QNA-MC: A Pefomance Evaluation Tool fo Communication Netwoks with Multicast Data Steams Gaby Schneide 1, Mako Schuba 1, Boudewijn R. Havekot 1 Infomatik 4 (Communication and Distibuted Systems), RWTH Aachen, D-5056 Aachen, Gemany mako@i4.infomatik.wth-aachen.de Laboatoy fo Distibuted Systems, RWTH Aachen, D-5056 Aachen, Gemany havekot@infomatik.wth-aachen.de Abstact. In this pape we pesent QNA-MC (Queueing Netwok Analyze suppoting MultiCast), a new pefomance evaluation tool fo the analytical evaluation of multicast potocols. QNA-MC is based on the QNA tool which (appoximately) analyses open netwoks consisting of GI G m nodes. We extend this method by allowing a moe geneal input in fom of multicast outes. These outes ae then conveted to seve as input fo standad QNA. Fom the esults deliveed by QNA ou tool deives seveal pefomance measues fo multicast steams in the netwok. We validate ou appoach by compaison to simulation esults. Moeove, we give an application example by evaluating diffeent multicast outing algoithms in the Euopean MBONE using QNA-MC. 1 Intoduction Point-to-multipoint (o multicast) communication has become an essential pat of today's communication potocols, because a numbe of applications equie the same data to be sent to a goup of eceives. Such applications ae most efficiently ealised via multicast. Moeove, the wold-wide deployment and use of the IP multicast potocols (in the MBONE [1, ]) has diven the development of new applications, which would not have been possible without a multicast infastuctue (e.g. distibuted simulation with seveal thousands of computes involved). Thus, pefomance investigations of existing and new multicast potocols become moe and moe impotant. Most pefomance investigations of multicast potocols so fa ae based on measuement (e.g. [3, 4]), potocol simulation (e.g. [5, 6]) o Monte Calo simulation on andom gaphs (e.g [7, 8]). Analytical evaluations ae athe seldom because thee is still a lack of appopiate analytical models fo multicast. Existing analyses make a lot of simplifying assumptions, e.g. they use static delays and do not conside queueing delays in intemediate nodes [9, 10]. Moeove, thei esults ae usually esticted to mean values. Measues like delay vaiation, which ae impotant fo instance in ealtime multicast communication, cannot be deived fom these models. Queuing netwoks (QN), which have poved to be vey useful fo the pefomance evaluation of communication systems, have not yet been applied to multicast potocols. The eason fo this lies in the fact that existing queueing netwoks assume communication between diffeent nodes to be point-to-point, i.e. copying of data as
equied fo multicast data steams is not suppoted. Moeove, well-known QN appoaches such as Jackson netwoks [11] have a numbe of limitations, e.g. the estiction to exponentially distibuted inteaival o sevice times, and thus may not model eal system behaviou vey well. One solution to ovecome the disadvantage of exponential distibutions is the Queueing Netwok Analyze (QNA), which is based on the ideas of Kühn [1] and was extended and tested extensively by Whitt [13, 14]. QNA allows inteaival and sevice times to be geneally distibuted (chaacteized by the fist two moments) and yields an appoximate analysis of the queueing netwok. Recently, QNA has been extended in a numbe of ways, e.g. by Heijenk et al. [15], El Zaki and Shoff [16] and Havekot [17]. This pape descibes QNA-MC, an enhancement of QNA to suppot the analysis of multicast data steams. An input consisting of one o moe multicast outes is conveted into the standad input equied by Whitt's QNA in ode to compute geneal pefomance measues. The output of QNA is finally conveted back to measues fo individual multicast steams, such as the maximum esponse time and its vaiation. The focus of this pape is to pesent the theoetical backgound of QNA-MC as well as its pactical application. Afte a bief summay of standad QNA in Section we will descibe in Section 3 the convesion fom multicast outes to the standad input and fom the QNA output to multicast oute measues. In Section 4 we will pesent some implementation aspects of QNA-MC and compae the appoximate esults deliveed by QNA-MC to those of simulations. In Section 5 we will show in a pactical application how QNA-MC can be used to compae diffeent multicast outing algoithms. Finally, Section 6 will give some concluding emaks. Standad QNA Below we vey concisely summaize the QNA appoach. Details of QNA not diectly elated to ou study ae omitted (see [13]). We biefly discuss the QNA model specification, the employed taffic equations and the pefomance measues computed, focussing only on the single-class case hee..1 Model specification With QNA it is assumed that thee ae n nodes o queueing stations, with m i seves at node i, i {1,..., n}. Sevice times at node i ae chaacteized by the fist moment E[S i ] = 1/µ i and the squaed coefficient of vaiation c si. Fom the extenal envionment, denoted as node 0, jobs aive at node i as a enewal pocess with ate λ 0i and squaed coefficient of vaiation c 0i. Intenally, customes ae outed accoding to Makovian outing pobabilities q ij ; customes leave the QN fom node i with pobability q i0. Finally, customes may be combined o ceated at a node. The numbe of customes leaving node i fo evey custome enteing node i is denoted by the multiplication facto γ i. If γ i < 1, node i is a combination node; if γ i > 1, node i is a ceation node. Finally, if γ i = 1, node i is a nomal node. Ceated customes ae all outed along the same path, o ae pobabilistically distibuted ove multiple outgoing outes. Note that this diffes fundamentally fom multicast outing whee also multiple customes ae ceated, but exactly one is outed to each of the successo nodes in the multicast tee (cf. Section 3).
