An Improved Rebroadcast Probability Function for an
Efficient Counter-Based Broadcast Scheme in MANETs
Aminu Mohammed1, Mohamed Ould-Khaoua2, and Lewis Mackenzie1
Abstract
Flooding is the simplest mechanism for broadcasting in mobile ad hoc
networks (MANETs), where each node retransmits a given broadcast packet
exactly once. Despite its simplicity, flooding can result in high redundant
retransmission, contention and collision in the network, a phenomenon referred
to as broadcast storm problem. Several probabilistic approaches have been
proposed to mitigate this inherent phenomenon. However, majority of these
schemes uses fixed rebroadcast probability which is quite unlikely to be optimal
in other simulation set up. In this paper, we propose an improved rebroadcast
probability function which takes into account immediate neighbor information
and some key simulation parameters (i.e. network topology size, transmission
range and number of nodes) in determining the appropriate rebroadcast
probability for a given mobile node. Simulation results reveal that this simple
adaptation achieves superior performance in terms of saved-rebroadcast,
number of retransmission node and end-to-end delay, without sacrificing
reachability in dense network.
Keywords: MANETs, Flooding, Broadcast storm problem, Saved-rebroadcast,
Simulation, Reachability, Latency.
1 Introduction
Mobile ad hoc networks (MANETs) are a special type of wireless network
characterized by dynamic topology without the aid of any fixed infrastructure,
centralized administration or standard support services [1]. This enables wireless
communications between participating mobile devices with the assistance of other
mobile devices. In such a network, each mobile node operates not only as a host but
also as a router so that it can send and receive messages as well as forward messages
for others. Scenarios that might benefit from MANETs technology include
rescue/emergency operations in natural or environmental disasters areas, special
operations during law enforcement activities, tactical missions in hostile and/or
unknown territories, and commercial/academic gatherings such as conferences,
exhibitions and workshops [1].
Broadcasting is a means of diffusing a message from a source node to all
other nodes in the network. It is a fundamental operation which is extensively used in
route discovery, address resolution and many other network services in a number
routing protocols[2]. For example Dynamic Source Routing (DSR) [3], Ad hoc On
Demand Distance Vector (AODV) [4], Zone Routing Protocol (ZRP) [5] and
Location Aided Routing (LAR) [6] use broadcasting or its derivative to establish
1
Department of Computing Science, University of Glasgow, Glasgow, G12 8RZ, U.K.,
[email protected]; [email protected].
2 Department of Electrical & Computer Engineering, Sultan Qaboos University, Al-Khodh 123,
Muscat, Oman, [email protected].
routes. Other routing protocols, such as the Temporally Ordered Routing Algorithm
(TORA) [7] use broadcasting to transmit an error packet for invalid routes. These
protocols typically assume a simplistic form of broadcasting known as flooding, in
which each mobile node retransmits every unique received packet exactly once.
Despite its simplicity and high success rate in reaching all nodes in the network, it can
result in high redundant retransmission of messages. In dense network, this redundant
retransmission can often causes high contention and collision in the network, leading
to loss of precious bandwidth and battery power, a phenomenon referred to as
broadcast storm problem [8].
Several broadcast schemes have been proposed [8-12] to alleviate this
inherent problem. These schemes are commonly divided into two categories;
deterministic schemes [8, 9] and probabilistic schemes [10, 12]. Deterministic
schemes use network topological information to build a virtual backbone that covers
all the nodes in the network [10]. To build a virtual backbone, nodes exchange
information about their immediate or two hop neighbors. However, this often incurs
large overhead in terms of time and message complexity for building and maintaining
the backbone, especially in the presence of mobility. Probabilistic schemes, in
disparity, rebuild a backbone from scratch during each broadcast [12]. Nodes make
instantaneous local decisions about whether to broadcast a message or not using
information derived only from overheard broadcast messages. Consequently these
schemes incur a smaller overhead and demonstrate superior adaptability in changing
environments when compared to deterministic schemes [10]. However, these schemes
have poor reachability as a trade-off against overhead [12].
Several probabilistic schemes have been proposed in the past [13, 14]. These
include probability-based, counter-based, location-based, distance-based and hybridbased schemes [12-16]. In probability-based scheme, a mobile node rebroadcasts a
message according to certain probability while in counter-based schemes messages
are rebroadcast only when the number of copies of the message received at a node is
less than a threshold value. Pure probabilistic schemes assume a fixed probability
value and it is shown in [13] that the optimal rebroadcast probability is around 0.65.
