we have a route within a spanning tree
(T) from multicast source (s) to a specific destination (d) such
as RT (s, d), then the multicast packets transmitted from a
source node may utilize a total available bandwidth such as:
( )
( )
( )
–
G
G
s
W s d M
D
s d M
P
TB L
D
SoL
→ ∈
→ ∈
=(18)
Determine the total delay for the multicast packets that
transmit from a source node to one of destinations within a
group and using the resulting expression in (18), we got,
( )
(
)
(
)
( )
( )
( )
,,
,
–
G
n Z T
s
W s d M
n Z
L R s d
D
P SoL
TB
SoL D L
L
→ ∈
∈
=∑(19)
where Ln, Z in (19) represents the total number of links (i.e.,
Z
∈
RT) that a packet needs to traverse in order to reach the
specific destination d along a path of RT within the tree T as
well as the number of links from source s to a multicast group
MG.
Since only one copy of the message is required in IP
multicast, we can say that a minimum bandwidth effort is
being used for the transmission of the message to all group
members connected in the network. This minimum bandwidth
is achieved due to the fact that a minimal transmission time is
required for the IP multicast which in turns reduces the overall
end-to-end delay as can be seen in the denominator of (19).
E. Bandwidth Efficiency Formulization for ESM
An ESM group can have at most N end-system nodes where
we focus on one of the end-system nodes (s) that multicast
information to the other participating nodes of a multicast end-
system group. From the source host point of view, this ESM
multicast group can be considered a group of destination
systems. For the sake of mathematical model, lets ESMG
denotes an ESM group that consists of one or more end-
system-destinations where as X represents the size of the
group such as X = | ESMG |.
In an overlay network, all participating end-system nodes
are fully connected to each other via the unicast links. Based
on the derived expression of unicast in the previous sections,
these unicast links that provide connection between end-
system nodes can not exceed to M such
as:
{
}
1 2
, ,........., y
m m m M
∈where one of the edges (m)
provides a unicast connection between the two end-system
nodes such as:
{
}
{
}
{
}
1 2
, ,
unicast link
m M n n s N
−
∈ → ⊂ .
The structure of the ESM is an overlay network in a sense
that each of the paths between the end-systems corresponds to
a unicast path. This implies that an overlay network consisting
of a set of N end-system nodes connecting though M number
of edges where one of the end-system is designated as a source
host (s) can be expressed as:
{
}
, ,
G s N M
=.
This also shows that an ESM is built on top of the unicast
services using a multicast overlay network that can be
organized in a spanning tree such as T = (NT, MT) rooted as an
ESM source (s) where the numbers of destinations in one
multicast group (ESMG) belong to the total number of nodes
present in the network such as: ESMG
∈
M. Here, the
membership and replication functionality is performed by the
ESM receivers, which connect together over unicast channels
to form a multicast tree (T), rooted at one data source (s). The
end receivers (i.e., the number of end-systems in ESMG) in a
multicast tree could be a parent or a child node depending on
the location of the node. In a multicast spanning tree (T), all
the non-leaf nodes can be both parent and child at the same
time where as all the leaf nodes are considered to be the child
nodes. In other words, the parent nodes perform the
membership and replication process where as the children
nodes are receivers that receives multicast packet from the
parent nodes.
Based on the above argument, one can say that a multicast
packet originated from the root (s) of a spanning tree (T) need
to traverse typically two links; source to non-leaf node (Pn, Cn)
and a non-leaf node to a leaf node (Cn). Lets RT (s, non-leaf
node) represents a route between a source node (s) and non-
leaf nodes that could be parent or child nodes such as:
{
}
T n n G
R P C ESM= ∨ ∈ where
{
}
,
n n
P C s N
∈U.
Similarly, RT (Pn, Cn) represents a route from a parent node
to a child node such as:
{
}
,
T n n G
R P C ESM
= ∈ . The above
arguments lead us to the following expression for computing
the total bandwidth available for transmitting a multicast
packet from a source node to one or more parent nodes (i.e.,
the bandwidth associated with the first link of transmission):
( )
{ }
( )
{ }
–
multiple unicast
n n
multiple unicast
n n
W s P C s N
s
D
s P C s N
TB
P
L
D
SoL
−
−
→ ∨ ∈
→ ∨ ∈
=
U
U
(20)
Determining the mathematical expression for the total delay
involve in transmitting a multicast packet from a source node
to one or more parent nodes and using the resulting expression
in (20) yields (21) for approximating the total bandwidth such
as:
( )
{ }
( )
( )
( )
,
1
,1
–
multiple unicast
n n
n n
n n
W s P C s N
s
y
D
i s P C
i
n
i
ii s P C i
TB
P
L
D R
SoL
where D R D L
−
→ ∨ ∈
→
=
→ =
=
∑
∑
U
(21)
where y in (21) represents the maximum unicast routes
between a source (s) and one or more non-leaf nodes and n
represents the maximum number of links a unicast route can
have.
Based on (21), we can derive a similar mathematical
expression for approximating the total bandwidth available for
a multicast packet which is transmitted from a parent node to a
child node as follows:
( )
( )
( )
( )
( )
( )
,
1
,
1
–
multiple unicast
n n
n n
n n
s
y
W P C
i P C
i
D
n
i
i P C
i
P
TB
D R
L
SoL
where D R D L
−
→
=
=
=
∑
∑
(22)
When comparing (21) with (22), it can be clearly evident
that only the total delay component in the denominator of both
equations differ with each other for both presented scenarios
which in turns change the amount of available total
bandwidth. By combining the above two equations, the total
bandwidth that a multicast packet may utilize when
transmitted from a source node (s) to a child node (Cn) can be
approximated as follows:
( ) { }
( ) ( )
( )
( )
( )
( )
,
,
1
,
1
–
+
multiple unicast
n n
n n
n n
W s C s P N
s
n
i s P C
i
D
n
i P C
i
TB
P SoL
D L
SoL L
D L
−
→ ∈
→
=
=
=
∑
∑
U
(23)
As one can see in (23) that the hosts are able to multicast
information and consequently use the same link to redirect the
packets resulting in an increase in the end-to-end delay of the
entire transmission process. The limitation in bandwidth and
the fact that the message needs to be forwarded from host-to-
host using unicast connection, and consequently incrementing
the end-to-end delay of the transmission process as can be
seen in the denominator of (23), contribute to the price to pay
for this new approach. For the ESM, this price will be paid in
terms of the second quantity of the denominator of (23). (i.e.,
( )
(
)
,
1n n
n
i P C
i
D L
=
∑). These reasons might make ESM slightly
less efficient than the IP multicast system.
IV. CONCLUSIONS
In this paper, we presented both analytical and mathematical
models for all the multicast approaches currently available for
multimedia applications. We first presented a mathematical
model for multiple unicast systems in which the source host
has to send a single copy of data to every single receiver. Our
proposed formulization suggested that this approach wastes a
lot of significant network bandwidth. Secondly, we presented
a mathematical model for the IP multicasting approach which
is a more efficient concept where the data source only sends
one copy of data which is replicated as necessary when
propagating over the network towards the receivers. Our
proposed formulization shows that the IP multicast
demonstrates some good bandwidth efficiency characteristics
than the other multicast approaches. Finally, we presented a
complete formulization of bandwidth efficiency for ESM
systems, which is an alternate to router-dependent multicast
service that allows end-systems participating in a multicast
group to replicate and forward packets to other receivers. Our
proposed formulization of bandwidth efficiency suggests that
the ESM is a feasible, especially for sparse, medium size
group.
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