Routing Multicast Streams in Clos Networks

Routing Multicast Streams in Clos Networks De-Ron Liang Institute of Information Science Academia Sinica Taipei Taiwan59 R.O.C. drliang@iis.sinica.edu.tw Chen-Liang Fang Jin-Wen College of Business and Technology Taipei Taiwan 3 R.O.C. fang@jwc.edu.tw Keywords: Multicast routing routing algorithms Clos networks ATM networks performance evaluation simulation. Abstract Multicast routing in high speed networks is one of the key issues in the design and implementation of applications such as video conferencing. Traditionally the problems are formalized as either the shortest path problem or Steiner tree problem where the objective function is either the shortest path or the minimum costs associated with the path. In this paper we study an on-line multicast routing problem in Clos network with minimizing internal blocking ratio. We propose three families of heuristics: Least busy LB) First t FF) and Minimal bandwidth MB). Our simulation shows LB and FF families perform well with unicast streams but they are not good enough to deal with multicast streams. The MB family has signicant performance on multicast streams. The MB family can be reduced to set covering problem which is well known NP-complete problem. We also propose heuristic minimal bandwidth HMB) family which are derived from greedy set covering algorithm to trade o computing time and routing performance. These algorithms perform almost as good as the expensive MB family and as simple as FF family. Therefore the HMB is the best algorithms we proposed for multicast routing. Introduction Multicasting is the simultaneous transmission of data to multiple destinations. The problem of multicast routing in high speed networks such as ATM networks has been recognized an important problem because of its application to video conferencing[?] and HDTV broadcasting[?]. The multicast routing problem has usually been formalized as an o-line allocation problem where the network is normally modeled by the graph representation and the set of multicast streams to be considered is known in prior. When a multicast stream arrives at the network the multicast routing algorithm is responsible for nding routes from the source to each of the destination; each route should have enough free bandwidth to support the stream. If multiple routes exist for a given multicast the routing algorithm will choose one so as to optimize a certain objective function. The existing algorithms can be classied into two categories in terms of objective functions: the Shortest- Path Algorithms and the Minimum Cost Algorithms. For the Shortest-Path Algorithms the routes are computed independently form the source to each of destination using Dijkstra's shortest path algorithms [?] then the multicast route is merged to form a multicast tree. For the algorithms in the second category the multicast routes is constructed in such as way that the sum of the costs associated with the used links is minimized. This problem is usually cited as the Steiner tree problem which is known to be NP-complete[??]. Numerous heuristic algorithms have been proposed to study this problem[??]. Kompella et al propose an extension to the Steiner tree problem where the minimum cost is searched with delay constraints[?]. Previous algorithms can handle a single multicast stream at a time Noronha and Tobagi study a routing problem with batch arrivals and propose an optimal algorithm using integer programming approach[?]. We study the on-line problem of multicast routing in Clos network with maximizing network throughput. In contrast to the o-line problems the arrival of the multicast streams to the Clos network is assumed to form a random process with known statistics. Furthermore the active duration of the multicast streams is characterized by a probability distribution. It is dicult to nd the optimal routing algorithms for the on-line problem. The objective of this problem is to develop ecient algorithms that can nd multicast routing trees to maximize the network throughput. We propose three families of routing algorithms each of which corresponds to a system measure using as the routing index; namely rst-t FF) leastbusy LB) minimal-bandwidth MB). The denitions