Recently, hybrid schemes [17, 18] were proposed which combines the advantages of
pure probabilistic and counter-based schemes to yield a significant performance
improvement. However, these schemes assume different fixed rebroadcast
probabilities of 0.65 and 0.5 respectively. Intuitively, these values are not likely to be
globally optimal because the performance depends also on other simulation
parameters like topology size, transmission range, number of neighbors etc.
Recently, a rebroadcast probability function has been proposed in [19] for
the hybrid broadcast scheme in [17] which takes into account key simulation
parameters like network topology size, transmission range and number of nodes to
determine the appropriate rebroadcast probability for a given node. The rebroadcast
probability of a node is computed based on these parameters. However, this
rebroadcast probability function does not take into account the value of the packet
counter associated with each node. This packet value gives an indication of the
number of duplicate packet received by a node which is used in rebroadcast decision
in any counter-based related schemes. Thus, a rebroadcast function needs to take this
value into account.
In this paper, we propose a rebroadcast probability function which takes into
account the value of the packet counter together with some key simulation parameters
(i.e. network topology size, transmission range and number of nodes) to determine the
appropriate rebroadcast probability for a given node. The rebroadcast probability of a
node is computed based on these parameters. We incorporate the rebroadcast
probability function into our hybrid scheme [17] and compare it against simple
flooding, fixed probability, counter-based and enhanced counter-based schemes[17].
Simulation results reveal that this simple adaptation achieves superior performance in
terms of saved-rebroadcast, reachability and end to end delay.
The rest of the paper is organized as follows. Section 2 introduces the related work
on probabilistic and counter-based broadcasting. An overview of efficient counterbased scheme with the plug-in rebroadcast probability function is presented in Section
3. We evaluate the performance of our scheme and present the simulation results in
Section 4. Finally, concluding remarks are presented in Section 5.
2 Related Work
Ni et al [13] have proposed a probability-based scheme to reduced redundant
rebroadcast by differentiating the timing of rebroadcast to avoid collision. The
scheme is similar to flooding, except that nodes only rebroadcast with a
predetermined probability P. Each mobile node is assigned the same forwarding
probability regardless of its local topological information. In the same work, counterbased scheme is proposed after analysing the additional coverage of each rebroadcast
when receiving n copies of the same packet.
Cartigny and Simplot [16] have proposed probabilistic scheme which
combine advantages of probability-based and distance-based schemes. The
probability p for a node to rebroadcast a packet is determined by the local node
density using “hello” packet and a fixed value k for the efficiency parameter to
achieve the reachability of the broadcast. However, the use of “hello” packet induces
more overhead and also the determination of an optimal efficiency parameter k is
difficult, since k is independent of the network topology.
In Ni et al’s follow-on work [14], the authors have described an adaptive
counter-based scheme in which each node dynamically adjust its threshold value C
based on its number of neighbors. Specifically, they extend the fixed threshold C to a
function C(n), where n is the number of neighbors of the node. In this approach there
should be a neighbor discovery mechanism to estimate the current value of n. This
can be achieved through periodic exchange of ‘HELLO’ packets among mobile
nodes.
Zhang and Agrawal [12] have described a dynamic probabilistic broadcast
scheme which is a combination of the probabilistic and counter-based approaches.
The scheme is implemented for route discovery process using AODV as base routing
protocol. The rebroadcast probability P is dynamically adjusted according to the value
of the local packet counter at each mobile node. Therefore, the value of P changes
when the node moves to a different neighborhood. To suppress the effect of using
packet counter as density estimates, two constant values d and d1 are used to
increment or decrement the rebroadcast probability. However, the critical question is
how to determine the optimal value of the constants d and d1.
In recent work, Alieza et al [10] proposed a color-based broadcast scheme in
which every broadcast message has a color-field, with a rebroadcast condition to be
satisfied after expiration of the timer similar to counter-based scheme. A node
rebroadcast a message with a new color assigned to its color-field if the number of
colors of broadcast messages overheard is less than a color threshold µ.
