From Individual to Collective Behaviour in CA Like Models of Data Communication Networks A.T. Lawniczak1 , K.P. Maxie1 , and A. Gerisch2

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1 Department of Mathematics and Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada, {alawnicz, kmaxie}@uoguelph.ca, FB Mathematik und Informatik, Martin-Luther-Universit¨ at Halle-Wittenberg, 06099 Halle (Saale), Germany.

Abstract. Understanding the collective behaviour of packets traffic in data networks is of vital importance for their future developments. Some aspects of this behaviour can be captured and investigated by simplified models of data networks. We present our CA like model describing processes taking place at the Network Layer of the OSI (Open Systems Interconnection) Reference Model. We use this model to study spatiotemporal packets traffic dynamics for various routing algorithms and traffic loads presented to the network. We investigate how additional links, added to a network connection topology, affect the collective behaviour of packets traffic. We discuss how some of the network performance aggregate measures reflect and relate to the emergence of packets collective spatio-temporal dynamics. We present selected simulation results and analyse them.

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Introduction

In recent years data communication networks have experienced unprecedented growth that will continue in the future. The dominant technology is the Packet Switching Network (PSN). Examples of PSNs include the Internet, wide area networks (WANs), local area networks (LANs), wireless communication networks, ad-hoc networks, or sensor networks. The widespread proliferation of PSNs has outpaced our understanding of the dynamics of packets traffics and their dependence on network connection topologies, routing algorithms, amounts and types of data presented to the networks. For the future development of PSNs it is important to understand what effects may have local changes in network connection topologies, changes in operation of single routers, or changes in routing algorithms on the collective behaviour of packets. The further evolution of PSNs and improvements in their design depend strongly on the ability to predict networks performance using analytical and simulation methods, see e.g. [1, 2,3,4] and the references therein. The need to have a simulation model closer to the real packet switching networks than static flow and queueing models in order to study which network features and parameters affect flow, congestion and collective spatio-temporal dynamics of packets in PSNs has motivated our P.M.A. Sloot, B. Chopard, and A.G. Hoekstra (Eds.): ACRI 2004, LNCS 3305, pp. 325–334, 2004. c Springer-Verlag Berlin Heidelberg 2004


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research. Our simulation model describes the processes taking place at the Network Layer of the OSI Reference Model. For this layer we have constructed a cellular automaton (CA) like model [5,6]. In order to simulate our CA model of PSNs we developed a C++ simulation tool, called Netzwerk-1 [7]. The benefits of our model are that, like real networks: it is concerned primarily with packets and their routing; it is scalable, distributed in space, and time discrete. Moreover, it avoids the overhead of protocol details present in simulators designed with different goals in mind. We have applied our model and its further refinements to study, among others, the effects of routing algorithms and network topologies connectivity, size and the addition of extra links with various preferentiality factors of attachment on the phase transition from free flow to congestion, throughput, average packet delay, packet average path length and speed of delivery, see [8,9,10,11]. The presented work is a continuation of our previous investigations. Its goal is to study how the packets collective spatio-temporal dynamics is affected by the network topology. Also, our aim is to isolate aspects of routing algorithms that lead to spatio-temporal self-organisation in packets traffics. We explore, for selected network performance aggregate measures, how they reflect and relate to the emergence of packets collective spatio-temporal dynamics. The paper is organised as follows. In Section 2 we describe our CA network model. In Section 3 we present selected simulation results and analyse them. In Section 4 we provide our conclusions and outline future work.

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Packet Switching Network Model

There is vast engineering literature devoted to PSNs, see [3] and the references therein. We limit our discussion to the development of our PSN model. Its detailed construction is described in [6,7]. The purpose of a PSN is to transmit messages from sender to receiver. In a real-world PSN, messages are broken into several equally sized pieces called packets. These packets are then sent across a network consisting of connected nodes, known as hosts and/or routers. In our model we restrict all messages to the length of one packet. Packets contain an information payload along with routing and other header information. Since our concern is strictly with the delivery of packets across the network, the information payload contained within each packet is irrelevant to our discussion. Hence, in the considered models we assume that each packet carries only the following information: time of creation, destination address, and number of hops taken. As in a real-world PSN, our network model consists of a number of interconnected switching nodes. Each node can perform the functions associated with a host and a router. The primary functions of the hosts are to generate and receive packets. Routers are responsible for storing and forwarding packets on their route from source to destination. Packets are created randomly and independently at each node. The probability λ with which packets are created is called source load. Nodes maintain incoming and outgoing queues to store packets. For this paper we consider one incoming queue and one outgoing queue per