of these routing algorithms are given in Section 3. We show that the problem to nd an optimal routing for a stream request according to each of these system measures can be formulated as an o-line routing problem. Our simulation shows that FF and LB families are both good enough for unicast streams. The MB family which can be reduced into set covering problem with complexity ON!) are the best one for multicast routing. We also propose a Heuristic Minimal Bandwith HMB) family which are derived from greedy set covering algorithm and can just loss limited performance to decrease most computing time. Problem Statement A multicast stream is specied by the quadruple: s k ; O k ; I k ; k ). We notice that s k is the switch input O k = fo ; :::; o nk g is the set of the switch outputs and n k is the multicast degree. I k = [t kb ; t ke ] is the active period starting from time t kb to time t ke. Finally k is the bandwidth requirement. Furthermore the switch inputs and switch outputs are assumed to be requested uniformly by the incoming streams and bandwidth requirement is characterized by a probability distribution. The route used to establish the multicast stream on the network is called multicast path. Thus a multicast path for stream S k can be viewed as the merge of n k point-to point path for s k to each of O k. We are interested in the design of ecient multicast routing algorithms capable of addressing the performance issues namely network throughput. The network throughput are contributed by individual multicast streams. The throughput of an active stream S k is dened as XS k ) = O k k. Suppose St) = fs k g K k= is the set of active streams at time t; the instantaneous throughput of the Clos network at time t is dened as XS k) = P s kst) XS k). The expected throughput of the network under given trac; E[X] =. As discussed the characteristics of the multicast streams are assumed statistically indistinguishable. To maximize the network throughput is equivalent to maximize the call acceptance rate. If the stream blocking is caused by the bandwidth shortage of external links then it is called external blocking. If the routing algorithm can not nd any feasible routing tree for current status of Clos network then it is callled internal blocking. In this paper we only discuss internal blocking. The problem of this research is to nd ecient routing algorithms to construct multicast path in Clos network based on the on-line information i.e. the existing active streams provided the topology of the Clos network and the statistic of the multicast streams are given. R T 0 lim Xt)dt T! T 3 The Routing Algorithms We dene the link residual capacity of link to be the available bandwidth of the link and denote it by x ij ; x i+k )). The residual capacity of the path P is dened as the minimum link residual capacity of the two links and we denote it as P ). The I/O residual capacity of [s o] is dened as the maximum path residual capacity among all paths in the path set of [s o]. The least busy path of [s; o]is a path which is in path set of [s; o] with maximal residual capacity. Furthermore the residual capacity of the switch is dened as the minimum of the I/O residual capacities among all [so]. The reachable third stage SE set of second stage SE i ROi) is a set of third stage SEs there is a destination outport on this third stage SE and there exists a path with enough residual capacity passing second stage SE i. In order to deploy the load balance eect of LB we also dene reachable least busy third stage SE set of second stage SE i ROLBi) as set of third stage SEs there is a destination outport j on this third stage SE and there exists a least busy path of [s; j]passing second stage SE i. The least-busy LB) routing algorithm is to nd a multicast routing tree in such a way that the residual capacity of the multicast tree is maximized. The rst- t FF) algorithm is to nd a multicast routing tree in such a way every path pass the lowest numbered SE in second stage for every outport. The minimumbandwidth MB) algorithm is to nd a multicast routing tree in such a way that total bandwidth of the multicast tree is minimized. The HMB family is basically derived from well known greedy set covering algorithm [?]. We also deploy two second selection criteria least-busy and rst-t to form two new routing algorithms named heuristic minimal-bandwidth least busy HMBLB) and heuristic minimal-bandwidth rst- t HMBFF). Consider a partial topology of the Clos network and the path residual capacities of all involved path as shown in Figure??a). Suppose there is a new arrival stream with s k = ; f; ; 3g; I k ; 0:). The randomly choose one available second stage SE for each of the outports f 3g and construct the routing tree as shown in Figure??b). The I/O residual capacity of [ ] [ ] and [ 3] are and respectively. The LB algorithm randomly choose SE x from two feasible second stage SEs for the Outport and choose x 3 for the Outport and 3. Such that routing tree has minimal I/O residual capacity for these I/O pairs. See Figure??c). The LB is trying to balance the load of all internal links. The FF chooses the rst feasible second stage SE x ; x 3 ; and x for the Outport and 3 respectively to construct the FF routing tree. The FF could use switch bandwidth as much as do. See Figure??d). That is why the FF is not good enough under multicast routing. Each

