Revisiting the notion of Origin-Destination Traffic Matrix of the Hosts that are attached to a Switched Local Area Network

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1 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 Revisiting the notion of Origin-Destintion Trffic Mtrix of the Hosts tht re ttched to Switched Locl Are Network Mondy Eyingho, Smuel Flki nd Anthony Atyero 3 Deprtment of Electricl nd Informtion Engineering, Covennt University, Ot, Nigeri eyimon@yhoo.com Deprtment of Computer Science, Federl University of Technology, Akure, Nigeri swoleflki@yhoo.com 3 Deprtment of Electricl nd Informtion Engineering, Covennt University, Ot, Nigeri tyero@ieee.org ABSTRACT There is widely held notion in the literture tht there is n origin-destintion trffic mtrix for ll the hosts (end nodes) tht re ttched to ny switched locl re network. This is usully in reltion to the need to clculte the end-to-end (or mximum end-to-end) delys between ll the end nodes tht re ttched to the network, for the sole purpose of using these delys vlues to design upper-bounded dely switched locl re networks, in efforts t solving the dely problems of this clss of networks. In this pper, we show tht this notion of origin-destintion trffic mtrix for ll the hosts tht re ttched to switched locl re network does not seem to be correct. KEYWORDS End-To-End Dely, Origin-Destintion Pir, Switched Locl Are Networks. INTRODUCTION It is widely held notion in the literture tht to determine the verge network dely of switched Locl Are Network (switched LAN), the trffic mtrix of the hosts tht re ttched to the LAN hs to be formulted. For exmple, in [], it ws stted tht, let T ij denote the informtion flow in pckets per unit time between end nodes i nd j of the network; we define the trffic mtrix of the network to be the n n mtrix T= (T ij ) where n is the number of end nodes. Moreover, in [], the verge end-to-end dely time of switched locl re network ws defined with respect to the verge trffic between end node i nd end node j for ll end nodes tht re ttched to the LAN nd the verge dely between the end nodes ws defined s the weighted combintion of ll end-to-end dely times. Elbum nd Sidi on the other hnd in [3] defined minimum verge network dely s the verge dely between ll pirs of users in the network. In this pper, we show tht this notion of origin-destintion trffic mtrix for ll hosts tht re ttched to switched locl re network with view to using it to compute end-to-end delys (or verge end-to-end dely) does not seem to be correct. DOI : 0.5/ijdps

2 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0. ANALYSIS OF SWITCHED LOCAL AREA NETWORKS ORIGIN-DESTINATION TRAFFIC ENUMERATION METHOD USING THE HOSTS THAT ARE ATTACHED TO THE NETWORK For pcket switched [4 p.366], rndom ccess [5], lossless [4, p.366], [5] networks, offered lod = throughput, but, mount of dt trnsferre d throughput = therefore, time tken to effect the trnsfer mount of dt trnsferred offered lod = time tken to effect the trnsfer For ny origin-destintion pir of nodes (hosts), time tken to effect the trnsfer = dely from origin node (host) to destintion node (host) mximum time to effect the trnsfer = mximum dely from origin host to destintion host mximum time to effect the trnsfer from origin host to destintion host = mximum time (dely) through the st switch + mximum time (dely) through the nd switch + + mximum time (dely) through the n th switch on the origin-destintion pth If n externl site presents to the origin-destintion pth (route) Z pckets every second (loding or offered lod), then it presents pcket every Z seconds. Since for lossless system, offered lod = throughput, mximum time to effect the trnsfer of pcket = mximum dely from origin host to destintion host of pcket = mximum time (dely) through st switch + mximum time (dely) through nd switch + + mximum time (dely) through n th switch Z () where n = the number of switches in the origin-destintion pth (route). The inequlity symbol in () implies tht Z seconds is n upper bound on end-to-end dely for the origin-destintion pir of hosts. Above Z seconds, we will hve loss system. Therefore, since we re interested in the mximum (upper bounded) dely, the symbol in () cn be replced by n = symbol nd hence, () becomes n eqution. We cn lso rtionlize the preceding ide in this wy. Consider n origin-destintion pir of hosts tht re involved in communiction session. Let us ssume tht the locl re network (LAN) is switched Ethernet LAN. Assume tht X Megbits/sec (Mbps) = mximum Ethernet port trnsfer rte of host, X = pckets/sec. () (minimum Ethernet pcket length) = Z pckets/sec, sy. Or, it trnsfers pcket every Z seconds. 8

