Net_CA: A Tool for Simulation and Modeling of Wired/Wireless Computer Networks based on Cellular Automata

Net_CA: A Tool for Simulation and Modeling of Wired/Wireless Computer Networks based on Cellular

Automata

Vasilios Mardiris1, Georgios Ch. Sirakoulis1, Charilaos Mizas1, Ioannis Karafyllidis1, and Adonios Thanailakis1

1 Democritus University of Thrace, Department of Electrical and Computer Engineering, Laboratory of Electrical and Electronic Materials Technology,

GR 67100 Xanthi, Greece mardiris, [email protected]

gsirak, ykar, [email protected]://vlsi.ee.duth.gr/index.html

Abstract. A tool for simulation and modeling of wired/wireless computer net-works based on Cellular Automata (CAs), is presented. More specifically, a two-dimensional NaSch CA model for computer network simulation has been developed and implemented in the proposed tool. Furthermore, algorithms for connectivity evaluation, system reliability evaluation and shortest path compu-tation in a wired/wireless computer network have also been implemented. The Net_CA system was designed and developed as an interactive tool that offers automated modeling with the assistance of a dynamic and user friendly graphi-cal environment. The simulation algorithms developed in the present work offer high flexibility. Furthermore, connection reliability and other important pa-rameters are inputs to the algorithms rendering Net_CA a very reliable and fast simulator for wireless networks, ad hoc networks and, generally, for low con-nection reliability networks. (Research Paper, Keywords: Computer Net-works, Cellular Automata, Simulation, Modeling, Connectivity, Reliability)

1 Introduction

The proliferation of computing and communications as well as the ever-increasing number of Internet users is driving a revolutionary change in our information society. The requirements for large scale wired and/or wireless computer networks character-ized by extremely complicated topologies and, in the same time, by huge bandwidth rates are truly skyrocketing [1]. It is obvious that such complicated computer net-works demand meticulous design in topology level, careful choice of the proper communication protocols as well as secured transition of extremely big data with high bandwidth rates. Furthermore, the determination of the parameters’ set applica-ble to the aforementioned computer networks that are responsible for their high speed functionality is an extremely difficult task. As a result, modeling the performance of computer networks is without any doubt a crucial and difficult interdisciplinary sub-ject. The principal idea behind simulation modelling is to represent accurately the

topology and functions performed in a network in order to obtain statistical results about the network’s performance under investigation [2].

Computer networks can be categorized based on the nature of data transmission in three different types; namely, the copper networks where the electric voltage is used as the transmission medium, the optical networks where the light is used and finally the wireless networks where electromagnetic waves are used for the transmission. Because of the pronounced freedom of node movement in a wireless computer net-work and the undersized connections found in it, a computer network model is indis-pensable [3]. Computer network models should be a simplified representation of the examined system and hence, only important features should be focussed on [4].

In order to model accurately real computer networks and allow predictive simula-tion during technology research and development, a system, named Net_CA, was constructed and is presented in this paper. The Net_CA is an interactive simulation tool that includes a Graphical User Interface [GUI] which has been implemented using Tcl/Tk facilities [5]. It simulates the network operation and provides perform-ance measurements. No programming is required as network descriptions are created using a graphical user interface, which is designed to model a wide variety of net-works.

The proposed system includes four different algorithms for computer network modelling all based on Cellular Automata (CAs). CAs are models of physical sys-tems, where space and time are discrete and interactions are local and they can easily handle complicated boundary and initial conditions, inhomogeneities and anisotropies [6]. CAs are very effective in simulating systems and solving scientific problems, because they can capture the essential features of systems where global behavior arises from the collective effect of simple components which interact locally [7-8]. As a result, first of all a two dimensional CA NaSch computer network model was de-veloped and implemented in Net_CA system. This algorithm is based on the NaSch CA highway traffic model and on the corresponding one-dimensional analogous NaSch CA computer network model found in literature [9-10]. A CA algorithm for network connectivity evaluation, a CA algorithm for network reliability evaluation and a CA shortest path network connectivity algorithm [11] were developed and implemented in the Net_CA system. The proposed Net_CA succeeds a mere numeri-cal agreement with the examined computer network and matches the simulated sys-tem’s own structure, its topology, its connectivity and every “deep” network property.

