Virtual Circuit Blocking Probabilities in an ATM Banyan Network with b×b Switching Element

Proceedings of the Applied Telecommunication Symposium(part of Advanced Simulation Technologies Conference)Seattle,Washington,USA,April22–26,2001Virtual Circuit Blocking Probabilities in an ATM Banyan Network with b×b Switching Elements∗Hongyuan Shi and Harish Sethu{hyshi@,sethu@}Department of Electrical and Computer EngineeringDrexel University3141Chestnut Street,Philadelphia,PA19104-2875Abstract—This paper presents the trade-offs that exist between the blocking probability,the size of the switching elements,the offered load and the link capaci-ties in an ATM-based Banyan multistage interconnection network for QoS-sensitive applications.We present a quasi-analytical study which indicates that,as the link capacity between switching elements is increased,the blocking probability reduces exponentially.We show that,in contrast to what is generally believed,larger switching elements are often,but not always the best design choice.Relevant design recommendations that arise from these results are discussed.The conclusions of this paper are also applicable in non-ATM contexts such as in circuit-switched networks with link-dilated topologies,or photonic switching networks using WDM. Keywords—ATM,blocking performance,Banyan networks,non-blocking networks,multistage intercon-nection networks.1INTRODUCTIONAsynchronous Transfer Mode(ATM)is a connection-oriented data transfer technology,in which communica-tion between any two end-points of the network is ac-complished through the establishment of a virtual circuit that allows variable-rate full-duplex transfer of data be-tween them.Each distinct data transfer application uses a separate virtual circuit for the transmission of cells be-tween the communicating entities.For applications such as telephony,achieving good availability of service is a critical component of the overall design objective.For such applications,therefore,a primary goal is to max-∗This work was supported in part by U.S.Air Force Contract F30602-00-2-0501imize the number of virtual circuits that can be estab-lished in the network in a manner that satisfies the perfor-mance and quality-of-service requirements of each con-nection.At the time a request is made for a connection, the network control has to determine if there is a route available with sufficient capacity for an appropriate level of quality of service required for the requested commu-nication.If a connection cannot be established,we say that the request for the virtual circuit is blocked.This paper presents a quasi-analytic study of the dependence of the virtual circuit blocking probability on the sizes of the switching elements,the offered load and the capacity of the physical links in an ATM-based Banyan network. The blocking behavior and the conditions for non-blocking in ATM-based multistage interconnection net-works have been analyzed for Clos,Benes and Cantor topologies[3,5,6].The results of these studies vali-date the widely held belief that the larger the switching element used to construct the network,the smaller the blocking probability.This paper shows that,in Banyan networks,a larger switching element is often,but not al-ways the better design choice as far as blocking probabil-ities are concerned.Relevant design recommendations that arise from these results are discussed.Section2presents the problem statement and derives the minimum link capacity required in an ATM Banyan network to ensure a zero blocking probability.Section3 describes the quasi-analytic simulation technique,first used recently in an analysis of blocking performance of photonic switching networks[1].Section4presents our simulation results with interpretations.Finally,Section5 concludes the paper with a brief discussion of the rel-evance of this work to other contexts in which switch-based multistage interconnection networks are used.150150(a)15(b)0123456701234567(c)Figure 1:16×16Banyan with (a)2×2and (b)4×4switching elements;(c)A subset of the 128×128Banyan network2REQUIREMENT FOR ZEROBLOCKINGGiven b ×b switching elements,one can build an N ×N network using the Banyan topology,if N =b s ,where s is the number of stages in the network.For purposes of illustration,Figures 1(a)and (b)show a 16×16Banyan topology using 2×2and 4×4switching elements,re-spectively.In this paper,we use the baseline network as the representative banyan topology.We define capacity of a link as the number of vir-tual circuits that can be simultaneously supported on the physical link.This capacity is a function of the physical bandwidth that the link can support and the bandwidth requirements of each virtual circuit.Let C be the ca-pacity of the physical links in an ATM network.When a switching element receives a virtual connection setup request over a certain physical link,it has to ensure that no more than C circuits have already been established.If C circuits are already established,the connection re-quest is said to be blocked.We assume that each inlet link or outlet link can support only one connection at any given instant of time,which is the case in many imple-mentations of ATM-based LANs and also in ATM-basednetworks used in videoservers for video-on-demand ap-plications.The following theorem,a special case of the result in [3],specifies the link capacity,C ,required to ensure that the network is non-blocking.Theorem 1Consider an N ×N Banyan network using b ×b switching elements,and physical links of capacity equivalent to C virtual circuits.The network is non-blocking if and only if C ≥K ,where,K = √N,if log b N is even . N b ,if log b N is odd .