Performance of data networks with random links

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1 Performance of data networks with random inks arxiv:adap-org/ v2 4 Jan 2001 Henryk Fukś and Anna T. Lawniczak Department of Mathematics and Statistics, University of Gueph, Gueph, Ontario N1G 2W1, Canada and The Fieds Institute for Research in Mathematica Sciences Toronto, Ontario M5T 3J1, Canada Emai: {hfuks,aawnicz}@fieds.utoronto.ca Abstract We investigate simpified modes of computer data networks and examine how the introduction of additiona random inks infuences the performance of these networks. In genera, the impact of additiona random inks on the performance of the network strongy depends on the routing agorithm used in the network. Significant performance gains can be achieved if the routing is based on geometrica distance or shortest path reduced tabe routing. With shortest path fu tabe routing degradation of performance is observed. Keywords: computer data networks, routing agorithms, network topoogy, network performance, congestion 1. Introduction Modes of computer data networks have attracted much attention in recent years. Generay, these modes assume either reguar topoogy of the network, in the form of a square attice [1, 2, 7] or a binary Cayey tree [11], or random graph topoogy [5]. On the other hand, it has been recenty demonstrated that many technoogica, bioogica, and socia networks are neither competey reguar nor competey random, being somewhere between these two From Juy 2000: Department of Mathematics, Brock University, St. Catharines, Ontario L2S 3A1, Canada, hfuks@brocku.ca 1

2 extremes [3]. Reguar attices rewired to introduce sma amount of random connections, termed sma word networks, offer many advantages over purey reguar or purey random topoogies. In particuar, modes of dynamica systems based on sma word attices can often exhibit enhanced signa propagation capabiities, as observed in epidemic modes or muti-payer prisoners diemma games payed on such attices [3]. The purpose of this work is to investigate if a simiar effect can be achieved in simpe data network modes, athough our approach is different than the approach taken in [3]. We do not rewire the network, but rather examine how introduction of additiona random inks infuences its performance. This is motivated by the question whether introduction of additiona inks can hep to decongest an existing network. 2. Network Modes Definitions The purpose of the network is to transmit messages from points of origin to destination points. In our mode, we wi assume that the entire message is contained in a singe capsue of information, which, by anaogy to packet-switching networks, wi be simpy caed a packet. In a rea packet-switching network, a singe packet carries the information payoad, and some additiona information reated to the interna structure of the network. We wi ignore the information payoad entirey, and assume that the packet carries ony two pieces of information: time of its creation and the destination address. Our simuated network consists of a number of interconnected nodes. Each node can perform two functions: of a hosts, meaning that it can generate and receive messages, and of a router (message processor), meaning that it can store and forward messages. Packets are created and moved according to a discrete time parae agorithm. The structure of the considered networks and the update agorithm wi be described in subsections which foow Connection Topoogies We wi consider two types of network connection topoogies: a two-dimensiona square attice L = L(L) and a two-dimensiona square attice L with additiona inks added randomy, denoted by L = L (L). The vaue of the subscript gives a number of an extra inks in a network and the vaue of L gives a number of nodes in the horizonta and vertica direction of the attice L. The attices L and L with periodic boundary conditions wi be denoted by L p and L p, respectivey, and with non-periodic boundary conditions by Lnp and L np, respectivey. Hence, with this notation, L p 0 = L p and L np 0 = L np. Most of our simuations wi be performed on attices with periodic boundary conditions. The network hosts and routers are ocated at nodes of the attice L. The position of each node on a attice L is described by a discrete space variabe r, such that r = ic x + jc y, (1) where c x, c y are Cartesian unit vectors, and i, j = 1,..., L. For each node r we denoted by C(r) the set of a nodes directy connected with the node r. Hence, for each r L p, the set C(r) is of the form C(r) = {r c x, r + c x, r c y, r + c y }. (2) 2