. Taffic equations Once the model paametes have been set, the analysis of the taffic though the QN poceeds in two main steps: Elimination of immediate feedback. Fo nodes with q ii > 0 it is advantageous fo the appoximate pocedue that follows to conside the possible successive visits of a single custome to node i as one longe visit. This tansfomation is taken cae of in this step. Calculation of the intenal flow paametes. In this step, the aival steams to all the nodes ae chaacteized by thei fist two moments. This poceeds in two steps: Fist-ode taffic equations. Similaly as in Jackson QNs, taffic steam balance equations ae used to calculate the oveall aival ate λ i fo node i. Next the node utilizations ρ i = λ i /m i µ i can be computed. They all should be smalle than 1, othewise the QN does not show stable behaviou. If this is the case, the second step can be taken. Second-ode taffic equations. QNA appoximately deives the squaed coefficients of vaiation of the custome flows aiving at the nodes ( c ai ). The squaed coefficient of the custome steam depating node i ( c di ) is computed with Maschall's fomula [18] using the Käme and Langenbach-Belz appoximation fo the expected waiting time at GI G 1 nodes..3 Pefomance measues Once the fist and second moment of the sevice and inteaival time distibutions have been calculated, the single nodes can be analysed in isolation. An impotant measue fo the congestion in a node is the aveage waiting time. To deive this value in geneal GI G 1 queues, only appoximate esults ae available, most notably, the Käme and Langenbach-Belz appoximation [19]. It delives an exact esult wheneve we deal with M G 1 and M M 1 queues. Fom the waiting time, othe congestion measues such as the mean esponse time, and (using Little's law) the aveage queue length and the aveage node population can be deived. Using a seies of appoximations, also the squaed coefficient of vaiation fo the waiting time at a node is computed (see Whitt [13]: (50)-(53)). Fom the pe-node pefomance measues, QNA calculates netwok-wide (end-toend) pefomance measues by appopiately adding the pe-node pefomance measues. Pefomance measues fo multi-seve nodes ae only computed appoximately, using the evaluation of the M M m queue, with aveage inteaival and sevice time as in the coesponding G G 1 queue, as a stating point (see Whitt [13]: (70)-(71)). 3 Integation of Multicast Routes QNA as defined in Section does only suppot the analysis of point-to-point connections, eithe given by the outing pobabilities, i.e non-deteministic, o by deteministic outes. In ode to allow customes o packets to be duplicated at cetain nodes QNA-MC genealizes the oute input of QNA to suppot multicast outes. The convesion of multicast outes to the standad input of QNA is quite simple, if the oute convesions given in [13] ae taken as an example.