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switching node. We assume that the outgoing queues are of unlimited length and operate in a first-in, first-out manner. Each node at each time step routes the packet from the head of its outgoing queue to the next node on its route independently from the other nodes. A discrete time, synchronous and spatially distributed network algorithm implements the creation and routing of packets [6, 7,8]. The considered network connection topologies, routing cost metrics and the PSN model algorithm are described in subsections that follow. Network Connection Topology. In a real-world PSN, while the physical means of communication links may vary, the underlying connection topology can always be viewed as a weighted directed multigraph L. Each switching node corresponds to a vertex and each communication link is analogous to a pair of parallel edges oriented in opposite directions. We associate a packet transmission cost to each directed edge, thus parallel edges do not necessarily share the same cost. For the presented set of simulations we consider network connection topologies L = Ll,2 (L) that are isomorphic to two-dimensional periodic or non-periodic square lattices with the number of nodes in the horizontal and vertical directions L = 16. We study both unperturbed networks with l = 0 additional links and perturbed ones by adding randomly l > 0 extra links. The first node to which an additional link attaches is selected with uniform probability from the set of all nodes in the network. The second node is selected randomly from the remaining nodes with bias towards the nodes with higher number of outgoing edges. The bias is called preferentiality factor (P F ) of attachment. If the bias is equal to zero then the second node is selected with uniform probability. All randomly generated links are added prior to a simulation run. All links in the network are static for the duration of the simulation. Static Routing and Adaptive Routing. In the network models, packets are transmitted from their source nodes to their respective destination nodes. The path across the network taken by each packet is determined by the routing decisions made independently at each node based on a least-cost criterion. Depending on the costs assigned to each edge of the graph, we consider routing decisions based on the minimum route distance or the minimum route length least-cost criterion [2,3]. We consider three types of edge cost functions called One (ONE), QueueSize (QS), and QueueSizePlusOne (QSPO) [6,7] that are described below. In each considered network model setup it is assumed that all edge costs are computed using the same type of edge cost function. Edge cost function ONE assigns a value of one to all edges in the lattice L. Applying a least cost routing criterion results in minimising the number of hops taken by each packet between its source and its destination node, minimum hop routing. Since the values assigned to each edge do not change during the course of a simulation, this type of routing is called static routing. The QS edge cost function assigns to each edge in the lattice L a value proportional to the length of the outgoing queue at the node from which the edge originates. A packet traversing a network using this edge cost function will travel from its current

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node to the next node along an edge belonging to a path with the least total number of packets in transit between its current location and its destination. The edge cost function QSPO assigns a summed value of a constant one plus a number proportional to the length of the outgoing queue at the node from which the edge originates. This cost function combines the attributes of the first two functions. Routing decisions made using QS or QSPO edge cost functions rely on the current state of the network simulation. Hence, they imply adaptive or dynamic routing in which packets have the ability to avoid congested nodes during a network simulation. Each switching node maintains a routing table of least path cost estimates to reach every other node in the network. This type of routing scheme is called full-table routing. When the edge costs are assigned by the function ONE the routing tables can be calculated once at the beginning of each simulation and they do not require updating, because edge costs do not change with time. In this case the cost estimates are the precise least-costs. When the edge costs in a PSN model are assigned by the function QS or QSPO, the routing tables are updated at each time step using a distributed version of Bellman-Ford least-cost algorithm [2]. Since only local information is exchanged at each time step, the path costs stored in the routing table are only estimates of the actual least path costs across the network. Packet Switching Network Model Algorithm. In our model we consider time as given in discrete time units of size one and perform simulations from time k = 0 to a final simulation time k = T . We can observe the state of the network model at the discrete time points k = 0, 1, 2, . . . , T only. At time k = 0, the network is initialised with empty queues and the routing tables are computed using the centralised Bellman-Ford least-cost algorithm [2]. One step of the discrete time, synchronous and distributed in space network algorithm advances the simulation time from k to k + 1. It is given by the sequence of five operations: (1) Update routing tables: The routing tables of the network are updated in a distributed manner; (2) Create and route packets: At each node, independently of the other nodes, a packet is created randomly with source load λ. Its destination address is randomly selected among all other nodes in the network with uniform probability. The newly created packet is placed in the incoming queue of its source node. Further, each node, independently of the other nodes, takes the packet from the head of its outgoing queue (if there is any), determines the next node on a least-cost route to its destination (if there is more than one possibility then selects one at random with uniform probability), and forwards this packet to this node. Packets, which arrive at a node from other nodes during this step of the algorithm, are destroyed immediately if this node is their destination, otherwise they are placed in the incoming queue; (3) Process incoming queue: At each node, independently of the other nodes, the packets in the incoming queue are randomised and inserted at the end of the outgoing queue; (4) Evaluate network state: Various statistical data about the state of the network at time k are gathered and stored in time series; (5) Update simulation time: The time k is incremented to k + 1.