The paper is organised as follows: In Section 2 a detailed mathematical description of Cellular Automata is given. The Net_CA simulation tool and all the corresponding CA algorithms are presented in full detail with examples in Section 3. Finally, the conclusions are drawn in Section 4.

2 Cellular Automata

In this section a more formal definition of a CA will be presented [12]. In general, a CA requires: 1. A regular lattice of cells covering a portion of a d-dimensional space;

2. A set ( ) ( ) ( ) ( ) trCtrCtrCtr , ..., ,,,,, m21=C of variables attached to each site

r of the lattice giving the local state of each cell at the time t = 0, 1, … ;

3. A rule R=R1, R2, …, Rm which specifies the time evolution of the states ( )tr,C

in the following way:

( ) ( ) ( ) ( ) ( )( )ttrtrtrRtrC qjj ,r ..., ,,,,,,1, 21 δδδ +++=+ CCCC

(1)

where kr δ+ designates the cells belonging to a given neighbourhood of cell r . In the above definition, the rule R is identical for all sites, and it is applied simul-

taneously to each of them, leading to a synchronous dynamics. It is important to no-tice that the rule is homogeneous, i.e. it does not depend explicitly on the cell position

r . However, spatial (or even temporal) inhomogeneities can be introduced by ascrib-

ing definite and permanent values for some states ( )rC j in some given locations of

the lattice. In the above definition, the new state at time t+1 is only a function of the previous state at time t. It is sometimes necessary to have a longer memory and intro-duce a dependence on the states at time t-1, t-2, …, t-k. Such a situation is already included in the definition, if one keeps a copy of the previous state in the current state.

The neighbourhood of cell r is the spatial region in which a cell needs to search in its vicinity. In principle, there is no restriction on the size of the neighbourhood, except that it is the same for all cells. However, in practice, it is often made up of adjacent cells only. For two-dimensional CA, two neighborhoods are often consid-ered: the von Neumann and Moore neighbourhoods [12]. Extending the neighbour-hood leads to various types of boundary conditions such as periodic (or cyclic), fixed, adiabatic or reflection [12].

3 Net_CA Simulation Tool

Net_CA is an automated computer network modelling system. In computer network modelling, graphic output is a necessary tool for rapid and clear understanding of computation results. Thus user-friendly tools allowing users to interact with the sys-tem is a basic part of the Net_CA’s environment. Net_CA’s user interface has been implemented using Tcl/Tk GUI’s facilities, enabling interactive simulation. The pro-posed tool is platform independent.

The simulation results are produced using the four aforementioned CA algorithms. More details regarding these algorithms are given later on this section. All algo-rithms’ codes were also developed with the usage of Tcl/Tk and interact instantly with the user’s defined parameters. These parameters are the topology of the exam-ined network, the nodes of the network and their maximum number of data packets and corresponding boundary conditions, as well as the links between the nodes of the network and their bandwidth, reliability and cost. Each algorithm is executed using

the corresponding button found in the top of the Net_CA’s initialization window. The aforementioned parameters remain unchanged during the simulation. It should be mentioned that Net_CA does not impose any a priori limitations on these parameters which are modifiable and can be changed interactively. Simulation results can be looked at in graphical form. Furthermore, the case of false values given to one or more parameters is examined by the system and the user is forced to correct these values. The Net_CA includes a help window that gives information related to the scope, the abilities and the usage of the system.

A demo version of the presented system would be available after the presentation of the paper from the http://infoman.teikav.edu.gr/~mardiris/net_ca.html.