3.1 Input Convesion In QNA-MC the netwok is descibed fist: n : numbe of nodes in the netwok, m i : numbe of seves at node i, i {1,..., n}, and : numbe of outes. Each multicast oute k {1,..., } is descibed in the fom of a tee with input: N k : set of nodes in oute k, fist k N k : fist node in oute k, λ outek : extenal aival ate fo oute k, and c outek : squaed coefficient of vaiation of extenal aival pocess fo oute k. Fo each node l N k of a multicast oute k the following additional infomation is equied (using the typical ecod notation l. ): l.numbe : node numbe, l.µ : node sevice ate, l. c s : squaed coefficient of vaiation of the sevice-time distibution of the node, l.γ : ceation/combination facto of the node (default value 1), l.λ : modified extenal aival ate of the node, i.e. the ceation/combination factos of pevious nodes ae taken into account (automatically calculated by QNA-MC), l.succ_no : numbe of successos of the node, l.succ i : ith successo of the node. Note that thee is a diffeence between a oute node l N k and a netwok node l.numbe {1,..., n}. The same netwok node may occu seveal times in a oute, i.e. as seveal diffeent oute nodes with possibly diffeent values l.µ, l.γ, etc. This input is conveted by QNA-MC to obtain the standad input fo QNA. Fist, the extenal aival ate λ 0i pe node i is deived as the sum of all oute aival ates: λ = λ 1 fist. numbe = i, 0i oute k k k = 1 { } whee the function 1{condition} denotes the indicato function, i.e. 1{A} = 1, if A = TRUE, and 1{A} = 0, if A = FALSE. The oveall flow ate λ ij between two nodes i and j is the sum of the flow ates in each oute going fom i to j. It can be witten as λij = l. λ l. γ 1 l. numbe = i and s { 1,..., l. succ _ no} : l. succs. numbe = j. { } Similaly, the depatue ate λ i0 fom node i, i.e. the flow fom the node out of the netwok, is given by the sum of the depatue ates of all oute nodes with numbe i and no successo (leaf nodes): λi0 = l. λ l. γ 1 { l. numbe = i and l. succ _ no = 0}. Based on these ates the Makovian outing pobabilities q ij, i.e. the popotion of the customes that go fom i to j to those leaving i, can be calculated by nomalizing the flow ate λ ij to the oveall flow ate out of node i: (1) () (3)
q (4) ij = λij λi0 + l. λ l. γ 1 { l. numbe = i and l. succ _ no 0}. Next the aveage ceation/combination facto γ i of node i can be computed. It depends on the aival ates and ceation/combination factos of the individual multicast outes and is given as a weighted aveage of the individual node factos l.γ (maked gey): Similaly, the aveage sevice-time µ i pe custome at node i and the coesponding squaed coefficient of vaiation can be deived as weighted aveages as follows: and Note that (7) uses the fact that the second moment of a mixtue of distibutions, i.e. c ( si +1) µ i, equals the mixtue of the second moments of the mixed distibutions. Finally, the squaed coefficient of vaiation c 0i fo all extenal aival pocesses is detemined (see Whitt [13]: (10) - (1)). If the extenal aival ate λ 0i = 0, then c 0i is set to 1 and thus has no futhe impact on the QNA calculations. If λ 0i 0 then c c0i = ( 1 wi)+ wi k = 1 whee the value fo w i is defined as γ i = 1 ES [ i] = = µ 1 ES [ i ] = ( csi + 1)= µ i i c si l. λ l. γ 1{ l. numbe = i} l. λ 1{ l. numbe = i} and ρ i = λ i /(µ i m i ). Afte calculation of all these values QNA-MC stats the standad QNA method as descibed in Section, which delives seveal netwok and node measues. 3. Output Convesion In addition to the esults deliveed by standad QNA, QNA-MC computes measues fo individual multicast outes and fo diffeent paths p within these outes, i.e. measues fo individual eceives of the steam. Note that we define a path to be the unam-. (5) 1 l. λ 1{ l. numbe = i} l. µ, (6) l. λ 1{ l. numbe = i} 1 l. λ ( l. cs + 1) 1{ l. numbe = i} l. µ. l. λ 1{ l. numbe = i} oute k λoute k λ0i w i = 1+ 41 ρ ( i ) ( v i 1) 1 with v i = k =1 1 { fistk. numbe = i}, λ outek λ 0 i 1{ fist k.numbe = i}. 1 (7) (8) (9)
