PACKET SWITCHING NETWORK PERFORMANCE INDICATORS AS FUNCTION OF NETWORK TOPOLOGY AND ROUTING ALGORITHMS Anna T. Lawniczak Xiongwen Tang Dept. of Mathematics & Statistics Dept. of Stat. & Actuarial Science University of Guelph University of Iowa Guelph, Ont., N1G 2W1, Canada Iowa City, Iowa 52242-1409, USA email: [email protected] email: [email protected] Abstract 2. Abstraction of the OSI Network Layer

We investigate how the network performance indicators “throughput”, “number of packets in transit” and “average delay time of all packets delivered” are affected by network connection topology and routing. We study how they capture the phase transition point (i.e., the critical load of a network). We investigate how additional links added to the network connection topology affect the network performance indicators and the phase transition point. We observe substantial differences in the behaviour of the network performance indicators among the networks with a static routing cost metric (i.e., when the cost of transmission of packets from one router to another is constant over time) and those with the dynamic routing cost metrics (i.e., when the costs of transmission of packets from one router to another incorporate the information about how congested the routers are). We observe, in accordance with other models, that throughput is maximal at the critical load of a network with adaptive routing. Keywords: network performance indicators, number of packets in transit, average packet delay, throughput, OSI Network Layer.

1. Introduction The Packet Switching Network (PSN) is the dominant technology of data communication networks; see [1] and references therein. In a PSN each message is partitioned into smaller units of information called packets. Packets are transmitted individually from their sources to their destinations via a number of routers (switching nodes) which are interconnected by communication links. Since the Network Layer of the OSI Reference Model is responsible for routing packets across the network and for control of congestion, then it plays an essential role in packet traffic dynamics [1], [2]. Thus, we focus on this layer and present an abstraction of the Network Layer used in our research. Using this abstraction we investigate how onset of traffic congestion is captured by network performance indicators. Continuing our work of [2] and [6]-[11] we study how addition of a communication link affects packet traffic dynamics for static and adaptive routing algorithms.

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In our research we use an abstraction of the Network Layer developed in [2] and [3], and a C++ simulator, called Netzwerk-1 described in [4] and [5]. This PSN model is concerned mainly with packets and their routing as the Network Layer in real packet switching networks. In our PSN model all messages are restricted to one packet carrying only the following information: time of creation, destination address, and number of hops taken. As a case study we consider a PSN model with a network connection topology that is isomorphic to Lnp†(16, l), that is a two-dimensional nonperiodic square lattice with 16 nodes in the horizontal and vertical directions and with l = 0 or 1 additional links added to this square lattice. Each node performs the functions of a host and a router and maintains one incoming and one outgoing queue which is of unlimited length and operates according to a first-in, first-out policy. At each node, independently of the other nodes, packets are created randomly with probability λ called source load. We use the following convention when we want to specify additionally what type of an edge cost function (ecf) the PSN model set-up is using, namely, Lp†(16, l, ecf), where l = 0 or 1, and ecf = ONE, or QS, or QSPO. The ecf ONE assigns a value of “one” to each edge in the lattice L. Thus, this results in a static routing. The ecf QS assigns to each edge in the lattice L a value equal to the length of the outgoing queue at the node from which the edge originates. The ecf QSPO assigns a value that is the sum of the ecfs ONE and QS. The routing decisions made using ecf QS or QSPO imply adaptive or dynamic routing where packets have the ability to avoid congested nodes during the PSN model simulation. In the PSN model time is discrete and we observe its state at the discrete times k = 0, 1, 2, …T, where T is the final simulation time. At time k = 0, the set-up of PSN model is initialized with empty queues and the routing tables are computed. Our time-discrete, synchronous and spatially distributed PSN model algorithm consists of the sequence of five operations advancing the simulation time from k to k + 1. These operations are: (1) Update routing tables, (2) Create and route packets, (3) Process incoming queue, (4) Evaluate

network state, (5) Update simulation time. The detailed description of this algorithm is provided in [2] and [3].

3. Behaviour of Network Performance Indicators We investigate, for various ecfs, what is the impact of addition of communication link on packet traffic dynamics and how network performance indicators detect this impact. The performance indicators that we consider are critical source load, throughput, number of packets in transit, and average delay time of all packets delivered. For their definitions see [2]. In our PSN model, for each family of network set-ups, which differ only in the value of the source load λ, values of λsub-c for which packet traffic is congestion-free are called subcritical source loads, while values λsup-c for which traffic is congested are called super-critical source loads. The critical source load λc is the largest sub-critical source load. Details about how we estimate the critical source load are in [2]. For the PSN model set-ups considered here the estimated critical source load values are, respectively, λc = 0.045 for Lnp†(16, 0, ONE), λc = 0.085 for Lnp†(16, 0, QS), λc = 0.085 for Lnp†(16, 0, QSPO), λc = 0.020 for Lnp†(16, 1, ONE), λc = 0.090 for Lnp†(16, 1, QS), and λc = 0.090 for Lnp†(16, 1, QSPO). We observe that for PSN model set-ups Lnp†(16, 0, ecf) where ecf = QS or QSPO, the critical source loads are equal (at the precision level of our estimation) and their value is higher than λc for Lnp†(16, 0, ONE). Also, the critical source loads are equal for PSN model set-ups Lnp†(16, 1, ecf) where ecf = QS or QSPO and their value is significantly higher than λc for Lnp†(16, 1, ONE). Thus, addition of an extra link increases λc for the PSN model set-ups using dynamic ecf QS or QSPO (i.e., adaptive routing) and significantly decreases λc for the PSN model set-ups using static ecf ONE. In the case of the considered ecfs the same holds true for PSN model set-ups using other types of network connection topologies, see [2], [6]-[9]. Fig. 1 and Fig. 2 display graphs of throughput, number of packets in transit, and average delay time of all packets delivered as a function of source load λ at simulation time T=6400. The blue graphs correspond to PSN model set-ups with ecf ONE, the green graphs to the ones with ecf QS and the red graphs to the PSN model set-ups with ecf QSPO. Fig. 3, Fig. 4 and Fig. 5 display spatial distributions of the outgoing queue sizes at nodes of PSN model set-ups. The x- and y- axis coordinates of each plot denote the positions of switching nodes and z-axis denotes the number of packets in the outgoing queue of the node located at that (x,y) position. Fig 1 and Fig. 2 show that the graphs of number of packets in transit are constant for source load values smaller than λc and they increase with the increase of source load values for source loads greater than λc. The same holds true for the graphs of average delay time of all packets delivered in the case of PSN model set-up Lnp†(16, 0, ONE) and the PSN model set-

