An Approximate Analytical Performance Model for Multistage Interconnection Networks with Backpressure Blocking Mechanism

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1 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH 47 An Aroximate Analytical Performance Model for Mltistage Interconnection Networs with Bacressre Blocing Mechanism John Garofalais eartment of Comter Engineering and Informatics, University of Patras and, Research Academic Comter Technology Institte Patras, Greece Eleftherios Stergio eartment of Information Technology and Telecommnications, ATEI of Eirs, Greece Astract Mltistage Interconnection Networs (MINs are sed to interconnect different rocessing modles in varios arallel systems or on high andwidth networs. In this aer an integrated erformance methodology is resented. A new aroximate erformance model for self-roting MINs consisting of symmetrical switches which are sject to a acressre locing mechanism is analyzed. Based on this, the steady-state distrition of the qee tilization is estimated and then all imortant erformance metrics are calclated. Moreover, a general evalation factor which hels in choosing a etter erformance MIN in comarison with other similar MIN architectre secifications is defined. The model was exemlified for the case of symmetrical single- and dole-ffered MINs. It rovides accrate reslts and converges very qicly. The otained reslts were validated y extensive simlations and were comared to existing related wor in the literatre. Index Terms mltistage interconnection networs, Banyan networs, locing, erformance analysis, switching networs I. INTROUCTION Mltistage interconnection networs (MINs are sed as an efficient interconnection medim for mltirocessors, interconnection rocessors, and memory modles. The ehavior of the interconnection networs lays an imortant role in the erformance of mltirocessors. Therefore, to ensre an otimal design, it is necessary to analyze varios configrations and constraints of the interconnection networs. A trade-off has to e made etween a MIN s comlexity and the erformance redction cased y conflicts that might occr when two or more tass occr simltaneosly. However, MINs remain a satisfactory commnication medim for arallel systems, in general. MINs are also a significant comonent of high seed networs sch as Asynchronos Transfer Mode (ATM networs. Manscrit received Jly 9, 9; revised Novemer 4, 9; acceted Ferary 5, In the indstry there are several commercial roters which are ased on the mltistage interconnection networing faric (e.g. the new CRS- Cisco Roter [9]. The erformance evalation of a MIN is of crcial imortance. Ths, a lot of research has een devoted to the stdy of how these networs erform nder varios conditions throgh analytical or simlation methods [, 4 6, 8, 4, 5]. etailed reslts can e fond for secific cases of MINs which rely mainly on aroximation methods for examle [, 8, 4]. The nmerical simlation model is ased on an analysis of the discrete time ehavior of the system. In this case, a formla was derived from analysis of the formla that was extracted y considering the steady state of the MIN. The steady state descries the MIN sitation in which the roaility of staying in a articlar state will not change. In contrast, the classic simlation model determines the system state of each state at each time slot. For instance, it determines how many acets are in a secific qee. If oth simlation and mathematical modeling are feasile, then the otimm techniqe deends on the ind of investigation erformed. The mathematical modeling method is a etter choice when a lot of tests are reqired. While a nmerical model is timeconsming to create, it can then e sed to generate reslts qicly. In this aer the focs is on a new analytical method which involves qeing theory, and moreover, is sed as a simlator for reslts validation. So far, several MIN architectres have een roosed in the literatre and a lot of wor has een devoted to the stdy and evalation of the MIN s erformance. The following showcases some of the revios wor which has een taen into consideration in the search for a new aroach. Most of the MIN analysis focses on niform traffic (i.e. acages coming to a networ with an eqal roaility of reaching ott [6,, ]. On the other hand, there are nmeros non-niform traffic atterns in real alications that reqire secial treatment. Sch non-niform aroximations can e seen in [3, ]. ACAEMY PUBLISHER doi:.434/jcm

2 48 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH The initial aroach in stdying MINs mainly considered the case where acets are lost when they try to enter the next stage. Boras et al. [6] rovided nearly tight er onds on the mean delays of the second stage and eyond (in the case of infinite ffers and validated their reslts y simlations. Their analysis indicated that after the second stage there is no notale difference etween the delay times, giving a artial ositive answer to the conjectre and exerimental reslts of [3]. Garofalais et al. [5] analyzed anyan networs with finite ffers and came with the exact soltion of the steady-state distrition of the first stage. They also aroximated the soltion for the sseqent stages and resented the exact soltion for all stages of MINs with single-ffered switches. This roved the well nown formla of [5]. Aroximations for the erformance of acet switched MINs ased on niform traffic can also e fond in [4]. Simlations of detailed contention-ased networ models (sed for redicting arallel erformance are still qite challenging, t, relative to one-rocessor arallel time in [4], decent seeds have een achieved. Mathematical aroaches [5, 8 ] were also sed as a gide in constrcting this new analytical model. In Ttsch and Hommel [, ], a system of eqations was set for erformance estimation. ring the set-, some rles emerged for ilding sch a system. These rles were created for atomatic generation of systems of eqations in Ttsch [, ], which coed with the mlticast erformance analysis of MINs consisting of switching elements larger than. Moreover, we have resented a soltion for single ffer size MINs in [8], while in this wor a general soltion for MINs with finite ffers is given. In the literatre there are a lot of lications that are ased on different traffic distrition assmtions. For examle, Lin and Kleinroc [5] roosed a model for secific hot sot atterns and niform traffic. Also, Raja et al. [5] sed a simlation aroach dealing with two tyes of traffic: traditional Poisson and self-similar traffic. Koelman et al. [] condcted an analysis ased on offered traffic that follows geometrical distrited message lengths on finite int ffered anyan networs. In 6, another new soltion sing simlation was resented y Vasiliadis et al. [7]. Parallel rocessing is an efficient form of information rocessing which emhasizes the exloitation of concrrent events in the comting rocess. To achieve arallel rocessing, it is necessary to develo more caale and cost-effective systems. Recently, new MIN designs have een introdced; for examle an irreglar class of falt tolerant MIN named a New For Tree (NFT Networ was resented in []. Also, in [9] a new class of irreglar falt tolerant MIN called Imroved For Tree (IFT was introdced. Besides this, single-chi arallel rocessing reqires high andwidth etween rocessors and on-chi memory modles. In [4] a hyrid Mess-of-Tree (MoT ffered networ that comines the MoT networ with the area efficient tterfly networ was introdced. Finally, in [3] the athors roosed a secific mltistage architectre that ses PC-ased roters as switching elements. This enales them to ild a high-seed, large-size, scalale, and reliale software roter. All the aovementioned mltistage systems reqire secial treatment in calclating erformance evalation isses. Frthermore, secial soltions have een develoed for very concrete rolems. One sch rolem exemlified y [] related to a new method develoed for evalating the residal roadcast reliaility of falt-tolerant MINs. One weaness of existing analytical methods is that they are strictly for very concrete MIN strctres and therefore are difficlt to adat to MIN architectre alterations. A new analytical method mst e develoed to evalate the erformance of similar MIN architectre s. With the aove isses in mind, an attemt was made to create an accrate and reliale calclation method. Frthermore, it has to e easily adated and alied with small changes in some MINs modifying constrction schemas. In this aer, a novel analytical model of a synchronos MIN with finite ffers is resented where this faric is imlemented to wor with a acressre locing mechanism. An iterative method is roosed for solving the recrrence relationshi that defines the eqilirim state roailities. Varios erformance measres are derived from the soltion and accrate reslts are resented. Or research contrites in the following ways:. The roosed analytical method rovides more accrate reslts than simlation exeriments which reqire a more time-consming rocess [8, 7, ].. In addition, or analytical method converges in a smaller nmer of iterations than revios ones (e.g. []; less than 6 iterations is enogh to ensre accrate reslts. 3. The roosed erformance analysis of MINs is rost and flexile. As sch, this analysis incldes all metrics sfficiently and accrately given varios networ sizes and ffer length configrations. This has the effect of maing their stdy more detailed and efficient. 4. A comined erformance factor for a mlti-criteria evalation of MINs is defined. 5. This methodology is going to act as the asis for the calclation evalating the erformance of MINs in secial modern MIN constrction alterations. Using this methodology, the more comlicated sject of networs as MINs with riorities or MINs which sort mlticast traffic, or even cominations of them, can e etter nderstood. The easy adatation of this analytical aroach constittes its sovereign advantage, articlarly comared with Marovian analytical methods that can face more limited readth in modern comlicated MIN erformance evalation isses. ACAEMY PUBLISHER