3 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 The minimum Ethernet pcket length should be used in () becuse, this will give us the mximum pckets/sec. (mximum loding) nd hence, it will give us n upper bounded dely sitution. Therefore, for lossless system, ech pcket should cross ll the switches in its pth from origin-to-destintion in Z seconds. This ide ws succinctly expressed by Berseks nd Gllgr in [4, p.5]; where it ws stted tht, strict implementtion of communiction session s rte of r pckets/sec. would be to dmit pcket every r seconds. As n exmple, for hosts (worksttions), X could be 0Mbps (meg bits per second) or the bsic Ethernet rte, 00Mbps (Fst Ethernet rte).for Servers, X could be,000mbps (Gigbit Ethernet rte). Consider network tht hs two () hosts connected by switch s shown in Figure. H H We cn see tht either of the hosts will be sending dt trffic to the other host or receiving dt trffic from it; therefore, we will hve the trffic mtrix shown in (3). (3) is Host sending dt trffic to Host nd Host receiving dt trffic from Host; similrly, is Host sending dt trffic to Host, nd Host receiving the trffic. is Host sending dt trffic to itself (which is not possible); nd is Host sending dt trffic to itself (which is lso not possible). Therefore, the digonl entries re not necessry, but we retin them so tht we cn get correct picture of the network trffic mtrix; the digonl entries re hence, crossed out. But we cn see from Figure nd from (3) tht end-to-end dely in the direction from Host to Host is the sme s the end-to-end dely from Host to Host; so for two () hosts network, we need end-to-end dely, since: end-to-end dely = end-to-end dely. Consider lso, network tht hs three (3) hosts. There re more thn one wys of connecting the hosts. It my be through switch or through multiple switches (this gin cn hve multiple configurtions).we illustrte just two of the configurtions in Figures nd 3. We should emphsize t this point tht, in our illustrtions (Figures.,, nd 3) nd indeed in ny LAN instlltion, one or more of the hosts my be server or servers (for exmple, file server, web server); but for the purpose of our nlysis, we regrd ll connected end devices (computing devices, printing devices nd others) s hosts. 9

4 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 H H H 3 H H 3 H Wht ever be the configurtion (connection) of the three hosts network does not mtter; wht is importnt in the context of our nlysis is tht, Host cn either be sending trffic to Host or receiving trffic from Host, it cn either be sending trffic to Host3 or receiving trffic from Host3. The sme scenrio holds for Host nd Host3. We therefore, hve the trffic mtrix for three (3) hosts connected through two () switches network s shown in (3) (4) Following our previous explntions, we cross out the digonl entries. Also, since the trffic from Host to Host will cross the sme number of switches from origin to destintion, s the trffic from Host to Host, it follows tht, end-to-end dely = end-to-end dely. Similrly, end-to-end dely 3 = end-to-end dely 3, end-to-end dely 3 = end-to-end dely 3.Therefore, for three (3) hosts network, we need 3 end-to-end delys. By following the preceding explntions, we cn proceed to write out the trffic mtrix for four (4) hosts network s shown in (5). By similr resoning, we cn see tht for four (4) hosts network, we need 6 end-to-end delys. Following the sme logic, we cn show tht for five (5) hosts network, we need 0 end-to-end delys, for six (6) hosts network, we need 5 end-to-end delys. 30

5 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 origin-destintion origin-destintion origin-destintion (5) We hve been ble to come-up with closed-form reltion for finding the number of end-to-end delys (number of origin-destintion pirs) to be used with inequlity () (in inequlity (), we hve considered only one () origin-destintion pir of two () hosts involved in communiction session) for ny switched LAN. For switched LAN, if, p = the LAN s number of end-to-end delys (with respect to ttched hosts or end nodes), k = number of end nodes (hosts) in the LAN, then, k x= origin-destintion 4 p = ( k x) (6) 3 (3 x= For exmple, if k =3, p = x ) = (3 x ) = (3 -) + (3 ) = + = 3 x = No mtter the number of hosts tht re ttched to the LAN, we cn clculte the number of end-to-end delys (number of origin host to destintion host delys) with respect to () by using (6). Consider the three (3) hosts, two () switches LAN shown in Figure 3. Since the ppliction of (6) gives us three origin-destintion pirs, nd from () the 3 origin-destintion pirs of hosts re -, -3, nd -3. For this network therefore, the three origin-destintion equtions (using () which hs n upper bound of Z ) re: origin-destintion 6 origin-destintion 5 For the - origin-destintion pir of hosts, mximum time to trnsfer pcket from H to H (mximum end-to-end dely) = mx. time (dely) through switch = (7) Z For the -3 origin-destintion pir of hosts, mximum time to trnsfer pcket from H to H 3 (mximum end-to-end dely) = mx. time (dely) through switch +mx. time (dely) through switch = (8) Z For the -3 origin-destintion pir of hosts, mximum time to trnsfer pcket from H to H 3 (mximum end-to-end dely) = mx. time (dely) through switch +mx. time (dely) through switch = (9) Z3 where Z, Z, nd Z 3 re the Ethernet port trnsfer rtes in pckets per second of ny of the two hosts tht re involved in communiction session in (7), (8), nd (9) respectively. We cn write out (7), (8), nd (9) together in mtrix form s: 3