biguous node sequence fom the sende to a single eceive in a multicast oute. Let P k denote the set of paths of multicast oute k. Fom the standad QNA output we know the mean and vaiance of the waiting time of an abitay custome in netwok node i (denoted E[W i ] and V[W i ], esp.). Thus, the esponse time E[R p ] of the eceive at the end of path p P k is the sum of the expected sevice times and expected waiting times of all nodes l in the path p: ER [ p] = 1 + EW [ l. numbe] (10) l p l. µ and the vaiance is calculated in a simila way as VR [ p] = 1 lc. s + V[ Wl. numbe ]. (11) l. µ Of paticula inteest fo many analyses (see e.g. the application of QNA-MC in Section 5) ae the maximum esponse times and vaiances within a multicast oute. Hence, these values ae computed as well. Finally, if we define the weight ω p as the popotion of the customes leaving the netwok at the end of path p to those depating fom the oveall oute then (simila to equations (5) - (7)) the mean esponse time of the oveall oute k, i.e. fo a geneal custome of the multicast steam, can be witten as ER [ P k ] = ω per [ p] (13) and the coesponding vaiance can be calculated fom the weighted second moments of the individual paths minus the squaed expectation value deliveed by (13): VR [ ] = ω ( VR [ ] + ER [ ] ) ER [ ]. (14) 4 Tool Suppot: QNA-MC ω p = l p l p p P k l p l. λ 1{ l. succ _ no = 0} l. λ 1{ l. succ _ no = 0} p P k P k p p p P k p P k We implemented QNA-MC in C++ on a Solais platfom based on the C code of QNAUT (QNA, Univesity of Twente). Afte ewiting most of the code in C++ we added the input and output convesions descibed in the last Section. We tested QNA-MC fo a numbe of diffeent paamete sets and netwok scenaios by compaing the QNA-MC esults to the esults of equivalent simulations. Hee, we pesent only those esults elevant fo ou example application in Section 5. Howeve, esults fo othe paamete combinations had the same quality (see [0]). The simple test scenaio is shown in Fig. 1. The investigated netwok consists of fou nodes. The only eal multicast oute (black) entes the netwok at node 1, passes on to node and then goes on to node 3 and 4, whee the customes leave the netwok. The othe outes (gey) model back- (1)
gound taffic and ae used to incease the oveall load ρ in the nodes. Customes of these outes leave the netwok diectly afte having passed thei entance node. 1 3 4 Multicast Route Backgound Taffic Fig. 1. Test scenaio fo QNA-MC validation The following paamete combinations have been selected accoding to ou application example (cf. Section 5), i.e. constant inteaival times fo the multicast data steam, constant sevice times at the nodes and backgound taffic with high vaiance. The espective paamete values ae given in Table 1. Table 1. Model paametes Load Multicast Route Backgound Taffic a Node Sevice b ρ λ oute_mc c oute_mc λ oute_bg c oute_bg µ c s 0.5 1 0 4 {, 3} 0 0 0.5 0 8 {, 3} 0 0 0.7 3 0 11 {, 3} 0 0 0.9 4 0 14 {, 3} 0 0 a. The same values ae used fo all backgound taffic outes in the netwok. b. The node paametes ae the same fo all nodes in all outes. Fig. compaes the esults of QNA-MC to those of simulation. Since the 95% confidence intevals of the simulations wee too small to be distinguishable fom the mean values (less than 0.0) they ae not explicitly shown. Because of the symmetic multicast tee the pefomance measues fo individual eceive esponse times (see (10) and (11)) and oveall multicast oute esponse time (see (13) and (14)) ae the same. 5 8 mean esponse time (seconds) QNA ccoute_bg = 3 oute_bg 4 Sim ccoute_bg = 3 oute_bg 3 QNA ccoute_bg = oute_bg Sim ccoute_bg oute_bg = 1 0 0. 0.4 0.6 0.8 1 ρ (oveall load pe node) esponse time vaiance 7 QNA coute_bg oute_bg = 3 6 Sim coute_bg oute_bg = 3 5 QNA coute_bg oute_bg = 4 3 Sim coute_bg = oute_bg 1 0 0. 0.4 0.6 0.8 1 ρ (oveall load pe node) Fig.. QNA-MC and simulation esults fo the test scenaio