3 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH 49 The remainder of this aer is organized as follows: In Section, all the reqired definitions and lemmas of or analysis are given as well as the first level MIN s analytical aroximation scheme. In Section 3, the aroximate analytical formlae for evalating the erformance of MINs are resented (MINs are exemlified throgh single- and dole-ffered switching elements. Section 4 rovides some of the nmerical reslts generated y or analytical aroximation model. These were in trn comared with the reslts otained y the simlation exeriments. In section 5 the comined erformance factor is defined and in section 6 the methodology s exandaility is discssed. Finally, in Section 7, or conclsions and anticiated ftre wor are resented. II. PROPOSE APPROXIMATE ANALYTICAL MOEL A. MIN analysis In general, an N N MIN is constrcted from L = log N stages of Switching Elements (SEs, where is the degree of the SEs. Let deict an aritrary nmer of stages, where can e escalated from to L. Generally, each SE consists of -int and -ott orts. In the faric, there are exactly (N / SEs at each stage, so the total nmer of SEs of a MIN is (( N / log N (Fig.. There are ( N log N interconnections among all stages, nlie the crossar networ, which reqires O( N SEs and lins. There is a niqe ath from each rocessor (sorce node to each memory modle (sin node, and therefore the stdied MIN elongs to the class of Banyan Networs (BNs. A -int, -ott switch can receive acets at each of its -int orts and send them throgh each of its -ott orts (Fig.. In each ott ort there is a ffer. We assme that the ffers may e of finite or zero length (single- or dole-ffered switches. Sch a networ can e modeled as a laeled digrah where nodes are of the following three tyes: sorce nodes (indegree, otdegree, sin nodes (indegree, otdegree, or switches (ositive indegree and otdegree. In this laeled digrah each edge reresents one or more lines going from a node to its sccessor. The whole networ oerates synchronosly, which means that the time cycles refer to gloal cloc tics. The networ cloc cycle consists of two hases. In the first hase, flow control information asses throgh the networ from the last stage to the first stage. In the second hase, acets flow from one stage to the next in accordance with the flow control information. The roting algorithm alied here, assmes that there is a fixed ath which has to e followed y a acet throghot the networ. The ath can e encoded as a seqence of laels of the sccessive switch otts of the ath (ath descritor. More concretely, the SEs in mltistage networs are digit-controlled crossars. This is done y inclding a control seqence in the acet, named a acet control seqence. The control seqence is a series of digits allocated for each stage of the networ. The digit indicates which ott of the SE is to e connected to the int. Therefore, the control seqence reresents the ath to e taen y the message throgh the MIN. Pacets are generated at each rocessor y indeendent, identically distrited random rocesses. In this analysis it is assmed that each rocessor generates a acet with roaility ( at each cycle and sends this with eqal roaility to any memory modle (niform access. The switches have a FIFO (First Int First Ott olicy for their servers (otts. Conflicts etween acets simltaneosly roted to the same ott ort are resolved y qeing the acet. Or analysis assmes that acets moving from stage i to stage ( i and finding the ott ffer of stage ( i fll will loc the server (ott of their origin s ott (stage i. The aove, of corse, does not aly to the rocessor s feeding stage i = (i.e. ( remains the same in every cycle or to the ffers of the last stage i = L, which are not sosed to ecome loced nder any condition. Blocing of an ott is interreted as stoing its oeration, that is, it cannot accet any acets for service (it cannot forward acets to the next stage. The service time of the ott qees of each switch is assmed to e constant and eqal to the networ cycle time. The niform access assmtion allows s to reresent any switch as a system of qees woring in arallel, each with a deterministic server (of service time eqal to. Any acet which enters any of the ints of the switch goes with roaility to any of the (ott qees of the switch. I n t s (N-/ (N- FigreAn... st Stage st Stage N N.. (L- Stage. L Stage (N-/ (N- single ffered MIN (elta tye with L stages constrcted y SEs with = In or analysis we assme that the ffer length ( does not inclde the server (ott. So, a single-ffered switch is assmed with =. We assme that arrivals haen at the end of each cycle (ths first the qee is O t t s ACAEMY PUBLISHER

4 5 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH served and then new acets arrive, if there are any. The roting logic at each switch is assmed to e fair, that is, conflicts are randomly resolved. In addition, it is worthy of reminder that or analysis is ased on homogeneity, and ths all switches in a stage and, liewise, all otts within a switch are statistically identical. B. Basic definitions Let s consider stages i and i (for i = to i = L. A articlar ott qee OQ of stage i will finally e ale to send a acet (when it has one only when it selects a qee of stage i which is not fll. efinition Let e the steady-state roaility that a articlar ott server of stage i of the switch networ is sy. An ott server is sy either ecase it is serving a acet or ecase it is loced. This is the tilization in steady state of an ott ffer of stage i of the switch networ. An aritrary qee of a MIN with ffer size ( has a nmer ( of ossile tilized states. The roaility of those distinct qee states is exressed as: j. The j exress the qee tilization y (j acet olation (j =,,...,. Also, ( j= i i the qee tilization is given: ( =. efinition Let e the steady-state roaility that a articlar ott server of stage i of the switch networ is loced. Oviosly, (L. serv = efinition 3 Let e the steady-state roaility that a articlar ott server of stage i of the switch networ is serving a acet. efinition 4 Let e the steady-state roaility that a articlar ott ffer of stage i of the switch networ is emty. Oviosly, =. C efinition 5 Let e the random variale denoting the nmer of acets arriving at an ott ffer of a switch of stage i ( i =.. L, of the networ at the ( end of a cycle and x i (, c = Pr( C i = c. Any qee in the system can e tilized y normal or loced acets. Normal acets are the acets that have jst arrived in the qee and are ready for service the next time, whereas loced acets are the acets that have already tried to e serviced t have een loced for any reason and therefore remain in the qee. Ths, the tilization in a qee can e exressed as: = ( serv C.. Performance metrics Throght of a MIN, T h, is defined as the nmer of acets delivered to their destinations er nit of time. j Nevertheless, ecase the qees of the last stage are never loced, the tilization of the last stage qees is eqal to the MIN s throght. So, = Th. Normalized throght of a MIN, T hn, is defined as a ratio of the average throght Th to the networ size N. Formally, T hn is exressed y T Th N hn =. Average latency is the average time a acet sends assing throgh the MIN. Formally, is exressed y n ( t d i = lim = ( t n( t where n(t denotes the total nmer of acets acceted within t time slots and d reresents the total delay for the i th acet. Recall from Section that the acets are roted y store and forward roting from one stage to the next y the acressre mechanism. The d is considered to e the sm of qi and ti, where qi denotes the total qeing delay for the i th acet waiting at each stage and t denotes the total transmission delay of the i th i acet at each stage of the MIN. Conseqently, taing into accont that all qees are single-ffered, the average delay of a acet traversing the networ can e calclated y, whereas the average roaility of a acet eing acceted n a qee (L of the first stage is eqal to the tilization of the last stage L, ecase no acets are lost within the networ and all acets are removed from their destinations immediately after arrival. Ths Eqation ( can e simlified to L i= =. L i= Normalized latency N of acets traversing a MIN with L stages can e defined as the ratio of the average latency of acets to the minimm delay reqired y a acet to traverse the MIN withot any locing. This minimm delay deends on the nmer of stages that have a MIN. So, the normalized latency N, can e exressed y the formla N = (3 L where is the average latency of acets traversing the MIN.. First level aroximation scheme The nmer of cycles needed for the ott qee OQ to sccessflly send a acet after j trials can e ACAEMY PUBLISHER