6 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 mx imum dely of origin/ destintion mximum end to ny dt pcket end dely 0 through switch origin/ destintion 3 mximum end to = end dely (0) origin/ destintion 3 mximum end to mx imum dely of ny dt pcket end dely through switch Where n entry in the bit mtrix is if pcket from or to ny of the hosts t either ends in the origin-destintion pth (the mximum end-to-end dely entry of the mximum end-to-end dely column vector) crosses the corresponding switch in trnsiting from origin host to destintion host; the entry is 0, if the pcket does not cross the switch. We cn re-write (0) s in (); where = = = 3 = 3 = nd = 0. Mtrix Eq. () cn be written for ny switched locl re network, with ny rbitrry number of m switches nd k hosts. Let y, y, y 3,,y p-, y p represent origin-destintion pir mximum end-to-end dely, origindestintion pir mximum end-to-end dely, origin-destintion pir3 mximum end-to-end dely,,origin destintion pir p- mximum end-to-end dely nd origin-destintion pir p mximum end-to-end dely respectively. mx imum dely of origin/ destintion mximum end to ny dt pcket end dely through switch origin/ destintion 3 mximum end to = end dely () origin/ destintion 3 mximum end to 3 3 mx imum dely of ny dt pcket end dely through switch Also, let x, x, x 3,, x m-, x m represent the mximum dely of ny dt pcket through switch, the mximum dely of ny dt pcket through switch, the mximum dely of ny dt pcket through switch 3,, the mximum dely of ny dt pcket through switch m-, the mximum dely of ny dt pcket through switch m respectively, then () cn be written for ny switched locl re network s in (). y x y y 3 3 = y p p yp p 3 p p m m 3m p m p m m m 3m p m pm x x 3 x m xm () 3

7 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 where y = Z, y = Z,,y p = nd x, x, x 3,., x m-, x m re the unknown. Z p m = number of switches in the locl re network; ij = /0, i =,,3,,p; j =,,3,,m re elements of the pth bit mtrix for the whole network.; ij =, if dt pcket tht trnsverses origin-destintion i psses through switch j; ij = 0, if dt pcket tht trnsverses origin-destintion i does not pss through switch j; p = the LAN s number of origin-destintion pirs of hosts, which is given by (6). Put in compct form, () cn be written s: y = AX (3) Where A is the p m mtrix, X = (x, x, x 3,,x m-, x m) is m column vector, nd y = (y, y, y 3,,y p-, y p) is p column vector in () respectively. On the surfce, () seems to support the notion tht there is n origin-destintion pirs trffic mtrix with respect to end-to-end dely computtion for ll the hosts tht re ttched to switched LAN s enuncited (we think, not correctly) in [], [], [6]. We now explin why this notion does not seem to be correct when pplied to switched LANs. If we look t (0), we see tht rows nd 3 of the bit mtrix hve the sme type of entries, indeed = 3 nd = 3 in reltion to (). This will mke the set of vectors resulting from () to be linerly dependent. If () is written out for (), we will hve (4). Crrying out mtrix multipliction on (4) will result in the system of equtions s shown in (5). y y y 3 = 3 3 x x (4) y = x + x y = x + x y 3 = 3 x + 3 x (5) (5b) (5c) The reson for the liner dependence of the system of equtions in (5) is tht, if we look t the network of Figure 3, for host H to communicte with host H 3, nd vice vers, dt pckets will trnsit through switch nd switch ; the sme thing s when host H is to communicte with host H 3 or H 3 with H. Therefore, the mximum delys of dt pcket in switch nd switch for the two communiction sessions re the sme; hence, the similrity of the entries in rows nd 3 of the end-to-end pths bit mtrix in (0). In the words of Kreyzig in [7, p.33], wht is the point of liner independence nd dependence? And he provided the following nswer: well, from linerly dependent set of vectors, we my often omit vectors tht re liner combintion of others until we re finlly left with linerly independent subset of the relly essentil vectors, which cn no longer be expressed linerly in terms of ech other [7, p.33]. Therefore, if we eliminte the linerly dependent set of vectors from (4), some of the y i s will vnish; mening tht the communiction pths of some of the hosts (origin-destintion pirs of hosts) will vnish. This is becuse pckets trvelling between two end hosts will suffer mximum delys in the sme set of switches long their end-to-end pths. This proves the fct (nd we re proposing new theory) tht there is no origin-destintion pirs trffic mtrix with respect to end-to-end dely computtion for ll the hosts tht re ttched to switched locl re network. 33