3 In this case, the node r is connected with its four nearest neighbours. However, for attices with non-periodic boundary conditions L np or square attices with additiona inks added randomy L p and L np the form of the set C(r) can be different, for some nodes r, from the one in (2). For exampe, for nodes r on the boundary of a attice L np the set C(r) can contain two or three eements ony, depending on where a node r is ocated on the boundary. In the case of a attice L p or L np the set C(r) can contain many non-nearest neighbours nodes depending on a number of additiona inks which originate from the node r. The extra inks are constructed using the foowing procedure. We first seect randomy a node r 1 on a square attice L p or L np. Next, we seect randomy another node r 2, different from the node r 1, and connect these two nodes with direct communication ink. By repeating this procedure independenty times we obtain a attice L p or L np, respectivey, with additiona random inks. It can happen that the nodes r 1 and r 2 can be seected again to form a new ink. Hence, in the network there can be severa inks connecting directy the same nodes. We want to emphasize that a the connections in our modes are static, during the simuation period they do not change. Additiona random inks are added before the simuation starts, and remain unchanged. In the networks considered here, each node maintains a queue of unimited ength where the arriving packets are stored. The number of packets in the queue at a node r at time step k wi be denoted by n(r, k), whie the tota number of packets in the system at time step k wi be denoted by N(k), N(k) = n(r, k). (3) r L Packets stored in queues, at individua attice nodes, must be deivered to their destination addresses. To assess how far a given packet is from its destination, we introduce the concept of distance between nodes. Depending on a network connection topoogy we wi use three metric functions to compute the distance between nodes r 1 = (i 1, j 1 ) and r 2 = (i 2, j 2 ). Namey, we wi use 1. for attices with non-periodic boundary conditions Manhattan metric d M (r 1, r 2 ) = i 2 i 1 + j 2 j 1, (4) 2. for attices with periodic boundary conditions periodic Manhattan metric d P M (r 1, r 2 ) = L i 2 i 1 L 2 j 2 j 1 L 2, (5) 3. and regardess of boundary conditions the shortest path metric d SP (r 1, r 2 ) defined as the number of inks in the shortest path joining r 1 and r 2. By shortest we mean the path with the smaest number of inks. Metric d P M wi be used on attices L p, whie d M wi be used on attices L np. Note that on a square attice with no extra inks d SP (r 1, r 2 ) = d M (r 1, r 2 ), and d SP (r 1, r 2 ) = d P M (r 1, r 2 ) for attices with non-periodic and periodic boundary conditions, respectivey. Furthermore, for each r L κ, where κ = p or np and {0, 1,...} 3

4 C(r) = {x L κ :d SP (x, r) = 1}. (6) When irreguarities such as extra inks are present, d SP (r 1, r 2 ) can be computed using one of the we known agorithms. In our simuations, we used shortest path backward tree agorithm [8] Routing Agorithms The dynamics of the networks are governed by the parae update agorithms shown Figure 1, simiar to the agorithm used in [7]. We start with an empty queue at each node, and with discrete time cock k set to zero. Then, the foowing actions are performed in seque: 1. At each node, independenty of the others, a packet is created with probabiity λ. Its destination address is randomy seected among a other nodes in the network with uniform probabiity distribution. The newy created packet is paced at the end of the queue. 2. At each node, one packet (or none, if the oca queue is empty) is picked up from the top of the queue and forwarded to one of its neighboring sites according to a one of the routing agorithms to be described beow. Upon arriva, the packet is paced at the end of the appropriate queue. If severa packets arrive to a given node at the same time, then they are paced at the end of the queue in a random order. When a packet arrives to its destination node, it is immediatey destroyed. 3. k is incremented by 1. This sequence of events, which constitutes a singe time step update, is then repeated arbitrary number of times. The state of the network is observed after sub-step 3 (cock increase), but before sub-step 1 (creation of new packets). In order to expain the routing agorithms mentioned in sub-step 2, we wi first describe one of its simpified versions. Let us assume that we measure distance using some metric d, where d coud be any of the previousy defined metrics d M, d P M, or d SP. To decide where to forward a packet ocated at a node r with the destination address r d, two steps are performed: 1. From sites directy connected to r, we seect sites which are cosest to the destination r d of the packet. More formay, we construct a set A (r) such that A (r) = {a C(r) : d(a, r d ) = min x C(r) d(x, r d)} (7) 2. From A (r), we seect a site which has the smaest queue size. If there are severa such sites, then we seect one of them randomy with uniform probabiity distribution. The packet is forwarded to this site. Using a forma notation again, we coud say that the packet is forwarded to a site seected randomy and uniformy from eements of a set B (r) defined as B (r) = {a A (r) : n(a, k) = min n(x, k)}. (8) x A (r) 4

