Using Traffic Regulation to Meet End-to-End Deadlines in ATM Networks

Transcription

1 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER Usng Traffc Regulaton to Meet End-to-End Deadlnes n ATM Networks Amtava Raha, Sanjay Kamat, Xaohua Ja, Member, IEEE Computer Socety, and We Zhao, Member, IEEE AbstractÐThs paper consders the support of hard real-tme connectons n ATM networks. In an ATM network, a set of hard realtme connectons can be admtted only f the worst case end-to-end delays of cells belongng to ndvdual connectons are less than ther deadlnes. There are several approaches to managng the network resources n order to meet the delay requrements of connectons. Ths paper focuses on the use of traffc regulaton to acheve ths objectve. Leaky buckets provde smple and userprogrammable means of traffc regulaton. An effcent optmal algorthm for selectng the burst parameters of leaky buckets to meet connectons' deadlnes s desgned and analyzed. The algorthm s optmal n the sense that t always selects a set of burst parameters whose mean value s mnmal and by whch the delay requrements of hard real-tme connectons can be met. The exponental sze of the search space makes ths problem a challengng one. The algorthm s effcent through systematcally prunng the search space. There s an observed dramatc mprovement n the system performance n terms of the connecton admsson probablty when traffc s regulated usng ths algorthm. Index TermsÐATM network, hard real-tme communcaton, network delay analyss, traffc regulaton. æ 1 INTRODUCTION THERE s a growng nterest n the applcaton of ATM networks for dstrbuted Hard Real-Tme (HRT) systems. In a dstrbuted HRT system, tasks are executed at dfferent nodes and communcate among themselves by exchangng messages. For successful operaton of the system, the messages exchanged by tme-crtcal tasks have to be delvered by certan deadlnes. Examples of such systems nclude supervsory command and control systems used n manufacturng, chemcal processng, nuclear plants, telemedcne, and warshps. Ths paper addresses the ssue of guaranteeng end-to-end deadlnes of tme-crtcal messages n ATM networks that support dstrbuted HRT systems. ATM s a connecton-orented technology n whch messages are packetzed nto fxed-sze cells. Therefore, guaranteeng message deadlnes s tantamount to ensurng that the worst case end-to-end delay of a cell does not exceed ts deadlne. To provde such guarantees, three orthogonal approaches can be taken: 1. route selecton for connectons; 2. output lnk schedulng at ATM swtches; 3. traffc regulaton at the User Network Interface (UNI). The frst approach selects approprate routes for connectons such that the delays are wthn bounds. The scope of. A. Raha s wth Fjtsu Software Corp., USA. E-mal: amtava@fsc.fujtsu.com.. S. Kamat s wth Bell Laboratores, 101 Crawfords Corner Road, Room 4C- 510, Holmdel, NJ E-mal: sanjayk@dnrc.bell-labs.com.. X. Ja s wth the Department of Computer Scence, Cty Unversty of Hong Kong, Kowloon, Hong Kong. E-mal: Ja@cs.ctyu.edu.hk.. W. Zhao s wth the Department of Computer Scence, Texas A&M Unversty, College Staton, Texas. E-mal: Zhao@cs.tamu.edu. Manuscrpt receved 2 Sept. 1997; revsed 28 May For nformaton on obtanng reprnts of ths artcle, please send e-mal to: tc@computer.org, and reference IEEECS Log Number ths approach s lmted because typcal ATM networks for HRT applcatons are LANs. The second approach focuses on schedulng at the ATM swtches' output lnks where traffc from dfferent connectons s multplexed. Smlar to CPU schedulng, classcal real-tme schedulng polces, such as the Frst Come Frst Serve (FCFS), Earlest Deadlne Frst (EDF), Generalzed Processor Sharng (GPS), and Far Queung (FQ), are employed [2], [12], [13], [16], [23], [24], [25]. However, most commercally avalable swtches use a hgh prorty queue for HRT connectons and ths queue s served n a FCFS manner. Ths paper focuses on the thrd approach: controllng the network delays by regulatng the nput traffc of each connecton. Regulatng the nput traffc can smooth the burstness of the traffc, whch tends to reduce the adverse mpact of burstness on the end-to-end delays of other connectons. Most of the exstng ATM networks provde for traffc regulaton at the UNI. It s relatvely easy to tune the regulaton parameters as desred. Ths s justfcaton for focusng on usng the traffc regulator as an access control mechansm for HRT ATM networks. It must be noted that all three approaches are mportant and that they are complementary. The results of usng traffc regulaton complement the prevous work on route selecton and output lnk schedulng. The dea behnd traffc regulaton s to regulate a connecton's traffc so that t has a lower mpact on the delays of cells of other connectons. When two or more connectons are multplexed on a sngle lnk, an ncrease n burstness of one connecton's traffc adversely mpacts the delays of the cells of others. Regulatng a connecton's traffc makes the traffc less bursty, thereby reducng the delays of other connectons' cells. However, regulatng a connecton's traffc may delay some of ts own cells. Therefore, t s mportant to choose an approprate degree of traffc /99/$10.00 ß 1999 IEEE