8 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 3. CONCLUSIONS We hve shown in the context of computing the end-to-end (or mximum end-to-end) delys of switched locl re networks tht there is no origin-destintion pir trffic mtrix for ll the hosts tht re ttched to ny of such networks. We hve therefore, proposed the theory tht, there is no origin-destintion pirs trffic mtrix with respect to end-to-end dely computtion for ll the hosts tht re ttched to switched locl re network. How then do we enumerte ll the end-to-end delys of ny switched locl re network, in order to be ble to clculte, for exmple, the network verge of mximum end-to-end delys? This cn be used to design n upper bounded dely switched locl re network. A methodology for enumerting ll the endto-end delys of ny switched locl re network will be reported in nother pper. REFERENCES [] P. Torb & E. Kmen (999) Lod Anlysis of Pcket Switched Networks in Control Systems, Proceedings 5 th Annul Conference of the IEEE Industril Electronics Society, Sn Jose, CA, vol. 3, pp. 7. [] E., Knem, P. Torb, K. Cooper & G. Custodi (999) Design nd Anlysis of Pcket Switched Networks for Control Systems, Proceedings IEEE Conference on Decision nd Control, Phoenix, AZ, pp [3] R. Elbum & M. Sidi (996) Topologicl Design of Locl Are Networks using Genetic Algorithms, IEEE Trnsctions on Networking, vol.4, no.5, pp [4] D. Bertseks & R. Gllger (99) Dt Networks, Prentice-Hll, Englewood Cliffs, USA. [5] M. Reiser (98) Performnce Evlution of Dt Communictions Systems, Proceedings of the 98 IEEE Conference, vol. 70, no., pp [6] N. Krommencker, E. Rondeu, & T. Divoux (00) Study of Algorithms to Define the Cbling Pln of Switched Ethernet for Rel-Time Applictions, Proceedings of the 8 th IEEE Interntionl Conference on Emerging Technologies nd Fctory Communictions, Antibes, pp.3-30 [7] E. Kreyzig (003) Advnced Engineering Mthemtics, 8 th Edition, John Wiley nd Sons, New York, USA. Mondy Ofori Eyingho obtined B. Sc. Degree in Electronic nd Electricl Engineering from the Obfemi Awolowo University, Ile-Ife, Nigeri in 988. He lso holds Post-Grdute Diplom in Computer Science from the University of Lgos, Lgos, Nigeri (99), n MBA (Generl Mngement) Degree from the River Stte University of Science nd Technology, Port-Hrcourt, Nigeri (003), n M.Sc Degree in Computer Engineering, from the Federl University of Technology, Owerri, Nigeri (004), nd Ph. D in Computer Engineering from Covennt University, Ot, Nigeri (0). Dr. Eyingho hs hd extensive industril work experience in number of corporte orgniztions. He is currently Lecturer in the Deprtment of Electricl nd Informtion Engineering, Covennt University, Ot, Nigeri. Dr. Eyingho hs published scholrly ppers in interntionl journls. He is member of The Nigerin Society of Engineers, COREN Registered Electronics Engineer, nd member Nigeri Computer Society. His principl reserches re in the res of developing nd implementing microprocessor-bsed systems nd performnce optimiztion of locl re networks. 34

9 Interntionl Journl of Distributed nd Prllel Systems (IJDPS) Vol., No.6, November 0 Oluwole Smuel Flki is Professor of Computer Engineering t the Federl University of Technology, Akure, Nigeri. He holds n M. Eng Degree in Electricl Engineering from Leningrd University, n M. Sc Degree in Computer Science from UCLA nd Ph.D in Electricl Engineering from the University of Lgos, Lgos, Nigeri. His reserch interests re in the res of Computer Architecture, Computer Communictions nd Networking, Network Security nd Digitl Signl Processing. He is Fellow of the Nigeri Computer Society, Member of the Nigerin Society of Engineers, Member of IEEE nd Member of ACM. Aderemi A. Atyero is Senior Lecturer in the Deprtment of Electricl nd Informtion Engineering, Covennt University, Ot, Nigeri. He grduted from the Moscow Institute of Technology (MIT) with Bchelor of Science Degree in Rdio Engineering nd Mster of Science Degree in Stellite Communiction Systems in 99 nd 994 respectively. He received Ph.D from the Moscow Stte Technicl University of Civil Avition (MSTUCA) in 000. Dr. Atyero is member of the Institute of Electricl nd Electronic Engineers. He hs published number of scientific ppers in Interntionl peer-reviewed journls nd proceedings. He hs served s the Hed of Deprtment of Electricl nd Informtion Engineering t Covennt University. He is on the editoril bord of the Covennt Journl of Science nd Technology. He is recipient of the 009 Ford Foundtion Teching Innovtion Awrd. His current reserch interests re in Informtion Theory nd Stochstic Self-Similr Processes nd their pplictions in telecommunictions.. 35