The left diagam of Fig. shows the load dependent mean esponse times fo both multicast eceives. Fo small load the esults of QNA-MC ae vey close to the simulation esults. When the load is inceased the absolute diffeence becomes slightly lage but is still within an (fom ou point of view) acceptable ange. Looking at the esponse time vaiance calculated by QNA-MC (Fig. ight) the values ae even bette. It should be mentioned that a futhe incease of the coefficient of vaiation leads to a moe significant oveestimation of the esponse time by QNA-MC. If a high coefficient of vaiation is used then the esults of QNA-MC have to be intepeted moe caefully because of the pessimistic appoximations deliveed by QNA-MC. Oveall (also if all othe validation esults ae taken into account, see [0]) QNA- MC has shown to be an appopiate tool fo the calculation of appoximate measues in geneal QNs with multicast data steams. Because of its shot un time QNA-MC can be applied to multicast QN models fo which simulation times ae no longe acceptable (see e.g. the example in Section 5). The un times of QNA on a Sun Ulta fo all examples in this pape wee less than a second. In contast to this even the simple model shown in Fig. 1 equied a simulation time between 15 and 30 minutes (30.000 seconds simulated time, simulation tool OPNET, Sun Ulta), i.e. QNA-MC yields a speed up facto of 1000 to 000 even fo simple netwoks. 5 Example Application: Multicast Routing Algoithms We will now show in an example how QNA-MC can be applied to analyse multicast potocols. Paticulaly, we descibe the application of QNA-MC to evaluate multicast outing algoithms in the Euopean Intenet. 5.1 Intoduction The most popula multicast potocol used in wide aea netwoks is IP multicast. Its outing algoithm calculates multicast outes based on shotest path tees between the souce and all membes of the multicast goup o as coe-based tees (e.g. in the Potocol Independent Multicast Achitectue [1]), whee all sendes shae a delivey tee ooted at a special node called coe. To ealize such a point-to-multipoint outing algoithm in the Intenet, whee most outes do not suppot multicast outing, an ovelay netwok topology between multicast-capable outes (called MBONE, see Fig. 3 left) is equied. Diect outing of multicast packets between multicast outes is achieved by inteconnecting these outes by so-called IP tunnels, i.e. point-to-point connections tanspoting encapsulated multicast packets. To compae shotest path multicast outing with coe-based tee outing we fist model a netwok, paticulaly a pat of the Euopean MBONE (MBONE-EU). 5. Modelling the Topology of the Basic MBONE-EU Stating fom the ovelay netwok MBONE-EU (Fig. 3 left) we efine this topology by adding all the nodes usually involved in a multicast tansmission (Fig. 3 ight). Note, that the topology efinements ae just made by ule of thumb and the netwok might look diffeent in eality. We add non-multicast outes (white nodes) fo tunnels in the backbone and local aea, local aea multicast outes (small black nodes), membes of the multicast goup (white squaes) and two diffeent coes (black squaes) used only
fo coe-based tee outing. The distinction between all the diffeent oute types is meaningful, because backbone outes usually have a highe pefomance than local outes and handling of a multicast packet usually takes moe time than handling of a unicast packet. Finally, we have to conside the constant popagation delay between the backbone multicast outes. We make the simplifying assumption that it is popotional to the distance as the cow flies and model this by GI/D/100 nodes (gey). Ou investigations showed that the queuing delay in nodes with 100 seves is negligible. The popagation delay in the local aea, i.e. between a eceive, sende, o coe and its neaest backbone oute is neglected. MBONE Euope (1996) Stockholm Basic MBONE-EU Model Stockholm Pais Pais Amstedam Coe Coe1 Amstedam CERN Stuttgat Bologna CERN Stuttgat Bologna Backbone Multicast Route Local Multicast Route Backbone Route Local Route Goup Membe (may act as sende) Popagation Delay Node Fig. 3. Euopean MBONE: ovelay netwok topology (left) and basic model (ight) 5.3 Paameteization of the Basic MBONE-EU In the next step we define the node sevice ates. Both, fo backgound and multicast packets, the sevice ates µ of all outes (pps = packets pe second) ae assumed to be constant (i.e. c s = 0) as given in Table. Table. Node sevice ates Node Type Backgound Sevice Rate Multicast Sevice Rate Backbone Multicast Route 00,000 pps 100,000 pps Local Multicast Route 10,000 pps 000 pps Backbone Route 00,000 pps 190,000 a pps Local Route 10,000 pps 9000 a pps Souce / Receive - (not equied) 500 pps Coe - (not equied) 10,000 pps a. Encapsulated packets have a lage packet length and thus a longe sevice time.