3.1 The two-dimensional NaSch network model

The original NaSch network model is a one-dimension CA model for computer net-work, which is similar to the NaSch model used in the research on road traffic [9-10]. In the proposed simulation tool a two-dimensional NaSch network model is presented for the first time. Unlike the NaSch traffic model, the NaSch network model is com-posed of two kinds of cells, node cells and link cells. The node cells represent com-puters, routers, and switchers in a computer network. The link cells represent the lines of communication between nodes. Node cells are allowed to have more than one data packets at any time, while link cells can be occupied by at most one packet at a given instant. The two-dimensional NaSch network model results in an extended Moore neighboohood, its size corresponding to the topology of the simulated network. It should be mentioned that every cell of the extended Moore neighbourhood is found in one of three possible states: (a) node cell, (b) link cell and (c) non active.

The number of link cells connecting two adjacent node cells represents the band-width of network. A large number of link cells results in more time for transferring packets from a node cell to the other. This directly leads to a lower bandwidth. The bandwidth reaches the maximum if no link cell exists between two adjacent node cells. In this case, packets can be directly transferred from one node to the other. This implies that the link cell is not necessary and the NaSch network model can consist of node cells only.

The state of cell can be characterized by the number of packets it has. Obviously, for a node cell, it can take an arbitrary value from 0 to N, where N is the maximum number of packets a node cell can hold. However, the state of a link cell can have only the value of 0 or 1. Each packet in the NaSch network model has a velocity property. This property holds information of how the packet “move ahead” in the next time step. For the sake of simplicity, the maximum velocity of a packet is limited to 1, that is, it must be taken 0 or 1.

The developed two-dimensional NaSch network model is assumed to obey the evolution rules as follows: 1. Acceleration: For packets that are in the link cells or at the front of the queue of

node cells, their velocities are set to be 1. The velocities of other packets remain unchanged.

2. Slowing down: The velocity of a packet will reduce to 0, if the packet satisfies any of the alternatives:

For any cell that holds one packet, the next adjacent one is a link cell and only one packet is occupied in this link node.

For any cell that holds one packet, the next adjacent one is a node cell and the number of packets in this link node has reached the maximum.

3. Randomization: For each node cell, slow down the packet lined at the front of queue and set its velocity to 0 with probability p, if this cell is not empty (or it holds at least one packet).

4. Exit of packets from node cells: In this case the followings result: If N>L, then L packages of the node attain to velocity 1. If N<L, then N packages of the node cell, attain to velocity 1 and move to ran-

dom departure cells; where, in both cases, N is the number of packages of node cell, L is the number of the links cells found in the exit of node cell and could possible receive data.

5. Entrance of packets in node cells: In this case the followings result: If N-Nmax>L, then L packages attain to velocity 1 and enter the node cell. If N-Nmax<L, then N-Nmax packages, chosen randomly from the L link cells

found in the entrance of the examined node cell, attain to velocity 1 and enter the node cell; where, in both cases, N is the number of packages of node cell, Nmax is the maximum number of packages that can be storied in node cell, L is the number of the links cells found in the entrance of node cell and could possi-ble send data (are occupied by data package).

6. Packet motion: Each packet goes ahead according to its velocity. If its destina-tion is a node cell, it will be added to the end of the queue. In step 3, the randomization, should be attracted special attention. The randomiza-

tion is regarded as the abstract of the forwarding packet process of switchers or routers in a real world network. Through the above steps, we can have dynamic be-haviors of network.

As mentioned above, we use several parameters, including the number of node cells N, the number of link cells L, the capacity of node cell Q, the random probabil-ity p, and the packet density ρ, for describing the NaSch network model. Thus the NaSch network model is generally written as CA(N,L,Q,p,ρ) for convenience.

An example of the proposed two-dimensional NaSch model for modelling a com-puter network through the Net_CA simulation tool is depicted in Figure 1. The Net_CA gives the opportunity to the user to define the minimum number of link cells corresponding to the maximum bandwidth connection. In other words, the user de-fines the connection with the minimum bandwidth which would be used by the sys-tem as the basis for the computation of the link cells for the rest connections depend-ing on the bandwidth of each one. Additionally, the white coloured link cells corre-spond to empty of data packets, while the red ones hold data packets.