5 To summarize, the routing agorithm R described above sends the packet to a site which is cosest to the destination (in the sense of the metric d), and if there are severa such sites, then it seects from them the one with the smaest queue. If there is sti more than one such node, random seection takes pace. It is cear that each packet routed according to the agorithm R wi trave to its destination taking the shortest possibe path (shortest in the sense of the metric d, not necessariy in terms of the number of time steps required to reach the destination). In rea networks, this does not aways happen. In order to aow packets to take aternative routes, not necessariy shortest path routes, we wi introduce a sma modification to the routing agorithm R described above. The modified agorithm R m, for each node r, wi use instead of the set A (r) a set A m (r) defined as foows. In the construction of the set A m (r) instead of minimizing distance to the destination d(x, r d ), as it was done in (7), we wi minimize Θ m (d(x, r d )), where Θ m (y) = { y, if y < m, m, otherwise, (9) for a given integer m. Thus, the definition of the set A m (r) is A m (r) = {a C(r) : Θ m (d(a, r d )) = min Θ m (d(x, r d ))} (10) x C(r) The above modification is equivaent to saying that nodes which are further than m distance units from the destination are treated by the routing agorithm as if they were exacty m units away from the destination. If a packet is at a node r such that a nodes directy inked with r are further than m units from its destination, then the packet wi be forwarded to a site seected randomy and uniformy from the subset of C(r) containing the nodes with the smaest queue size in the set C(r). It can happen that the seected site can be further away from the destination than the node r. Therefore, introduction of the cutoff parameter m adds more randomness to the network dynamics. One coud aso say that the destination attracts packets, but this attractive interaction has a finite range m: packets further away than m units from the destination are not being attracted. It is aso possibe to reate various vaues of the cutoff parameter m to different types of routing schemes used in rea packet-switching networks. Assume that each node r maintains a tabe containing a possibe vaues of d(x, r d ), for a possibe destinations r d and a nodes x C(r), and that packets are routed according to this tabe by seecting nodes minimizing distance, measured in the metric d, traveed by a packet from its origin to its destination. Such a routing scheme is caed tabe-driven routing [8] and it is equivaent to the routing agorithm R. In this case, construction of the set A (r) woud require ooking up appropriate entries in the stored tabe. Let us now define D max to be the argest possibe distance between two nodes in the network. When m < D max, then for a given x, we need to store vaues of d(x, r d ) ony for nodes r d which are ess than m units of distance away for a other nodes distance does not matter, since it wi be treated as m by the routing agorithm. Hence, at each node r the routing tabe to be stored is smaer than in the case when m = D max. The routing scheme based on this smaer routing tabe is caed the reduced tabe routing agorithm [8] and it is 5