2 918 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 regulaton so that all connectons can meet ther end-to-end deadlnes. In ths paper, we study the mpact of nput traffc regulaton on the worst case end-to-end delays experenced by cells of a set of hard real-tme connectons. In partcular, we consder the leaky bucket regulator. The degree of regulaton of a connecton depends on the leaky bucket parameters assgned to t. The degree of traffc regulaton chosen for a connecton affects not only the delays of that connecton, but also those of others sharng resources wth that connecton. Thus, the leaky bucket parameters for the entre set of connectons must be assgned carefully to ensure that every connecton meets ts end-to-end deadlne. Berger proposed an analyss model of leaky bucket regulators n [1]. In hs work, Berger examned a sngle leaky bucket, analysng the job blockng probabltes (or system throughput) versus parameters such as job arrval patterns, capacty of token bank, and sze of job buffer. In contrast to Berger's work n [1], we consder the nteractons of a set of leaky bucket regulators, searchng for a vector of burst parameters of the leaky bucket regulators to meet the end-to-end deadlnes of all HRT connectons. Our algorthm for searchng burst parameters of leaky buckets s optmal n that t can fnd the vector of burst parameters whose mean value s mnmal and by whch the delay requrements of all HRT connectons can be met whenever such an assgnment exsts. Our algorthm s computatonally effcent and can be utlzed durng connecton setup. The results presented n ths paper are drectly applcable to ATM networks that are avalable currently, wthout makng any modfcatons to the hardware. The results are also compatble wth the proposed ATM standards. We evaluate the system's capablty to support hard real-tme connectons n terms of a metrc called admsson probablty [18]. Admsson probablty s the probablty of meetng the end-to-end deadlnes of a set of randomly chosen connectons. We have observed that the admsson probablty ncreases wth a proper choce of leaky bucket parameters at the UNI. Although we focus on traffc regulaton for meetng endto-end deadlnes, our work also complements many of the prevous studes that concentrate essentally on desgnng and analyzng schedulng polces for ATM swtches [2], [5], [6], [8], [10], [11], [12], [13], [16], [20], [21], [23], [24], [25]. A modfed FCFS schedulng scheme called FIFO was proposed and studed n [2]. The swtch schedulng polcy called ªStop and Goº s presented n [8]. A vrtual clock schedulng scheme n whch cells are prortzed by a vrtual tme stamp assgned to them, s dscussed n [25]. The use of the Earlest Deadlne Frst schedulng n wde area networks has also been nvestgated [6]. Zhang and Ferrar [23] dscuss schedulng at the output lnk by ntroducng a regulator at each lnk of an ATM swtch. Kweon and Shn [12] use the rate-monotonc schedulng polcy n whch the nput to the scheduler s constraned by regulatng the traffc of each connecton traversng the scheduler. Schedulng polces based on far queueng and ts dervatons are dscussed n [5], [16]. In our analyss of network delays, we assume the output lnk schedulng polcy used s FCFS. However, our analyss and methodology can be appled easly to systems usng other schedulng polces. The outlne of the rest of the paper s as follows. A glossary of terms s shown n Table 1. Secton 2 defnes the system model. Secton 3 develops a formal defnton of the traffc regulaton problem for HRT ATM networks. Secton 4 presents our algorthm to select the leaky bucket parameter values. Performance results are presented n Secton 5. Secton 6 provdes a summary of results and conclusons. 2 SYSTEM MODEL Ths secton presents the network model, the connecton model, and the traffc descrptors used to specfy the worst case traffc pattern of HRT connectons. 2.1 Network Model Fg. 1 shows a typcal ATM LAN. In ATM networks [4], [9], [22], messages are packetzed nto fxed-sze cells. The tme to transmt a sngle cell s a constant denoted by CT. We assume that tme s normalzed n terms of CT. That s, n ths paper, tme s consdered as a dscrete quantty wth the cell transmsson tme CT beng taken as one tme unt. ATM s a connecton-orented transport technology. A connecton has to be set up between two hosts before they can begn communcaton. Fg. 2a shows a sequence of network components that consttute a typcal connecton (llustrated by a connecton from Host 1 to Host 2 n Fg. 1). Cells of a connecton pass through a traffc regulator at the entrance to the network (the User Network Interface or UNI) and then traverse one or more ATM swtches, nterconnected by physcal lnks, before reachng ther destnaton host. In most ATM networks, the traffc s regulated at the source usng leaky buckets. A leaky bucket regulator conssts of a token bucket and an nput buffer. The cells from the source assocated wth the leaky bucket are buffered at the leaky bucket. A pendng cell from the nput buffer s transmtted f at least one token s avalable n the token bucket. There are two parameters assocated wth each leaky bucket regulator: the burst parameter and the rate parameter. The burst parameter, denoted by, s the sze of the token bucket,.e., the maxmum number of tokens that can be stored n the bucket. The rate parameter, denoted by, s the token generaton rate n the bucket. The number of cells that may be transmtted by a leaky bucket regulator n any nterval of length I s bounded by b Ic. An ATM swtch multplexes a number of connectons onto a physcal lnk. The cells of connectons beng multplexed are buffered at the controller of the output lnk. In most commercally avalable swtches, cells of connectons wth strngent delay requrements (.e., Class A traffc) are buffered n a hgh-prorty queue and served n an FCFS order. Hence, n ths paper, we consder FCFS schedulng polcy for HRT connectons. 2.2 Connecton Model To support HRT applcatons, the network must guarantee that all cells from a gven set of connectons are transmtted before ther deadlnes. We wll use the followng notatons

3 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 919 TABLE 1 Glossary of Terms concernng a set of HRT connectons. Hereafter, we wll omt the qualfer ªHRTº for connectons snce we only deal wth HRT connectons.. N denotes the total number of connectons n the system.. M s the set of N connectons competng for resources wthn the ATM network: MˆfM 1 ;M 2 ;...;M ;...;M N g; 1 where connecton M s specfed by the followng nformaton: - Source address, - Destnaton address, - Connecton route, 1 - End-to-end deadlne D, - An upper bound on average cell arrval rate, - Worst case nput traffc characterstcs.. ~D s a vector that specfes the end-to-end deadlnes of connectons n M: ~D ˆ <D 1 ;D 2 ;...;D ;...;D N >; 2 where D s the end-to-end deadlne of a cell of connecton M. That s, f a cell arrves at the source 1. To ensure stablty, we assume that connecton routes do not form loops [3], [16].

