A STUDY OF THE SOURCE TRAFFIC GENERATOR USING POISSON DISTRIBUTION FOR ABR SERVICE

Transcription

1 A STUDY OF THE SOURCE TRAFFIC GENERATOR USING POISSON DISTRIBUTION FOR ABR SERVICE Department of Computer Engneerng Wad Internatonal Unversty Syra-Homs E-Mal (correspondng author): Abstract: Ths paper descrbes modelng of the Avalable Bt Rate (ABR) source traffc n Asynchronous Transfer Mode (ATM) network usng BL / GTEXP traffc generator, whch employs Posson dstrbuton for modelng the burst length ( BL ) and onental dstrbuton for modelng the gap tme ( GT ). Ths traffc generator nherts the advantages of both Posson and onental dstrbuton functons to acheve enhanced lnk performance. Analytcal and smulaton results for BL / GTEXP traffc generator have been presented and compared. 1 INTRODUCTION The Posson process s an extremely useful process for modellng purposes n many practcal applcatons, such as, e.g. to model arrval processes for queueng models or demand processes for nventory systems. It s emprcally found that n many crcumstances the arsng stochastc processes can be well approxmated by a Posson process. From the standpont of the network users, the messages wthn a sesson are typcally trggered by partcular events. But from the standpont of the network, these message ntatons are somewhat arbtrary and unpredctable. Therefore, the sequence of tmes at whch messages arrve for a gven sesson s a random process. Further, the

2 arrval process of one sesson has to be ndependent of others. Poson process has been proved to be the most approprate for such purpose and therefore has been consdered n the present work. In ths paper source traffc modelng and smulaton study have been carred out for ABR servce n ATM networks usng Posson dstrbuton for modelng the Burst Length ( BL ) and onental dstrbuton for modelng the Gap Tme GT ). The correspondng traffc generator s desgnated as BL / GT. ( 2 BL / GT Traffc Generator EXP BL / GT traffc generator generates cells sent at a fxed rate (ACR) EXP durng BL and no cells are sent durng GT. BL s assumed to be Posson dstrbuted ( BL ) whereas GT exponental dstrbuted ( GT ). BL / GT traffc generator was created and parameterzed as explaned below. EXP When events occur accordng to an onental Dstrbuton, they are sad to occur completely at random. Thus the arrval tme of the next event s not affected by the tme elapsed snce the prevous event [1-5]. An exponental random varable wth a mean value of (where 0) s gven by the followng Probablty Densty Functon (PDF). f ( x) - x e x 0 0 x<0 (1) and the cumulatve densty functon (probablty that tme between events s x) can be obtaned by ntegratng equaton (1): 2

3 Fx ( ) - x 1-e x 0, (2) 0 x<0 The mean and varance of the onental dstrbuton are: x 1 E ( x ) x e dx (3) 0 2 x (4) 2 0 Var ( x ) x e dx ( ) A Posson random varable x wth parameter by [1, 2] (where >0 ) has PDF gven x f ( x ) e, x 0,1,2,... (5) x! The mean and varance are dentcal for the Posson dstrbuton [1]: E ( X ) Var ( X ) (6) where a Posson dstrbuton can be developed by nvolvng the defnton of a Posson Process for a fxed tme (t ), so that t. For Posson dstrbuted events, the tme between successve events s exponentally dstrbuted wth mean 1/. For an event that occurs at a tme nterval wth an exponental dstrbuton, the rate of occurrence of the event s Posson dstrbuted wth mean. Requred dstrbuton modelng nvolves a transformaton functon for convertng a random varable of unform dstrbuton nto the requred dstrbuton. Consderng the fundamental transformaton law of probabltes for two probablty densty functons f (x) and p (u) f ( x) dx p( u) du or du f ( x) p( u) (7) dx where p (u) s the PDF of random varable u and f (x) s another PDF of random 3