6 equivaent to the routing agorithm R m. In the case when m = D max the routing agorithm R m = R. Finay, when m = 1, the distances between hosts and destinations are not considered in the routing process of packets. Therefore, there is no need to store any tabe of possibe paths at nodes of the network. This case corresponds to the tabe-free routing agorithm [8] in which packets are routed randomy. Hence, this agorithm can send packets on circuitous and ong routes to their destinations. The anaysis of this routing agorithm has been done in [4], where some anaytica resuts are aso presented. At present such resuts are not avaiabe for routing agorithms with m > Performance of networks with square attice connection topoogy (L p 0, d P M), (L np 0, d M ) In order to asses the performance of a network, graphs of deay as a function of presented oad are frequenty used in network performance iterature [9]. In our case, deay τ wi be defined as the number of time steps eapsed from the creation of a packet to its deivery to the destination address. We wi aso use average deay τ(k), where the average is taken over a packets deivered to their destination from the beginning of the simuation (k = 0) up to time k. Probabiity of a packet creation λ wi be used as a measure of a presented oad Fu tabe routing We assume that the network topoogy is a square attice L p 0(L) with d P M metric and the network routing agorithm is the fu tabe routing agorithm R m, with m = D max, i.e. R m = R. Figure 2a shows graphs of the average deay τ(k) versus presented oad λ, as measured during simuation performed on a attice L p 0(50). The three curves shown there correspond to different times. It is cear that beyond a certain critica vaue of λ = λ c, the average deay drasticay increases. Moreover, the average deay grows with time, which suggests that for λ > λ c there is no equiibrium state. In fact, when λ > λ c, a typica queue size and consequenty, the number of packets in the system N(k), grows without bounds, as shown in Figure 2b and Figure 3. It is possibe to find an approximate vaue of the critica oad λ c by the foowing argument. For λ < λ c, the system reaches steady state, and in the steady state the number of packets created per unit time (given by L 2 λ) must be equa to the number of packets deivered per unit time. Since the average time spent in the system by a packet is τ(k), we can reasonaby assume that N(k)/τ(k) packets are deivered to their destinations per unit time, hence N(k) τ(k) = L2 λ. (11) This reationship, known as Litte s aw in queuing theory [6], hods ony beow the critica point, as shown in Figure 4. For the routing agorithm R m, with m = D max, when the number of packets in the network is sma, an individua packet is aways routed in such a way that it foows the shortest path to its destination avoiding a occupied nodes. This means that for sma 6

7 N(k), the average packet deay is approximatey equa to average distance from the packet s origin to its destination, which wi be caed free packet deay τ 0 After some agebra, this eads to τ 0 = 1 L 4 L 1 i 1,i 2,j 1,j 2 =0 τ 0 = 1 L 4 { L r 1,r 2 d P M (r 1, r 2 ) (12) i 2 i 1 L 2 j 2 j 1 L } 2 = L 2. (13) Obviousy, when the oad increases, at some point the number of packets in the network wi be so arge that it woud not be possibe to find a route to a destination competey avoiding other packets. Assuming that packets are approximatey uniformy distributed over the entire attice, this wi happen when a sites are occupied, i.e. when N(k) = L 2. Using (11) this gives an estimate of λ c : λ c = 1 (14) τ 0 For L p 0(50) we obtain τ 0 = 25 and λ c = 0.04, in good agreement with the vaue obtained from simuations λ c = ± Quite simiar cacuations can be performed for a square attice with non-periodic boundary L np 0 (L). In this case, τ 0 = 1 L 4 r 1,r 2 d M (r 1, r 2 ) = 2 3 L 2 1 L 2 L, (15) 3 yieding λ c = 0.03 for L np 0 (50). The measured vaue of λ c for L np 0 (50) is ± 0.001, i.e., much ower. The discrepancy is mainy due to the fact that for the attices L np 0 (L) packets are not uniformy distributed on the attice, having a tendency to custer at the center. Consequenty, jamming occurs earier than one woud expect assuming uniform distribution of packets Partia tabe routing Decrease in vaue of the cutoff parameter m has a profound effect on the critica oad. Smaer m means that packets which are ocated further than m inks from their destination move with a high degree of randomness, and as a resut, their average deay is arger. This increase of the deay can be aso seen in a pot of a singe packet deay as a function of m (Figure 5). Whie vaues of m cose to D max do not significanty change τ 0, vaues of m cose to 1 resut in an increase of τ 0 by up to two orders of magnitude. 4. Performance of networks based on square attices (L p, d P M),, d M ) with additiona random inks (L np 7