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The above described time step from k to k + 1 of the network algorithm can be repeated an arbitrary number of times. Step (3) of the algorithm simulates a sort of processing delay and ensures that a packet, which arrives at a node in one time step, does not leave the node before the next time step. Further, randomising the incoming queue simulates that the packets arrive at each node in random order.

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From Individual to Collective Behaviour of Packets in Transit

In this section we present selected simulation results and discuss the emergence of collective behaviour in packets traffic for the described adaptive routing algorithms. We explore how changes in the network connection topology affect critical source load values and packets spatio-temporal collective behaviour. For a discussion of how other network performance aggregate measures reflect and relate to the emergence of packets collective behaviour we refer to [8,9,10,11,12]. Critical Source Load. We classify host source loads λ as sub-critical, critical, and super-critical source loads. A source load is said to be sub-critical if the average number of packets entering the network does not exceed the average number of packets leaving the network. This results in free flowing traffic in a network where packets reach their destinations in a timely manner. A supercritical source load is just the opposite of this: the average number of packets entering the network exceeds the number of packets exiting the network. The result of applying a super-critical source load is a congested network. Finally, the critical source load λc is the largest sub-critical source load, or the phase transition point from free flowing to congested network states. This phase transition has been observed in empirical studies of packet switching networks [13]. For details concerning the calculation of λc for a given PSN model and analysis how various aspects of a PSN affect the critical source load value see [6,7,8,9,10, 11]. We consider how the random addition of links affects the critical source load. The results for periodic (left figure) and non-periodic (right figure) square lattices of size L = 16 with P F = 0 are displayed in Fig. 1. The lower graphs in both figures correspond to edge cost function ONE, and the upper ones, that almost coincide, to the edge cost functions QS and QSPO. We notice that for a periodic lattice, in the absence of additional links, all three edge cost functions result in similar values for λc . For the non-periodic lattice, λc for the edge cost ONE is significantly lower than those for QS and QSPO in this case. The reason is that the static minimum hop routing does not route packets around congested regions of the network. The critical source loads for the three edge costs of the otherwise same model setups are significantly lower for the non-periodic lattice without or with relatively small number of extra links than those for the equivalent periodic lattices. We attribute this to the increased average distance between any two nodes in the non-periodic lattice. This implies, that on average many

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Fig. 1. λc versus l for Ll,2 (16) with P F = 0. The left figure shows results for periodic lattices, the right for non-periodic lattices. The lower graphs correspond to edge cost ONE, the upper ones to QS and to QSPO. The graphs for QS and QSPO almost coincide

more packets remain longer in transit resulting in faster build up of congestion for lower source load values. Further, many more packets must transit through the centre region of the network, in particular for static routing, i.e. edge cost ONE; in the case of QSPO, see Figs. 2 and 3. For periodic and non-periodic lattices and adaptive routing, i.e. edge cost QS or QSPO, additional links increase the value of λc . In fact, for both, QS and QSPO, there is very little difference in the values of λc (this may be due to the granularity of the applied source loads). For edge cost ONE (static routing) the addition of a small number of links results in a dramatic drop in the critical source loads. This is likely the result of the local congestion occurring at the ends of these few additional links as is seen in [12]. As more links are added, the critical source loads begin to recover towards their original levels. However, for the periodic lattice the number of additional links needed to fully regain this value is much higher than the one presented in the graph. With regards to the preferentiality factor of attachment of extra links, our study showed that an increase in the P F -value has an adverse effect on the critical source load values [10,11]. Spatio-temporal Packets Traffic Dynamics. Here, we discuss the effects of the lattices periodicity and addition of extra links on the collective spatiotemporal behaviour of packets in transit in PSN models with edge cost function QSPO for connection topology L = Ll,2 (16). The presented results are for preferentiality factor P F = 0 and numbers of additional links l = 0, 1, and 51. For the periodic lattices, the source load values λ are 0.125, 0.130, and 0.175, respectively. For the non-periodic lattices they are 0.090, 0.095, and 0.160, respectively. The values of the considered source loads are about 0.005 higher than those of the corresponding λc values. For a discussion of results for other cost functions, topologies and source loads, see [12]. In Figs. 2, 3 and 4 the x- and y-axis coordinates denote the positions of switching nodes in L = Ll,2 (16). The z-axis denotes the number of packets in the outgoing queue of the node at that (x, y) position. In Fig. 2 we present results for periodic lattices and in Fig. 3 for non-periodic ones. In both figures, the left column corresponds to the lattice without an extra link, the right column to the lattice with one extra link. In Fig. 4 we show results for lattices with l = 51