4 varable x. Snce u s a random varable of a unform dstrbuton n the range 0 to 1 therefore p (u) s a constant (=1) and hence x du f ( x) and therefore u F( x) dx f ( z) dz (8) 0 Equaton (8) can be used to fnd source random varable x G(u) through nverse transformaton of u F(x). For the requred dstrbuton, the nverse can easly be found from equaton (8) wth f (z) correspondng to the requred dstrbuton. u s unformly dstrbuted n the range ( 0 u 1). It can be generated by usng the functon rand ( ) provded by the standard Lnux lbrary or usng Mersenne Twster (MT) [6]. For modelng the GT, equaton (8) s used wth u F ( x ) 1 e x or 1u e x (9) Therefore the requred transformaton s x 1 ln(1 u) (10) By changng the varables x and (1 u) of equaton (10) wth X and U respectvely, we get X 1 ln( U) (11) 2.1 Estmaton of the Load ( L ) for the Traffc Generator The load varaton of the traffc can be realzed by syntheszng predefned load such that the resultng load L N L 1, where L s the traffc load due to th source. Therefore, the aggregate traffc from N sources wll generate the load L on a 4

5 lnk wth rate R Mbps gvng average throughput of R L Mbps. The load L generated by an ndvdual source can be expressed as L BL K BL ( K P ) GL r (12) where BL, GL, K, and P r are the mean BL, mean gap length, cell sze, and mnmum nter-cell gap length (Preamble) respectvely n bytes, then the load be found from equaton (12). 2.2 Estmaton of the Mnmum Gap Tme ( M ) GT L can The mnmum desred load, GT ( M GT ) s a secondary parameter dependent on load. Gven a M GT s calculated by the source automatcally usng onental dstrbuton. Usng equaton (12) the GT can be expressed as 1 L GL BL K Pr L (13) The BL can be wrtten as BL E ( x ) M (14) where BL BL M BL s the mnmum BL. GL can be wrtten as GL E ( x ) M (15) GL GL Substtutng the values of n equaton (13), BL and GL from equatons (14) and (15) respectvely M GL can be wrtten as K M M K P GL BL r L (16) 5

6 Consderng the lnk rate and usng the followng relaton Byte Tme Byte Sze( bts ) or Lnk Rate ( bts / sec) Byte Tme b (17) R The M GT now can be computed as b K M M K P GT BL r R L (18) Now the value of P r =1/ACR s readly avalable, dependng upon the selected value(s) of ACR that can be separately taken as varable, and thus equaton (18) can be re-wrtten as M GT K b M R BL 1 L 1 (19) Therefore, equaton (19) can be used for computng the value of M GT that would result n lnk load closer to L usng the selected values of L of the th source, K, and Posson dstrbuton parameters. Consderng the same parameter values for burst and gap lengths and M 1, equaton (19) can be smplfed as BL M GT K b M R BL 1 L 1 (20) 2.3 Generatng the GT EXP Bearng n mnd that ACR, snce n ABR servce a source sends ts data wth a rate equal to ACR. Defnng as the exponental mean arrval rate for GT, GT consderng equaton (15) n terms of tme, we get equaton (21) E( x) GT 1 M GT (21) GT 6

7 Now the BL / GT traffc generator can compute the GT usng the relaton EXP GT M ln( U ) (22) GT 2.3 BL Generatng BL can be modeled as follows: For ABR servce n ATM networks the value of ACR should be wthn the range MCR ACR PCR. The source frst starts transmttng a random-szed burst of cell at ICR. It then wats for a random amount of tme, whch follows onental Dstrbuton. The source wll go on repeatng ths cycle BL / GT untl completng transmsson. In every cycle the number of cells s determned by ACR, whch s changed due to the feedback used n the swtch. The source repeats the calculaton of Posson dstrbuton, equaton (5) wth ACR, k tmes, where k s changng from 0 to Peak Cell Rate (PCR). The results wll be saved n an array wth sze k. The number of cells nsde the burst can be found by generatng a random number u [6], applyng the cumulatve dstrbuton functon (cdf) for all array values, so that the cumulatve values are smaller than or equal to a random number u. The followng relaton gves the BL that s used by the BL / GT traffc generator BL e ACR x u x, x 0 x! (23) where the number of cells nsde the burst should be at least one ( M BL 1). 3 ANALYTICAL RESULTS Consderng the values of L, R, K, M BL, Posson mean arrval rate 7

