COMPLETE BUFFER SHARING WITH PUSHOUT THRESHOLDS IN ATM NETWORKS UNDER BURSTY ARRIVALS Ozgur Aras and Tugrul Dayar Abstract. Broadband Integrated Servces Dgtal Networks (B{ISDNs) are to support multple types of trac such as voce, vdeo, and data. The Asynchronous Transfer Mode (ATM) s the transport technque of choce for B{ISDNs by the standards commttees. In ths mode of operaton, all nformaton s carred usng xed sze packets (called `cell's) so as to share the network among multple classes of trac. Snce multclass trac wll be carred on B{ISDNs, derent qualty of servce requrements wll be mposed by derent applcatons. One type of congeston control for ATM networks deals wth dscardng cells at ATM buers n order to guarantee a prespeced cell loss rate. One bt n each ATM cell header s reserved to assgn the space prorty of cells. Ths bt ndcates whether the gven cell s hgh prorty or low prorty. Prorty cell dscardng s a buer management scheme n whch hgher prorty cells are favored n recevng buer space. An ecent technque for determnng the cells to be dscarded when congeston occurs s the complete buer sharng scheme wth pushout thresholds. In the system under consderaton, there are two classes of trac arrvng to an ATM buer of sze K. Tme s dvded nto xed sze slots of length equal to one cell transmsson tme. The arrval of each trac class to the buer s modeled as an ndependent Interrupted Bernoull Process (IBP). The eects of usng complete buer sharng wth pushout thresholds versus partal buer sharng wth nested thresholds are nvestgated under loads of varyng burstness through smulaton. Key words. Asynchronous transfer mode, buer management, pushout thresholds Introducton Broadband Integrated Servces Dgtal Networks (BISDN) wll support multple types of trac such as voce, vdeo, and data. The Asynchronous Transfer Mode (ATM) s the transport technque of choce for BISDNs by the standards commttees. ATM s a connecton orented transport technque n whch all nformaton s conveyed usng a xed sze packet (called a `cell') for swtchng and transmsson purposes so as to share the network among multple classes of trac. It s well known that for a lmted buer system supportng derent classes of trac, such as an ATM queue, ecent buer management schemes are necessary to mnmze loss rates. One mechansm for buer management s the ntroducton of space prortes among the ncomng trac [], [5], [4]. Ths mples that hgher prorty cells are favored n recevng buer space. One bt n each ATM cell header s reserved to assgn the space prorty of cells. Strateges for determnng whch cells to dscard when congeston occurs are termed space prorty buer management schemes or prorty cell dscardng schemes n the lterature. Here we Department of Computer Engneerng and Informaton Scence, Blkent Unversty, 65 Blkent, Ankara, Turkey (aras@ug.bcc.blkent.edu.tr, tugrul@blkent.edu.tr).
study two space prorty buer management schemes, namely partal buer sharng and complete buer sharng, for the IBP arrvals case. In the partal buer sharng scheme wth nested thresholds [], a common buer s provded for all classes and sharng of the buer s controlled by a set of dscardng thresholds. Let T denote the dscardng threshold for the class trac and assume that class has prorty over class +. If the number of cells n the buer s less than T then buer access for class cells s granted. For the hghest prorty cells, T = K where K s the total buer space. In the complete buer sharng scheme wth pushout thresholds [5], the space prorty s determned accordng to a set of overwrte thresholds [T; T] where T + T = K. A class cell arrvng to a full buer can overwrte a class cell f the number of class cells n the buer s greater than the T threshold, otherwse the class cell s dscarded. Smlarly, a class cell arrvng to a full buer can overwrte a class cell f the number of class cells n the buer s greater than the T threshold, otherwse the class cell s dscarded. Performance measures of nterest for a lmted buer system are average number of cells n the buer at steady state and the assocated loss probabltes. The second secton ntroduces the Markovan model used for the threshold pushout scheme when cells of two derent classes arrve accordng to ndependent Bernoull processes. Secton two also descrbes the IBP arrvals employed n the smulaton of the buer management schemes of ths paper. The motvaton behnd nvestgatng an arrval process such as IBP s that cell arrvals to