This paperanalyzes the blocking probability ofmultistageBanyan networks in a combined space and time domain. Theanalytical solution provides unique tradeoff among variousparameters and the corresponding blocking probability.Specifically, this work presents combinatorial basedapproachesto analyze Banyan in the time domain, which caneasily be extended to the WDM domain. The analytical resultsshow that adding time drastically reduces the blocking of(space optimal) Banyan networks. This is significant ingeneral, and in particular, in the optical domain, where thecost ofeach switching element is high.
1. Introduction
Multistage interconnection networks (MINs) arewidely used in various networking and computingapplications, such as switching equipment, as well as indistributed and parallel systems. In the context of highspeed networks the problem of constructing a low costnon-blocking switching structure is important. Low costmeans the switching structure requires low number ofswitching elements. The classical non-blocking crossbarrequires N-by-N (or NxN) with N2 switching elements orcrosspoints. Non-blocking connectivity means that anyinput can be connected to any output independently fromany other connection. MIN is an alternative switchingstructure decreasing the number of switching elements(SEs) by using multiple stages of small SEs with a properinterconnection between the stages. However, not alltypes of MINs provide a dedicated path to all inputoutput pairs. That means some connections share thesame path , which cause a certain level of space blocking.
I The work described in this paper was carried out with the support ofthe BONE-project ("Building the Future Optical Network in Europe"), aNetwork of Excellence funded by the European Commission throughthe 7th ICT-Framework Program and also by funds from the EuropeanCommission (contract N° 002807 for the IP-FLOW project) .
This is in contrast to the crossbar structure. One type ofthe MIN kind is a Banyan [1][2] . There are differenttypes of Banyans such as Delta, Omega, Flip , Cube,Baseline and Shuffle Exchange [3]. The classical BanyanNxN matrix is based on bxb SEs and has m stages , wheremvlogiN. Each bxb SE is built as a crossbar switch withfull connectivity. Throughout this paper the followingnotation will be used: m for number of stages, b for SEsize, N for number of switch inputs /outputs.
The Banyan switching structure has an optimalcomplexity, which is on the order of O(Nlog~) andmakes it relevant to various network applications; e.g.,all-optical. One of the main properties of Banyan is thatdifferent input-output pairs share interstage links;therefore, blocking can occur when different input-outputconnections require the same interstage linksimultaneously. For instance, Fig. 1 shows a Banyan withb=2, m=2, N=4. Only one of inputs {O, I} can beconnected to one of outputs {2,3} . The secondconnection from SEll to SE22 is conflicting andtherefore it is blocked.
Input Interstage path side Outputside side
0 0
I SEll SE2 1I
2 2SEI2 SE223 3
Figure I. 2-stage 4x4 Banyan, based on crossbar 2x2 SE
A number of studies have been done on various Banyanstructures, e.g., [2][7]. However, little is known about apossible implementation of Banyan and its blockingbehavior in the time domain.
Blocking performance analysis in general has a longhistory in networking. Traditionally, the term callblocking was used in previous works on blocking
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analysis (e.g., [4][5][6]). Call rejection is considered asan event when no more network resources (e.g., circuitsin telephony or radio channels in wireless) can beallocated to successfully establish a newly arrived call.Thus, an analysis of call rejection probability is calledcall blocking probability analysis. When analyzing callblocking probability, traffic patterns and stochasticdistributions are taken into account.
In this work we focus on another method to analyzeblocking using combinatorial approach. The objective isto show how to reduce blocking by adding timedimensions. Specifically, this work focuses on how toadd the time dimension while using a technique calledtime driven switching (TDS) [8][9]. Note that in theoptical domain TDS is also called FAS (fractional lambdaswitching) [8]. The main principles of TDSIFAS will beexplained in Section 2.
Banyan with TDSI FAS is highly suitable for alloptical network solutions minimizing the number of SEswhile reducing blocking, which are important sinceoptical SEs are still very costly. Furthermore, all-opticalSEs have high transfer rate (potentially more than 100Gbit/s) and multiple wavelengths. All-optical switch of asmall dimension was recently implemented [13] in ourlab. In the context of this work we use the term "flow"(of packets) instead of "call", "connection" or "session".The use of flows is convenient when discussing thetransmission of video streams [11]. Generally speaking aflow consists of one or more time frame in respect to timecycle. More details on time frame and time cycle areprovided in Section 2.