Recently, in [18] an efficient counter-based scheme was proposed which
combines the merits of probability-based and counter-based algorithms using a
rebroadcast probability value of around 0.65 as proposed in [13, 15] to yield a better
performance in terms of saved-rebroadcast, end-to-end delay and reachability.
Furthermore, in follow-on work [17], they showed that a better rebroadcast
probability value was around 0.5, which achieve better performance than their earlier
scheme. However, majority of these schemes uses fixed rebroadcast probability which
is quite unlikely to be optimal in other simulation set up.
In Mohamed et al follow-on work [19], a rebroadcast probability function
was proposed for their previous scheme [17] which takes into account key simulation
parameters like network topology size, transmission range and number of nodes, in
determining the appropriate rebroadcast probability for a given node. However, this
function does not take into account a key decision parameter of the scheme (i.e.
packet counter value).
In this paper, we propose a function for setting rebroadcast probability value
of a nodes for our counter-based scheme [17]. The function takes into account the
packet counter value of each node together with key simulation parameters for the
computation of the rebroadcast probability which makes it a better candidate in any
simulation set up. This simple adaptation yields better performance in terms of savedrebroadcast, end-to-end delay and reachability. The detail of the function together
with counter-based scheme is described in the next section.
3 Efficient Counter-Based Scheme (ECS)
In this section, we discuss briefly the efficient counter-based broadcast scheme that
aims to mitigate the broadcast storm problem associated with flooding through the use
of appropriate rebroadcast probability (details in [17]).
In this algorithm, a node upon reception of a previously unseen packet
initiates a counter c that will record the number of times a node receives the same
packet. Such a counter is maintained by each node for each broadcast packet. After
waiting for a random assessment delay (RAD, which is randomly chosen between 0
and Tmax seconds), if c reaches a predefined threshold C, the packet is drop.
Otherwise, the packet is rebroadcast with a rebroadcast probability P which is a fixed
constant determined based on extensive simulation. However, this value might not be
globally optimal and so the selection of appropriate rebroadcast probability is vital to
the performance of our scheme.
In this work, we propose a function that will dynamically assign a rebroadcast
probability value to a mobile node. Therefore, as in our earlier scheme [17], if c
reaches a predefined threshold C, the packet is dropped. Otherwise, the packet is
rebroadcast with a rebroadcast probability P which is computed from the function f(c)
which is discussed in the next section. The use of this modification for broadcasting
enables mobile nodes to makes localized rebroadcast decisions on the rebroadcast
probability to use with respect to network topology size, transmission range and
number of neighbours. Essentially, this adaptation provides a better broadcast solution
in MANETs. An outline of the scheme is presented in Figure 1.
The Algorithm:
On hearing a broadcast message m at a node X
- initialize the counter c to 1;
- set and wait for RAD to expire;
- While waiting:
o for every duplicate message m received
o increment c by 1;
- if (c < C) (counter threshold-value)
o rebroadcast probability P = f(c)
else{
o Drop the message;
o Go to Exit }
- Generate a random number Rn over [0, 1]
- If
Rn ≤ P
o Rebroadcast the message;
else
o Drop the message
Exit
Figure 1: An outline of the efficient counter-based broadcast scheme
3.1 The Rebroadcast Probability Function
The most important factor in our scheme is the selection of the rebroadcast probability
P. A larger P incurs more redundant rebroadcast while a smaller P leads to lower
reachability. Our scheme adjusts the rebroadcast probability dynamically by the used
of the function f (c). In this section, we present the analysis of average number of
neighbors to provide the basis for presenting the rebroadcast probability function f (c).
Let A be the area of the MANET, N be the number of mobile nodes in the
network, and R the transmission range. The average number of neighbor (λ) can be
computed by the following formula:
λ = ( N − 1)
πR 2
A
Figure 2 shows the average number of neighbors for different nodes with the network
area of 800m x 800m, 1000m x 1000m, and 1200m x 600m, respectively. It is clearly
shown from the figure that the higher is the average number of neighbors, the denser
the network and the lower the average number of neighbors is, the sparse the network.
Average Number of Neighbors
60
A(800mX800m)
50
A(1000mX1000m)
40
A(1200mX600m)
30
20
10
0
0
20
40
60
80
100
120
Number of Nodes
Figure 2: Relationship between average number of neighbours and number of nodes
in the network.