5 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH 5 ( i ( i ( i aroximated y Pr( OQ = ; that is, OQ oerates with a geometric service time ( rocess of sccess roaility i. ( i j ( ( In the sirit of the aroximation for solving the stages eyond the first in Boras et al. in [5], let s assme that stage i ehaves as rocessors with acet generation ( i roaility =. The ondary conditions ( are = for the rocessors (stage feeding the first (L (L stage of the MIN, and = for the destination of the acets eyond the MIN (i.e. the last stage is never loced. Becase of the aove assmtions, we now have: Remar A. A articlar ott qee of stage i (for i = to i = L can e aroximated y a discrete qee of size, of geometric service time, with exit roaility (, and of l arrivals, where the nmer of arrivals in any cycle is a Bernolli of trials ( and sccess roaility ( i /. Let s call sch a qee a Be/G//. For the last stage L, the qees are Be///, that is, the service time is assmed to e constant and eqal to the networ cycle time (with vale, since the last stage is never loced. Notice, however, that in the general case, in order to get the arameters of the arrival rocess and service time of a qee at stage i, one has to now the soltions of stages ( i and ( i. Ths, or aroximation scheme is now a convergence rocedre where the following two hases are reeated ntil the qee tilizations do not change any more. The scheme is initialized y letting all qee tilizations e eqal to. Iterative algorithm PHASE A (acward soltion of the MIN. Starting from (L the last stage L, solve for to get the arameter of the geometric service rocess of stage ( L. Reeat ntil stage is reached. PHASE B (forward soltion of the MIN. Starting from the first stage, with int arameter and geometric service as fond from hase A, find its tilization. Use this as int arameter for stage, and so on, ntil the last stage is reached. In Sections 4, 5 and in Aendix A, we resent this mathematical convergence method for the single- and dole-ffered MINs with variale networ sizes. E. Additional efinitions For the general case of an L-stage MIN, consisting of switches, with ott ffers of length ( < in all stages, we have: efinition 6 The arrival rocess of acets at the ott qees of the first stage of the networ is given y a inomial distrition in(, /, where is the fixed roaility of a acet eing generated y a rocessor in each cycle. Therefore: c c (, for c x = (4, c c,, otherwise efinition 7 The arrival rocess of acets at the ott qees of stage i (for i = to i = L of the networ, is aroximated y a inomial distrition ( i (i in(,, where is the tilization of an aritrary qee of stage i, which we assme to lay the role of the fixed roaility of acets which are generated y rocessors at each cycle, feeding stage i. Therefore: ( i c ( i c i x, for c and i L (5 (, c c,, forallothervales of c efinition 8 The state of an aritrary ott qee of stage i at the end of cycle n is a two-dimensional variale, with ossile vales: {(,, (,, (,, (,, (,, (,,, (,}, where in (x,y x is the nmer of acets in the ott ffer, and y can tae two vales: when the ott qee is not loced, or when it is. ( n efinition 9 Let ( q, s e the random variale denoting the state of an aritrary ott qee of stage i at the end of cycle n, where q is the nmer of acets in the ott ffer and s is if the ott qee is not loced, or when it is. Let ( q, s e the steady-state ( n limit of ( q, s. efinition Let ( n v e the nmer of acets that are entering an aritrary ott qee of stage i at the end of cycle n, and let v e the steady-state limit of ( n v ( n v C. It holds that at each cycle n. efinition Let = Pr[( q, s = ( q, s], q, s =, q, s when the qee is not loced and when it is loced ( n e the distrition of ( q, s in the steady state. So, =,, = is the tilization of an aritrary qee of stage i. H. Lemmas Lemma For < m min(, and for all stages i ( n of the networ: For q, s from ( q, s it holds that: Pr( v x, m m = x, m x, m ( n = L x, if q Δ( q Δ( s m< if q Δ( q Δ( s m= otherwise where: Δ ( q Δ ( s is the deartre of a acet from an aritrary ott qee of stage i at the end of cycle n if there is a acet and if the ott server is not loced. (6 ACAEMY PUBLISHER

6 5 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH ( n For m =, Pr( v = = x, for any q., ( ( Oviosly, for m > min(,, Pr( v i n. = m The roof of this lemma is similar to the roof of the related lemma in Boras et al. [5]. In smmary, it states that an ott ffer of stage will accet as many acets as there are vacancies in the ffer. The remaining acets will loc their origin s ott qees. Lemma Relating locing roailities with tilization. In a MIN with locing, for all stages excet the last one, the roaility of locing in stage i (where i = (L is eqal to the difference in the roailities of tilization in stage i and the tilization in the last stage (L. = (7 Proof In every qee excet the qees of the last stage we have from Eqation (: = serv (8 We se the following oerational argment: Let S e the total service time sent in stage y all acets traversed throgh the MIN; i =,..., L; that is i S ( = N T, where NT is the total nmer of acets generated dring T which were not lost on entering stage, since service time =. e to homogeneity, for a = N T qee of stage, the total service time is x = M, where M is the nmer of int orts to the MIN. x NT Ths, serv = = T M T for i =,, L. x So, = serv =, where T is the time dring which the qee is loced. For the last x x stage, = = = serv, where i =,,..., L. T T So, ecase all the entering acets in the MIN are not lost: ( ( serv = serv =... = serv (9 Bt ecase in the last stage we do not have locing, serv =, and = ; ths = ( III. THE APPROXIMATE SOLUTION FOR MIN WITH SES A. The general aroximate soltion for -ffered MIN with x SEs In order to demonstrate the aroximation scheme and the nderlying analytical assmtions and techniqes, we start y alying the scheme to the general case, with MINs consisting of switches, with finite ffer size (. Let state (s reresents the state of an aritrary qee of MIN when its acet olation is eqal to s (where s=... The total nmer of ossile states for each qee is (. These distinct states of an aritrary i-stage are denoted y the roailities: and s. - is the qee roaility of eing emty, and - s is the tilization of qee when it holds ( s nmer of acets (where s =... Conseqently, the aggregate tilization of an i-stage qee is given y: s= s s =. The aggregate roaility of all states is: ( x, ( x, ( i ( i x, x, ( x, ] ( x, x, x, ( x, ] x, ( x, ],3 ( x, 3 x, ] x, ( x, ] 3 x, ( x, ] 4 ( ( ( 3 x, ] ( x, ( x, ] ( x, ( x, ] x ] ( x x ( x ] x ( ( x x = s= According to lemma, ( eqations can e alied; one for each distinct state, roviding the following system of eqations: ( (, (,,, (,, ( In fact, the aove system ( of ( eqations is a linear and homogenos system. Comining the first ( eqations of the system ( with the eqation ( forms a new linear system (t not homogenos of ( eqations with ( nnowns. This general system of ( eqations (for ffer size= has the following linear strctre: a a a a a3 a3 a a3 a33 a a34 3 a a a(... ( i.e. = i ( i a x, a = ( x, 3 4,... All coefficients are exressions of and x a ij (3 where: ( - i is the locing roaility of the sccessive, j stage and - x is the roaility of acet arrivals at the crrent, j stage, where j={,,} denotes the roaility of that j acets arrive from the revios stage. Ths, all the coefficients are exressions, a, = x,, etc. Frthermore: - According to definition 8, the acet arrivals x can a ij, j e exressed as a fnction of the tilization of the ( i recedent stage x = f (, j j ACAEMY PUBLISHER