8 ( BL ), and for BL / GT traffc generator as gven n Table 1. The value GT of M GT was determned usng equaton (20), (Table 1) The analytcal result of BL / GT traffc generator for 1000 count values of U, generated by the unform dstrbuton, are shown n Fgs. 1 and the correspondng computed values of mean, varance, maxmum and mnmum values of BL and GT and are gven n Table 2. Table 1: The Evaluated Parameters for the BL / GT Traffc Generator. The parameters The Values L 0.3 R K Mbps M BL 1 cell bts BL 1 cell/sec 30 cells/sec GT M GT sec 8

9 BL U for GT GT U for BL 9

10 Fg. 1: BL / GT Traffc Generator for 1000 Count Values of U. Table 2: BL (cells) and GT ( sec ) for BL / GT Traffc Generators. Mean Varance Maxmum Mnmum BL GT The varatons n BL, GT as functons of BL and GT for BL / GT traffc generator for 100 count values of U are shown n Fgs. 2 and 3 respectvely. The ncrement steps for BL (1-110) cells/sec and GT (1-110) cells/sec are 10 for each. Referrng to Table 2 t s seen that the mnmum values of BL / GT are greater than ther correspondng values of M / M. Referrng to Fg. 2 t can be concluded that the Posson mean arrval parameter BL shouldn t BL BL 10

11 be a very large value, because BL wll, consequently, be very large as well, and the source wll spend most of ts tme sendng only the burst cells wth a smaller number of gap ntervals for BL / GT traffc generator resultng n less-bursty traffc. Referrng to Fg. 3 t can be concluded that the onental mean arrval parameter GT Exo should be selected between 2 and 30 cells/sec for smulaton of real bursty traffc because t offers hgher peak values of GT. Ths s further supported by the observaton that for GT Exo n the range 30 to 100 cells/sec, the peak values of GT has the least varaton ndcatng smoothest traffc. Fg. 2: BL versus BL and U. 11

12 Fg. 3: GT versus GT Exo and U. 4 SIMULATION RESULTS The ATM network smulaton was carred out under Lnux network programmng. The Parameters specfed n Table 1 were used for ths smulaton. Sx sources S ( =1, 2, --6) sendng ther data at the rate ACR ( =1, 2, --6) between Mnmum Cell Rate (MCR) and Peak Cell Rate (PCR) were consdered. The performance of the Relatve Rate Markng (RRM) swtch was evaluated for traffc generator wth respect to the Allowed Cell Rate (ACR), Swtch Input Rate (SWIR)/Swtch Output Rate (SWOR), Memory Access Tme (MAT), Queue Length (Q ), and Cell Transfer Delay (CTD). The ntal value of ACR for sources S was 12

13 taken as PCR 2 whereas the fnal ACR value was kept between 200 to 700 cells/sec n ncremental steps of 100 for =1, 2, --6 and takng buffer sze=1000 cells, Hgher Queue Threshold ( Q H ) = 200 cells, Lower Queue Threshold ( Q L ) = 100 cells, and assumng that each source has to send a total of 1000 cells. The varatons n ACR, SWIR/SWOR, MAT,Q, and CTD as a functon of tme are gven respectvely n Fgs The smulaton results for mean, varance, maxmum and mnmum values of ACR, SWIR, SWOR, MAT, Q and CTD are gven n Tables 3-5 respectvely. It can be seen from Fgs. 4 that ACR changes due to the feed back receved from the swtch. When the swtch s heavly loaded (congeston) there s a decrease n the ACR and when t s lghtly loaded (no mpedng congeston) the ACR s ether kept constant or t s ncreased. The relatve changes between the SWIR and SWOR are shown n Fg. 5 are responsble for the varatons n the MAT (Fg. 6) and Q (Fg. 7). When the SWIR becomes greater than SWOR the buffer starts fllng and reaches a specfed threshold level. The swtch then sgnals the source to start reducng ts data rate. Consequently, source ACR reduces, and ts effect appears at the queue, whch causes a reducton n the rate of ncrease n the queue sze. For a SWIR smaller than a SWOR, the buffer starts becomng empty and when Q reaches ts mnmum value, the source s sgnaled to start ncreasng ts data rate. There s a tme lag between the swtch experencng a traffc load varaton, effect of swtch feedback control, and the occurrence of the new load due to the feedback. Referrng the Fg. 8, and Table 5 t s notced that the CTD between any source and ts correspondng destnaton changes from mnmum value of 0.2 sec to a 13