4 920 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 Fg. 1. ATM network archtecture. at tme t, then t should reach the destnaton by t D. In a connecton (see Fg. 2a), each of the network components traversed by a connecton's cells can be modeled as a server. Thus, a connecton can be consdered to be a stream of cells beng served by a sequence of servers [3], [18]. Servers can be classfed as constant delay servers and varable delay servers. A constant delay server s one that offers a fxed delay to every arrvng cell. For example, physcal lnks are consdered constant delay servers. On the other hand, cells may be buffered n a varable delay server and, hence, suffer queung delays. The leaky bucket traffc regulator and the FCFS output lnk schedulers at ATM swtches are examples of varable delay servers. Fg. 2b shows the logcal representaton of the connecton n Fg. 2a. The traffc pattern of a connecton at a pont n the network s characterzed by a traffc descrptor. The traffc at the source of a connecton s the raw traffc (unregulated) generated by applcatons. It s descrbed by the perodc descrptor C;P, whch means that a maxmum of C cells may arrve at the connecton n any nterval of length P. The perodc descrptor s very general to descrbe real-tme traffc at the applcaton level. The classcal perodc or synchronous traffc (.e., C contguous cells arrvng at the begnnng of every perod of length P ) s a specal case of ths knd of traffc. Most hard real-tme traffc (at source) s assumed to be synchronous [14], [15] and, hence, s specfed adequately by ths traffc descrptor. The raw traffc of a connecton s regulated by the leaky bucket regulator before t gets nto the network. After beng regulated by a leaky bucket wth parameters (, ), the traffc pattern becomes b Ic for any nterval of length I. After the regulaton, the traffc traverses through ATM swtches. The traffc pattern wll become ncreasngly rregular as cells are multplexed and buffered at the swtches. For the descrpton of more general traffc patterns, we use rate functon descrptor [17], [18], I, to descrbe the traffc after leaky bucket regulaton. I specfes the maxmum arrval rate of cells n any nterval of length I. That s, a maxmum of I I cells of the connecton may arrve n any nterval of length I. I s general enough to descrbe any traffc pattern. For example, the traffc pattern descrbed by b Ic can be expressed by the rate functon n (61) n Appendx A.1. The followng notatons are used n the rest of the paper to descrbe traffc at dfferent network ponts:. C;lb ;P;lb : nput traffc to the leaky bucket of connecton M. out server. ;j I and ;j I : nput and output traffc at FCFS server j of connecton M. 2.3 Delay Computatons The set of connectons M s admssble n an ATM network f and only f the worst case delays of cells do not exceed ther deadlnes. Let ~ d be a vector whose components are the worst case end-to-end delays for connectons n M; that s,

5 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 921 Fg. 2. Connecton decomposton nto servers. (a) The devces and lnks traversed by the connecton. (b) The sequence of servers traversed by the connecton. ~d ˆ <d 1 ;d 2 ;...;d ;...;d N >; 3 where d s the worst case end-to-end delay experenced by a cell of connecton M. Defne the relaton ª º on vectors as follows: Let ~x ˆ <x 1 ;x 2 ;...;x N > and ~y ˆ <y 1 ;y 2 ;...;y N >. ~x ~y f and only f 8; 1 N; x y : 4 Wth ths relaton, M, the set of HRT connectons, s admssble f and only f ~d ~D: 5 Hence, to check whether M s admssble, we need a systematc method of computng the worst case end-to-end delay experenced by a cell of each connecton. Consder connecton M, M 2M. To compute d, we need to nvestgate the delays n every network component traversed by a cell of M. Now, we can decompose d nto three parts as follows: 1. d const denotes the summaton of the delays a cell suffers at all the constant delay servers n ts 2. connecton route. denotes the worst case queung delay experenced by a cell at the leaky bucket whch regulates

6 922 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 Fg. 3. Expermental setup. Fg. 4. Expermental results. M 's traffc. When M 's traffc s not regulated, s d net denotes the summaton of the worst case delays a cell suffers at all the varable delay servers after the leaky bucket. d net can be obtaned by decomposng t further as follows: Suppose the route of connecton M traverses swtches (FCFS servers) 1; 2;...;K. Let d fcfs ;j be the cell delay of connecton M at FCFS server j. d net can thus be expressed as: d net ˆ d fcfs ;1 d fcfs ;2... d fcfs ;K : d, the worst case end-to-end delay for M, can now be obtaned as d ˆ d const ˆ d const d net d fcfs ;1 d fcfs ;2... d fcfs ;K : Snce d const s a constant, we focus on obtanng upper bounds on and d fcfs ;j. 3 PROBLEM DEFINITION In ths secton, we formally defne the problem of leaky bucket parameter selecton for HRT ATM networks. Frst, we examne some expermental data. We consder a smple 6 7 network consstng of two ATM swtches (see Fg. 3). Each ATM swtch has two nput lnes and two output lnes. There are three connectons n the system, M 1, M 2, and M 3. All three connectons share the same output lnk at the second swtch (swtch B). Connectons M 1 and M 2 enter the network at swtch A and traverse through swtches A and B. Connecton M 3 enters the system at swtch B and traverses swtch B only. The three connectons carry dentcal streams of vdeo data. The vdeo source under study s a two-hour encodng of the move ªStarwarsº [7]. The vdeo source provdes a total of 171,000 frames, at the rate of 24 frames per second. In ths experment, we vary 1 (the leaky bucket burst parameter of M 1 ) whle keepng 2 and 3 (the leaky bucket burst parameters of M 2 and M 3, respectvely) constant. Fg. 4 shows that, as 1 ncreases, the delay (measured n CT unts) experenced by M 1 tends to decrease. However, when 1 s ncreased, the delay experenced by M 3 tends to ncrease. Furthermore, we observe that, for 1 > 1; 400, the delays experenced by M 1 and M 3 reach constant values. An ntutve explanaton of these results s provded below. An ncrease n 1 tends to ncrease the burst sze of M 1 's traffc nto the network. When a larger burst sze s allowed at the output of M 's leaky bucket, the need to buffer M 's cells wthn the leaky bucket decreases. Ths tends to lower