x x 0. x x3 3 x3 x3 x3 x33 RO) = RO) = RO3) = a) x x3 x 0. x x x x x x x x x x x3 P ) ) ) ) ) ) ) ) x x x x3 x3 x3 x33 x3 x3 x33 x3 x x3 x3 x3 x x3) x3 x33 x3 x3 x3 x33 b) 0. x 0. x3 0. SE x x and x 3 can not cover all Outports and 3 alone; that means we must choose at least two of them to build up the routing tree. There are two possible solutions fx ; x 3 g and fx ; x 3 g for MB the example shows it randomly chooses the rst solution. See Figure??e). The jroj of the second stage SE x x and x 3 are and respectively. The HMB randomly chooses x to cover third stage SE x 3 and x 33 for rst run. The uncover set U becomes fx 3 g. The current jroj of x and x 3 are 0 and respectively. The HMB chooses x 3 to nish the routing process and forms the routing tree shown in Figure??f). Th jrolbj of x x and x 3 are 0 and respectively. The HMBLB chooses x 3 to cover third stage SE x 3 and x 33 for rst run. The uncover set U becomes fx 3 g. The current jroj of x and x are 0 and respectively. The HMB chooses x to nish the routing process and forms the routing tree shown in Figure??g). There are two candidates x and x 3 in rst run of HMB. The HMB just randomly choose one. In contrast the HMBFF applies second criteria rst-t can choose x in rst run. The current jroj of x and x 3 are 0 and respectively. The HMB chooses x 3 to nish the routing process and forms the routing tree shown in Figure??h). x x3 x x3 4 Simulation study x x x3 x33 x c) 0. x3 x3 x3 x33 e) x3 x33 g) x3 x3 0. First Run RO) =0 RO) = RO3) = Second Run RO) =0 RO) = x x3 x33 d) x3 x3 x x3 x33 f) x3 x3 x3 x33 h) First Run RO) = RO) = RO3) = Second Run RO3) = First Run RO) = RO) = RO3) = Second Run RO3) = Figure : Example for illustrating all routing algorithms' solutions; a) all feasible paths b) 's c) LB's d)ff's e)mb's f)hmb's g)hmblb and h)hmbff's solutions The simulation experiment consists of two parts: the unicast routing and the multicast routing; where the former concerns routing the point to point and is a special case of the multicast routing. As discussed later the performance of the routing algorithms varies from the unicast routing to the multicast routing. A random algorithm ) is also implemented to serve as the benchmark algorithm. is to nd a multicast routing tree in such way every path is randomly chosen from all feasible paths of each outport and then merge them together. We have designed a simulator to compare the performance of those on-line routing algorithms we have discussed in the previous sections. The simulator is implemented using C++ with a general propose simulation package called CSIM[?]. All the simulation are obtained within 95% of the condence interval with % in halfwidth. In the simulation we study N N Clos networks for N= 4 6 36 64 00 and 44 with expansion factor f set to. The bandwidth of the external and internal links are 50 Mbps. All streams request the same bandwidth or homogeneous streams) in the simulation i.e. k = ; 8k. We dene the link capacity k to be the total number of streams can be carried in a link. For example a 50 Mbps link can carry at most 3 of 50 Mbps streams thus k=3. The steam arrives are assumed to be Poisson and the life time is exponentially distributed with

mean set to 00 minutes. The system load is calculated as output port load and set from 0% to 00%. For the performance comparison we dene the performance metrics internal blocking ratio B i and performance gain for homogeneous simulation. The internal blocking ratio B i is calculated as b B i = i S total?b e where b i is the total number of internal blocking S total is the total number of arrival streams and b e is the total number external blocking. We deploy algorithm as our benchmark algorithm. The performance gain to benchmark algorithm is dened as b 0 = b r bi where b r is the internal blocking rate of. 0. 