8 4.1. Fu tabe routing Let us now consider the network dynamics governed by the routing agorithm R taking pace on attices L p and Lnp which in addition to norma nearest neighbor connections, feature additiona inks, where > 0. Figure 6 shows how addition of random inks changes the graph of deay vs. oad for both non-periodic and periodic case. We are sti using d M and d P M metric for L np and L p attice, respectivey, which means that the distance between two points r 1 and r 2 is sti computed using d M (r 1, r 2 ) or d P M (r 1, r 2 ) metric, respectivey, even if r 1 and r 2 are directy connected by some extra ink. As expected, addition of extra inks improves performance of the network, shifting the critica point λ c to the right (see Figures 6 and 7). This means that the network can carry more oad without experiencing congestion. Performance improvement is more pronounced for attices with non-periodic boundaries, as shown in Figure 7. For exampe, by adding 100 random inks to attice, which increases tota number of inks by 2%, we increase the critica oad by over 25%. Increasing the number of inks by 8% doubes the critica oad. This can be attributed to the fact that some packets can bypass congested centra area by using shortcuts, and their deay decreases not ony because they have shorter distance to trave, but aso because they have avoided congestion. In the case of attices with periodic boundaries, packets are more uniformy spread even in the presence of additiona random inks. Thus, the performance improvement is ony caused by the decrease in the distance traveed, but not by bypassing congestion, since congestions are aso uniformy spread in the case of attices with periodic boundaries. In the remainder of this artice, we wi focus our discussion on attices with periodic boundaries ony Partia tabe routing When a routing agorithm R m, with m < D max, is used, additiona random inks can significanty increase critica oad, and the reative performance gain is much arger than in the case of the fu tabe routing agorithm. Figure 8a,b shows the reative change of the critica oad, defined by λ c = λ c(m, ) λ c (m, 0), (16) λ c λ c (m, 0) where λ c (m, ) denotes the critica oad at a given m and, for two different vaues of m, m = 50 and m = 20. One can immediatey notice that the impact of additiona inks on performance of the network is much stronger in the case of partia tabe routing (m = 20) than in the case of fu tabe routing (m = 50). For exampe, about 50 extra inks are sufficient to doube the critica oad corresponding to m = 20, whie the same number of inks has amost negigibe impact on the critica oad when m = 50. 8