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Fig. 2. Evolution of queue lengths for edge cost QSPO, λ > λc , periodic lattice L = Ll,2 (16) without an extra link (left column) and with an extra link (right column), for k = 1600 (top row) and k = 3200 (bottom row)

extra links; in the left picture for the periodic lattice, in the right picture for the non-periodic one. The distribution of packets is presented for two separate snapshots in time corresponding to k = 1600 and 3200 in Figs. 2 and 3, and in Fig. 4 only at k = 3200. The graphs of Fig. 2 demonstrate spatio-temporal self-organisation in packets traffic for periodic lattices with l = 0 and l = 1 and emergence of wave-like pattern with peaks and valleys in packet distribution for congested networks (λ > λc ) using adaptive routing with the QSPO edge cost function. As the number of packets in the network model increases, the wave-like pattern emerges in spite the routing scheme attempts to distribute the packets across the network queues evenly. One notices that for each node that is part of a peak (i.e. with a relatively large number of packets in its queue), its directly attached neighbours are part of a trough (i.e. with a relatively small number of packets in their queue). The converse also holds true. The addition of an extra link hastens the formation of the wave-like pattern. The pattern is similar to the one obtained for QS for periodic lattice [12]. However, the addition of many extra links destroys the formation of this pattern, see Fig. 4. Large queues of more or less the same size appear at the ends of the extra links with very small number of packets elsewhere.

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Fig. 3. Evolution of queue lengths for edge cost QSPO, λ > λc , non-periodic lattice L = Ll,2 (16) without an extra link (left column) and with an extra link (right column), for k = 1600 (top row) and k = 3200 (bottom row)

The graphs of Fig. 3 demonstrate that connection topology periodicity plays an important role in building up of congestion and emergence of packets collective behaviour. For the lattice with l = 0 we observe across the network an essentially random distribution of packet queue lengths with their variability increasing over time. The landscape of queue sizes resembles the one observed in the case of the edge cost ONE [12], with the difference that here the queues are significantly larger at the centrally located nodes. That means that, for the QSPO edge cost function, the ONE-part of the cost plays a much more significant role in routing of the packets in the case of non-periodic lattices than in the case of the periodic ones. We observe that the addition of one extra link dramatically changes packets collective dynamics. It destroys the increasing build up of the queue sizes of large variability in the interior of the network. The extra link provides a “short-cut” across the network and plays the surrogate role of the missing links due to the lattices lack of periodicity. The extra link attracts much larger numbers of packets than links elsewhere in the network resulting in local congestion growing at its ends with time. The remaining packets are distributed more evenly among the other queues of the network, then as it was observed for l = 0. Initially, the queues are located mostly in the centre of the network. As time progresses this local congestion spreads out almost evenly across the

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Fig. 4. Packet queue lengths for k = 3200, edge cost QSPO, λ > λc , lattice L = Ll,2 (16) with 51 extra links: left picture for the periodic lattice, right picture for the non-periodic lattice

network and a wave like pattern emerges among these queues. The sizes of these queues stay always significantly smaller than that at the ends of the extra link. For times k > 1600, the distribution of queue sizes resembles that of the superposition of the distribution of packets for the edge cost ONE with that for QS presented in [12]. The addition of a large number of extra links destroys wave like pattern and packets queue mostly at the ends of the extra links. Fig. 4 shows that the variability of the queue sizes is higher for the non-periodic lattices than for the periodic ones.

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Conclusions

We described our CA like model of the OSI Network Layer for PSNs. The presented simulation results demonstrate that the network connection topology plays an important role for the amount of traffic that can be presented to the network and for the emergence of packets in transit collective behaviour. For static minimum hop routing, the addition of one extra link decreases drastically the critical source load value and in order for the network to recover its performance many links must be added. For adaptive routing, the critical source load values increase with the increase in the number of extra links. We observe that the addition of one extra link can hasten the spatio-temporal self-organisation in the distribution of packets in transit and lead to the emergence of a wave-like pattern of queue lengths across the network. However, addition of many links destroys this process. The presented study is a continuation of the authors work in [12]. We plan further investigation into the factors responsible for spatio-temporal selforganisation of queue lengths. Discovering and understanding correlations between the traffic patterns and network parameters has the potential to impact decisions made with regards to topological and routing scheme designs for realworld networks.

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Acknowledgements. The authors acknowledge support from the University of Guelph, The Fields Institute and the use of the SHARCNET computational resources. They thank B. Di Stefano and P. Zhao for helpful discussions. A.T.L. acknowledges additional support from NSERC and IPAM.

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