ATM buers at tmes may follow bursty patterns and the demands of such an arrval process may be qute derent than Bernoull arrvals. Based on the nformaton gven n secton two, a smulaton model s devsed and results are collected usng IBP arrvals. The thrd secton presents these results for the two alternatve buer management schemes. The last secton s comprsed of concludng remarks. The Model The analyss and smulaton results of the complete buer sharng scheme wth pushout thresholds for the Bernoull arrvals case s gven n [5] and therefore we have not ncluded them. In the system under consderaton, there are two classes of trac arrvng to an ATM buer of sze K. Tme s dvded nto xed sze slots of length equal to one cell transmsson tme. The arrval of trac class l (=,) to the buer s modeled as a Bernoull process wth probablty of cell arrval p l n a slot. The states of the correspondng queueng system may be represented by the ordered par (; ), where and are the number of class and class cells n the buer, respectvely. Let k(= + ) denote the total number of cells n the buer at state (; ). Then, a natural state space orderng that places the states wth the same number of total cells n the buer (.e., k) consecutvely, gves rse to a block matrx wth P K k=(k + ) = (K + )(K + )= states. The rst block conssts of the state (,) (.e., the state n whch the buer s empty), the second block has states (,), (,), the thrd block has states (,), (,), (,), and so on. The kth block has k + states. That s, we have the followng orderng: (; ) (; ) (; ) (; ) (; ) (; ) (; ) (; ) (; ) (; ) (K; ) Durng a tme slot, no cells, one cell, or two cells may arrve. If one or two cells arrve, ths happens at the begnnng of a slot. A cell departure occurs by the end of the slot
f the buer has at least one cell at the begnnng of the slot. Hence, an arrvng cell cannot be transmtted before the end of the next slot. Wth these assumptons, a cell s dscarded two cells arrve to a full buer. The pushout threshold for class cells s gven by T and the pushout threshold of class cells s gven by T(= K T). If two cells arrve to a full buer (.e., + = K), then a class cell s dscarded f > T, otherwse a class cell s dscarded f < T. When = T, the lower prorty trac class cell s dscarded. One may vew the system as f there s temporary space to store up to two arrvals whle the buer s full and a decson as to whch class of cell wll be dscarded s made. The state transtons correspondng to complete buer sharng wth pushout thresholds n ATM networks are gven n Table. To smplfy the model, t s assumed that the head of the queue (.e., the cell that wll be leavng the buer at the end of the current tme slot f there was one to begn wth) s a class cell wth probablty =( + ) and t s a class cell wth probablty =( + ). The DTMC correspondng to these assumptons s block trdagonal (wth the excepton of the rst row of blocks) where each dagonal block s trdagonal and has a derent block sze. Dependng on the selected threshold, the nonzero elements n the last row of blocks change makng t very dcult to apply analytcal soluton technques to such a system wth control. The performance measures of nterest n a buer management scheme are average number of class and class cells n the buer and correspondng loss probabltes at steady state. These measures may be found by computng the steady state probablty vector (whch turns out to be the statonary vector n ths case) through numercal soluton technques []. Generatng the DTMC of the model wth IBP arrvals s not as natural and easy as t s for the case of Bernoull arrvals. For ths reason, we conne ourselves to a smulaton study and analyze the IBP arrvals case. IBP s governed by a two{state Markov chan, see Fgure. These two states are the busy state and the dle state. p busy - q dle q - p Fgure : Markov chan for IBP: Tme s thought of as beng slotted and the state changes may only occur at ponts that are multples of a slot duraton, whch s bascally the constant nterarrval tme durng a busy perod. No arrvals occur f the process s n the dle state. Arrvals occur n a Bernoull fashon f the process s n the busy state. If the process s n the busy state, then n the next tme slot t remans n the busy state wth probablty p or changes to the dle state wth probablty p. Smlarly, f the process s n the dle state, then n the next tme slot t remans n the dle state wth probablty q or changes to the busy state wth probablty q.