7 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH 53 - According to lemma, the locing roaility of the sccessive stage (i is also a fnction of the tilization according to the lemma. Ths, all the coefficients are a tilization fnction: a ij = f (, ( i, (L a ij That means all factors inclde exclsive tilization metrics. The aove system of eqations (3 can e solved y alying Cramer s theorem, as follows: =, and similarly, all other state roailities can e estimated y: s s =, for s=... (4 Where:, and are Cramer s matrices. s The aggregate qee tilization can e calclated y: = = ` (5 The formla (5 is in fact a recrsive formla ecase oth matrices,, and, inclde only tilization metrics. In articlar, the tilization of the revios, crrent, sccessive and last stage qees are inclded. ( = f ( Ths,, and = f (, ( i, and then the aggregate tilization of an i-stage qee is: = f (, i,,,,,, ( i., This is the reason why an iterative algorithm is sed for aroaching the soltion of the general recrrence relationshi (5. The convergence of this recrsive algorithm will define the eqilirim state tilization s roailities. Ths, alying this convergent algorithm (which is demonstrated in the following section, 4., a convergence at a fixed oint is reqired. In order to evalate the roailities aove, we mae the assmtion of aroximate interstage indeendence (which seems to e more accrate, as is getting smaller. Actally, Krsal and Snir in [3] derive the same eqation for, i =,,.., L, as we do, for the single ffered MIN withot locing (a case clearly with interstage indeendence, giving evidence that or assmtion is aroximately tre for small, when acets are lost. In or case of locing, there is of corse a stronger deendence among stages, which is taen into accont in some extent y adoting Remar A. Comarison to simlation reslts later here, show that this assmtion is a reasonale one. Bondary conditions: The reqirements for the first and last stage are as follows: For the first stage, i = : Since there is no receding stage, the roaility of acet arrivals to the ints, is the offered load to the networ ints. So, ( For the last stage, i = L : ( = A acet at an ott ort of the last stage can always roceed. However, ffers in the SEs of the last stage can not roceed in the loced state. Ths: L = = = ( serv The convergence algorithm: Using a fixed-oint iteration (ε< 4 over the state tilization, a steady state is reached from which erformance metrics of interest are determined. The convergent algorithm is resented in Aendix A. The illstrated algorithm in Aendix A incldes the formlas for the tilization. Evalation of the locing roailities can e calclated similarly. Β. Case Stdies: Single and dole ffered MINs Β. Aroximate soltion for single ffered MIN with x SEs The aove demonstrated aroximate convergent method is exemlified here for MINs, consisting of single ffered ( = SEs. The steady-state distrition in this case consists of two distinct states:, the roaility of the qee eing emty and =, is the roaility that the ott qee tilizes a acet. When we have a small ffer size, the aove general eqation (5 has a soltion which is exressed y a closed formla, as shown here. Using the analysis derived from the aforementioned s-section 4. (which is ased on lemma, it can e aroximated y the following: x, ( x, (6 By solving the aove set of eqations (6, we get: x, = (7 x, Using eqation (3 of definition (8, we arrive at: ( i x, ( (tilization s exression and alying lemma for ( i stage, we arrive at: = (tilization s exression For stage, the aove eqations (7, can e relaced y: ( i ( ( i, (8 ( i ( i ( L ( ( ACAEMY PUBLISHER

8 54 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH The formla (8 (also nown as the tilization s exression is the recrsive formla for an single ffered case of MIN. For stages i = and i = L, the same formla (8, sing the relevant ondary conditions, is sed. Bondary Conditions: remains the same as is sggested in s-section 4., aove. The convergence algorithm: is the same algorithm as is demonstrated aove in s-section 4.. However, the crrent formla (8 is sed to demonstrate this secial case stdy, instead of the general formla (5. Β. Aroximate soltion for dole ffered MIN with x SEs Also the aove demonstrated aroximate convergent method is exemlified here for MINs, consisting of dole ffered ( = SEs. The steady-state distrition in this case consists of two distinct states:, the roaility of the qee eing emty, Bondary conditions: remains the same as is sggested in s-section 4., aove. The convergence algorithm: is the same algorithm as is demonstrated in Aendix A. However, the crrent formla ( is sed to demonstrate this secial case stdy, instead of the general formla (5. IV. APPLYING THE ARITHMETIC CONVERGENT METHO AN SIMULATION This arithmetic convergent method is resented and exemlified throgh its alication on a MIN consisting of single- or dole-ffered ( = or SEs nder varios networ sizes. A nmer of different exeriments were erformed. In each exeriment, less than 6 iterations were sed to achieve a convergence (. The roaility ( of acets arriving at the ε < 4 ints of the MIN raned from. to. A simlator was also constrcted and alied nder the same MIN conditions in order to validate the reslts given y the analytical method. All erformance metrics, is the roaility that the ott qee tilizes a 5 acet and otained from the simlation ran for cloc cycles. The nmer of simlation rns was adjsted to ensre a,is the roaility that the ott qee tilizes two steady-state oeration condition for the MIN. acets. Ths, = All erformance metrics sch as tilization, latency, Using the analysis derived from the aforementioned service, and locing roailities were recorded for each s-section 4. (which is ased on lemma, it can e qee in all stages. It is clear that all statistics otained y aroximated y the following: simlation exeriments verified the nmerical reslts of or novel aroximate mathematical soltion. Also, all = the data that are resented in s-sections 5. and 5.3 are ( x, ( x, (9 ( ( i ( i ( i ( ] ( = evalated sing the Relative Statistical Error (RSE x, x, x, x, = indicator. Δ y RSE =. y The definition of RSE is: % Setting on the system: A = x, ( x,, The difference etween the measred or inferred vale Β = ( x of a erformance qantity y and its actal vale y is, By solving the aove set of eqations (9, we get: B ( x, B and ( i = A B = B = x, A B = ( x x B B, ( A B, B given y ( i = = ( x, ( A B x, B B n, ε / ( x, x, A, B and i are tilization s Where:,, exressions. Ths, ( ( x =, ( ( x, (, (, (,],,( (, ( i x x x x x x And finally, = ( The formla ( (also a tilization s exression is the recrsive formla for the dole ffered case of MIN. Δy = y y = ε s / n where t n, / y is the erformance metric nder stdy (e.g. normalized throght, average acet latency, ε gives the exected error (here, ε = 3, s exresses the variance of a finite nmer of vales, and t gives the qantile of the t -distrition with n degrees of freedom. The RSE gives the asolte statistical error. The estimated RSE is closely related to the confidence level. Or simlations were erformed sing an accracy of 3 % and RSEs in all cases of or exeriments were less than 4% ( RSE < 4. A. Model verification Comare normalized throght The roosed novel analytical model was also validated y for older classic models: Jenq s model [7], ACAEMY PUBLISHER