From the average number of neighbors (λ) and given nodes counter value c, we can
present our rebroadcast probability function as follows:
f (c ) = e
−(
c
λ
)
(1)
We opt for a rebroadcast function that uses exponential function because a high
rebroadcast probability incurs more redundant rebroadcast while a low rebroadcast
probability leads to low reachability. Moreover, nodes with few numbers of neighbors
should be assigned a high rebroadcast probability while those with many number of
neighbors be assign a low rebroadcast probability. Therefore, as the number of
neighbors increases, the rebroadcast probability should decreases. Based on the above
features of the rebroadcast probability an ideal mathematical function that can fit into
these requirements is the exponential function. Moreover, the function takes into
account mobile node 1-hop information (i.e. c value) and an approximation of the
global network density information (i.e. λ) to arrive at an appropriate rebroadcast
probability value for a given node.
Our motivation for this rebroadcast function is to enhance rebroadcast
decision by taking into account key network parameters and node local information
through counter value. This tends to give a better estimate of nodes’ neighbor
information and likewise the key network parameters (i.e. topology, transmission
range, etc) are readily available to each node at simulation start up. On the other hand,
one might say that the use of “HELLO” packet might be much easier in determining
exact number of neighbors. However, “HELLO” messages themselves consume
channel bandwidth and therefore affect the overall performance.
4 Performance Analysis
In this section, we provide the details of our simulation environment, performance
metrics used and simulation results.
4.1. Simulation Set up and Metrics
We evaluate the performance of our scheme using ns-2 packet level simulator (v.2.29)
[20]. The performance analysis is based on the assumptions widely used in
literature[4].
The radio propagation model used in this study is based on similar characteristic to
commercial radio interface, Lucent’s WaveLAN card with a 2Mbps bit rate. The
distributed coordination function (DCF) of the IEEE 802.11 protocol [21] is utilized
as MAC layer protocol while the random waypoint[22] model is used as the mobility
models.
In a random waypoint mobility model, each node at the beginning of the simulation
remains stationary for a pause time t seconds, then chooses a random destination and
starts moving towards it with a randomly selected speed from a uniform distribution
[0, Vmax]. After the node reaches its destination, it again stops for a pause-time interval
t sec and chooses a new random destination and speed. This cycle repeats until the
simulation terminates. As it takes time for the random way point model to reach a
stable distribution of mobile nodes, the modified random waypoint mobility model
[22] used take care of this node distribution problem.
The simulation is allowed to run for 900 seconds for each simulation scenario with
both models. Other simulation parameters that have been used in our experiment are
shown in Table 1 and the broadcast schemes are evaluated using the following
performance metrics:
•
Reachability (RE) – This is the percentage of nodes that received the
broadcast message to the total number of nodes in the network.
•
Saved Rebroadcast (SRB) – This is the percentage of nodes that have
received but not rebroadcast the message. Thus, SRB is defined as ((r –
t)/r)*100, where r and t are the number of nodes that received the broadcast
message and the number of nodes that transmitted the message respectively.
•
End-to-end delay - is the average time difference between the time a data
packet is sent by the source node and the time it is successfully received by
the destination node.
Table 1: Summary of simulation parameters used in the experiment
Parameter
Value
Simulator used
Transmission range
Bandwidth
NS-2 (version 2.29)
100 m
2Mbps
Interface queue length
50 packets
Packet size
512 bytes
Topology size
1000 x 1000 m2
Number of Nodes
Mobility Model
20, 40, .., 120
Random waypoint [22]
Maximum speed
5 m/s
Pause times
0 sec
Simulation time
900 sec
MAC Protocol
Traffic rate
Confidence level
IEEE 802.11
10 packets/sec
95%
4.2 Simulation Results
We present the performance results of the new scheme (ECS-function) side by side
with those for counter-based, fixed probability and flooding. The simulation output is
collected using replication mean method [23] where each data point represents an
average of 30 different randomly generated mobility scenarios with 95% confidence
intervals. The error bars on the graphs represent the upper and lower confidence limits
from the means and in most cases they have been found to be quite small. For the
sake of clarity and tidiness, the error bars have not been included in some graphs.