This study analyzes a Banyan in the time domain inan analytical way that in turn gives us a mechanism toevaluate the blocking probability without anysimulations. Thus, it becomes possible to find tradeoffsbetween the number of SEs and the blocking probability.The number of SEs is directly connected with the costindex [12]. The study for the general case of m stages isprovided by formal mathematical analysis. We also showthat the uniform model we assume is reasonable bycomparing and validating simulation with analyticalresults. We would like to emphasize that introducing timeand spectral domains dramatically reduce the spaceblocking probability of Banyan networks. Some relatedTDS blocking probability analysis can be found in[15][16]. However, this study focuses on analyzing theinternal blocking of Banyan structure and improving it byintroducing the time domain.The structure of the rest of the paper is as follows.Section 2 discusses the model that has been used and theanalysis methodology. Then Section 3 presents theanalysis for the general case of m stages. In Section 4some comparison of simulation results vs. analytical iscarried out in order to show that the chosen analytical
model is realistic. Finally, Section 5 gives conclusionsand summarizes the paper contributions, while discussingsome future research directions.
2. Modeling and Analysis Methodology
This section presents the model and the correspondinganalysis methodology that is used in this work. We havealready mentioned that TDS is proposed in order todecrease the blocking probability of Banyan networksand to increase the bandwidth provisioning flexibility(Le., sub-lambda provisioning). Since TDS is used, theinputs/outputs/interstage paths are shared in the timedomain by multiple flows. Coordinated universal time(UTC), which is used in TDS, provides phasesynchronization with identical frequencies everywhere.However, traditional TDM (time division multiplexing)systems, such as SONET/SDH, represent anothersynchronous solution, based only on frequencysynchronization with known bounds on clock drifts. Inorder to overcome possible data loss, SONET/SDH usesrather complex overhead information to accommodate:the accumulation of delay uncertainties and continuousclock drifts from the nominal value. Multiplexing inSONET/SDH is based on time slots (TS) organized in aframes. Due to the lack of phase synchronization amongnodes, incoming TS might be stored up to the duration ofa whole frame before being sent out. In order to keep thedelay introduced by each node small, the frame durationis defined as 125 JlS and, consequently, the TS has a verysmall size data of l-byte, The l-byte TS duration in10 Gb/s signal (OC-192/STM-64) is about onenanosecond.
TDS avoids collision by means of scheduling eachflow in a time frame (TF) of predefined duration thatserves as virtual container for multiple IP packets. Thestandard UTC second is used as a common time reference(CTR) and is divided into TFs that are grouped into timecycles (TC). CTR is required in order to ensurecontinuous phase alignment of both TFs and TCs on allswitch inputs. Otherwise, the switch operation is morecomplex (e.g., SONET) and the combinatorial-basedanalysis methodology presented in this work wouldn't bepossible. The duration of the TF may be different on linkswith different capacities [14]. The whole TF is reservedfor each flow. In case a flow lasts more than one TF, thenthe number of necessary TFs is reserved for the flow (1TF per TC). Time-driven switching is based on pipelineforwarding of data units within each TF that areforwarded through the network one hop every predefinedinteger number of TFs. The sequence of TFs on each link
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Banyan blocking structure
LXJ busy TF 0 available TF - schedulable TF IFigure 2.Banyan with N=4, b=2 - with IF schedulable TF in position 4
As can be seen from the immediate forwarding (IFexamples in Fig. 2, no buffers are used between stagesand the TF in which the flow is scheduled has the sameposition index in the TC. In this study we consider onlyimmediate forwarding. The case of adding one buffer inthe interstage path will be reported in a future paper.
During the study of the Banyan with IDS withdifferent scenarios we noticed some interestingdependences. Specifically, the blocking probability doesnot seem to vary with increasing the number of inputsand outputs (i.e., with N). Increasing the number of inputsleads to increasing the SE-size . However, the blockingprobability depends on the number of stages (i.e., m) andis independent from the number of inputs N and the SEsize. In order to verify this, a simulation tool wasdeveloped. The same tool will be used further to validatethe model for comparing simulation and analyticalresults . The simulator is developed in OMNet++ and C++environments. The simulator is a tool for the modelingBanyan based on bxb SE with TDS with C TFs per TC,where band C are user-defined parameters. Every inputhas its own source. All sources are independent. Thegenerated traffic is uniformly distributed from N inputs toN outputs with a given load.
