A Performance Model of Space-Division ATM Switches with Input and Output Queueing *

Transcription

1 A Perforance Model of Sace-Dvson ATM Swtches wth Inut and Outut Queueng * Guogen Zhang Couter Scence Deartent Unversty of Calforna at Los Angeles Los Angeles, CA 94, USA Wlla G. Bulgren Engneerng Manageent Progra Unversty of Kansas Lawrence, KS 6645, USA Vctor L. Wallace Deartent of Electrcal Engneerng and Couter Scence Unversty of Kansas Lawrence, KS 6645, USA Abstract An analytc odel for the erforance evaluaton of sace-dvson Asynchronous Transfer Mode (ATM) swtches s resented. Ths odel assues that the swtch has a fxed caacty of, where ( s the nuber of trunks). Other ortant araeters nclude arrval rate and buffer szes. uercal solutons for the axu throughut, cell delay, and cell loss robablty are gven wth sulaton beng utlzed n order to valdate the analytc odel. For ndeendent and dentcal Bernoull arrvals, the study shows that the contenton rocesses can be odeled as dscrete M/D/ (FIFO or Rando) queues, whle nut queues can be odeled by Geo/G/ queues, and the outut queues are G [x] /D/ queues. A closed-for aroxaton for cell delay when > s gven. The result shows that the erforance of swtches wth a sall caacty can aroach that of outut queueng. The odel and result can be used for swtch desgn analyss and hgher layer erforance odels.. Introducton. Analytc erforance odels: background The erforance analyss of ATM swtches [,,3] has been attractng great attenton n the recent years, and a nuber of odels have been roosed and analyzed [4-5], wth new odels stll aearng. The ortant odel araeters of the current research on ATM swtches nclude caacty, nut/outut bufferng, buffer sze, selecton (arbtraton) olcy, synchronous or asynchronous oeraton ode, and varous traffc atterns. Ths aer utlzes classcal queueng theory to odel the behavor of sace-dvson ATM swtches. Fgure. shows a general odel of sace-dvson ATM swtches.... Swtch Fabrc (Caacty ).. : nuber of trunks : caacty of the swtch Fgure. General odel of an ATM swtch In a sace-dvson ATM swtch fabrc, a luralty of aths s rovded between the nut and outut orts. These aths oerate concurrently so that any cells ay be transtted across the swtch fabrc at the sae te. A sace-dvson fabrc can be further classfed as sngle-ath or ultle-ath accordng to the nuber of aths between any gven nut and outut ar (ths nuber s often called the caacty of the swtch). Sacedvson swtches usually oerate n synchronous ode, all cells n the nut orts arrvng at the swtch fabrc wth ther headers algned to slfy the desgn. In a te-dvson based swtch fabrc, cells flow across a sngle councaton hghway shared by all nut and outut orts. Ths research s aed at sace-dvson odels. The rary erforance crtera of an ATM swtch are throughut, cell delay, and cell loss robablty. Defntons of each are as follows: Throughut of an ATM swtch s the ercentage of total cells leavng the outut to the total te slots. Cell delay s the te fro a cell arrval to the nut ort, to the cell dearture fro the outut ort. Cell loss robablty s the robablty that a cell wll be lost n the swtch fabrc due to the lossy contenton resoluton echans. * The work was done when G. Zhang was at Unversty of Kansas.