9 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH 55 Mn s model [6], Theimer s model [], and Yoon s model. i Comare the reslts of a single-ffered MIN. The roosed novel analytical model was also validated y three older classic models: Jenq s model [7], Mn s model [6], and Theimer s model []. Figre deicts the normalized throght of a sixstage single-ffered MIN (64 64 verss the offered load. It is worth noting that all models are accrate at low loads, t their accracy decreases as the acet arrivals at ints increase. According to Figre, the accracy of Jenq s model is less sfficient nder moderate and high traffic conditions ( >.4 ecase many acets are loced, mainly at the first stages of the MIN, esecially at high traffic rates. Mn s model imroves the accracy y introdcing a loced state. Moreover, Theimer et al. introdce the deendencies etween the two ffers for each switching element, imroving their model and aroaching the simlation exeriments etter than Mn s model. Normalized Throght.5.45 single-ffered (64x64 MIN.4.35 Jenq's model Mn's model.3 Theimer's model.5 Or model Or simlation ( - Proaility of arrivals on ints Figre. Normalized throghts verss roaility of acet arrivals for a six-stage MIN Finally, or novel analytical method achieves etter aroaches than all revios models (Fig. sing a very fast convergence (less than 6 iterations. ii Comare the reslts of a dole-ffered MIN. The roosed analytical model was also validated y two older classic models: Mn s model [6] and Yoon and colleages model [5]. Figre 3 deicts the normalized throght of a six-stage dole-ffered MIN (64 64 verss the offered load. It is worth noting that all models are accrate at low loads, t their accracy decreases as the acet arrivals at ints increase. Normalized Throght dole-ffered (64x64 MIN Proaility of arrivals on ints Yoon's Model Mn's Model Simlation Or Model Figre 3. Normalized throghts verss roaility of arrivals at ints in a six-stage dole-ffered MIN The lots clearly verify that or model is more accrate than the other two models. The Yoon model is the worst case since it does not consider the loced state and the rest of its assmtions are simle. Also, Mn s model gives less accrate reslts owing to the roailistic comlexity of the model. Both models give a throght overestimation in final stages. That overestimation haens ecase in the later stages oth models calclated vales of locing roailities nderestimate their real vales. Actally, with high traffic, many acets can e loced even from the first stage. In conclsion, comarisons with other existing models revealed that the roosed model is consideraly more accrate, irresective of the networ size, ffer size, or offered load. Finally, according to Figres and 3 the normalized throght of a six-stage MIN is close to 4% for singleand 56% for dole-ffered MIN configrations resectively, nder fll offered load conditions. Conseqently, the extra ffer availaility leads in trn to far fewer locings, and ths the throght gain was fond to e very significant (4%. Comare average acet latency of single- and dole-ffered MIN. Figre 4 deicts the average acet latency of a sixstage MIN (64 64 verss the offered load. The solid crves illstrate reslts for the single-ffered case while the dotted crves deict reslts for the corresonding case of a dole-ffered MIN. It is worth noting that all models are accrate at low loads, t their accracy decreases as the acet arrivals at ints increase. According to this figre, the reslts otained y or analytical model were in close agreement with those of or corresonding simlation exeriments for oth configration set-s ( =,, again demonstrating the accracy of or roosed analytical method. Average acet Latency Single- and dole- ffered 6-stages MIN =-Yoon's model =-Mn's model =-Or model =-Or simlation =-Yoon's model =-Mn's model =-Or model =-Or simlation (~ Proaility of arrivals at ints Figre 4. Average acet latency verss roaility of acet arrivals for single- and dole-ffered six-stage MINs It is also noticed that sing dole-ffered qees leads to more delays. This ehavior ecomes ercetile even at low loads ( =.4, while the delay increment ecomes aarent at medim and high loads (.6. B. Performance of single-ffered MINs Normalized throght for single-ffered MINs Figre 5 reresents the normalized throght of a single-ffered i-stage MIN, where i = 3, 4, 6, 8,, verss the roaility of acet arrivals. In the diagram, ACAEMY PUBLISHER

10 56 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH crves Nm-L = i and Sim-L = i deict the normalized throght of a single-ffered i-stage MIN estimated y the analytical model and y a simlation resectively. From Figre 5 it is ovios that the normalized throght deteriorates as the networ size increases. Normalized Throght (-Proaility of arrivals on ints Sim-L=3 Nm- L=3 Sim-L=4 Nm- L=4 Sim-L=6 Nm- L=6 Sim-L=8 Nm- L=8 Sim-L= Nm- L= Figre 5 Normalized throght of an i-stage MIN verss roaility of arrivals according to the analytical model and the simlation It is worth noting that the nmerical reslts of oth methods have een fond to e in close agreement (differences were less than %. Blocing roailities for single-ffered MINs Figre 6 illstrates the locing roailities (P er stage verss the roaility of acet arrivals ( at ints. In the diagram, crves L = X-Nm and L = X- Sim deict the locing roailities (P at layer X, where X =,,, 8 of an single-ffered eight-stage MIN estimated y the analytical model and simlation resectively. According to this diagram, the locing roailities (P in the first layers are greater, while in the last layer there is no locing. The nmerical reslts of the two methods have een fond again to have the same close agreement. The locing roaility decreases with the nmer of stages. So, the se of an asymmetric ffer size can e roosed. An imlementation that is with a ffer size that is larger in the first layer and ecomes gradally smaller dring the following stages can e sed as an otimal cost-effective soltion. This techniqe may also imrove the erformance of the MINs. So, this configration can e alied in the design of large scale MINs, in order to develo high-seed networs. e achieved more easily. In this case it is only necessary to write the tilization eqation er stage. Then, tting them in the forward and acward sections of the iterative method easily otains the steady-state of qees tilization. 3 Normalized acets latency on single-ffered MINs Similarly, Figre 7 reresents the normalized acet latency of a single-ffered i-stage MIN, where i = 3, 4, 6, 8,, verss the roaility of acet arrivals according to oth analytical model and simlation. It is seen that the normalized latency ecomes higher as the networ size increases. Normalised Throght (-Proaility of arrivals on ints Sim-L=3 Nm- L=3 Sim-L=4 Nm- L=4 Sim-L=6 Nm- L=6 Sim-L=8 Nm- L=8 Sim-L= Nm- L= Figre 7. Normalized acets latency of an i-stage MIN verss roaility of arrivals according to analytical model and simlation Low vales of acet latency are oserved for relevant low vales of acet arrivals. This haens ecase the acet olation is low in nmers and therefore the nmer of locing acets oserved is also low. Then, as the offered load rises, the acet latency follows this agmentation de to the increment in the acressre henomenon. The reslts otained y the two methods were again fond to e in the same close agreement. 4 Utilization er stage in single-ffered MINs Finally, Figre 8 resents the tilization ( er stage verss the roaility of arrivals ( at ints for an eight-stage (56 56 MIN, where the nmerical reslts otained y the two methods have again een fond to e in the same close agreement. It is worth noting that the tilization of the last stage deicts the throght of the MIN ecase there is no locing at the last stage. Blocing Proaility L=-Nm L=-Sim L=-Nm L=-Sim L=3-Nm L=3-Sim L=4-Nm L=4-Sim L=5-Nm L=5-Sim L=6-Nm L=6-Sim L=7-Nm L=7-Sim L=8-Nm-Sim (-Proaility of arrivals on ints Figre 6. Blocing roailities (P /stage of a eight-stage (56 56 MIN verss roaility of arrivals according to the analytical model and simlation Altering this secial analytical method, the calclation of the erformance evalation of the aove descried asymmetric with resect to the ffer size MIN can Utilization/stage Nm-L= Nm-L= Nm-L=3 Nm-L=4 Nm-L=5 Nm-L=6 Nm-L=7 Nm-L=8 Sim-L= Sim-L= Sim-L=3 Sim-L=4 Sim-L=5 Sim-L=6 Sim-L=7 Sim-L= (- Proaility of arrivals on ints Figre 8. ivergence of tilization/stage verss roaility of acet arrivals for an eight-stage MIN The throght of a MIN is one of the two most significant erformance factors the other is the latency maing the mltistage faric sitale for the core and acone networs which tyically rovide high caacity ACAEMY PUBLISHER