The different configurations such as 2-stage Banyanof various SE-size were simulated. The simulationresults are shown in Fig. 3. The results are given for 2stage Banyan of l6x16, 64x64, 256x256 with b=4, 8, 16SE-size correspondently and with C=I and C=4 TFs perTC. The results report almost the same blockingprobability value for all three configurations. Fig. 4shows how the blocking probability decreases withincreasing number of TFs per TC. The results are shownfor Banyan 16x16, where C varies from 4 up to 128. It iseasy to notice that the blocking probability decreasesquite fast with increasing the number of TFs per TC, andunder low loads it even reaches zero for C=128.
Since the independency of the blocking probabilityfrom number of inputs Nand SE-size b has been shown,we introduce the model that does not take into accountthese two parameters (i.e., Nand b).
i+lth TC234
t-T--+-:-:-t->-i""'l- 0I
-+-+-11-+-+ 2
i+lt h TC2 3
x
x
Internal PathsInputs
-----ith TC i+1 th TC (
4 1 2 3 41o x x
Call(1;0)~-+-~.-..4
2-+--+--+--+--1-13 -l.......J....<.>l..-..L..-r-I
along a path from source to destination can be chosenaccording to two primary pipeline forwarding modes :
Immediate Forwarding (IF): IP packets withinTFs arriving to the switch are switchedimmediately to a predefined output according to apredefined schedule;
- Non-Immediate Forwarding (NI-F): theforwarding of IP packets within TFs can bedelayed up to D TFs later, where D is an integernumber.
When a flow arrives to an input of Banyan switch insome TF, it is to be scheduled in some available TFwithin a TC. It means that the flow has maximum C(where C is the number ofTFs per TC) possibilities to bescheduled. The other limit case is when all C TFs arealready reserved and no TF is available. In this case theflow will be dropped or rejected and considered asblocked. Note that availability of free TF must besimultaneously in the same TC position on input,interstage path and output. Some basic definitions areintroduced below.DeU. A TF is considered free ifno connection or flow isscheduled within it.DeC2. A TF is considered busy if a flow is scheduledwithin it.DeC3. A TF is called schedulable if it is simultaneouslyfree (in the same position within the TC) on input, outputand interstage path.Def4. Scheduling is a procedure of assigning andreserving a schedulable TF within a TC for a givenincoming flow.DeC5. Blocking probability is the ratio between blockedflows and all arrived flows .
For example, Fig. 2 presents a 4x4 Banyan based onSE-size b=2, with 4 TFs per TC. A flow arrives to input 1and requires output O. The corresponding interstage pathhas index 0 (the route is drawn with bold line) . The aim isto find a free TF that is in the same position on all threelines. At input 1 TFs 1,2 and 4 are free . However TFs 1and 2 can not be considered as schedulable, since they arebusy at the interstage path. Thus only TF 4 remains beingalso free at output side. Thus the TF 4 is schedulable.
A schedulable TF is searched within a TC. When theTF is found it is reserved and gets status of busy TF oncorresponding input, output and interstage paths. If morethan one TF is schedulable, different TF assignmentstrategies can be implemented. The most reasonablemethod is selecting a schedulable TF at random from theset of found schedulable TFs . This method provides moreuniform distribution of busy TFs . The random selectionmethod was implemented in our simulations. Thus , thebusy TFs are assumed to be distributed uniformly atinput , interstage paths and output sides .
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Banyan based on different SE-size, m=2 3. Analytical Model
0.9
Banyan: N=16, m=2
Figure 4. Banyan ltix l ti with various numbers ofTFs per TC
In other words our model for 2-stage Banyan is describedby "three lines" that are input line, interstage path andoutput line and the number of TFs per TC. These twoparameters cause the blocking due to the overlapping ofbus and free TFs at different lines/sta es.
All three lines are independent. The B busy TFs aredistributed uniformly on each line (specifically, theprobability of all combinations of selecting B out of C isthe same); then: "find the probability of a flow to beblocked or, in other words, the probability of not findingat least one free vertical within a TC."
Method 1 - 2 StagesSince the busy TFs are distributed uniformly it followsthat there is an unordered set of TFs on each line. Thedistribution of the TFs can vary for each line. Theprobability ofany distribution is the same.
Let NFV denotes the event of No Free Vertical, thatis: there is at least one busy TF at each vertical i, where
I::; i ::;C. Then the required blocking probability is theprobability of the event NFV.