7 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 923 connecton M 1 's delay as 1 s ncreased. On the other hand, the ncreased burstness of M 1 as 1 s ncreased adversely mpacts connecton M 3 's traffc, ncreasng M 3 's worst case cell delay. However, for large values of 1 ( 1 > 1; 400), the delays of ether connecton are unaffected by any ncrease n 1. Ths s because, at such large values of 1, the burst sze allowed for M 1 s so hgh that no cells of M 1 are queued at the leaky bucket nput buffer,.e., there s vrtually no traffc regulaton on M 1 's traffc. In ths experment, M 2 's traffc was not montored because t s expected to exhbt the same trend as M 3 when 1 ncreases. The expermental results shown n Fg. 4 ndcate clearly that the choce of s for dfferent connectons plays a crtcal role n the worst case cell delays experenced by all the connectons. Before the formal defnton of the problem, we need some notatons:. ~ s the rate vector,.e., ~ ˆ < 1 ; 2 ;...; ;...; N >; 8 where s the rate parameter assgned to the leaky bucket regulatng connecton M at ts network nterface. We assume that s assgned a value equal to the long term average cell arrval rate of M.. ~ s the burst vector,.e., ~ ˆ < 1 ; 2 ;...; ;...; N >; 9 where s the burst parameter assgned to the leaky bucket regulatng M 's traffc.. m ~ s the norm of the burst vector ~,.e., m ~ ˆXN ˆ1 : 10 Snce we analyze a slotted system, takes postve ntegers only. Therefore, the mnmal value of m ~ s N.. ~ d ~ s the worst case end-to-end delay vector for a gven selecton of ~,.e., ~d ~ ˆ<d 1 ~ ;d 2 ~ ;...;d ~ ;...;d N ~ >; 11 where d ~ s the worst case cell delay of connecton M when ~ s the chosen burst vector. Usng (7), d ~ can be expressed as d ˆd ~ const ˆ d const ~ d net ~ ~ d fcfs ;1 ~ d fcfs ;2 ~... d fcfs ;K ~ ; 12 where ~ and d fcfs ;j ~ are the worst case queung delays at the leaky bucket and at jth FCFS server, respectvely, when the burst vector s. ~ The computaton of ~ and d fcfs ;j ~ are gven by (60) and (63), respectvely, n Appendx A.. A M s the set of burst vectors for whch the connecton set M s admssble,.e., A M ˆf ~ j ~ d ~ ~Dg: 13 Our man goal s to ensure that the end-to-end deadlnes of all the connectons n a gven set are met,.e., to meet (5). We have chosen traffc regulaton as our means of achevng ths objectve. In terms of the above notatons, gven a set of HRT connectons, M, the method must fnd vector ~ that belongs to A M. We wll desgn and analyze a -selecton ~ algorthm for ths purpose. Such an algorthm wll take connecton set M as ts nput and return vector ~ as ts output. When A M s empty, t s clear that no assgnment of ~ can make the connecton set admssble. Our parameter selecton algorthm wll return an all-zero ~ when A M s empty. Furthermore, snce the degree of regulatng the traffc stream from M s hgher for smaller values of (.e. the regulated traffc s less bursty and thus has less mpact to others), t s desrable to select a vector ~ havng a small norm, m. ~ Let ~ be the mnmum n A M n terms of ts norm. That s, ~ satsfes: m ~ ˆ mn ~2A M m ~ : 14 We defne a -selecton ~ algorthm to be optmal f t always produces the ~ whenever A M s nonempty. We now prove that ~ s unque f t exsts for a set of connectons M. The followng lemma s ntroduced to llustrate the relatonshps between ~ and d net. ~ Lemma 3.1 Consder a connecton set M. d net ~ does not decrease as ~ ncreases. Lemma 3.1 s ntutvely vald. As ~ ncreases, more cells of those connectons whose values are ncreased wll get through the leaky buckets and be njected nto the network. Ths wll add more cells to the followng FCFS servers. Therefore, d net, ~ the total delays of an M 's cell at all the FCFS servers, wll ncrease or reman the same f the servers stll have the capacty to transmt the ncreased amount of cells wthout bufferng them. So, d net ~ wll never decrease as ~ ncreases. The formal proof of Lemma 3.1 s gven n [17]. The next theorem proves the unqueness of ~. Theorem 3.1 For a gven system, ~, whch satsfes m ~ ˆ mn ~2A M m ~ ; 15 s unque f A M 6ˆ ;. Proof. We prove the theorem by contradcton. Assume that A M 6ˆ ; and that there are two dstnct vectors ~ 0 ˆ< 1 0 ;...;0 N > and ~ 00 ˆ< 1 00;...;00 N > whch satsfy (15). Thus, m ~ 0 ˆXN ˆ1 0 ˆ XN ˆ1 00 ˆ m ~ 00 ˆ mn ~2A M m ~ : 16 We can construct a new vector ~ ˆ < 1 ;...; N > from ~ 0 and ~ 00 such that for j ˆ 1;...;N, j ˆ mn 0 j ;00 j : 17