11 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH 57 commnication facilities. The crves shown in Figres 6 and 8 clearly show that the increment in the offered load rovides higher tilization and ths locing roailities. It is also noteworthy that these roailities have lower vales at later stages de to the fact that the last stages are sject to lighter loads, when locing is heavier. 5 Reslts for lost and serviced acets roailities The lost acets at ints of the MIN are correlated with the serviced acets which have finally een acceted y the system in comarison with the total nmer of acets arriving at ints. The roailities of serviced acets (or the olation of serviced acets remain constant in each stage, as all the acets lead from the int to the ott, ecase acets cannot e lost in the intermediate stages. Figre 9 resents the service roailities for a single-ffered i-stage MIN where i = 4, and for variale cases of arriving traffic. As can e seen, the service roaility remains constant in all stages and that confirms the analysis of Lemma, Formla (9. Moreover, the loss roaility of acets at the MIN s ints is stdied. Ths, Figre 9 illstrates the lost acets at ints of the MIN verss the roaility of acets arriving at ints of an i-stage MIN where i = 4,. As can e seen, the loss roaility is increased as the arrival rate of acets increases. In the case of low traffic, the vales of lost acets remain low. On the other hand when traffic is high ( >.7 the roaility of acets eing lost is over 3%. Frthermore, the eqation = serv lost is confirmed y arithmetic soltion, as exected. Proailityies of Arrivals, serviced and lost acets P arrived=p lostp serviced Proaility of arrivals at ints Arrivals Serv:Nm- L=4 Serv:Sim-L=4 Lost:Sim-L=4 Lost:Nm- L=4 Serv:Nm- L= Serv:Sim-L= Lost:Nm- L= Lost:Sim-L= Figre 9. Proailities of serviced and lost acets for a single-ffered MIN consisting of SEs Ths in Figre 9, if we add the vales of the crves to the corresonding nmers of the crves lost serviced, then we otain the nmers of the crve, which is an indirect confirmation of or reslts. arrived throght of a dole-ffered i-stage MIN estimated y the analytical model and y simlation resectively. From Figre it is ovios that the normalized throght deteriorates as the networ size increases. Normalized Throght (-Proaility of arrivals on ints Sim-L=3 Nm- L=3 Sim-L=4 Nm- L=4 Sim-L=6 Nm- L=6 Sim-L=8 Nm- L=8 Sim-L= Nm- L= Figre Normalized throght of an i-stage dole-ffered MIN verss roaility of arrivals according to the analytical model and simlation Comaring the vales of Figre with the corresonding vales of Figre 3, it is ovios that the throght vales of dole-layer MINs are higher than those of single-ffered MINs with the same configration set-. V. COMBINE PERFORMANCE FACTOR A. Comined erformance factor for mlti-criteria evalation of MINs In general, erformance evalation factors can e divided into two major sets: factors to e maximized (e.g. throght and factors to e minimized (e.g. latency, cost, etc.. Let the first maximized set e x max = { x,max, x,max,..., xμ, max} of normalized erformance metrics and let the minimized set of normalized erformance metrics e y max = { y,max, y,max,.., yν, max}, where μ, ν are the nmers of factors to e maximized and minimized resectively. Nevertheless, it is interesting to have a general evalation sing only one factor. This factor mst sggest etter overall erformance, that is, when the first factor s set is maximized and the second factor s set is minimized simltaneosly. We call this factor the Comined Performance Factor (CPF and it is given y the following formla: CPF = μ ν x i,max i= i= yi,min C. Performance of dole-ffered MINs 6 Normalized throght for dole-ffered MINs Figre reresents the normalized throght of a dole-ffered i-stage MIN, where i = 3, 4, 6, 8,, verss the roaility of acet arrivals. In the diagram, crves Nm-L = i and Sim-L = i deict the normalized In any mlti-criteria decision-maing rolem, however, the imortance of each criterion is a design rolem. Therefore, when it is of interest to give a weight (concerning the imortance in the networ to each searate metric then the aove formla can e relaced y: ACAEMY PUBLISHER

12 58 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH w i CPF ( w, w w j i μ ν w i ( xi,max w j i= j= = y j,min j μ ν w i i= j= where, are the corresonding weights of the normalized system s arameters. According to this eqation, when the metrics ecome larger and/or y x i,max the i,min metrics ecome smaller, the CPF ecomes larger. The reference vale domain of CPF ranges from to. The main condition which mst e satisfied when the CPF factor is alied is the assmtion that y j, min. Besides this, all the measred factors mst e calclated and manilated as inter-individal metrics. In this aer, we se the most imortant erformance indicators of normalized thoght and normalized latency ( N. It is ovios that the erformance of a MIN is considered otimal when ( T hn is maximized while N is minimized. Conseqently, the formla for comting the CPF acts so that the overall erformance metric follows that rle. Formally, CPF can e simlified to: CPF ( w w Th, w = w Th. T Th w hn w N w w j T ( hn where and w denote the corresonding weights of Th the two erformance metrics articiating in the overall erformance factor CPF, designating its imortance for the cororate environment. According to this eqation, when the throght metric ecomes larger and/or the latency ecomes smaller, the CPF ecomes larger. The reference vale domain of CPF ranges from to. Conseqently, as the CPF ecomes higher, the erformance of the MIN is considered to imrove. B. Alying the Comined erformance factor The role of ffer size in MINs Figres and deict the ehavior of the CPF for single- and dole-ffered MINs correlated with the offered load nder varios networ sizes, where different weights for each factor articiating in the CPF are considered, ths designating that factor s imortance in the cororate environment; for examle, for atch data transfers throght is more imortant, whereas for streaming media the latency mst e otimized. According to these figres, solid crves SB-L = i reresent the overall erformance metric CPF for singleffered i-stage MINs, while dotted crves B-L = i stand for the corresonding dole-ffered configrations where i = 4, 6, 8., CPF(, ( - Proaility of offered load on ints B-L=4 B-L=6 B-L=8 SB-L=4 SB-L=6 SB-L=8 Figre. CPF verss ( roaility of acet arrivals at i-stage MINs ( w th =, w d = In the first diagram the throght factor is considered to e of dole imortance ( w th =, w d =, while in the second diagram the latency factor is assmed to e of twofold significance ( w th =, w d =. In Figre, where throght is more imortant, two areas may e identified: the first one sans the light int load segment of the x-axis in which single-ffer configrations offer slightly etter overall erformance, and the second one sans the medim- and high-load segment of the x-axis in which the gain for the CPF metric of dole-ffered MINs is considerale. CPF(, ( - Proaility of offered load on ints B-L=4 B-L=6 B-L=8 SB-L=4 SB-L=6 SB-L=8 Figre. CPF verss ( roaility of acet arrivals at i-stage MINs ( w th =, w d = On the other hand, when the latency is assmed to e of twofold significance (Figre, it is seen that all single-ffered set-s exhiit imroved overall erformance comared with the corresonding doleffered ones. Conseqently, the findings of this metric can e sed y networ designers for drawing otimal configrations while setting MINs to est meet the overall erformance and cost reqirements nder the anticiated traffic load and qality of service secifications, where erformance rediction efore actal networ imlementation can also minimize deloyment cost and rollot time. VI. METHOOLOGY S EXPANABILITY This methodology can e extended to deal with erformance calclations in modern, more comlicated MIN architectres [9, ] which are directed at new tyes of alications. The general idea of erformance calclations is as follows. Becase MINs have a comond strctre, they can e analyzed in stages or modles (sally ffers. Every modle can e stdied in an aritrary time cycle of ACAEMY PUBLISHER