Pblk =P(NFV) (1.1)
This section presents the TDS analysis of 2-stageBanyan with immediate forwarding (IF) in Part A, andthen its extension for the general case of m stages with IFin Part B. The second approach is presented in Part B.Before formulating the problem we introduce somedefinitions. The following notation is used: C - numberofTFs per TC, B - number of Busy TFs per TC.Det:6. A vertical is a column of TFs with the same indexwithin a TC. For example, Fig. 5 shows 9 verticals.Det:7. Afree vertical occurs if all TFs of the vertical arefree (e.g., in Fig. 5: verticals 4 and 9 are free).Det:8. A busy vertical occurs if at least one TF of thevertical is busy (e.g., in Fig. 5: vertical 1, 2, 3, 5, 6, 7, 8are busy).Det:9. A flow is called blocked if there is no free verticalwithin a TC.
Figure 5. Three lines with 9 TFs per TC and 3 busy TFs per TC
c::::::J free TF c::::::J busy TF
Problem formulation for 2-stage Banyan: There arethree lines: input, interstage path, and output. Each line isdivided into C TFs in which the TC is aligned. B TFs outof C are busy in each TC. Band C are the same for eachline. Formula (1.0) defines how the load L is connected tothe maximum number of busy TFs B.
L=!!...C
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o.?
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-.- N 16. e I TF--+- N (,.I.e I TF__ N 256.C I TF
-A- N 16. e HF-?- N (,.I.e HF-0- N 256. e-l TF
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-r-t
H::H-~-:uttE}1~1111 W61
II W2f
... I TF- -0- 4TF -~ 32 TF --- 12ST F
Figure 3. 2-stage Banyan: blocking probability vs. load
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The following section contains the analytical modeland some additional necessary definitions. We map theanalysis of Banyan with N inputs based on SE with SEsize b into the analysis considering number of"lines" (thenumber of "lines" depends on number of stages) with CTFs on each and all possible distributions of busy TFs.Since the busy TFs are distributed uniformly acombinatorial approach can be used. With this approachblocking is calculated by counting all possiblecombinations of distributions of busy TFs on input line,interstage path and output line. It will be shown later thatsuch an analysis matches the simulation results thatvalidates the model and fits the case ofBanyan very well.
Now we focus on the cases of filing each line with busyTFs in order that the event NFV occurs (Fig. 6).
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Figure 6. An example ofa blocking configuration - NFV
There are (~) possible combinations to distribute B
busy TFs on the input line of C TFs . Thus C-B free TFsremain on the interstage path line, lout of C-B places canbe occupied with busy TFs, where I E [O,B]. Hence there
are (C ~B) combinations to choose these ITFs from C-B. Then B-1 busy TFs remain and there are ( B ) ways
B-1
to distribute them. On the third output line C-B-l placesmust be occupied in order to have NFV. There are B-(CB-l) = 2B+l-C busy TFs remaining and they can beplaced on C-(C-B-l) =B+l positions. Hence there are
(B + I ) combinations to place 2B+l-C busy TFs and
2B+l-C
the obvious (C-B-I) combinations to place theC-B-l
remaining C-B-l TFs. The sample space of event NFV
is(~r.Observe all combination must be summed by I,
wherel E [O,B]. Hence the required probability can befound using equation (1.2):
~(~)(C~BXB~/X2B~~~CX~=~ = ~)P(NF) = (~)3 ' (1.2)
with the following conditions:C - B ~ I~ O
2*B+I- C ~O
r---r--r-"""T"".:...lplacesL......J'---L----L1_I I ~
C-B-l places
(2.5)
(2.4)
Observe that the events FV/ are from the same samplespace and are not mutually exclusive. Thus we can applythe inclusion-exclusion principle to find the probabilitythat either of event occurs . The subtraction follows to benecessary in order to avoid multiple counting of the sameevent/vertical. The well-known inclusion-exclusionprinciple is used here.
The event FV/ gives i free verticals. For instance, FV;is the event of having exactly two free verticals . They canbe first and second , second and third, first and thirdverticals and so on. Thus we introduce another event Ai'Let Aidenote the event of ,-tll vertical is free. So A/denotesthat the first vertical is free, A; denotes that the secondvertical is free and so forth . Then Ai Aj denotes the eventthat the ,-tll and r verticals are free etc. Applying theprinciple to our case we get (2.3):C-B c
In (2.4) all summands are equal to the same number FV/and since there are C summands in the sum then:
AI +A2 +...+Ac =~c FV =C*FV; =(C)*FV;L..,:I I 1
In this approach we concentrate on finding probabilityof the event FV. The event FV of finding at least one freevertical takes place when one, two, three and so on up toC-B verticals are free. Let FV/ be defined as an event offinding exactly i free verticals . In that case the event FVcan be presented as the union of events FV/:
FV =FV. U FV2 U ... U FVC _B =U;=~B FV; (2.2)
Free vertical, in which there is at least one free verticalover C possible verticals .