We assess the impact of network density on the performance of the four
schemes by varying the number of nodes from 20 to 120 deployed randomly on a
fixed area of 1000 x 1000 m2. Figure 3 demonstrate the effects of density on the saved
rebroadcasts for the schemes. The figure shows that saved rebroadcast increases with
higher densities and the amount of saved rebroadcasts increases as density of the
nodes increases, i.e. as the number of nodes covering a particular area increases. The
scheme using rebroadcast probability function (ECS-Function) achieves better saved
rebroadcast compared to counter-based, fixed probability and flooding.
Figure 4 illustrates the degree of reachability achieved by the schemes. The
result shows that reachability improves with higher density because as node density
increases, the number of node covering a particular area also increases. Flooding has
the best performance in terms of reachability. The efficient broadcast scheme using
the rebroadcast function (ECS-Function) has better reachability than fixed probability
in sparse network. However, in medium to dense network (60 -120 nodes) its
reachability is comparable to flooding.
Figure 5 depicts the effects of density on end-to-end delay using different
network densities. It shows that the delay is largely affected by network density and it
increase with increase in density. This might be as result of contention and collision
increases with increasing network density. The modified schemes (ECS-Function)
achieved better end-to-end delay compared to counter-based, fixed probability and
flooding schemes. This is due to the low number retransmitting nodes achieved by the
scheme.
80
Flooding
Counter-Based
ECS-Function
Fixed Probability
Saved Rebroadcast (%)
70
60
50
40
30
20
10
0
20
40
60
80
100
120
Number of Nodes
Figure 3: Saved Impact of density on saved rebroadcast using 5 m/s node speed and
10 packets per second broadcast rate.
.
100
Reachability (%)
80
60
Flooding
Counter-Based
40
ECS-Function
20
Fixed Probability
0
20
40
60
80
100
120
Number of Nodes
Figure 4: Impact of density on reachability using 5 m/s node speed and 10 packets per
second broadcast rate.
0.08
Flooding
Counter-Based
ECS-Function
Fixed Probability
End to end delay (sec)
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
20
40
60
80
100
120
Number of Nodes
Figure 5: Impact of density on end-to-end delay using 5 m/s node speed and 10
packets per second broadcast rate.
5 Conclusions
In this paper we introduce a rebroadcast probability function for an efficient counterbased broadcast scheme for MANETs that mitigate the broadcast storm problem
associated with flooding. The rebroadcast probability function takes into
consideration key networks parameters like number of nodes; transmission range; and
network topology to assign rebroadcast probability to a mobile node. Compared to
the other three schemes, our simulation results have revealed that ECS-Function
achieved superior saved rebroadcast (about 20% better than its closest competitor i.e.,
counter-based scheme, in dense network) and end-to-end delay (around 26% better
than counter-based scheme in dense network) without sacrificing reachability in
medium and dense networks. Nevertheless, this work is to stimulate more research in
this direction in order to come up with a rebroadcast probability function that will be
suitable for dynamically changing networks like MANETs.
One area that is worth investigation is the effect of varying these key
network parameters (topology, transmission range) on the performance of the scheme.
This can definitely bring out the superiority of the scheme or otherwise. In fact
another area that is worth exploring is the network in which each node can leave and
join during the simulation period. Thus, this type of network requires a distributed
system that can communicate and update these key parameters to each node in the
network. Another area for future work is to implement and evaluate the performance
of the scheme as a route discovery mechanism in an on-demand routing protocol like
AODV.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
S. Basagni, M. Conti, S. Giordano, and I. Stojmenovic, Mobile Ad Hoc Networking:
IEEE Press, 2004.
M. D. Colagrosso, "Intelligent broadcasting in mobile ad hoc networks: Three classes
of adaptive protocols," EURASIP Journal on Wireless Communication and
Networking, vol. 2007, pp. 25 - 40, 2007.
D. B. Johnson and D. A. Maltz, "Dynamic source routing in ad hoc wireless
networks," in Mobile Computing, vol. 353, T. Imelinsky and H. Korth, Eds.: Kluwer
Academic Publishers, 1996, pp. 153 - 181.
C. E. Perkins and E. M. Moyer, "Ad-hoc on-demand distance vector routing," in
Proceedings of 2nd IEEE Workshop on Mobile Computing Systems and Applications.
New Orleans, L.A, 1999, pp. 90 - 100.
Z. J. Haas, M. R. Pearlman, and P. Samar, "Determining optimal configuration for
zone routing protocol," IEEE Journal on Selected Areas in Communications, vol. 17,
pp. 1395 - 1414, 1999.