13 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH 59 its oeration. From this stdy an eqation is extracted which descries the state and the state transitions in qestion (e.g. a tilization eqation. Afterwards, the eqations of the seqence stages (with their relevant ondary conditions are t into the sections of the iterative method. The convergence of the algorithm gives vales of erformance indicators that nderin the system when it is in a state of eqilirim. Some examles which exloit this fast arithmetical convergent method are resented in the following cases: MINs that sort traffic with two or more riority classes In this case the acets entering the faric are distinct in two or more riority classes. The higher riority classes always earn the memory sace in comarison with other acets with lower class riority. The acet riority rocesses of acet forwarding can e modeled y arallel qees ielines. There are as many arallel rocesses as there are riority classes. Each arallel rocess ehaves as a single riority model (lie the resented model. The crrent Lemma remains the same for each riority class inasmch as the acets entering the faric cannot e lost while they are forwarded to the otts. A noteworthy oint here is the locing roailities at the last stage. In the last stage, the acets with highest class riority do not sffer from locing. Contrary to this, the acets of a lower riority class may have een loced owing to the existence of higher riority traffic. The locing roaility of a lower class of riority traffic in the last stage is calclated y the roliferation of qee tilization of each last-stage er class s riority. This is the relation which connects the acet riority classes. Afterwards, the eqations are formed on the asis of the s-section 4. analysis, taing into accont the locing roailities which aeared in the last stage. The extracting eqations are t within the iterative algorithm s sections. Rnning the algorithm ntil it converges rovides indicators aot the state of eqilirim. This method gives a soltion to the isse of a large nmer of riority classes. MINs sorting mlticast traffic Most imortant in this case is that in a stage of the MIN the acets increment which aears is cased y the mlticasting oeration. Ths, the asic condition of Lemma cannot e tre ecase the faric does not act as a ieline, since the nmer of entering acets does not remain the same as they are forwarded from stage to stage. The last stage has a high density of acets and this amont is redced to that of the receding stages y a factor which is eqal to, where (w is the w mlticast ratio which denotes the mlticast acets olation divided y the total acet olation in a stage. This factor is considered to e fixed for all stages. Taing into accont the acet redction from the last stage to the first and woring in the same way as in Lemma of this aer, we extract a modifying Lemma. Then, in the same way, as shown in Section 4., a tilization formla can e extracted and in conseqence the same iterative algorithm can e alied. 3 MINs with variale ffer sizes among the stages In this case the tilization formlae do not remain the same for all stages. Therefore, following the analysis descried aove in s-section 4. we can otain relevant tilization formlae for each stage. In conseqence, the ( forward and acward sections of iterative algorithms are comonded stage y stage, tting the stage s tilization eqations with their ondary conditions. The convergence of the iterative algorithm also gives the eqilirim vale of the tilization metric. Finally, Marov rocesses are often roosed for modeling and evalating MINs in arallel or distrited systems. Simlation ased on Marov chains rovides a owerfl method for erformance evalation. Bt it comes with a hge drawac: it often reqires long rn times ntil accrate reslts are determined with high confidence levels. On the other hand, the adatation, accracy, and fast convergence are the main advantages of the method resented aove, articlarly comared with Marovian analytical aroaches. The exemlification of the crrent arithmetical method is not limited. This aroximate method can e alied even in other cases of modern MIN architectres, maing their erformance evalation attainale. VII. CONCLUSIONS AN FUTURE WORK Today s gigait Ethernet and ATM switches, terait roters, mltirocessor systems, and general arallel systems are tyical alications of interconnection networs which have een identified as efficient comonents in commnication strctres. In this aer, a erformance methodology for Mltistage Interconnection Networs (MINs is resented. The erformance methodology goals were threefold. Firstly, it incororates an analytical method which gives fast and accrate reslts ased on an iterative algorithm which converges qicly, giving erformance metrics as searate factors in the state of eqilirim. Secondly, it is accomanied y a general evalation factor which hels s in choosing MINs which erform etter in comarison with other similar MIN architectral secification and design goals. Thirdly, it resents exandaility of several MIN architectral reqirements. The methodology was exemlified for the case of symmetrical MINs comrising single- or doleffered SEs. This model reresents a real tye of locing (acressre which is a very common henomenon for SEs. This new aroximate analytical method verifies the anticiated fact that the locing roaility and the tilization will get smaller when moving from the first stage to the last one (the last stage has zero locing roaility. The reslts otained y a thorogh stdy are confirmed y simlation. It was fond that the reslts of or aroximate method are in close agreement (differences are less than % with the corresonding simlation ACAEMY PUBLISHER