For instance, Fig. 5 reports one FVevent: verticals 4and 9 contain only free TFs . In other words, the event FVis a complementary event to the event NFV. Hence theprobability of occurrence NFV is defined by (2.1):
P(NFV) =1-P(FV) (2.1)
oI I:c::::::::::::
LE;9 I
BInput
InterstagePathOutput
(2.7)
It is important to note when B is less than one third of C,no blocking occurs . In other words given uniformlydistribution of TFs it is always possible to find a freevertical.Method 2 - m Stages
The second approach is presented in this part. Sincethe demonstration rationale is the same as in the previouscase for three lines, it is given immediately for generalcase of m stages . Let FV denotes the event of having a
The number of elements with exactly two free verticals :
A]A2 =A]A) =...=AC_]AC=FV2 (2.6)
In (2.6) all summands are equal to the same number FV;
and since there are (~) summands in the sum then :
4A, +4A3 +···+4--14-=(~)*FV2
And so forth, consequently we obtain the followingresult :
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The probability of event FV is the sum of probabilities ofthe events of having exactly one free vertical, minus thesum of probabilities of having exactly two free verticals,plus the sum of probabilities of the events of havingexactly three free verticals and so on. The formula (2.9)becomes:
(2.14)C-B (CJ (CJWFV; = *FVI- *FV2 +1=1 1 2
...+ (_I)C-B-t(C=B}FVC_B
The probability ofoccurrence FV is:
P(FV) = p(U;=~BFV;)
C-B
P(FV) = P(LFV;) = P(FV.) - P(FV2 ) +;=1
+P(F~) +... + (_I)C-B-l P(FVC_B )
(2.8)
(2.9)
(2.10)
(C;T+'P(FV;)= (~rl
Thus modifying (2.12) and considering (2.13-2.14), forgeneral case of Banyan NxN with SE-size b and numberof stages m= logiN given C TFs per TC, where B out ofthem are busy, then the blocking probability is defined byformula (2.15):
(2.15)
Then the probability of occurrence NFV defined from(2.1) becomes:
The sum is bounded by maximum number of freeverticals that is C-B. Observe that the probability of anevent P(FV;) is defined:
The number of event outcomes is given by (2.8). The
sample space is given again With(~)3 . Combining (2.11),
Formula (2.16) is for the general Banyan case withany number of stages. A formal proof by induction offormula (2.16) will be provided in the full paper. Notethat both 1.2 and 2.17 give the same blocking probabilityfor the case of 2-stage Banyan. It can be shownmathematically that (1.2) equals to (2.16) with m=2.However, the demonstration is not trivial due to thedifferent methodologies that were used. Thus, we usednumerical validation by using the Matlab tool in order toverify that both (1.2) and (2.16) provide exactly the sameblocking probability.
4. Analytical vs. Simulation Results
(2.11)Numberof event outcomesP(F~)=--------
Totalnumberof outcomesin samplespace
P(FVj ) is the probability of having exactly one freevertical. It means the busy TFs are to be chosen from C1, where 1 is the number of free verticals. In case of threelines nominator and denominator are cubed, thatcorresponds to 2-stage Banyan case, where m=2.
(C
B
- I]m+l(2.13)
P(FV\)= (~r+\
P(FV;) is the probability of having exactly i free verticals.It means that the busy TFs are to be chosen from C-i,where i is the number of free verticals.