14 maxmum value of 1.04 sec. Examnng the smulaton results for ACR, SWIR, SWOR, MAT, Q and CTD, whch ndcate the performance of RRM swtch under BL / GT traffc generator, t s seen that the swtch offers the best performance. 14

15 Fg. 4: The Varaton of ACR. 15

16 16

17 Fg. 5: The Varaton of SWIR/SWOR. 17

18 Fg. 6: The Varaton of MAT. 18

19 Fg. 7: The Varaton of Q. 19

20 Fg. 0.1: CTD Usng BL / GT Traffc Generator. 20

21 Table 3: The Values of the ACR (Cells/Sec). ACR1 ACR2 ACR3 ACR4 ACR5 ACR6 Mean Varance Maxmum Mnmum Table 4: The Values of the SWIR, SWOR, MAT, and Q. Mean Varance Maxmum Mnmum SWIR(Cells/Sec) SWOR(Cells/Sec) MAT(Sec) Q Table 5: The Values of the CTD (Sec). CTD (S1_D1) CTD (S2_D2) CTD (S3_D3) CTD (S4_D4) CTD (S5_D5) CTD (S6_D6) Mean Varance Maxmum Mnmum CONCLUSION In ths paper a mathematcal modelng of the BL / GT source traffc 21

22 generator has been carred out. Analytcal results showed that the mnmum values of BL / GT are greater than ther correspondng values of M / M and the Posson mean arrval parameter BL shouldn t be a very large value, because BL BL BL wll, consequently, be very large as well, and the source wll spend most of ts tme sendng only the burst cells wth a smaller number of gap ntervals for BL / GT traffc generator resultng n less-bursty traffc. It can be concluded also that the onental mean arrval parameter GT Exo should be selected between 2 and 30 cells/sec for smulaton of real bursty traffc because t offers hgher peak values of GT. Ths s further supported by the observaton that for GT Exo n the range 30 to 100 cells/sec, the peak values of GT has the least varaton ndcatng smoothest traffc. The analytcal results have been verfed through smulaton for sx sources. References [1] Wllam, H. and Douglas, M. (1990) Probablty Statstcs n Engneerng and Management Scence. John Wley and Sons, Inc. 3 rd ed., New York. [2] Das. T., Keerthpala, W., and Murray, I. (2000) Performance Analyss of Aloha Channels wth Self Smlar Input Traffc. Proceedngs of ATS02, pp , San Dego. [3] Ghanbar, M., Hughes, C., Snclar, M., and Eade, J. (1997) Prncples of Performance Engneerng for Telecommuncaton and Informaton Systems. The Insttute of Electrcal Engneers, London. 22

23 [4] Leon-Garca, A. (1994) Probablty and Random Processes for Electrcal Engneerng. 2nd ed. Addson Wesley Pub. Co. Inc., U.S.A. [5] Sadku, M., and Ilyas, M. (1995) Smulaton of Local Area Networks. CRC Press, Florda. [6] Matsumoto, M., and Nshmura, T. (1998) Mersenne Twster: A 623- Dmensonally Equdstrbuted Unform Pseudo-Random Number Generator. ACM Transactons on Modelng and Computer Smulaton, 8, pp [7] Fahmy, S., Jan, R., Goyal, R. and Vandalore, B. (2202) Far Flow Control for ATM-ABR Multpont Connectons. Journal of Computer Communcatons, 25, pp