0 Sensitivity Analysis of Link Stream Capacity N=64 M= L=.0) LB FF 00 0 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 00 a) Sensitivity Analysis of Link Stream Capacity N=64 M= L=.0) /LB /FF 4. Unicast We notice that MB and HMB are both identical to under the unicast condition. Similarly HMBLB is equivalent to LB. Therefore only results of LB FF and are reported in this section. We vary the link stream capacity k from to 0 under varies system load. The Figure?? and?? show the results of unicast routing. Figure?? depicts the impact of the link stream capacity k to the system performance of the routing algorithms with N=64 and L=.0. As shown in Figure??a) the impact of k to the internal blocking ratio is signicant in particular the internal blocking ratio drops dramatically from k= to k=4 for all algorithms. Furthermore the internal blocking ratio remains relatively stable for FF when k 5 where it decreases continuously for LB as k increases. See Figure??b)).The similar results are obtained for all other tested cases with various system loads L) and network sizes N). The Figure?? shows the sensitivity analysis of network size. Figure??a) shows that the network size have signicant impact over the internal blocking ratio for all algorithms. Furtheremore the performance gain of FF over ) improves steadily as N increase as shown in Figure??b). Though the performance improvement of N is not as signicant as the other system parameters such as k. 4. Multicast In order to get more precise comparison on multicast degree we redene the multicast degree as the number of third-stage SEs involved and only one destination outport in each involved third-stage SE for all arrival streams. We set multicast degree M from to 5 and vary the link stream capacity k from to 6. The Figure???? and?? show the sensitivity analysis of link stream capacity k multicast degree M and network size N respectively. Notice that all Y-axes are in log scale. The condence interval is worse when the internal blocking ratio < 0?6. The Figure?? shows 0 0 3 4 5 6 7 8 9 0 3 4 5 6 7 8 9 0 b) Figure : The sensitivity analysis of streams capacity k against the routing algorithms for N=64 and L=.0; a) internal blocking b) performance gain. 0. 0 Sensitivity Analysis of Network Size M= k=3 L=.0) LB FF 00 0 0 40 60 80 00 0 40 9 8 7 6 5 4 3 a) Sensitivity Analysis of Network Size M= k=3 L=.0) /LB /FF 0 0 40 60 80 00 0 40 b) Figure 3: The sensitivity analysis of network size N against the routing algorithms with k=3; a) internal blocking b) performance gain.

0 Sensitivity Analysis of Network Size M=3 k=3 L=.0) 0. 0 00 e-05 e-06 Sensitivity Analysis of Link Stream Capacity N=64 M=3 L=.0) LB FF MB HMB HMBLB HMBFF e-07 0 3 4 5 6 7 8 00000 0000 000 00 0 a) Sensitivity Analysis of Link Stream Capacity N=64 M=3 L=.0) /LB /FF /MB /HMB /HMBLB /HMBFF 0. 0 3 4 5 6 7 8 9 b) Figure 4: The sensitivity analysis of streams capacity k against the routing algorithms for N=64 M=3 and L=.0; a) internal blocking b) performance gain. 0. 0 00 e-05 e-06 Sensitivity Analysis of Multicast Degree N=64 k=3 L=.0) MB HMB HMBLB HMBFF e-07 0 3 4 5 Multicast Degree M 00000 0000 000 00 0 a) Sensitivity Analysis of Multicast Degree N=64 k=3 L=.0) /MB /HMB /HMBLB /HMBFF 0. 0 3 4 5 Multicast Degree M b) Figure 5: The sensitivity analysis of multicast degree M against the routing algorithms for N=64 k=3 and L=.0; a) internal blocking b) performance gain. 00 e-05 e-06 MB HMB HMBLB HMBFF e-07 0 0 40 60 80 00 0 40 00000 0000 000 00 a) Sensitivity Analysis of Network Size M=3 k=3 L=.0) /MB /HMB /HMBLB /HMBFF 0 0 0 40 60 80 00 0 40 b) Figure 6: The sensitivity analysis of network size N against the routing algorithms for M=3 k=3 and L=.0; a) internal blocking b) performance gain. that LB and FF can only have at most 50% of the MB and HMB families' performace improvement when stream link capacity k=. The larger k is the performance of FF and LB are more incomparable to MB and HMB's. Therefore we will not discuss the FF and LB families under multicast routing cases furthere. Figure?? depicts the performance of the routing algorithms with N=64 M=3 and L=.0. The Y-axis of Figure??a) is raw internal blocking ratio and Figure??b)'s is performance gain. All algorithms perform better than the benchmark algorithm under all k. It clearly demonstrated that MB and HMB families are the clear winners and they follow by FF and LB. It also shows that the linear-complexity HMBLB performs even better than simple MB does. The Figure?? also demonstrates that the link stream capacity k) has major impact on the performance of all algorithms. For example the internal blocking ratio is below 0?5 when the link stream capacity is greater than 4 for N=64 M=3 Load=.0 as shown in Figure??. The Figure?? shows the sensitivity analysis of multicast degree M with N=64 k=3 and L=.0. The Figure?? shows the performance of the MB and HMB increases as the multicast degree increases. For example the internal blocking ratio of the HMB decreases from 00 to 0?5 as the multicast degree varies from to 4 as shown in Figure??a). The Figure?? also points out the fact that MB and HMB are both very eective in avoiding internal blocking as M 4. Figure?? shows the impact of the network size N on the performance of various algorithms with N=6 to 44 k=3 and L=.0. The internal blocking ratio