2 The artcle by Hluchyj, et al. [5] contans an excellent revew of queueng n ATM swtches. In general, the tycal odels of ATM swtches are categorzed nto three broad classes, naely, outut queueng, nut queueng, and nut and outut queueng. Ths research exlores a odel of nut and outut queueng.. Analytc erforance odels: our research Based on research resented, our study was desgned to nvestgate a general analytc odel of an ATM swtch. The fxed caacty of the swtch and the buffer szes are ortant araeters that dstngush ths research fro ror research. The ortant erforance crtera nclude cell delay, throughut and cell loss robablty. Our odel s dfferent fro the nut and outut queueng odel resented n [6] n that backressure due to outut buffer beng full s not consdered (contenton olcy), and cell loss robablty s calculated. In [6], the caacty of the swtch s not fxed and backressure s aled when the outut buffers are full (no cell loss s ossble). It s our belef that our odel reflects the realty of any ATM desgns and therefore s of ractcal sgnfcance.. A odel for nut and outut queueng For ths odel, the followng assutons are ade: () Cell arrvals on the nuts are ndeendent and dentcal Bernoull rocesses. In any gven te slot, the robablty that a cell wll arrve on a artcular nut s, where <. () Each cell has equal robablty / of beng addressed to any gven outut, and successve cells are ndeendent. (3) The swtch caacty s, where. That s, f there are k cells at the heads of nut queues and destned for the sae outut ort, only cells can be routed to the outut queue at one te f k>. The others wat for next chance. The selecton olcy can be rando or FIFO. (4) There are B buffers at each outut, where B can be nfnty. There are nfnte buffers at each nut. It s reasonable to assue that B. otce that assutons () and () are the sae as the revous odel assutons. Assuton (3) dfferentates the selecton (arbtraton) olcy between rando and FIFO. References [4,6,7] dscuss the dfferences between these two olces. Ther study and our sulaton result [6] show that the dfference between these two olces s neglgble and we wll focus our analyss on FIFO. The nut queueng s a secal case of our odel wth =, and the outut queueng, =. The analyss s dvded nto two searate cases. Case I s when both and B are large (the ltng case). Case II s when s large and outut buffer sze B s fnte (lted). In both of these stuatons, the contenton rocess, the outut queue and the varous erforance crtera, whch nclude the cell delay, the throughut, and ossble cell loss are consdered.. Case I: ltng analyss (FIFO wth and B nfnte) In ths secton, we resent an aroxate odel for large and large outut buffer sze B. The analyss s dvded nto three sectons: the contenton rocess; the outut queue; and the erforance crtera. Contenton rocess. Durng each te slot, cells at the heads of nut queues and destned for the sae outut contend to be routed to the outut. These contendng cells for a logcal queue. There s one such contenton queue for each outut and n steady-state each s syetrc and statstcally dentcal. For a tagged outut contenton queue, defne: A : the nuber of cell arrvals to the heads of nut queues and destned for the tagged outut durng the th te slot; C : the nuber of cells n the tagged queue at the end of the th te slot. We have C = ax( C,) + A The ltng dstrbuton of A s Posson [4,6]: k e ak = Pr[ A = k ] =, k =,,, L () k! Usng ths result, the contenton rocess can be odeled as an M/D/ queue. The M/D/ queue s a Markov chan wth ts state reresented by the value of C. The transton robablty fro state j to state l n adjacent slots s gven by a l-j+ for j, or a l for j<. Denote Q the state transton atrx of the Markov chan, whch s a a a a3 L M M M M M O a a a a3 L Q = a a a a3 L + a a a L + a a L M M M M M O Let the steady-state robablty = l Pr[ C = j ], and = (,,,...) j. We have the balance equatons

3 ( I Q) =, and = () where I s the dentty atrx and s the colun vector wth all eleents equal to. For the syste under steady state, there s a unque soluton for ths equaton syste. Let n = (,,,3,...,,...) and n = (,,4,9,...,,...), the frst and second oents of nuber n the contenton rocess are C = n T (3) and C = n T. (4) The frst oent of te n the contenton rocess s gven by Lttle's result as C S = (5) and the second oent of te s gven by the generalzaton of Lttle's result [7] as S ( C C) =. (6) Outut queue. For the outut queue, the arrval s the dearture fro the contenton rocess, whch s odeled by a dscrete-te M/D/ queue wth average dearture rate. Therefore, the nuber of cell arrvals E to the outut queue has rando batch sze fro to wth robablty dstrbuton gven by k k =,,..., ; d k = Pr[ E = k ] = j k = ; (7) j = k > ; Denote V the nuber of cells n the outut queue at the end of the th te slot. We have V = ax( V, ) + E. (8) Therefore, the outut queue s a dscrete-te G [x] /D/ queue. It can be odeled by a dscrete-te Markov chan wth ts state reresented by the value of V. The transton robablty fro state j to state l n adjacent slots s gven by d l-j+ for j, or d l for j=. Denote R the state transton atrx of the Markov chan, whch s d d d L d L d d d L d L R = d d L d d L. (9) 3 d L d d L M M M M L M M O Let the steady-state robablty v = l Pr[ V = j ], and j v = ( v, v, v,...). We have the balance equatons v( I R) =, and v = () For the syste under steady state, there s a unque soluton for ths equaton syste as follows. v = ; d v = v; d d d d j v = v v d d d v j, For j= () The average nuber n the outut queue s gven by T V = vn. By Lttle's result, the ean te n the outut queue, denoted by D out, s V D out =. () Perforance crtera. For a tagged cell, the total delay n the syste contans three coonents: watng n the nut queue untl ovng to the head of the queue; te n the contenton rocess at the head of the queue; and te n the outut queue. We denote these three delay coonents by D n, D c, and D out, resectvely. For the nut queue, the arrval rocess s Bernoull, that s, the nterarrval te s geoetrcally dstrbuted. The servce te s the te n the contenton rocess, whch has general dstrbuton wth the frst and second oents of te obtaned as equatons (5) and (6) above. Therefore the nut queue s a dscrete-te Geo/G/ queue. The average watng te s gven by ( S S) D n = (3) ( S) And we have D c = S. The average te n the outut queue s gven by (). The average total delay becoes ( S S) D = D D D S V n + c + out = + +. (4) ( S) In order to obtan the axu throughut, steady state s assued. It s requred that both the nut queue and the outut queue be n steady state. For the nut queue, fro (3), the condton s S >. (5) Snce S s a functon of, whch cannot be exressed n closed for, we need to use nuercal ethod to get the value. For the outut queue, whch s G [x] /D/, the requreent s, because s the ean arrval rate. Ths condton s subsued by the assuton and (5). Therefore the axu throughut s deterned by solvng (5) for the uer bound of.. Case II: analyss for large and fnte B In ths secton, we resent an aroxate odel for