8 924 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 Usng (17) n (10), we have m ~ ˆXN ˆ1 <m ~ 0 ˆm ~ 00 : 18 Snce a leaky bucket regulator s at the entrance of each connecton, ~ s ndependent from any j where j 6ˆ. Thus, we have: ~ ˆ : Then, because of (17), we have, for 1 N ~ ˆ 8 < : ~ 0 f ˆ 0 : ~ 00 f ˆ 00 : Also, because ~ ~ 0 and ~ ~ 00, from Lemma 3.1, we have, d net ~ mn d net ~ 0 ;d net ~ 00 : 21 Further, because ~ 0 and ~ 00 are dstnct, for a gven, 1 N, we have two cases: Case 1: < 0 That s, ˆ 00. Hence, from (20), we get and, from (21), we get Therefore, ~ ˆ ~ d net d ~ net ~ 00 : 23 ~ d net d ~ lb ~ 00 d net ~ 00 : 24 But, ~ 00 satsfes (15). Therefore, Case 2: ˆ 0 Hence, from (20) we get and, from (21), we get Therefore, ~ d net ~ D : 25 ~ ˆ ~ 0 26 d net d ~ net ~ 0 : 27 ~ d net d ~ lb ~ 0 d net ~ 0 : 28 But, ~ 0 satsfes (15). Therefore, ~ d net ~ D : 29 Because of (25) and (29), ~ s also a feasble assgnment for whch the connecton set s admssble. However, because of (18) and the defnton of ~, ~ 0 and ~ 00 cannot be ~. Ths s a contradcton. Hence, the theorem holds. tu 4 ALGORITHM DEVELOPMENT In ths secton, we develop an optmal and effcent algorthm. We frst formulate the problem as a search problem and nvestgate some useful propertes of the search space. These propertes help us n the development of the algorthm. 4.1 Parameter Selecton as a Search Problem By defnton, an optmal algorthm takes M as ts nput and returns ~ whenever A M s nonempty and returns < 0; 0;...; 0 > as ts output when A M s empty. We can vew the problem of selectng ~ as a search problem, where the search space conssts of all the ~ vectors. Let AM ~ be the set of all ~ vectors. A M, the set consstng of ~ vectors wth whch the deadlnes of M are met, s a subset of AM ~. Let! be a relaton on A ~ M defned as follows: Gven ; ~ ~ 0 2 A ~ M,! ~ ~ 0 f and only f 9j; 1 j N, such that 8 < 1 f ˆ j 0 ˆ 30 : f 6ˆ j: Note that m ~ 0 ˆm 1 ~ and ~ 0 dffers from ~ only n the jth component. Let 4 ; ~ ~ 0 denote the ndex j. For example, f ~ ˆ < 1; 4; 2; 7; 5 > and 0 ~ ˆ < 1; 4; 2; 8; 5 >, then! ~ ~ 0 and 4 ; ~ ~ 0 ˆ4. The relaton! allows us to defne an acyclc drected graph G over A ~ M, wth a node set V and an edge set E gven by. Vˆ ~A M,. Eˆf ; ~ ~ 0 j! ~ ~ 0 g. Thus, G s a graph representaton of AM ~, the search space. Graph G can also be consdered as a rooted leveled graph; vector < 1; 1;...; 1 > s the root and levelp conssts of all ~ vectors havng norm p (p N). Fg. 5 llustrates such a graph when N ˆ 3. In graph G, let L p be the set of all ~ vectors at level p. Note that a node at level p can have edges only to nodes at level p 1. Based on ths representaton of the search space, a smple breadth-frst search method can be constructed to fnd ~. Fg. 6 shows the pseudocode for such a method. Ths search method frst examnes all ~ vectors n L p before proceedng to those n L p 1, shown by the dotted search path n Fg. 5. For each ~ vector consdered, the method uses the procedure 2 Compute d ~ ~ to evaluate d ~. ~ The frst ~ encountered n the search path that satsfes the deadlne constrant d ~ ~ ~D s clearly the ~ vector. At ths pont, the followng questons about the method shown n Fg. 6 may be rased: 1. For a gven connecton set M, set A M may be empty. In such a case, ~ s not defned and the algorthm of searchng ~ wll not termnate. 2. Detals of procedure Computer ~ d ~ are n Appendx A.3. The fnal verson of Computer ~ d has one more parameter, whch wll be dscussed later.