Table State transtons for the threshold pushout scheme n ATM networks (Bernoull arrvals case) Block State transton Condton Event Probablty transton from (; ) to k! k ( ; ) > No arrvals, class departure (; ) > No arrvals, class departure k! k ( ; + ) > Class arrval, class departure ( ; + ) > ; < T ; Class, arrvals, + = K class departure ( ; + ) > ; = T ; Class, arrvals, + = K; T < T class departure + ( p )( p ) + ( p )( p ) + ( p )p + p p + p p (; ) = ; = ; K Class arrval, ( p )p no departure (; ) > Class arrval, class departure (; ) > Class arrval, class departure (; ) > ; > T ; Class, arrvals, + = K class departure (; ) > ; = T ; Class, arrvals, + = K; T T class departure (; ) < < T ; + = K Class, arrvals, + p ( p ) + ( p )p + p p + p p + p p class departure (; ) > ; = T ; Class, arrvals, + p + = K; T < T class departure (; ) = ; = ; K = Always ( + ; ) > Class arrval, class departure ( + ; ) > T ; + = K Class, arrvals, class departure ( + ; ) > ; = T ; Class, arrvals, + = K; T T class departure + p ( p ) + p p + p p k! k + (; + ) = ; = ; K Class arrval, ( p )p no departure (; + ) > ; + < K Class, arrvals, class departure + p p (; + ) = ; = ; K = Class, arrvals, p p T = no departure ( + ; ) = ; = ; K Class arrval, p ( p ) no departure ( + ; ) > ; + < K Class, arrvals, class departure + p p ( + ; ) = ; = ; K = Class, arrvals, p p T = departure k! k + ( + ; + ) = ; = ; K > Class, arrvals p p no departure IBP s characterzed by the arrval rate (r) and the square coecent of varaton of the nterarrval tme (C ). Most of the tme, these parameters are suppled as nput parameters and one can obtan the state transton probabltes p and q by substtutng r and C n the followng formulae (see [], pp. {4):
where q q = b b a[r(r ) + C + ] ; a q( r) + r p = ; r a = r + C + ; b = r(r ) + C + : For example, r = :6, C = gves p = :969 and q = :954. The hgher the arrval rate (r), the larger the number of arrvals durng a xed tme nterval happens to be. The hgher the coecent of varaton of the nterarrval tme (C ), the burster the arrval process becomes. In the next secton, we gve smulaton results for the IBP arrvals case. Smulaton Results Ths secton presents smulaton results for complete buer sharng wth pushout thresholds (Model ) and partal buer sharng wth nested thresholds (Model ) under IBP arrvals wth varyng degrees of burstness. In the followng, r denotes the arrval rate of class cells and r denotes the arrval rate of class cells. One can thnk of the arrval rate of a gven class as beng the load oered to the ATM buer for that class of cells. T and T refer to the thresholds of class and class cells. The smulaton s run for 5 mllon tme slots.. Model The rst model we consder s the complete buer sharng scheme wth pushout thresholds for ATM networks dscussed n x, but ths tme havng IBP cell arrvals. Here we use the followng parameters: K = 7; r = :5; r f:; :; : : : ; :9g; C = ; ; : In ths model, the value of C s set to,, and to vary the burstness of the arrval process. The arrval rate of class cells s xed at.5, whereas the arrval rate of class cells assumes values that are multples of. wthn the nterval [,]. T + T = K. Fgures {4 show average number of class and class cells n the buer for C = ;, and. In Fgure, one can see that the results obtaned for C = are n good agreement wth the results obtaned for the Bernoull arrvals case n [5]. The reason s that C = corresponds to low burstness mplyng a Bernoull lke behavor. When the plots n Fgures {4 are crosscompared, we see that for xed r, the derence between average number of class cells for `adacent' threshold choces of T ncreases wth ncreasng burstness. Wth adacent threshold choces of T, we mean two choces of T that der by. In other words, the choce of a sutable threshold value becomes more mportant, yet an easer task, as the arrval process gets ncreasngly bursty. The same observaton also holds for class cells. As a result, the choces of threshold values T and T are more determnstc for hgher burstness. We should remark that the threshold parameter s not an nuental factor when arrval rate of class cells s less than.5.