(1)Proof:A given physical link in the network will never block a connection request,if it is able to support all the virtual circuits that may possibly use this link.Define the weight of a link as the maximum number of connections that may use this link at any given instant.The minimum link capacity required for the non-blocking property is the largest weight of all the links in the network.When log b N is even,it is easily observed that links with the largest weights are those between the middle two stages of the network.When log b N is odd,the links with the largest weight are the ones connected to the middle stage switches.For example,the links shown with thicker lines in Figure 1(c)are two of the links with the largest weights in a 128×128Banyan network.This figure shows only the switches and links in the paths that may use these two links.Since only one connection can be established from an inlet to any outlet,the weight of a link is the smaller of the total number of inlets and the total number of outlets that can be reached from the link.It is obvious from this figure that the weight of the left one of the two links is just the total number of inlets in the network that can be reached from this link.Similarly,the weight of the right one of these two links is the total number of outlets of the network that can be reached from this link.Consider a tree with its root as one of the links with the largest weight in the network.Assume,without loss of generality,that the number of inlets reachable from this link is less than or equal to the number of outlets reach-able from this link.Let the leaves of this tree be the inlets of the network that can be reached from this link.Let all the intermediate nodes in the network be the links on the paths between the inlets of the network and the link with the largest weight.Clearly,this is a tree in which the root and all the intermediate nodes have exactly b children.The partial network in Figure 1(c)provides a pictorial analogy with the tree for b =2and N =128.Now,the minimum capacity required for the non-blocking prop-erty is the number of leaves in this tree.Table 1:Capacity Requirements for a Nonblocking N ×N Network Using b ×b Switching Elementsb N 816326412825651210242244881616324-4-4-16-168---8--8-16-----16--32-------32Not counting the root of the tree,the total number oflevels of the tree is (log b N )/2when log b N is even,and (log b N −1)/2when log b N is odd.When log b N is even,therefore,the number of leaves in the tree is b (log b N )/2=√N .When log b N is odd,the numberof leaves is b (log b N −1)/2= .2Table 1lists the minimum link capacities required forthe non-blocking property for various values of N and b .Note from the table,that a 64×64network with link ca-pacities equivalent to 4virtual circuits,has zero blocking probability with 4×4switching elements,but a non-zero blocking probability with 8×8switching elements.A similar phenomenon of smaller switching elements lead-ing to a lower blocking probability is observed in the 1024×1024network with 4×4switching elements.In the above theorem,we have assumed the same link capacity at each stage of the network.This is because,in order to achieve cost-effectiveness with the least pos-sible number of different parts,most commercial switch-ing systems use identical switching elements and,there-fore,also have the same link capacity at each stage.A few specific design recommendations are easily de-rived based on the entries in Table 1.For example,when the link capacities are equivalent to 4virtual circuits,a 64×64network is best designed using 4×4switching el-ements rather than 8×8switching elements.When the link capacities are smaller,the size of switching elements to use is not quite obvious and can only be determined based on evaluations of blocking probabilities described in the next two sections.In the vast majority of cases,use of larger switching elements is a better design choice since it improves the throughput and thus the capacity of the links.However,as can be observed from Table 1for 64×64and 1024×1024networks,a larger switch-ing element as the building block is not always the better choice.3METHODOLOGYIn this section,we describe the quasi-analytic simula-tion technique,first used in [1],to evaluate the block-ing probabilities.We model connection setup requests by a Poisson process with an arrival rate of λfrom each source.The utilization factor,ρ,of the network is given by (λ/µ)(1−P B ),where µis the service rate,and P B is the blocking probability of the network.In our study,we use a normalized service rate equal to 1,i.e.,µ=1.The offered load,λ/µ,therefore is equal to λ.For an N ×N network,P B =N −1 k =0P B |k P kwhere,P k is the probability that k virtual circuits are established in the network,and P B |k is the conditional probability that a connection is blocked while k virtual circuits are already established.The quantity P k is given by,P k = N k ρk(1−ρ)N −k ,∀0≤k <NRandom permutations of source-destination pairs are generated in random order.Two counters