Y.-B. Ko and N. H. Vaidya, "Location aided routing (LAR) in mobile ad hoc
networks," in Proceedings of the 4th ACM/IEEE International Conference on Mobile
Computing and Networking (Mobicom). Dallas, Texas, 1998, pp. 66 - 75.
V. Park and S. Corson, "Temporally-Ordered Routing Algorithm (TORA) Version
1," IETS Internet darft, July 2001.
J. Wu and W. Lou, "Forward-node-set-based broadcast in clustered mobile ad hoc
networks," Wireless Communication and Mobile Computing, vol. 3, pp. 155 - 173,
2003.
P. Rogers and N. Abu-Ghazaleh, "Towards reliable network wide broadcast in mobile
ad hoc networks," [online], Available: http://arxiv.org/abs/cs/0412020, 2004.
A. Keshavarz-Haddad, V. Ribeiro, and R. Riedi, "Color-Based Broadcasting for Ad
Hoc Networks," in Proceeding of the 4th International Symposium on Modeling and
Optimization in Mobile, Ad Hoc, and Wireless Network (WIOPT' 06). Boston, MA,
2006, pp. 1 - 10.
M. S. I. Stojmenovic and J. Zunic, "Dominating sets and neighbor elimination-based
broadcasting algorithms in wireless networks," IEEE Transaction on Parallel and
Distributed Systems, vol. 13, pp. 14 - 25, 2002.
Q. Zhang and D. P. Agrawal, "Dynamic Probabilistic Broadcasting in MANETs,"
Journal of Parallel and Distributed Computing, vol. 65, pp. 220-233, 2005.
S. Ni, Y. Tseng, Y. Chen, and J. Sheu., "The broadcast storm problem in a mobile ad
Hoc networks," in Proceeding of the ACM/IEEE International Conference on Mobile
Computing and Networking (MOBICOM), 1999, pp. 151-162.
Y.-C. Tseng, S.-Y. Ni, and E.-Y. Shih, "Adaptive approaches to relieving broadcast
storms in a wireless multihop ad hoc networks," IEEE Transaction on Computers,
vol. 52, pp. 545-557, 2003.
B. Williams and T. Camp, "Comparison of Broadcasting Techniques for Mobile Ad
Hoc Networks," in Proceeding MOBIHOC. Lausanne, Switzerland: ACM, 2002, pp.
194-205.
J. Cartigny and D. Simplot, "Border node retransmission based probabilistic
broadcast protocols in ad hoc networks," Telecommunication Systems, vol. 22, pp.
189-204, 2003.
[17]
[18]
[19]
[20]
[21]
[22]
[23]
A. Mohammed, M. Ould-Khaoua, L. Mackenzie, and J. Abdulai, "Improving the
Performance of Counter-Based broadcast Scheme for Mobile Ad Hoc Networks," in
Proceedings of 2007 IEEE International Conference on Signal Processing and
Communications (ICSPC 2007), 2007, pp. 1403 - 1406.
A. Mohammed, M. Ould-Khaoua, and L. Mackenzie, "An Efficient Counter-Based
Broadcast Scheme for Mobile Ad Hoc Networks," in Proceedings of the Fourth
European Performance Engineering Workshop (EPEW 2007) Lecture Notes in
Computing Science Volume 4748, K. Wolter, Ed. Berlin, Germany: Springer-Verlag,
2007, pp. 275 - 283.
A. Mohammed, M. Ould-Khaoua, and L. M. Mackenzie, "On the Rebroadcast
Probability of an Enhanced Counter-Based Broadcast Scheme for Mobile Ad Hoc
Networks," Communications of SIWN, vol. 4, pp. 101 - 105, 2008.
"The Network Simulator ns-2, http://www.isi.edu/nsnam/ns/."
IEEE standard 802.11-1997, Wireless LAN medium access control (MAC) and
physical layer (PHY) specifications: IEEE, 1997.
W. Navidi, T. Camp, and N. Bauer, "Improving the accuracy of random waypoint
simulation through steady-state initialization," in Proceedings of the 15th
International Conference on Modeling and Simulation (MS'04). Marina Del Rey,
California, USA, 2004.
A. M. Law and W. K. David, Simulation Modeling and Analysis. Boston: McGrawHill Higher Education, 2000.