9 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 925 Fg. 5. Example search space graph. 2. Even f A M s nonempty, the exhaustve nature of the breadth-frst search results n exponental tme complexty. In the next subsecton, we overcome the frst dffculty by boundng the search space. In the subsequent subsecton, we reduce further the search complexty by prunng the search space and adoptng a search method that s more effcent than the breadth-frst search. 4.2 Boundng the Search Space In the experment descrbed n Secton 3, an nterestng observaton was that when 1 was ncreased beyond 1; 400, there was no change n the worst case end-to-end delays of any of the connectons. The followng theorem asserts that such behavor s to be expected n any ATM network and leaky bucket regulators. Let max be the mnmum value of for whch s zero. Theorem 4.1. For a connecton M whose traffc s descrbed by C ;P and regulated by a leaky bucket wth ;lb ;lb parameters ;, max s bounded by C;lb. Proof. Let Q ;lb be the maxmal queue length at the leaky bucket of M. From Lemma A.2 n Appendx A.1, we have Q ;lb ˆ C;lb : 31 Snce max s defned to be the smallest nteger value for whch the maxmum watng tme of a cell at the leaky bucket queue becomes zero,.e., Q ;lb becomes zero, solve for the value of whch makes (31) zero. We have max ˆ C;lb : 32 Theorem 4.2. For any connecton M, ncreasng the value of beyond max has no mpact on the worst case end-to-end cell delay of M or any other connecton. ut Proof. For a connecton M, snce max s the mnmal value of for whch s zero, no cell of M s queued at ts leaky bucket when reaches max. That s, any cell that arrves at the leaky bucket wll get through the leaky bucket wthout beng buffered. There s vrtually no traffc regulaton on M 's traffc n ths case. Therefore, has no mpact on the worst case end-to-end cell delay of M or any other connecton. tu ncreasng the value of beyond max An mportant consequence of Theorem 4.2 s that ~ AM, the search space of canddate ~ vectors, can be bounded; we only need to consder ~ vectors that satsfy ~ ~ max : 33 Note that ~ max can be precomputed for a gven set M. Consder the example n Fg. 5. If we assume that ~ max ˆ < 1; 2; 3 >, then, after applyng Theorem 4.2, we get another graph shown n Fg. 8. The shaded regon n Fg. 8 s automatcally elmnated from consderaton. Usng Theorem 4.2, we modfy the breadth-frst search procedure shown n Fg. 6 to take nto account the bounded search space. The resultng pseudocode s shown n Fg. 7. However, snce the sze of L p ncreases exponentally, the complexty of the algorthm s stll exponental. In the next subsecton, we wll prune the search space to desgn an effcent algorthm. 4.3 Search Space Prunng The breadth-frst search algorthm defned n Fg. 7 has an exponental tme complexty. Now, we consder an alternatve to the exhaustve breadth-frst search method. As an alternatve to the breadth-frst search path, we desre a search path that begns at the root node < 1; 1;...; 1 > and follows the drected edges n graph G to locate ~ (f ~ exsts). Such a search path would allow us to examne only one vector ~ at each level. Consder the example n Fg. 9 whch depcts a case wth N ˆ 3. The shaded regon has already been elmnated from consderaton based on ~ max beng < 1; 2; 3 >. Assume that

10 926 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 Fg. 6. Pseudocode of the breadth-frst search method. ~ ˆ < 1; 2; 2 >. We would lke our search to be guded along one of the two paths: 1. < 1; 1; 1 >, < 1; 2; 1 >, < 1; 2; 2 >,or 2. < 1; 1; 1 >, < 1; 1; 2 >, < 1; 2; 2 >. To gude the search along the drect path from the root to ~ n G, we must choose an approprate chld node at each level. For example, n Fg. 9, at node < 1; 1; 1 >,wemay choose ether < 1; 2; 1 > or < 1; 1; 2 > as our next canddate node. However, f we are at node < 1; 1; 2 >, we must gude our search to choose < 1; 2; 2 > and not < 1; 1; 3 > as the next canddate node. In order to select canddate nodes at each level, we need to know whether a partcular node s an ancestor of ~ (f ~ ndeed exsts). A node ~ s sad to be an ancestor of node ~ 0 (denoted ~! ~ 0 )f ~ 0 can be reached from ~ (by a drected path n G). For example, n Fg. 9, the ancestor nodes of < 1; 2; 2 > are: < 1; 2; 2 >, < 1; 2; 1 >, < 1; 1; 2 >, and < 1; 1; 1 >. To formally defne the ancestor relatonshp, we proceed as follows: Frst, each vector n A ~ M s consdered an ancestor of tself. Let ~ p and ~ p k be two vectors n A ~ M such that m ~ p ˆpand m ~ p k ˆp k, k >0. Now, ~ p! ~ p k f 9 ~ p 1 ; ~ p 2 ;...; ~ p k 1 2 A ~ M such that ~ p! ~ p 1! ~ p 2!...! ~ p k 1! ~ p k : The next theorem states an mportant result that wll help us construct a drected path from the root node (.e., < 1; 1;...; 1 > ) to ~ for any gven M. We need some notatons frst. Let ~s ˆ<s ~ 1 ;s 2 ;...;s N > be the status vector assocated wth a node, ~ where 8 s ˆ ~ < 1 f d D ~ 34 : 0 otherwse: Vector ~s ~ ndcates the status of each stream (.e., whether the deadlnes of ndvdual streams are met or not) when ~ s selected as the burst parameter vector. The followng lemmas are ntroduced to help the proof of the theorem whch defnes the drect search of ~. Lemma 4.1. Consder a connecton set M. d j ~ does not decrease as the ncrease of s for any where 6ˆ j. Lemma 4.1 s vald. As we have sad before, d j ˆd ~ const j j ~ d net j. ~ d const j s a constant and j ~ s ndependant from f j 6ˆ. Only d net j ~ changes as the ncrease of s for any where 6ˆ j. From Lemma 3.1, we have that d net j ~ does not decrease as the ncrease of. ~ Therefore, Lemma 4.1 holds. The formal proof of ths lemma can be found n [17]. Lemma 4.2. Consder a connecton set M and assume that ~ exsts for M. If ~ s an ancestor of ~, for any y (1 y N) whch makes s y ˆ0, ~ then y <y. Proof. Snce ~ s an ancestor of ~, we have < ~ ~. That s, 8y; 1 y N; y y : 35 Fg. 7. Pseudocode of the bounded breadth-frst search algorthm.