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ups with ecf QS (except for very small values of λ) and QSPO. In the case of PSN model set-ups Lnp†(16, l, QS), where l=0 or 1, there are only few queuing packets for very small source load values. Thus, the costs of all paths are very similar and packets perform almost random walks on their routes from their sources to their destinations. This result in higher values of the graphs of average delay time of all packets delivered when the source loads are small. With the increase of source load values there are more queuing packets in a network and the costs of the paths become more differentiated. Thus, packets are delivered more efficiently to their destinations and the graphs of average delay time of all packets delivered decrease and stay constant until λc .

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(c) Figure 1. Plots of throughput (a), number of packets in transit (b), average delay time of all packets delivered (c) at simulation time T= 6400 of the PSN model set-up with network connection topology Lnp†(16, 0) and edge cost ONE (blue graphs), QS (green graphs) and QSPO (red graphs). In the case of PSN model set-up Lnp†(16, 1, ONE) we observe that the first increase in the graph of average delay time of all packets delivered corresponds to λc and it is followed by the plateau at the end of which the graph increases with the increase of λ. This plateau corresponds to the values

of source load λ for which the network is locally congested, that is, it is only congested at the end nodes of an extra link, see Fig. 3. For the higher values the network becomes globally congested. Looking at the Fig. 1 and Fig. 2 we observe that the graphs of throughput of PSN model set-ups Lnp†(16, 0, ecf), where ecf = ONE, QS, QSPO and those of Lnp†(16, 1, ecf), where ecf = QS, QSPO attain their maximum at values almost equal to the corresponding critical source load values. From Fig. 2 we see that the graph of throughput of PSN model set-up Lnp†(16, 1, ONE) attains its maximum at the value much higher than λc. This is because the network is only locally congested for λc and packets are still delivered timely through the paths that do not include the extra link, see Fig.3.

packets pass through centrally located nodes and these nodes become quickly congested, see Fig. 4. This is not the case when the network uses an adaptive routing, see Fig. 5 (a). In this Figure we see that packets are more evenly distributed among the network nodes. Also, on this figure we see that addition of an extra link may lead to self-organization in queue

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Figure 3. Spatial distribution of outgoing queue sizes in PSN model set-up Lnp†(16, 0, ONE) for source load in (a) λc = 0.020, (b) λsup-c. = 0.025, (c) λsup-c = 0.050, (d) λsup-c = 0.060 at simulation time T = 8000.

(c) Figure 2. Plots of throughput (a), number of packets in transit (b), average delay time of all packets delivered (c) at simulation time T= 6400 of the PSN model set-up with network connection topology Lnp†(16, 1) and edge cost ONE (blue graphs), QS (green graphs) and QSPO (red graphs). (a)

Looking at the plots of network performance indicators on Fig. 1 one notices that the performance of PSN model set-up with Lnp†(16, 0, ONE) is worse than that of the PSN model setups Lnp†(16, 0, ecf), where ecf=QS or QSPO. This is because, when the network uses the static routing algorithm most of the

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Figure 4. Spatial distribution of outgoing queue sizes in PSN model set-up Lnp†(16, 0, ONE) for source load λc = 0.045 in (a) and λsup-c = 0.050 in (b) at simulation time T = 8000.

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sizes distribution among the network nodes. Such behaviour was observed in other PSN model set-ups with adaptive routings, see [2], [12], [13].

(a) (b) Figure 5. Spatial distribution of outgoing queue sizes in PSN model set-up (a) Lnp†(16, 0, QS) for source load λsup-c = 0.090 and in (b) Lnp†(16, 1, QS) for source load λsup-c = 0.095 at simulation time T = 8000.

4. Conclusions The described qualitative behaviours of the discussed network performance indicators were observed for PSN model set-ups with other network connection topology types. The performance of PSN model set-ups with non-periodic network connection topologies was always worst when static routing algorithm was used instead of the adaptive ones. Furthermore, in the case of static routing algorithm addition of small number of extra links always negatively affected PSN model performance regardless of the type of network connection topology. However, it always improved the performance of the PSN models when adaptive routings were used. Furthermore, in the case of PSN models using one of the adaptive routings and a periodic or non-periodic square lattice connection topology we observed that addition of an extra communication link enhances self-organization in the distribution of outgoing queue sizes. We have not observed this phenomenon when periodic or non-periodic triangular network connection topology was used instead. We intend to study further this phenomenon.

Acknowledgements A.T.L. acknowledges partial financial support from Sharcnet and NSERC of Canada, and X.T. from Sharcnet and the University of Guelph. The authors thank B. Di Stefano, A. Gerisch and K. Maxie for helpful discussions.

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