DEVELOPMENT AND PERFORMANCE OF CELLULAR AUTOMATON MODEL OF OSI NETWORK LAYER OF PACKET-SWITCHING NETWORKS A.T. Lawniczak A. Gerisch B. Di Stefano Dept. of Math. and Stat. Dept. of Math. and Comp. Sc. Nuptek Systems, Ltd. University of Guelph University of Halle Toronto, Ont M5R 3M6 Guelph, Ont N1G 2W1, Canada 06099 Halle (Saale), Germany Canada [email protected] [email protected] [email protected] Abstract We present a Cellular Automaton model of the OSI Network Layer. Our focus is on parameters that can affect flow and congestion in the network, such as randomly inserted additional links, used to model additional wired and wireless connections in LANs and WANs. We present selected simulation results and observe, in accordance with other models, that throughput is maximal at the critical load of the network. Further, the addition of links increases the critical load of a network if queueing costs are taken into account in routing decisions for packets. Keywords: Packet-switching networks, routing, network performance indicators, cellular automata

1. INTRODUCTION The global Internet, wireless communication systems or ad-hoc networks are examples of packetswitching networks (PSNs). These PSNs experience unprecedented growth and understanding their dynamics of flow and congestion is of vital importance. Some of the aspects of this dynamics can be investigated by studying simplified models of PSNs [1, 2, 3, 4]. Models of PSNs have been proposed and studied at different levels of abstraction and with different objectives in mind. We focus our attention on the identification of those features and parameters that can affect flow and congestion in PSNs. For the purpose of our study we have identified that the OSI network layer is the most important layer of the OSI reference model. For this layer we developed a cellular automaton (CA) model [5]. Here we study the effects of additional links and of three cost functions on the critical load and throughput of the network models; for other simulation results CCECE 2003 - CCGEI 2003, Montr´eal, May/mai 2003 c 2003 IEEE 0-7803-7781-8/03/$17.00

see [3, 5]. We view the additional links in our model as an abstraction of additional wireless or wired links in LANs, WANs, and similar networks that are added in an adhoc fashion to the initial network connection topology. We present selected simulation results. In order to simulate our CA model of PSNs we developed a C++ simulation tool, called Netzwerk-1 [6].

2. PSN MODEL There exists a vast amount of literature about PSNs [7, 8]. Here we briefly review some facts needed for the development of our CA model of PSNs. The purpose of a PSN is to transmit messages from points of origin to destination points. In our model, we assume that the entire message is contained in a single “capsule” of information called a packet. We ignore the information “payload” that a single packet carries in a real PSN. We assume that each packet carries the following pieces of information: time of its creation, its destination address and some information to assess the performance of a model. Our CA network model consist of a number of interconnected nodes. Each node can perform two functions: that of a host, (i.e., it can generate and receive packets), and that of a router (i.e., it can store and forward packets). An incoming queue and an outgoing queue are maintained by each node to store packets on this node. We assume that each queue can be of unlimited size and observes a first-in first-out policy. The creation and routing of packets is implemented by a discrete time, synchronous and distributed in space network algorithm [5].