11 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 927 Fg. 8. Example of bounded search space. Fg. 9. Example graph of search space after prunng. Now, we prove: For any y (1 y N) fs y ~ ˆ0, then y < y. Because of (35), we only need to prove y 6ˆ y. We prove t by contradcton. Assume y ˆ y. Snce s y ˆ0, ~ we have d y ~ >D y. From Lemma 4.1, we have that d y ~ does not decrease no matter how much we ncrease sf6ˆ y. We now ncrement the correspondng components of ~ to make ~ equal to ~. d y ~ >D y stll holds. Ths contradcts the defnton of ~ and the exstance assumpton of ~. Therefore, y 6ˆ y. So, y <y. tu Theorem 4.3. If ~ exsts, then the followng are true: 1. < 1; 1;...; 1 > s an ancestor of ~, and 2. If ~ s an ancestor of ~, then there exsts a ~ 0 such that a)! ~ ~ 0, b) s 4 ; ~ ~ 0 ~ ˆ0, and c) Ether ~ 0 s ~ or an ancestor of ~. Proof. Statement 1 s true snce < 1; 1;...; 1 > s an ancestor of all ~ s. Let ~, ~ 6ˆ ~ ; be an ancestor of ~. Therefore, for all x, 1 x N we have x x : 36 Snce ~ s unque, there exsts y, 1 y N such that s y ~ ˆ0. Then, from Lemma 4.2, we have y < y : 37 Now, construct ~ 0 such that, for all z, z 6ˆ y 0 z ˆ z and 0 y ˆ y 1. From the defnton of ª!,º we have, obvously, Thus, a) holds. ~! ~ 0 : 38

12 928 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 From the defnton of ª4,º we have 4 ~ ; ~ 0 ˆy. Also, because s y ~ ˆ0, that s s 4 ~ ; ~ 0 ~ ˆ0: Thus, b) holds. By (36) and (37), for all x, 1 x N, we have Thus, 0 x x : ~ 0 ~ : 41 Hence, ether ~ 0 ˆ ~ or there exsts a drected path n G such that ~ 0! ~ : 42 So, ~ 0 s an ancestor of ~. Thus, c) holds. tu Now, consder the clams made by the above theorem. The frst clam n Theorem 4.3 s trval. It states that f ~ exsts, then there must be at least one path from the root node < 1; 1;...; 1 > to ~. The second clam n Theorem 4.3 mples that f ~ s an ancestor of ~ (.e., ~ exsts) and the assgnment of ~ does not make M admssble (.e., ~d ~ > ~D), then another assgnment ~ 0 derved from ~ such that! ~ ~ 0 and s 4 ; ~ ~ 0 ˆ 0 s also an ancestor of ~. The frst clam helps us to begn the search from the root node. Once we are at level p examnng a node ~ 2L p, the second clam helps us to choose the chld node of ~ f ~d ~ > ~D. The theorem states that we can choose a chld node ~ 0 of ~ such that s 4 ; ~ ~ 0 ˆ 0. The theorem ensures that such a chld node must also have a drected path to ~ f ~ exsts. Hence, ~ can be found by our search startng from < 1; 1;...; 1 > and usng the status vector ~s to gude the search along a drected path leadng to ~. 4.4 The Effcent Algorthm and Its Propertes In ths subsecton, we frst present an effcent and optmal algorthm, and then prove ts propertes. Fg. 10 shows the pseudocode of the algorthm. The algorthm s derved from the one n Fg. 7 by prunng the search space. The algorthm s an teratve procedure, startng from the root,.e., < 1; 1;...; 1 >. Durng each teraton, the algorthm selects a node from the next level. The node s selected (lne 9) wth the help of status vector ~s (computed by functon Compute ~s d; ~ ~D n lne 8). Ths teratve process contnues untl ether ~ s found or, for some j, j >j max. 3 The followng two theorems assert the correctness property and the complexty of the algorthm from Fg. 10. Theorem 4.4. For a connecton set M, the algorthm n Fg. 10 s optmal. Proof of ths theorem follows from Theorem 4.3 and the pseudocode of the algorthm. 3. Note that lne 9 n the algorthm can be modfed to select the connecton whose deadlne s mssed and for whch max has the mnmum value. Ths modfcaton mproves the average case tme complexty of the algorthm wthout changng the worst case one. Theorem 4.5. The tme complexty of the algorthm n Fg. 10 s O N P N ˆ1 max. Proof. In the algorthm shown n Fg. 10, the maxmum number of teratons s P N ˆ1 max N 1. Durng each teraton, the algorthm calls three procedures (lnes 4, 8, and 9). The worst case tme complexty of the procedure Compute d ~ ; ~ j (lne 4) s a functon of the network sze,.e., the longest path n the network. Hence, for a gven network, the tme complexty of Compute d ~ ; ~ j can be bounded by a constant. The procedure Compute ~s d; ~ ~D (lne 8) requres N steps of comparsons. Therefore, ts tme complexty s O N. Fnally, the worst case tme complexty of the procedure Fnd_ndex_j() (lne 9) s O N. Hence, the tme complexty of the algorthm of Fg. 10 s O N P N ˆ1 max. tu 5 PERFORMANCE EVALUATION In ths secton, we present performance results to evaluate the mpact of leaky bucket regulaton on HRT systems. We consder the sample network archtecture shown n Fg. 11. It conssts of two stages, wth a total of 11 ATM swtches. Each ATM swtch has 10 nput lnes and 10 output lnes. The connectons n the network form a symmetrcal pattern. There are 100 connectons n the system and each connecton goes through two swtches. The connectons are arranged n such a way that 10 connectons share one output lnk at each stage. At the frst stage, connectons M 00 ;M 01 ;...M 09 are multplexed n Swtch 0 and are transmtted over lnk 0. At the second stage, connectons M 00 ;M 10 ;...M 90 are multplexed n Swtch 10 and are transmtted over a lnk to Host 100. We evaluate the performance of the system n terms of the admsson probablty, AP U, whch s defned as the probablty that a set of randomly chosen HRT connectons can be admtted, gven the traffc load n terms of the average utlzaton of the lnks U. To obtan the performance data, we developed a program to smulate the above ATM network and the connectons. The program s wrtten n the C programmng language and runs n a Sun/Solars envronment. In each run of the program, 200 connecton sets are randomly generated. For each connecton, the total number of cells per perod s chosen from a geometrc dstrbuton wth mean 10. The worst case cell arrval rates (C, P) of the connectons sharng a partcular lnk at the frst stage are chosen as random varables. They are dstrbuted unformly between 0 and U, subject to ther summaton beng U, the average utlzaton of the lnk. Smlar results have been obtaned wth dfferent settngs of parameters. For each connecton set generated, the followng systems are smulated:. System A. In ths system, connecton traffc s unregulated,.e., the burst vector selected for the connecton set s ~ max.. System B. In ths system, constant burst vectors are used for all the connectons. In partcular, System B1 sets the burst vector to be < 3; 3;...; 3 > and System B2 sets the burst vector to be < 9; 9;...; 9 >.