decreases as N increases for both MB and HMB families. We observe that both MB and HMB families are in eective decreasing rates when N increases. Their internal blocking ratio become almost negligible 0?5 for N 00). The HMB performs also as good as the expensive MB family for all network size. 4.3 Summary When the arrival streams are all heavy chunk request link stream capacity is below 5 the simplest heuristic FF is the best method to handle streams with high bandwidth requirement. The linear-time algorithm LB have better performance over FF when the arrival streams are small. That is the LB is the best routing algorithm for high speed network with tiny unicast stream request. Based on above observations the qualitative and quantitative eect of network size and link stream capacity are similar in both unicast and multicast routing. The LB has similar performance improvement over the even when the multicast degree increases. That is the LB can not help in multicast routing. The expansive MB is the best one in our simulations. Though the three linear-complexity HMB family can not beat MB's performance HMB can performs almost as good as MB does for all the cases we simulated. If we can stand the slight performance lost the HMB family is our best routing algorithm for multicast routing. 5 Conclusion This paper has focused on the on-line multicast routing problem where the call requests arrive and depart in random fashion. We study the multicast routing problem in Clos network with optimizing criteria network call acceptance rate. It is usually dicult to nd the optimal solution for real-life online problem such as multicast routing problem. The objective of this problem is to develop ecient algorithms that can nd multicast routing trees to maximize the network throughput. We propose three families of algorithms LB FF and MB. The simulation results show that the LB and the FF are good under unicast problem; the MB group are the best for multicast cases but very expensive. The performance of both MB and HMB are vary close; the HMB family are our best routing algorithm for multicast cases due to the advantage of their linear complexity. None of the proposed algorithms can be the best for both unicast and multicast routing in our simulation. Therefore the LB and HMB can be used together depended on whether the arrival stream is unicast or multicast. It has been recommended to use backpressure mechanism to alleviate the trac congestion at the output of SE[?]. It has been shown that the cell loss ratio signicantly decreases when the backpressure mechanism is deployed in point-to-point communication. It is not clear whether this remains the same in the case of multicast communication. Our routing algorithms can be extended to solve wide-area network routing problem. Multicast applications such as MBone have grown rapidly in recent years. We are currently working on this area. References [] J. Moy. \Multicast Routing Extensions for OSPF." Communication of ACM 378):6-66 994. [] S. E. Deering and D. R. Cheriton. \Multicast Routing In Datagram Internetworks and Extended LANs". IEEE transaction on Computers 8):85-0 990. [3] E. W. Dijkstra. \ A note on two problems in connexion with graphs". Numerische Mathemtik :69-7 959. [4] P. Winter. \Steiner problem in networks: A survey." Networks 7:9-67 987. [5] T. S. Yum and M. S. Chen. \Multicast Source Routing in Packet-Switched networks.\ IEEE Transaction on Communication 4/3/4): -5 994. [6] B. K. Kadaba and J. M. Jae. \ Routing to multiple destinations in computer networks." IEEE Transactions on Communications. Com-33):343:35 983. [7] S.C. Liew. \Multicast Routing in 3-Stage Clos ATM Switching Networks." IEEE Transaction on Communication 4/3/4):380-390 994. [8] V. P. Kompella J. C. Pasquale and G. C. Ployzos. \Multicast Routing for Multimedia Communication." IEEE Transaction on Networking vol. no.3 pp.86-9 993. [9] Ciro A. Noronha Jr. and Fouad A. Tobagi. \Optimum routing of multicast streams." In Proc. of IN- FOCOMM pp. 856-864 994. [0] V. Chvatal \A greedy heuristic for the setcovering problem." Mathematics of Operating Research 43):33-35 979. [] H. Schwetmen CSIM Users' Guide. Microelectronics and Computer Technology Corporation Austin TX USA 990. [] Daniel Dais and J. R. Jump. \Analysis and Simulation of Buered Delta Network." IEEE Transaction on Computers C-30:73-8 April 98.