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2.1 Connection Topology and Routing We consider here network connection topologies, L = Ll (L) that are isomorphic to two-dimensional periodic square lattices with l ≥ 0 additional, randomly generated links. The parameter L stands for the number of nodes in the horizontal and the vertical direction of the lattice. The networks hosts and routers are located at the nodes of the lattice L and the communication links between nodes are represented by edges of L. With each direction of a link (an edge of L) is associated the cost, defined by an edge cost function, of transmission of a packet. The l randomly generated links are added before the simulation starts. We allow that some nodes can be connected directly by several links. We emphasise that all the connections in our model are static, they do not change during the simulation period. We note that the links of the underlying lattice topology can be thought of as local contacts whereas the randomly generated links can be thought of as short-cuts. In the network model each packet is routed from its source node through various links and packet switches to its destination node according to routing decisions. Depending on the costs assigned to the edges of L, the routing decisions base on the following least-cost criteria: minimum route distance or minimum route length [8]. We consider three types of edge cost functions called One (One), QueueSize (QS), and QueueSizePlusOne (QSPO) [3, 5]. These cost functions differ in the way they take into account the cost caused by traversing a link and the cost caused by the time spent queueing at a node before a link can be traversed. In each model studied here we assume that all edge costs are computed by the same edge cost function. In the case of the cost function One all edges in L are assigned a cost equal to one and by applying the least-cost criterion, the number of hops on the route from source to destination is minimised for each packet. This minimum-hop routing is independent of the current state of the network. A routing scheme having this property is called static. In the case of the cost function QS each edge in L is assigned a cost proportional to the number of packets awaiting transmission in the node from which the edge originates and at any given time, a packet routed according to the least-cost criterion will travel from its current node to the next one along the first edge of a route connecting the packet’s current node with its destination node and, additionally, this route is at this time one with a minimum sum of numbers of packets at the packet’s current node and all intermediate nodes. In the case of the cost function QSPO each edge in L is assigned a cost proportional to the number of packets awaiting transmission in the

node from which the edge originates plus the cost of a single hop equal to one, and a packet routed according to the least-cost criterion will travel on a path which combines the properties of the first two cases. For the cost functions QS and QSPO, routing decisions depend on the current state of the network. The corresponding routing schemes are called adaptive or dynamic. In the case of adaptive routing, one usually reconsiders the route choice toward the destination node of a packet after an edge has been traversed because the state of the network could have changed and another route might now be more cost effective. Adaptive routing schemes imply the ability to avoid congested areas in the network. Nodes of a network maintain routing tables in order to perform routing efficiently. We assume that each node in a network stores estimates of the least path costs from itself to all destination nodes in the network (full-table routing); other types of routings are considered in [2]. In order to keep the routing table up to date with the state of the network we must update them on a regular basis for adaptive routing schemes. This is not necessary for static routing schemes, for which it is sufficient to compute only once the routing table. Hence, at the initialisation time of the network the least-cost values, not just estimates, are computed and stored in the routing table entries for all ordered pairs of network nodes. A suitable algorithm for the computation of the initial routing table and for the computation of its centralised update (a case considered in [3, 5]) is the Bellman-Ford algorithm [7]. Here we consider a distributed routing table update. In this case a distributed version of the Bellman-Ford algorithm, see [7], is executed simultaneously and independently at every node in the network [5]. After the completion of one distributed routing table update the newly calculated values of the routing table are not necessarily equal to the least path costs of paths connecting pairs of network nodes, they are only locally computed estimates of these least costs. In this type of update nodes exchange information about their link costs in time in order to determine the least-cost routes from one node to another one. The least path cost routing tables are not built up in one cycle as in the case of the centralised routing table update. They are build up gradually in time.

2.2 PSN Model Algorithm At time k = 0, the network is initialised with empty queues and a centralised routing table update is performed to initialise the routing tables. One step of the discrete time, synchronous and distributed in space network algorithm advances the simulation time from

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3. SIMULATION RESULTS In this paper we consider simulation results from PSN models based on the network connection topologies Ll (25) for values l in the inclusive range [0, 200]. The cost function is either One, QS, or QSPO, and we use the distributed routing table update in the latter two cases. The source load is taken from the set {0.005, 0.01, 0.015, . . . , 0.2}. The simulations are run for T = 2000 time steps for all models. In real data networks a continuous phase transition between the state of free packet flow in the network and the congested state of the network can be observed. This phase transition is due to the numbers of pack-

ets in transit being below or above a certain critical threshold. In our models, we can control the number of packets in transit by the number of packets created at the hosts and therefore use the source load λ as a control parameter. For a given network model we can determine the critical source load λc , which is defined to be as large as possible under the restriction that the same network with any source load λ < λc will remain in the state of free packet flow for all time. In our investigations we are particularly interested how the addition of randomly generated links to a given network connection topology changes the values of λc of these models. We have plotted the (estimated) critical source load λc as a function of the number l of randomly generated links for the network models under consideration in Fig. 1. The procedure to estimate λc of a given network is discussed in detail in [5]. 0.16