Simila to Fig. 1 we model backgound taffic in outes as independent coss data steams. Accoding to aguments given in [] the coefficients of vaiance of the backgound steams ae chosen to be lage than 1 (namely c bg = {3, 5, 7}), i.e. they epesent taffic with high bustiness. The oveall netwok load ρ is modified by popotionally inceasing o deceasing the backgound taffic aival ate at all nodes. 5.4 Definition of Multicast Routes Finally, we have to define the extenal aival pocess fo the multicast data steam unde investigation. We assume the sende to be the souce of an audio steam (e.g. a PCM encoded steam of the MBONE tool vat [3]), which poduces packets with ate λ oute_mc = 50 pps and constant inteaival time, i.e. c oute_mc = 0. We estict ou investigations to the thee multicast tees shown in Fig. 4. Stockholm SOURCE Stockholm Stockholm Pais Amstedam Pais Amstedam Pais Amstedam Stuttgat Stuttgat Stuttgat CERN CERN CERN Bologna Bologna Bologna Shotest Path Coe1 Based Coe Based Fig. 4. A shotest path and two coe-based multicast tees fo the same goup (black aows indicate the path fom the souce to the coe) The maximum esponse times fo an individual path in the tee (hee always fom the souce nea Stockholm to a eceive nea Bologna at the ight bottom) and the mean esponse times fo the oveall tee (calculated accoding to (10) and (13) esp.) ae shown in Fig. 5 fo c bg = 3. 150 150 maximum path mean esponse time (msec) 100 50 Shotest Coe1 Coe oveall tee mean esponse time (msec) 100 50 Shotest Coe1 Coe 0 0 0. 0.4 0.6 0.8 1 ρ (oveall load pe node) 0 0 0. 0.4 0.6 0.8 1 ρ (oveall load pe node) Fig. 5. Response times fo the maximum path (left) and oveall tee (ight) fo c bg = 3. In both diagams the values ae vey simila independent of the outing tee used. Even
the longe paths via coe numbe do not influence the oveall delay vey stongly. Futhemoe, if we take 150 ms as the maximum acceptable delay fo audio tansmissions (see e.g. [4]) then this delay bound is eached by the mean esponse time only if the load is highe than 95%. Let us now take a close look at the influence of the backgound taffic bustiness on the esponse time measues (shown fo the shotest path tee only) in Fig. 6. In ode to obtain taffic with highe bustiness we set the coefficients of vaiation c bg to 3, 5 and 7. Although the absolute esponse time and vaiance values ae oveestimated by QNA-MC fo coefficients of vaiation lage than 3 (cf. Section 4) we think that the esults shown in Fig. 6 accuately epesent the geneal tendency of esponse time and vaiance fo inceasing bustiness. 150 1000 maximum path mean esponse time (msec) cbg=3c bg = 3 cbg=5c 100 bg = 5 cbg=7c = 7 50 maximum path esponse time vaiance cbg=3 bg = 3 800 cbg=5 bg = 5 600 cbg=7 = 7 400 00 0 0 0. 0.4 0.6 0.8 1 ρ (oveall load pe node) 0 0 0. 0.4 0.6 0.8 1 ρ (oveall load pe node) Fig. 6. Maximum path esponse time and vaiance fo busty taffic The influence of the backgound taffic bustiness on both mean esponse time and esponse time vaiance is athe stong. The highe the bustiness the ealie the citical delay bound of 150 ms fo audio samples is eached. Because the vaiance values incease athe fast with the oveall load the esponse time of individual packets will often be lage than 150 ms even fo less load. Fo an audio data steam this leads to the dopping of audio samples and thus to quality loss. 6 Conclusions In this pape we have pesented QNA-MC, which can be used fo the pefomance evaluation of lage communication netwoks with multicast data steams. We have descibed the theoetical backgound and have shown by compaison to simulations that QNA-MC delives accuate esults at low cost. Moeove, we have shown in an example based on the Euopean MBONE, how QNA-MC can be used to compae diffeent multicast outing algoithms. Wheeas simila simulations cannot be pefomed in easonable time, QNA-MC calculates the pefomance measues of inteest immediately. Moeove, these measues include esponse time vaiances. Thus it is possible to use e.g. Chebyshev s inequality to detemine the pobability that packets aive within a cetain delay bound, what is vey impotant all sots of eal time multicast applications. Oveall QNA-MC shows itself to be a pomising altenative to existing analysis and simulation methods of multicast potocols.
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