4 large and lted outut buffer B. In ractce, and B are always lted. Assue that there are only a fnte nuber of B buffers avalable at each outut ort. The outut queue becoes dscrete-te G [x] /D//B queue. The cells wll be lost f not enough outut buffers are avalable to accoodate all cells. Contenton rocess. When s large, the arrval to the contenton queue can be aroxated by the sae Posson arrval dstrbuton as n () [6]. The contenton rocess s odeled by an M/D// queue. To calculate the steady-state robabltes, we odel ths M/D// queue by a fnte state Markov chan. The state transton atrx Q of ths Markov chan becoes a a a a3 L a a Q = M + + M M M M M O M M a a a a3 L a a a a a a3 L a a a a a L a a 3 a a L a a 3 M M M M O M M L a a ( + ) ( + ) Let the steady-state robablty vector be = (,,,..., ). The balance equatons are ( I Q) =, and = (6) Fro ths equaton syste, can be solved nuercally. The frst and second oents of the te n the contenton rocess are S = (7) and S = ( ) (8) Outut queue. The nuber of cell arrvals E to the outut queue has rando batch sze fro to wth robablty dstrbuton gven by k k =,,..., ; dk = Pr[ E = k] = j k = ; (9) j= k > ; The outut queue s a dscrete-te G [x] /D//B queue. The state transton atrx R of ths fnte state dscretete Markov chan becoes d d d L d L d d d L d L d d L d d L R = 3 d L d d L d M M M M O M M O M B L L d ( B+ ) ( B+ ) Let the steady-state robablty vector be v = ( v, v, v,..., v B ). The balance equatons are v( I R) =, and v = () We can get the nuercal soluton for v. The ean nuber of cells n the outut queue s B V = v. () Perforance crtera. The robablty of the outut queue beng dle s v. Therefore the effectve throughut fro the outut queue s T = - v. () When the outut buffers are full or free buffers are not enough, all or art of the new cell arrvals to the outut queue wll be lost. The cell loss robablty s therefore equal to T v Pr[ cell loss] = =. (3) By Lttle's result, the average watng te n the outut s V Dout = (4) T The average cell delay s gven by ( S S) V D = Dn + Dc + Dout = + S + ( S) v. (5) where S, S, and V are gven by (7), (8) and (), v s the robablty of the outut queue beng dle obtaned fro ()..3 Sle nuercal ethod for lted case We need to solve equaton systes (6) and () to get the values for throughut, cell loss robablty, and cell delay.