are maintained for each value of k ,0≤k <N .The pass counter ,p k is incremented whenever a (k +1)-th connection is established,and the reject counter ,r k is incremented whenever a (k +1)-th connection request is rejected (i.e.,blocked).The following equation then gives us the con-ditional blocking probability given k connections are al-ready established.P B |k =r k p k −1As described in [1],P B and ρcan now be computed iter-atively starting with an assumed value of ρ.4SIMULATION RESULTSThis section presents simulation results using a 64×64interconnection network in a Banyan topology,using 2×2,4×4and 8×8switching elements.Figure 2(a)plots the blocking probabilities in this network against the nor-malized offered load.This figure is plotted assuming that each physical link between switching elements can support no more than 2virtual circuits.As can be ob-served from this figure,use of larger switching elements,in this case,reduces the blocking probabilities at all of-fered loads.Figure 2(b)plots the blocking probabilitiesFigure2:(a)P B vs.offered load,C=2;(b)P B vs. offered load,C=3.for the same64×64Banyan network,but assuming link capacities of3virtual circuits per link.Thisfigure is dif-ferent from Figure2(a)with respect to the relative per-formances of networks using4×4and8×8switching el-ements.The blocking probability using4×4switches, as indicated in thisfigure,is slightly better than that with 8×8switch chips.This further illustrates the fact that the use of larger size switching elements does not necessar-ily lead to an improvement in the blocking probability. Figure3plots the blocking probabilities for the same 64×64Banyan network,against the link capacity,at an offered load of0.7.Recall that we measure link capacity by the number ofvirtual circuits that can be supported on the link.As can be observed from Figure3,the block-ing probabilities decrease exponentially as the link ca-pacity increases.For example,a64×64network using 2×2switches with link capacities equal to5virtual cir-cuits has a blocking probability about34times as small as with link capacities equal to4virtual circuits.Thisfig-Figure3:P B vs.link capacity,λ/µ=0.7.ure suggests that increasing link capacities,as opposed to using larger switching elements,is almost always a better design choice(except when the switch size is in-creased to b×b and log b N is odd).This is simply be-cause using switch architectures that can be implemented in today’s VLSI chips,high throughputs of95%or more can already be achieved with2×2ATM switching ele-ments[7].A larger switching element therefore can only lead to a small increase in the throughput,resulting in only a small increase in the link capacity.5CONCLUDING REMARKSThis paper shows that an N×N ATM Banyan network using b×b switching elements is non-blocking if and only if each physical link between the switching ele-ments can support K virtual circuits,where K=√N if log b N is even,andN/b otherwise.In block-ing networks,using a recently developed quasi-analytic method,we have quantified the trade-offs between the blocking probability,the size of switching elements,the offered load and the link capacities for a representative 64×64network.These results can be used to develop design recommendations on the bandwidth of the links and the size of switching elements to use in the design of an ATM-based Banyan network.The most interesting aspect of these trade-offs is that the use of larger switch-ing elements in the design of a network does not always guarantee a lower blocking probability.Finally,the work presented in this paper is also rel-evant in some non-ATM contexts.For example,each link with capacity C may be thought of as multiple links of capacity1.Thus,the results of this paper can alsobe applied to circuit-switched networks with link-dilated topologies.In addition,these results are also valid in the analysis of blocking probabilities in optical Banyan-based networks using Wavelength Division Multiplexing (WDM)and switching devices that include wavelength converters.References[1]M.M.Vaez and C.-T.Lea,“Blocking performancewith crosstalk consideration of the photonic switch-ing networks based on electro-optical directional couplers”,Journal of Lightwave Technology,vol.17,no.3,pp.381–387,March1999.[2]C.-T.Lea,“Multi-log2N networks and their ap-plications in high-speed electronic and photonic switching systems”,IEEE Transactions on Commu-nications,vol.38,no.10,pp.1740–1749,October 1990.[3]R.Melen and J.S.Turner,“Nonblocking Networksfor Fast Packet Switching”,Proceedings of IEEE INFOCOM,vol.2,1989.[4]C.P.Kruskal and M.Snir,“The performance ofmultistage interconnection networks for multipro-cessors”,IEEE Transactions on Computers,vol.82, no.12,pp.1091–1098,December1983.[5]S.C.Liew,M.-H.Ng and C.W.Chan,“Block-ing and Nonblocking Multirate Clos Switching Net-works”,IEEE/ACM Transactions of Networking, vol.6,no.3,pp.307–318,June1998.[6]E.Valdimarsson,“Blocking in Multirate Intercon-nection Networks”,IEEE/ACM Transactions on Communications,vol.42,no.2/3/4,pp.2028–2035, February/March/April1994.[7]M.Katevenis, D.Serpanos and E.Spyridakis,“Credit-Flow-Controlled ATM for MP Interconnec-tion:The ATLAS I Single-Chip ATM Switch”,Pro-ceedings of4th Int’l Symp.High Performance Com-puter Architecture,1998.[8]A.Pattavina,Switching Theory,John Wiley&Sons,Inc.,New York,NY,1998.。