.5 7 Average number of class '' cells.5.5 [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7] Average number of class '' cells 6 5 4 [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7].5....4.5.6.7.8.9....4.5.6.7.8.9 Arrval rate of class '' cells Arrval rate of class '' cells Fgure : Average number of cells for the threshold pushout scheme; C = : Average number of class '' cells.5.5.5 [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7] Average number of class '' cells 7 6 5 4 [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7].5....4.5.6.7.8.9 Arrval rate of class '' cells....4.5.6.7.8.9 Arrval rate of class '' cells Fgure : Average number of cells for the threshold pushout scheme; C = :
Average number of class '' cells.5.5.5 [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7] Average number of class '' cells 7 6 5 4 [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7].5....4.5.6.7.8.9 Arrval rate of class '' cells....4.5.6.7.8.9 Arrval rate of class '' cells Fgure 4: Average number of cells for the threshold pushout scheme; C = : An analogous stuaton does not exst n the Bernoull arrvals case. Another observaton s that, when C ncreases and r :7, average number of class cells decreases; but, for r :4, average number of class and class cells both ncrease for ncreasng burstness. In Fgures 5{7, we gve the loss probabltes of class and class cells n the buer for the threshold pushout scheme when C = ;, and. All three plots correspond to arrval processes burster than Bernoull, hence t s not surprsng to see hgher class cell loss probabltes than those of Bernoull arrvals [5]. Recall that r s xed at.5. Interestngly, the loss probabltes of class cells ncrease lnearly wth the rate of class cell arrvals for C = (see Fgure 7). A smlar observaton s made for lower bursty IBP arrvals (see Fgures 5{6) when r :6. When C =, the loss probabltes of class cells are consderably hgh (but stll less than.) even for lower arrval rates of class cells. If T s closer to K, the loss probablty of class cells approaches one as r ncreases. Ths s because even though class cells are assgned a lower pushout threshold (.e., T = K T s small), they contnue to arrve at a rate.5. On the other hand, loss probabltes of class cells are slghtly hgher than those of Bernoull arrvals for lower bursty IBP arrvals (see Fgures 5{6). However, when C = and r s low, loss probabltes of class cells are extremely hgher than those of Bernoull arrvals and lower bursty IBP arrvals (see [5] and Fgure 7). Hgh burstness seems to have a detrmental eect on the loss probablty of class cells. The probablty curves for C = are almost horzontal lnes ntersectng the y{axs at hgher values for nonzero T (see Fgure 7). An ntutve explanaton s the followng. When C s very hgh, there are long bursts of class cell arrvals and even though the arrval rate of class cells s
.9.5 Loss probabltes of class ''cells.8.7.6.5.4.. [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7] Loss probabltes of class ''cells.45.4.5..5..5. [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7]..5....4.5.6.7.8.9 Arrval rate of class'' cells....4.5.6.7.8.9 Arrval rate of class'' cells Fgure 5: Cell loss probabltes for the threshold pushout scheme; C = : Loss probabltes of class ''cells.9.8.7.6.5.4.. [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7] Loss probabltes of class ''cells.5.45.4.5..5..5. [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7]..5....4.5.6.7.8.9 Arrval rate of class'' cells....4.5.6.7.8.9 Arrval rate of class'' cells Fgure 6: Cell loss probabltes for the threshold pushout scheme; C = :
Loss probabltes of class ''cells.9.8.7.6.5.4.. [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7] Loss probabltes of class ''cell.5.45.4.5..5..5. [7,] [6,] [5,] [4,] [,4] [,5] [,6] [,7]..5....4.5.6.7.8.9....4.5.6.7.8.9 Arrval rate of class'' cells Arrval rate of class'' cells Fgure 7: Cell loss probabltes for the threshold pushout scheme; C = : xed at.5, the buer stll lls up for lower r causng class cells to be dscarded wth hgh probabltes. The loss probabltes of class cells for r :4 and the loss probabltes of class cells for r : are almost zero. Fnally, as expected, the loss probabltes of class and class cells are both zero for T = and T =, respectvely.. Model The second model we consder s the partal buer sharng scheme for ATM networks dscussed n x, but ths tme havng IBP cell arrvals. Here we use the followng parameters: K = 7; r = :5; r f:; :; : : : ; :9g; C = ; ; : The value of C s set to,, and to acheve arrval processes of varyng burstness. The arrval rate of class cells s xed at.5, whereas the arrval rate of class cells assumes values that are multples of. wthn the nterval [,]. T = K and T( K) s the nested threshold value of class cells. Fgures 8{ show average number of class and class cells n the buer for C = ;, and. In these gures, we see that an ncrease n burstness for r :4 results n an ncrease n the derence between average number of class cells for adacent T choces. One can also see that the derence between average number of class cells for adacent T choces decreases as burstness ncreases. As a result, for low r, the choce of a sutable threshold value becomes crtcal especally for average number of class cells n the buer as burstness ncreases. For T >, the ncrease n average number of