This section compares simulation results withanalytical results. In Section 2 we presented the TDSblocking probability and its dependency on the number ofTFs per TC and the number of stages. The same tool usedin Section 2 is used to obtain the following results. Somedetails on model have been already given in Section 2.Now we give additional details regarding the Banyanmodel used in this simulation model: each input has itsown source, all sources are independent and loaded withthe same load L; each point-to-point flow is characterizedby a pair of input/output and a flow duration; each flow isscheduled in a schedulable TF that is searched within aTC each time a flow arrives, then a TF is assigned for theflow for the needed number of TCs; each flow that cannot be served is considered blocked and is dropped; each
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Figure 7. Blocking probability vs. load. Banyan with C=128, 256
Figure 8. Comparison of2-stage vs. 3-stagefor Banyan N=64
0.9
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1'''-:;
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Load
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A nalytical Result sBanyan: m ~2,3 ,4, C~256TF
0.3
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Banyan: m=2, average call duration=260 TC
<++++l ;.cr -" ·9-+=+=t=+
_ Analytical, C 256
···0· · Simulation, C 256......- Analytical, C 128~ Simulation, C 128
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Q.=tl 0.4
=~ 0.30;
Banyan: m=2, C= 128TF, average call duration=260 TC
0.9 - Analytical: m 2···0·· Simulation: m 2
0.8 --y- Analytical: m 3-fil- Simulation: m 3
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flow's route is static and is defined by structure of theBanyan, that is interstage paths for a given input-outputpair; each schedulable TF assignment is based on arandom algorithm; the number of TFs per TC is C and isfixed for every simulation.The flow is accepted by the system only if the routingpath is available. In case of pure Banyan thecorrespondent routing path is checked, meanwhile in thecase of Banyan with TDS the routing path availabilitiesare checked for all TFs within a TC. As was described inSection 2, the aim is to find a free TF in the same positionalong the routing path. Once the flow is scheduled and ,-tl1
TF is assigned for it for t TCs then simulator keepstracking of those TFs already busy until the flow iscompleted.
First, we simulated 2-stage Banyan: 64x64 with b=8and with 128 TFs and 256 TFs per TC. Fig. 7 reports theblocking probability versus the load for the mentionedstructures. The graphs present analytical againstsimulation results. The obtained simulation results matchvery well the analytical ones. The tendency of decreasingof the blocking probability while increasing the numberof TFs per TC is observed. This matching also validatesthe uniform distribution of busy TFs within the TC.
In general, in order to simulate a big structure, forinstance 256x256, a lot of time and computationalresources are needed. Due to this fact the analyticalapproach is preferable. Especially it is reasonable for bigstructures. We simulated Banyan 64x64 with 3 stagesbased on SE-size b=4. Fig. 8 reports the results for theblocking probability versus load, and simulations arecompared with analytical results for 2 stages and 3 stages.The simulation results again match the analytical ones.Fig. 9 shows some analytical results for the blockingprobability versus the load for Banyan in the time domainwith different number of stages. It is worth noting to notethat with the same number of TFs per TC the blockingprobability increases with the increasing number ofstages. Meanwhile for the same number of stages theblocking probability decreases with increasing thenumber ofTFs per TC.
There is a trade off between the number of stages,that depends on the size of the switch with a given SEsize and the number ofTFs per TC. For instance, Banyanof size 64x64, can be realized as 2-stage structure on thebasis of SE-size b=8 or as 3-stage one on the basis of SEsize b=4. Knowing the number of TFs needed to providea low blocking for each stage and taking intoconsideration the cost index of each structure, an optimalswitch design can be selected.
Load
Figure 9, Analytical result: blocking probability vs, load
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5. Discussion
In this paper the blocking probability performance ofBanyan with TDS was analyzed. It was shown that TDSapplied to Banyan significantly reduce its blockingprobability, and consequently, may be viewed as analternative to various space-based approaches to decreasethe blocking probability. In contrast to space solutions,introducing TDS does not increase the index cost [8].TDS is required in order to ensure phase alignment ofboth TFs and TCs on all inputs, and in tum to enable thepresented analysis. The analysis was conducted by meansof two different combinatorial-based approaches. Thefirst approach can be extended for the case of nonimmediate forwarding that is based on adding somelimited buffers into the interstage path. The secondapproach analyzes the general case of m-stage Banyan,which may be possible to extend to the multi-hop case infuture studies. Then both combinatorial analysisapproaches can be generalized for different WDMconfigurations.The series of simulation were carried out in order toverify the uniform assumption and modeling of thecombinatorial-based analysis. We would like to note thatthe analytical model gives the mechanism to calculate theblocking probability and find tradeoff between the cost ofthe switch and the number of TFs per TC. We alsowould like to emphasize that no further simulations arerequired as the blocking probability can be calculateddirectly using the analytical results.
We outline extension of the analysis for the case ofnon-immediate forwarding as a future work that we aregoing to present in the next paper. This case will requireadding switching elements, and therefore, it will bereasonable to compare it with other space approaches,such as, vertical replications and adding extra stages.
References
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