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k to k + 1 and is given by the following sequence of operations. (1) Update routing tables: The routing table of the network is updated in a centralised (considered in [5]) or distributed manner. (2) Create and route packets: At each node, independently of the other nodes, a packet is created with probability λ, called the source load. 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 select 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 and 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. The statistical data are described in more detail in Sec. 3. and also in [5]. (5) Update simulation time: The time k is incremented to k + 1. 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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Fig. 1. λc vs. l for models with connection topology Ll (25). The graphs in Fig. 1 show two striking features. (1) The critical source load for networks with edge cost functions QS or QSPO is steadily increasing with l. Both graphs are almost indistinguishable. However, we note that networks with these two edge cost functions show a different behaviour in the low load regime. In this case, the edge cost function QS provides little guidance for delivering packets to their destination because only queue sizes matter in their computation. In contrast, the function QSPO brings packets quickly to their destinations also for small values of λ. (2) The critical source load for networks with edge cost functions One drops sharply as soon as the first link is added to the connection topology. The critical source load recovers from this drop only slowly and the value of λc for l = 0 is not reestablished with up to l = 200 additional links. For more additional links this will eventually happen, see [2]. The graphs in Fig. 1 demonstrate clearly the ability of routing schemes with edge cost functions QS or QSPO to route packets around congested regions of the network; a property not shared by routing schemes with edge cost function One. More insight into the networks performance can be

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obtained by studying the throughput. The throughput Θk at time k is defined as the number of delivered packets up to time k divided by k and hence measures the average number of packets delivered per time step. Plots of ΘT vs. λ are given in Fig. 2 for edge cost function One and QSPO; the plot for edge cost function QS is visually indistinguishable from the plot for QSPO and is therefore omitted. 100

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4. SUMMARY We have described a CA model of the OSI Network Layer in PSNs. The presented simulation results demonstrate that by adding links at random to a topology, the critical source load can increase if the queue sizes are taken into account in the computation of the edge costs. If those sizes are neglected in the edge cost computation then a sharp drop in the critical source load is observed if only few links are added. This drop of the critical source load value can be tight to local congestion of the network by considering the networks throughput as a function of source load. The model presented in this paper can be modified and expanded easily to incorporate more features of a real PSNs. Acknowledgements: A.T.L. and A.G. acknowledge partial support from the University of Guelph and The Fields Institute. A.T.L. acknowledges additionally partial support from NSERC of Canada. B. Di S. acknowledges total financial support from Nuptek Systems Ltd. The authors acknowledge the use of the SHARCNET computational resources at the University of Guelph.

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be considered as the source load value at which local congestion builds up. This can be confirmed by looking at the time evolution of particular queues in the network [5].

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Fig. 2. ΘT vs. λ for models with edge cost function One (top) and QSPO (bottom). The markers ◦, ×, 4, ∗, , ♦ correspond to l = 0, 1, 10, 50, 100, 200. The vertical dotted lines indicate the λc -values corresponding to the values of l (same markers as above).

In these plots we observe that the throughput is highest near the critical source load in the case of edge cost function QSPO. This is to be expected (and also present in real networks) because an increase of λ beyond λc leads to a build up of queues at the nodes and the, with time increasing, size of the queues prevents more and more packets from reaching their destination. This behaviour is not present in the plot for edge cost function One (except l = 0). This indicates that these networks are only partially congested for values λ ≈ λc , namely near the end points of the additional links, and full congestion of the network sets in only for even larger source load values. In this case λc can

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