Average number of class '' cells.5.5.5 [7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7] Average number of class '' cells.5.5.5 [7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7].5.5....4.5.6.7.8.9 Arrval rate of class '' cells....4.5.6.7.8.9 Arrval rate of class '' cells Fgure 8: Average number of cells for partal buer sharng; C = : Average number of class '' cells.5.5.5 [7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7] Average number of class '' cells.5.5.5 [7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7].5.5....4.5.6.7.8.9....4.5.6.7.8.9 Arrval rate of class '' cells Arrval rate of class '' cells Fgure 9: Average number of cells for partal buer sharng; C = :
Average number of class '' cells.5.5.5 [7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7] Average number of class '' cells.5.5.5 [7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7].5.5....4.5.6.7.8.9 Arrval rate of class '' cells....4.5.6.7.8.9 Arrval rate of class '' cells Fgure : Average number of cells for partal buer sharng; C = : class cells becomes more dramatc as burstness ncreases. Hence, a small ncrease n r causes a relatvely large ncrease n average number of class cells when arrvals become burster. Snce the nested threshold value of (lower prorty) class cells cannot exceed the nested threshold value of (hgher prorty) class cells, loss probabltes of class cells for all combnatons of r and C are zero. That s, class cells are never dscarded n partal buer sharng wth nested thresholds as long as T s set to K and class cells have lower prorty. For that reason, we only provde three plots, those correspondng to the loss probabltes of class cells, n Fgures {. As the rate of class cell arrvals ncreases, the loss probablty of class cells approach.5 except for T =, for whch the loss probablty levels o around.7. For C = (see Fgure ), loss probabltes of class cells for all values of nested thresholds are larger than.4. In all three of the plots, loss probabltes of class cells are larger than. for r :5. 4 Concluson In ths paper, we have smulates and analyzed the performance of partal buer sharng wth nested thresholds and complete buer sharng wth pushout thresholds n an ATM buer for IBP arrvals. In the experments, we have used a class cell arrval rate of.5. We observe that average number of class and class cells n the buer under bursty arrvals for both schemes der consderably from the Bernoull arrvals case. Furthermore, when we use partal buer sharng wth nested thresholds, loss probabltes of class cells are adversely aected. For the threshold pushout scheme wth bursty and lower rates of class cell arrvals, average number of class cells n the buer s greater than those of Bernoull
[7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7] [7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7].8.8.7.7 Loss probabltes of class '' cells.6.5.4.. Loss probabltes of class ''cells.6.5.4........4.5.6.7.8.9....4.5.6.7.8.9 Arrval rate of class'' cells Arrval rate of class'' cells Fgure : Cell loss probabltes for partal buer sharng; C = and C = : [7,] [7,] [7,] [7,4] [7,5] [7,6] [7,7].8.7 Loss probabltes of class '' cell.6.5.4.......4.5.6.7.8.9 Arrval rate of clas'' cells Fgure : Cell loss probabltes for partal buer sharng; C = :
arrvals; but, for hgher rates of class cell arrvals, average number of class cells n the buer s less than those of Bernoull arrvals. Besdes, average number of class cells n the buer s greater than those of Bernoull arrvals for both lower and hgher rates of class cell arrvals. In terms of loss probabltes, class cells seem to be much more aected by ncreasng burstness n the IBP arrvals. Nevertheless, there s stll a combnaton of pushout threshold values for whch the loss probablty of class cells may be kept below.. As for partal buer sharng, bursty and lower rates of class cell arrvals result n average number of class cells n the buer beng greater than those of Bernoull arrvals. Furthermore, for hgher rates of class cell arrvals, average number of class cells n the buer for both Bernoull and IBP arrvals s almost the same. Average number of class cells n the buer s less than those of Bernoull arrvals for lower rates of class cell arrvals. However, for hgher rates of class cell arrvals, average number of class cells n the buer for both Bernoull and IBP arrvals s almost the same. Loss probabltes of class cells are zero whereas those of class cells are larger than.4 for all nested threshold values when IBP arrvals are extremely bursty. Furthermore, t s not possble to obtan class cell loss probabltes that are less than. when the arrval rate of class cells s above.5. We recommend complete buer sharng wth pushout thresholds as the buer management scheme of choce, because t provdes a better balance between loss requrements of class and class cells. References [] T. Dayar, Resequencng of Messages n a Queueng System wth Heterogeneous Servers Under Varous Schedulng Polces, Master's Thess, North Carolna State Unversty, Ralegh, NC, 99. [] D. Petr and V. Frost, Nested threshold cell dscardng for ATM overload control: optmzaton under cell loss constrants, Proceedngs of IEEE INFOCOM '9, pp. 4{ 4. [] W. J. Stewart, Introducton to the Numercal Soluton of Markov Chans, Prnceton Unversty Press, New Jersey, 994. [4] S. Sur, D. Tpper and G. Meempat, A comparatve evaluaton of space prorty strateges n ATM networks, Proceedngs of IEEE INFOCOM '94, pp. 56{5. [5] D. Tpper, S. Pappu, A. Collns and J. George, Space prorty buer management for ATM networks, n Asynchronous Transfer Mode Networks, Yanns Vnots and Raf O. Onvural, eds., Plenum Press, New York, 99, pp. 57{66.