5 Soluton for the contenton rocess. For the balance equatons of the contenton rocess (6), one sle way to get the soluton s fro the relatonsh + = Q = l (6) f an arorate ntal vector s gven, for nstance, = (,,..., ) Because =, by nducton, + = Q = =. The sle teraton of the frst equaton n (6) wll converge radly and result n the accurate soluton for (6). Soluton for the outut queue. We can wrte out the recurrence relaton of the soluton to the balance equaton of the outut queue () as follows. = ( d), d = (( d) d d j ), for ; d j j= = (( d) d ), for < B; d j j j= + where s deterned by B =. (7) Fro (7), we see that = c for soe constants c. We can easly get c by settng = fro (7) and B = + c. (8) We can also use the sae ethod for (6) above to get nuercal results for (). 3. Results Sulaton odels [6] were used to valdate the analytc odel. The result shows that the analytc odel s very accurate under ts assutons. In the followng, soe of the results are lsted. They are dvded nto three sectons: throughtut, cell delay, and cell loss robablty. 3. Throughut The axu throughut for varous values of fro both nuercal analyss and sulaton s lsted n Table. Table. Caacty versus ax. throughut (for =) Caacty () Throughut (uercal result) Throughut (Sulaton) (not avalable) (not avalable) (not avalable) For exale, consder trunks wth 6.8 Mb/s (STM-3) rate. In order to suort a 9% throughut,.e. a rate of 6.8 * 9% = 56 Mb/s, the swtch caacty ust be greater than. Slarly, f we consder the swtch cycle for trunks wth 55.5 Mb/s (STM-) rate, and we want to suort 9% throughut (4 Mb/s) wth seed-u swtch fabrcs, then the swtch cycle needs to be /3 of the te slot (>),.e., about.9 sec. For another exale, f a sace-dvson swtch s to suort 6 Mb/s rate on a 6.8 Mb/s rate trunk (throughut=.9645), the swtch caacty should be at least Cell delay Delay te n the nut queue (ncludng contenton te), n the outut queue, and total cell delay fro nuercal analyss for dfferent and s lotted n fgures, 3, and 4. When becoes large, the swtch erforance wll aroach that of outut bufferng. The ean cell delay n the syste for > can be aroxated by delay n an M/D/ syste wth arrval rate and constant servce te (one slot er cell),.e., D + ( ) The coarson of ths aroxaton wth nuercal result s lsted n Table and sketched n Fgure 5. Snce one te slot s 53 bytes, the te slot for 55.5 Mb/s rate s about.73 s, and.68 s for 6.8 Mb/s rate. If swtches n the network are under noral load (about 8% of axu throughut), then the delay ntroduced by a swtch s only several te slots long, and coarable wth cell roagaton delay n several kloeters dstance.

6 3.5 = = =3 =4 = = = =3 =4 =5 Te.5 4 Te Arrval rate Fgure. Te slots n nut queues(ncludng contenton te) Arrval rate (B=5) Fgure 4. Total cell delay (te slots) = = =3 =4 = Ar. =3,B=5 =4,B=6 Te 3 4 Te Arrval rate (B=5) Fgure 3. Te slots n outut queues 3.3 Cell loss robablty Cell loss robablty for varous values of and B fro nuercal analyss s lsted n Table 3. If the cell loss robablty needs to acheve -7 under the load of axu throughut of 8%, the outut buffers should be at least 6, where s the caacty of the swtch. And the buffer sze needs to be uch larger for eak rate n bursty traffc to revent cell loss Arrval rate () Fgure 5. Cell delay fro aroxaton and nuercal analyss 4. Dscusson and future research Our odel s a straght forward odel of the oeraton of swtches. The one exceton s the arrval rocess, whch s confned to Bernoull. The sulaton result shows that the analytc odel s an accurate odel of swtches under the assutons. Because the exact closed-for soluton are very

7 dffcult to obtan, an aroxate forula for cell delay when > s gven and can be aled easly. The alcablty of the odel deends on the valdty of the assutons of arrval rocess and rando destnaton of cells. In a networked envronent, the arrval to a trunk of a swtch wll be a erger of a large nuber of connectons fro any dfferent sources, to any dfferent destnatons. By vrtue of Pal- Khntchne's theore, ths erged rocess wll aroach a Posson rocess n contnuous stuaton, whle n dscrete te, we conjecture that Bernoull s the ltng rocess. The result fro [] s addtonal evdence that the assutons are arorate n any cases. The odel can be easly extended to non-unfor traffc. And odels wth other traffc atterns wll be exlored further. In artcular, we wll be usng algebrac queueng theory and sulaton odels to better odel the traffc atterns and analyze the statstcal and dynac behavor of swtches. Further studes wll nvestgate ore erforance ssues wthn the hgher layers of ATM and B-ISD systes. Table. Mean te fro dscrete M/D/ queue vs. cell delay fro nuercal result for =,3,4 Table 3. Rate() Ar. Te =, B=4 =3,B=5 =4,B= (ot Alcable) Cell loss robablty wth varous buffer szes(b) caacty () arrval rate () buffer sze B= buffer sze B=4 buffer sze B=6 all

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