9 5. Performance of networks based on square attice (L p, d SP ) with additiona random inks and d SP metric As stated before, for a square attice without additiona inks, metric d SP is identica to d M or d P M metric. This is no onger true for a square attice with additiona random inks. Routing based on d SP metric fuy utiizes shortcuts provided by additiona inks, significanty decreasing free packet deay τ 0. One woud expect that a decrease in free packet deay wi decrease aso average deay, as it was in the case of networks with d M and d P M metric. In reaity, we observe just opposite effect (Figure 8) Fu tabe routing Figure 8c shows how the critica oad λ c (m, ) changes when additiona random inks are introduced. This is shown for the network dynamics governed by R, i.e. the fu tabe routing agorithm with m = D max, on a square attice with periodic boundaries One can ceary see that if the number of additiona random inks is beow some critica vaue c (m) the critica oad λ c (D max, ) is actuay smaer than λ c (D max, 0), in spite of increased connectivity between nodes of the network. The performance of the network is at its worst when just a few additiona random inks are added. However, it improves with the increase of a number of additiona random inks and at some critica vaue c (m) it becomes the same as of the network without any additiona random ink. When the number of additiona random inks is greater than c (m) an improvement in the network performance is observed. For the network L p 500(50) the critica oad λ c (50, 500) is amost equa to the critica oad λ c (50, 0) of the network L p 0(50), i.e. λ c (50, 500) λ c (50, 0), and the improvement of the L p (50) network performance is observed for greater than 500. This rather unexpected phenomenon can be understood as foows. When additiona inks are introduced, and their number is ess than c (m), they provide a shortcut between distant parts of the network. Since packets are forwarded to their destinations via the shortest path, it often happens that one ink serves as a shortcut for many packets from the neighborhood. One coud say that additiona inks attract most of the traffic and quicky become congested, even though sites which are not cose to extra inks are amost empty. This is we iustrated in Figure 9, which shows snapshots of dynamics of the network with R routing agorithm, nodes and periodic boundary conditions. The presented oad is λ(50, 0) = 0.025, just beow the critica vaue λ c (50, 0) = If there are no additiona random inks the network dynamics remains in the steady state, as is showed by the eft coumn of Figure 9. The number of packets in the network fuctuates sighty over time, but remains at the same eve: at k = 100 there are 2106 packets in the network, whie at k = 1000 there are 2127 packets. When additiona 100 random inks are introduced, which is ess than c (50), keeping a other network parameters unchanged, the network dynamics enters the congested phase, as it is iustrated by the right coumn of Figure 9. The number of packets in the network increases rapidy over time from 2238 packets in the network at k = 100, to packets at k = Congestions occur mainy at inputs and exists from the extra inks. At these nodes the queue sizes are substantiay arger than in other nodes of the network. This is iustrated by the dark spots in the figures of the right coumn of 9

10 Figure Partia tabe routing The performance of a network changes from the one described above when the vaue of the cutoff parameter m is ess than D max, i.e. m < D max. For the routing agorithm R 20 appied on the attice L p (50) for various vaues of the parameter the performance of the network is shown on Figure 8d. From this figure we observe that the critica vaue c (m) beow which the performance of the network with the routing agorithm R 20 is worse than the performance of the network without random inks added is rather ow. This vaue is ower than the corresponding vaue when R routing agorithm has been used. For exampe, adding more than about 15 inks increases the critica oad λ c. Adding about 50 inks, just a 2% increment in the number of inks, increases the critica oad by 100%! The performance of the network improves significanty further with the increase of the number of random inks. The expanation of this behavior is straightforward. As we have aready mentioned, for a reguar square attice without random inks added, the vaues of the critica oad λ c (m, 0) strongy decrease with decrease of the cutoff parameter m. This resuts from the fact that packets which are further away than m units from their destinations are not being attracted to the destinations and trave randomy through the network. However, addition of random inks significanty decreases the average distances between network nodes in the metric d SP. Therefore, it does not matter what is the exact vaue of the cutoff parameter m. Most of the time distances in the metric d SP are way beow m and packets can be attracted to their destinations much faster. This attraction increases with the increase in the number of additiona random inks. Hence, when additiona inks are present, the critica oad λ c (m, ) is not very much dependent on m, uness m is very sma and increases with increase in vaue of the number of the random inks added. Let λ c (m, ) be the critica oad of a network with the cutoff parameter m and with extra inks added. Let m 1 = D max and et m 2 be smaer than m 1, but not too cose to 1, for exampe, m 1 = 50, m 2 = 20, as in Figure 8. The performance of the networks with the routing agorithms R m1 and R m2, as in the Figure 8, can be summarized as foows for i = 1, 2 when < c (m i ) and c (m 2 ) < c (m 1 ), λ c (m 2, 0) < λ c (m 1, 0), λ c (m i, ) < λ c (m i, 0), λ c (m i, 0) < λ c (m i, ), for i = 1, 2 when > c (m i ), and for sufficienty arge λ c (m 1, ) λ c (m 2, ). Figure 10 shows how the introduction of 100 additiona inks to a network with nodes, routing agorithm R 20 and presented oad λ = (which is above the critica oad for = 0), affects the network dynamics. When = 0, the number of packets in the network 10