3.2 The s-t connectivity evaluation algorithm

The problem of s–t connectivity evaluation consists in finding if there is a path from a source node s to a terminal node t. Ascertaining the connectivity of the network re-quires knowledge of the cut sets or path sets of the system [9] or a depth-first proce-dure [10–12]. To address the problem, a generalised class of Boolean networks will

be used: each node has a binary OR state machine as local transition function with k inputs and m outputs.

Fig. 1. The Net_CA simulation tool for the modeling of computer network with the usage of NaSch model after the initialization of the computer network.

Let G=(N, A) be a graph, where N is the set of n nodes, NNA ×⊆ is the set of arcs and Li is the neighbourhood of the node i.

That is, each node i is mapped to a cell whose neighbourhood is the set of nodes connected to it by its input arcs. Source node s has one input arc and terminal node t has one output arc. Note that in a computer network, each node corresponds to a computer.

The state w(i) of each node is binary, with w(i)=1 corresponding to the activated state and w(i)=0 to the quiescent state. Each node i performs an OR Boolean transi-tion function over the states of the nodes in its neighbourhood. That is:

[ ] ikjkwjwORiw Λ∈= ,..., where,)(),...,()( (2)

To determine if there is a path from a source node s to a terminal node t, the CA propagates information from one node to another: each node is activated if there is at least one activated node in its neighbourhood. In this way, a path of activated nodes from node s to node t could be generated.

Initially, all nodes are in the quiescent state and the source node s is activated. Then each node executes its transition function, just waiting for the activation of any of its neighbors. The computation ends either when node t is activated or the process

stagnates. The longest s–t path in G consists of n-1 nodes, therefore the s–t connect-edness can be computed in O(n) time.

The basic algorithm is: 1. Set all nodes in the quiescent state 2. Activate the input node s 3. Iteration=1 4. Update each node state in parallel by transition formula (2) 5. Iteration=iteration+1 6. If node t is activated, then stop: a path between s and t is found 7. If iteration < n-1 go to 4, else 8. If node t is in the quiescent state then s–t path does not exist.

An example of the proposed s-t connectivity evaluation algorithm using the Net_CA simulation tool is depicted in Figure 2. In the beginning, the starting node s is coloured red while all the other nodes are coloured yellow. In the following time step the nodes connected to the starting node s, with connectivity direction pointing from the s node to them are coloured red and the process keeps on going till the end-ing node t is coloured red. Then a message of success is shown in the upper right corner of the Net_CA simulation tool screen. Otherwise a proper message informs instantly the user about the inability of finding a connectivity path between the exam-ined nodes of the computer network.

Fig. 2. The final screen of the Net_CA simulation tool for the s-t connectedness evaluation algorithm applied to the given topology of the considered computer network.

3.3 System reliability evaluation algorithm

Let us consider a system with n components and its network representation. Each component has two states (failure and success) and component failures are independ-ent events.

Each arc, between node i and j, represents a component with associated reliability rij. The state of each arc xij can be evaluated using its reliability and a random number U, distributed uniformly between [0, 1], as:

⎩⎨⎧

≤≤≥

=ij

ijij rU

rUx

0 if state) (failure 0 if state) (success 1

(3)

To take into account the probabilistic nature of each arc, the Boolean transition function for each node Eq. (2) is written as:

w(i)=OR [AND(w(j),xji), …, AND(w(k),xki)], ikj ∆∈,..., (4)

In order to perform a Monte-Carlo reliability evaluation with the Number of Evaluation (NE) given, we proceed as follows: 1. . Set iteration=1 2. Generate a random system state X. 3. Using CA algorithm evaluate if there is a path. 4. If there is a path then update number of “good” system states. 5. Iteration=iteration+1. 6. If iteration < NE go to 2, else STOP.

At the end, we can estimate the system reliability as:

NEtyconnectivi with States System ofNumber yReliabilit =

(5)

The presented system reliability evaluation algorithm has been implemented in the proposed Net_CA simulation tool and tested with a simple example shown in Figure 3. After the definition of the starting node s and the ending node t of this computer network topology, the previously described reliability evaluation algorithm is used. First, the presented network connections are examined about the status of their func-tion according to the aforementioned probabilistic model. Then, the s-t connectedness evaluation algorithm is executed and the user has the opportunity to examine if there is a physical connection between the nodes of the specified network. Finally, the reliability is calculated using Eq. (5). It should be mentioned that the working con-nections are displayed with the help of continuing arrows and the non-working con-nections are displayed with dashed arrows. The final number of the execution time steps of the proposed system reliability evaluation algorithm is determined by the user.