14 6 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH exeriments. Additionally, the reslts in some cases were BEGIN validated y existing related wor in the literatre. Calclate: (formlae (5 /*for stages (L-,.., */ Moreover, the erformance factor which is defined for EN FOR mlti-criteria evalation of MINs can lay a significant ( role in decision-maing for MIN selection. Calclate: (formlae (5, sing acet The rocess and reslts which are otained y this arrivals ( /*for stage */ erformance analysis can e a sefl tool for analyzing /* End of PHASE A (Bacward Soltion */ commnication, esecially in the area of arallel systems, and moreover it can e a sefl tool for designers or UNTIL ([ m] [ m ] < ε for all stages i = to L teams evalating, monitoring, or otimizing systems. Set to the vales of for all stages The main advantage of the roosed erformance i = to L evalation methodology is its flexiility, owing to its Calclate,, N aility to e adated easily to MINs varios architectral reqirements and their oerations. Therefore, it cold e REFERENCES the main latform for tested and erformance analysis in some secial modern sjects lie MINs sorting traffic with riorities or mlticast traffic, or MINs which oerate with retransmission acets. In ftre wor we will consider sch cases and will mae efforts to rovide MIN designers with metrics that will sort them in choosing the est MIN configration, taing into accont the alications (e.g. mltimedia streaming verss file transfer that the MIN will sort. APPENIX A CONVERGENT ALGORITHM Let e the vale of dring the m-th iteration of the following algorithm: Algorithm I BEGIN m := /*Start of PHASE A (Initialize Bacward Soltion*/ O BEGIN ( Initialize: i [] : = /* for stages L,,*/ EN FOR /* End of PHASE A */ [] Ranjan Kmar ash, Nalini Kanta Baranda, and Chita Ranjan Triathy, A New and Efficient method to Evalate Residal Broadcast Reliaility of Falt-tolerant Mltistage Interconnection Networs, IJCSNS International Jornal of Comter Science and Networ Secrity, VOL.8 No.9, Setemer 8. [] Kmar, S., Mathematical Modelling and Simlation of a Bffered Falt Tolerant ole Tree Networ, Advanced Comting and Commnications, 7. ACOM 7. International Conference on Volme, Isse, 8- ec. 7 Page(s:4 433 [3] Atiqzzaman M. and M.S. Ahatar, Efficient of Non- Uniform Traffic on Performance of Unffered Mltistage Interconnection Networs, IEE Proceedings Part-E, 994. [4] Boras C., Garofalais J., Sirais P., Triantafillo V., A general erformance model for mltistage inteconnection networs, Ero-Par 97. Agst 5-9. [5] Boras C., Garofalais J., Sirais P., Triantafillo V., An analytical erformance model for mltistage interconnection networs with finite, infinite and zero length ffers, in Performance Evalation 34( [6] Boras C., Garofalais J., Sirais P., Triantafillo V., Qeing delays in differed mltistage interconnection networs, in Proc. 987 ACM Simetrics Conf., May - 4, 987, Banff, Alerta, Canada,. -. [7] Y.-C. Jenq, Performance analysis of a acet switch ased on single ffered anyan networs, IEEE Jornal Selected Areas of Commn. SAS-(6 (983, 4- REPEAT [8] Garofalais J., P. Sirais, The erformance of mltistage m:= m interconnections networs with finite ffers, in :Proc. /* Start of PHASE B (Forward Soltion */ ACM SIGMETRICS Conf., 99, short aer. ( Calclate: (formlae (5, sing acets arrivals [9] Cisco Systems, htt://newsroom.cisco.com/dlls/4/next_generation_net ( /*for stage */ wors_and_the_cisco_carrier_roting_system_overview. FOR i = TO L- O df. BEGIN [] Hsiao S.H. and Chen R. Y., Performance Analysis of Calclate: Single-Bffered Mltistage Interconnection Networs, (formlae (5 /*for stages,, L- */ 3rd IEEE Symosim on Parallel and istrited EN FOR Processing, , ecemer -5, 99. Calclate: (formlae (5, with = [] I-Pyen Lyen, avid M. Koelman, An Analysis of Banyan Networs Offered Traffic with Geometrically /*for stage L */ istrited Message Lengths, IEE Proceedings /* End of PHASE B (Forward Soltion */ Commnications Volme 4, Isse 5, Octoer 995. m := m /* Start of PHASE A (Bacward Soltion */ [] Sandee Sharma, P.K.Bansal, Karanjit Singh Kahlon On Calclate: a class of mltistage interconnection networ in arallel (formlae (5, with = rocessing, IJCSNS International Jornal of Comter /*for stage L */ Science and Networ Secrity, VOL.8 No.5, May 8 FOR i = L- OWNTO O ACAEMY PUBLISHER

15 JOURNAL OF COMMUNICATIONS, VOL. 5, NO. 3, MARCH 6 [3] Krsal C.P., Sinir M., The erformance of mltistage interconnection networs for mltirocessors, IEEE Trans. Comt. C-3 ( [4] G. Zheng, T.Wilmarth, P. Jagadishrasad, and L. V. Kale. Simlation-ased erformance rediction for large arallel machines, In International Jornal of Parallel Programming, nmer to aear, 5 [5] Lin T., Kleinroc L., Performance Analysis of Finite- Bffered Mltistage Interconnection Networs with a General Traffic Pattern, Joint International Conference on Measrement and Modeling of Comter Systems, Proceedings of the 99 ACM SIGMETRICS conference on Measrement and modeling of comter systems, San iego, California, United States, Pages: 68-78, 99. [6] H. Mn and H.Y. Yon, Performance analysis of finite ffered mltistage interconnection networs, IEEE Trans. Comt. 43( (994, [7].C. Vasiliadis, G.E. Rizos, C. Vassilais Performance Analysis of locing Banyan Switches, Proceedings of the IEEE sonsored CISSE 6, ecemer, 6. [8] John Garofalais, El. Stergio An analytical erformance model for mltistage interconnection networs with locing Sixth Annal Conference on Commnication Networs and Services Research (CNSR8 Halifax, Nova Scotia, Canada. May 5-8, 8. [9] Sandee Sharma, K.S. Kahlon, P.K. Bansal and Kawaljeet Singh, Irreglar Class of Mltistage Interconnection Networ in Parallel Processing, Jornal of Comter Science, -MAR-8 [] Theimer T.H., Rathge E. P., and Her M.N., Performance Analysis of Bffered Banyan Networs, IEEE Transactions on Commnications, vol. 39, no., , Ferary 99. []. Ttsch, M. Brenner. A Mltistage Interconnection Networ Simlator. In 7th Eroean Simlation Mlticonference: Fondations for Sccessfl Modelling & Simlation (ESM 3; Nottingham, SCS,.. 6, 3. []. Ttsch, G. Hommel. Generating Intrconnection Networ Simlator. Generating Systems of Eqations for Performance Evalation of Bffered Mltistage Interconnection Networs. Jornal of Parallel and istrited Comting, 6, no. :. 8..4, [3] Bianco Andrea, Finochietto Jorge, Mellia Marco, and Neri Faio, Mltistage Switching Architectres for Software Roters. IEEE Networ Jly/Agst 7. [4] Aydin O. Balan, Gang Q, Uzi Vishin, An Area- Efficient High-Throght Hyrid Interconnection Networ for Single-Chi Parallel Processing, esign Atomation Conference, 8. AC 8, 45 th ACM/IEEE Plication ate: 8-3 Jne 8 [5] Raja J., S. Shanmgavel, Performance Stdies of Banyan ATM Switching Networs sing RS Codes, IE Jornal- CP, Vol 84, May 3.. He is resonsile and scientific coordinator of several recent Eroean and national IT and Telematics Projects (ICT, INTERREG, etc.. His lications inclde more than articles in refereed International Jornals and Conferences. His research interests inclde We and Moile Technologies, Performance Analysis of Comter Systems, Comter Networs and Telematics, istrited Comter Systems, Qeing Theory. Eleftherios Stergio is lectrer in the deartment of Information Technology and Telecommnications, at Eirs Institte of Technology in Greece since. He is also a research fellow at the University of Patras. He received the B.S. degree in electrical engineering from NTUA, Athens Greece, and he finished his ostgradate stdies at the comter science deartment of the University of Sheffield (998. His research interests on erformance evalation of networs integrate y lishing aers in international jornals. Among these interests, comting analytical methods, interconnection networs, arallel and distrited systems, high-seed networs, are inclded. Mr E. Stergio is memer of IEEE Comter Society. John Garofalais (htt://athos.cti.gr/garofalais/index_e n.htm is Associate Professor at the eartment of Comter Engineering and Informatics, University of Patras, Greece, and irector of the alied research deartment "Telematics Center", of the Research Academic Comter Technology Institte (RACTI. ACAEMY PUBLISHER