11 grows with time, from 1248 at k = 100 to 6750 at k = 1000, indicating that the system is in the congested state. The eft coumn of Figure 10 shows that the queue sizes grow amost uniformy over a nodes of the network. Hence, the congestion is distributed uniformy over a nodes of the network. However, when = 100 additiona random inks are introduced, the right coumn of Figure 10 shows that congestion is eiminated. The number of packets in the network remains amost steady and it fuctuates around 230 (Figure 10). If an occasiona sma congestion occurs near the entrance to one of the shortcuts, it quicky disappears. For exampe, dark square visibe in the right coumn of Figure 10 at k = 100 is not visibe at k = 1000, demonstrating that oca congestions are not permanent. 6. Concusion We found that the impact of additiona random inks on the performance of the network strongy depends on the routing scheme used in the network. Critica oad of a network can be notaby improved if the routing is based on a geometrica distance. Adding sma number of additiona inks can decrease the average deay and shift the transition to the jammed phase toward higher oad vaues. This, in genera, is not true for routing schemes based on the shortest path metric. In this case, if the number of additiona inks is sma, one can actuay observe degradation of performance: many packets attempt to utiize shortcuts introduced by additiona inks, causing congestion which in effect pushes the network to a jammed phase. Reduced tabe routing can, to some extent, eiminate this probem. If packets ocated further than m inks from the destination are routed randomy (other packets taking the shortest possibe path), performance gains obtained by adding sufficient number of extra inks can be quite significant. In order to reate our findings to data network protocos used in practice, more research is ceary needed. In particuar, congestion contro mechanisms buit into protocos such as TCP/IP wi certainy affect phenomena reported here. This issue, as we as other possibe modifications of the mode, is currenty under investigation. Furthermore, the authors beieve that some of the issues raised in [10], reated to sef-simiar traffic modeing and anaysis, and performance modeing of modern high-speed networks can be addressed by the methodoogy of this paper. Acknowedgment The authors acknowedge partia financia support from the Natura Sciences and Engineering Research Counci (NSERC) of Canada and The Fieds Institute for Research in Mathematica Sciences. They express their gratitude to Bruno Di Stefano and Murad S. Taqqu for hepfu discussions. References [1] I. Campos, E. Tarancón, F. Cérot, and L. A. Fernández. Therma and repusive traffic fow. Phys. Rev. A, 52(6): , [2] J. H. B. Deane, C. Smythe, and D. J. Jefferies. Sef-simiarity in a deterministic mode of data transfer. Internationa Journa of Eectronics, 80(5): ,

12 [3] D. J. Watts and S. H. Strogatz. Coective dynamics of sma-word networks. Nature, 393: , [4] H. Fukś, A. T. Lawniczak, and S. Vokov. Packet deay in modes of data networks. To appear in ACM Transactions on Modeing and Computer Simuation, Juy [5] J. Kadirire. Minimising packet copies in muticast routing by expoiting geographica spread. Comput. Commun. Rev., 24(3):47 62, [6] R. Neson. Probabiity, stochastic processes, and queueing theory: the mathematics of computer performance modeing. Springer-Verag, New York, [7] T. Ohira and R. Sawatari. Phase transition in computer network traffic mode. Phys. Rev. E, 58(1): , [8] T. N. Saadawi, M. H. Ammar, and A. E. Hakeem. Fundamentas of Teecommunication networks. John Wiey and Sons, New York, [9] W. Staings. High-speed networks: TCP/IP and ATM design principes. Prentice Ha, New Jersey, [10] M. S. Taqqu, W. Wiinger, and A. Erramii. A bibiographica guide to sef-simiar traffic and performance modeing for modern high-speed networks. In F. P. Key, S. Zachary, and I. Ziedins, editors, Stochastic Networks: Theory and Appications, pages , Oxford, Carendon Press (Oxford University Press). [11] A. Y. Tretyakov, H. Takayasu, and M. Takayasu. Phase transition in a computer network mode. Physica A, 253: ,