Fig. 3. The Net_CA Simulation tool for the modeling of computer network with the usage of system reliability evaluation algorithm.

3.4 Shortest path computation

The problem of finding the shortest path (SP) from a single source to a single destina-tion in a graph arises as a sub-problem to many broader problems. In general, differ-ent path metrics are used for different application. For example, in communication systems, if each link cost is 1, then the minimum number of hubs is found. However, cost can also represent the propagation delay, the link congestion or the reliability of each link. In the latter case, if the individual communication links operate independ-ently, then the problem can be stated as to find what path has the maximum reliabil-ity.

Let G=(N, A) be a graph, where N is the set of n nodes, NNA ×⊆ is the set of arcs and Li is the neighbourhood of the node i. That is, each node i is mapped to a cell whose neighbourhood is the set of nodes connected to it by its input arcs. Associated with each arc is a non-negative number. Cpq stands for the cost of arc

from node p to node q. Non-existing arc costs are set to infinite. Let Pst be a path from a source node s to a destination node t, defined as the set of consecutive con-nected nodes: Pst=s; n1; n2;…; ni; t. The state of each node, at each time step ts, is

represented by a vector with two entries

Aqp ∈),(

)(),()( 21sisisi tVtVtV >= : the first com-

ponent is a pointer to the previous node in the path, while the second is the cost of the

partial path up to node i. is not necessary for evaluating the shortest path

length, but it is used only for indicating the shortest path itself.

)(1si tV

The basic algorithm is: 1. Initialise the state of the node as Vs(0)=s,0 and Vi(0)=i,∞, i≠s, ts=0.

2. Synchronously update each automaton i, i≠s finding iLk∈ that minimise

( )ikskski CtVtU ,2

, )()( += or, in case of conflict, minimise and set )(2sk tV

( ) iksksi CtVktV ,2 )(,)1( +=+ .

3. ts=ts+1. 4. If ts=n-1 or the process stagnates stop, else go to 2.

The minimum s–t path cost is the second component of the state vector of node t. In case this value is infinity, then there exists no s–t path.

The results of the implementation of the presented shortest path computation algo-rithm to the proposed Net_CA computer network simulation system are depicted in Figure 4. As before the user defines the starting node s and the ending node t of the given computer network topology. During the execution of the aforementioned algo-rithm each node is described by a pair of values, one for its number (name) according to the cost of the connection up to it, and the other for the minimum cost of connec-tion of the starting node to the examined node. After the execution of the algorithm the shortest connectivity path between nodes s and t is colored yellow.

Fig. 4. The final screen of the Net_CA system during the execution of the shortest path algo-rithm in order to find the shortest connectivity path between nodes s and t.

4 Conclusions

Net_CA, a tool for simulation and modeling of wired/wireless computer networks based on CAs, was presented. More specifically, a two-dimensional NaSch CA model for computer network simulation has been developed and implemented in the pro-posed tool. Furthermore, algorithms for connectivity evaluation, system reliability evaluation and shortest path computation in a wired/wireless computer network have also been implemented. The Net_CA system was designed and developed as an inter-active tool that offers automated modeling with the assistance of a dynamic and user friendly graphical environment. Net_CA’s user interface has been implemented using Tcl/Tk enabling platform independence and possible cooperation with other well known computer network simulation tools based on Tcl/Tk, like ns2. Furthermore, connection reliability and other important parameters are inputs to the algorithms rendering Net_CA a very reliable and fast simulator for wireless networks, ad hoc networks and, generally, for low connection reliability networks. The proposed sys-tem can be easily modified enabling the modeling of computer network protocols like TCP/IP. Finally, the inherent parallelism of the proposed CA algorithms and its easy VLSI implementation make it suitable for real-time applications in the field of com-puter networks.

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