13 START Procedure CREATE k:=0 For every r, Q(r)=0 (empty queue) Procedure ROUTE Generate random number q [0,1) q<λ Yes Randomy seect one attice node r d Create packet with destination address r d, pace it at the end of Q(r) RETURN No simutaneousy appy procedure CREATE to a nodes r simutaneousy appy procedure ROUTE to a nodes r k:=k+1 Is Q(r) empty? No Pickup one packet from the top of queue Q(r), determine its addrerss r d Construct set C(r) of a sites directy connected to r Find A m (r), set of sites beonging to C(r) which are coses to r d Construct set B m (r) containing sites from A m (r) with smaest queue size Randomy seect one eement in B m (r) and denote it a Yes No r d =a Yes Pace the packet at the end of queue Q(a) Destroy the packet (packet deivered to its destination) RETURN Figure 1: Network update agorithm. Symbo Q(r) denotes the queue at node r. 13

14 τ (a) λ (b) 80 N 60 L λ Figure 2: (a) Average ifetime of a packet τ(k) as a function of λ, for L p 0 (50) with d P M and m = D max after k = 1000 ( ), k = 1500 ( ), and k = 2000 ( ) iterations. (b) Number of packets in the system N(k) as a function of λ after k = 1000 ( ), k = 1500 ( ), and k = 2000 ( ) iterations. 14

15 λ = N(k) λ = k Figure 3: Number of packets in the system N(k) for subcritica and supercritica vaues of λ (λ = and λ = 0.042, respectivey). L p 0(50) with d P M metric and m = D max N 0.04 τl λ Figure 4: Verification of Litte s aw for a attice L p 0(50) with d P M k = Continuous ine corresponds to L 2 ) = λ. and m = D max, at 15

16 τ m Figure 5: Free packet deay τ 0 as a function of m for a attice L p 0 (50) with d P M metric. 16

17 τ (a) λ τ (b) λ Figure 6: (a) Average ifetime of a packet τ(k) as a function of λ for k = 1500 and for L np (50) attice with d M metric and m = D max, and with the number of extra random inks = 0 ( ), = 100 ( ), = 200 ( ), and = 400 (+). (b) The same pot for L p (50) attice with d P M metric and m = D max. 17

18 λ c Figure 7: Critica oad λ c as a function of a number of extra inks for the attice with periodic ( ) and non-periodic ( ) boundaries, using d P M and d M metric, respectivey. In both cases, m = D max. 18

19 λ λ c λ λ c (a) d P M metric, m = 50 λ λ c (c) d SP metric, m = λ λ c (b) d P M metric, m = (d) d SP metric, m = Figure 8: Change in critica oad for a attice L p 0(50) using metrics d P M and d SP with m = 20 and m = 50. Vertica axis corresponds to (λ c (m, ) λ c (m, 0))/λ c (m, 0), where λ c (m, ) is a critica oad at a given m and. 19

20 = 0 = 100 k = 0 k = 0 k = 10 2 k = 10 2 k = 10 3 k = 10 3 Figure 9: Comparison of dynamics of the network L p (50) with d SP and R agorithm for = 0 (eft coumn) and = 100 (right coumn). Queue sizes are represented as shades of gray, from the highest queue size of 20 or more represented by back coor to the empty queue represented by white coor. In order to preserve carity, additiona inks are shown for k = 0 ony. 20

21 = 0 = 100 k = 0 k = 0 k = 10 2 k = 10 2 k = 10 3 k = 10 3 Figure 10: Comparison of dynamics of the network L p (50) with d SP and R 20 agorithm for = 0 (eft coumn) and = 100 (right coumn). Queue sizes are represented as shades of gray, from the highest queue size of 20 or more represented by back coor to the empty queue represented by white coor. In order to preserve carity, additiona inks are shown for k = 0 ony. 21