13 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 929 Fg. 10. Pseudocode of the effcent algorthm. Fg. 11. Example network used n smulaton.. System C. In ths system, the burst vector produced by our optmal algorthm s used. Fgs. 12 and 13 show the performance fgures correspondng to the cases where D s set to be 2P and 1:5P, respectvely. In HRT systems, t s common for deadlnes to be assocated wth perods [14], [15]. From these fgures, the followng observatons can be made:. System C, where our optmal algorthm s used to set the burst vectors, performs the best. The performance gan s partcularly sgnfcant when the lnk utlzaton becomes hgh, n comparson wth systems A, B1, and B2. For example, n Fg. 12, at U ˆ 0:5, AP U s close to 1 for System C, but 0 for systems A and B2. Ths justfes our early clam that the burst vector must be properly set n order to

14 930 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 Fg. 12. Admsson probablty for perodc traffc (D ˆ 2P ). acheve the best system performance wth HRT applcatons.. In general, the admsson probablty s senstve to the average lnk utlzaton. As the utlzaton ncreases, the admsson probablty decreases. Ths s to be expected because the hgher the network utlzaton, the more dffcult t s for the system to admt a set of connectons. We can also fnd the decreasng speed of the admsson probablty n systems A, B1, and B2 s much faster than n System C as the utlzaton ncreases. It suggests that, n the stuatons where the lnk utlzaton s hgh, the proper selecton of burst vector becomes more mportant to the system performance.. Comparng the performance of systems A, B1, B2, and System C, we can see clearly the dfference between no-regulaton, regulaton wth a large ~, regulaton wth a small ~, and regulaton wth an optmal ~. System A, whch does not have any traffc regulaton, performs the worst among all the systems. System B1 wth a small ~ performs better than system B2, whch has a larger ~. System C, wth the optmal ~, performs the best. It s another evdence supportng the drecton of our work: choosng the mnmal ~ whch can stll meet the end-to-end deadlnes. The smulaton results further strengthen the need for a good ~ selecton algorthm.. The performance of all systems s very stable as D changes. The curves n Fg. 12 all have smlar shapes as those n Fg. 13, demonstratng that the smulaton results are stable. They have not been affected by any system dynamc factors, such as system loadng, other system traffcs, and so on. 6 CONCLUSIONS In ths paper, we have addressed the ssue of guaranteeng end-to-end deadlnes of HRT connectons n an ATM Fg. 13. Admsson probablty for perodc traffc (D ˆ 1:5P ).

15 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 931 network. Much of the prevous work n ths area has concentrated on schedulng polces used n ATM swtches. Our approach to ths problem s to regulate the nput traffc at the network nterface. In partcular, we consder leaky bucket traffc regulators. Ths s the frst study that uses traffc regulaton (n partcular, wth leaky buckets) as a method of guaranteeng the end-to-end deadlnes of HRT connectons. Tradtonally, a leaky bucket has been used as a polcng mechansm when the source traffc at the nput of the network does not conform to ts negotated characterstcs. We have desgned and analyzed an effcent and optmal algorthm for selectng the burst parameters of leaky buckets n order to meet connectons' deadlnes. Our algorthm s optmal n the sense that f there exsts an assgnment of burst parameters for whch the deadlnes of a set of HRT connectons can be met, then the algorthm wll always fnd such an assgnment. Our algorthm s also effcent. We smulated and compared the performance of ATM networks wth dfferent regulaton polces. We observed that there s a dramatc mprovement n the system performance when the burst parameters were selected by our algorthm. Our soluton for guaranteeng end-to-end deadlnes n HRT ATM networks s effectve and generc. It s ndependent of the swtch archtecture and the schedulng polcy used at the ATM swtches. It can be used for admsson control n any ATM network that uses leaky bucket traffc regulators. APPENDIX A COMPUTATION OF d ~ Recall that a connecton s served by a leaky bucket regulator and a sequence of network servers (ATM swtches). Hence, the end-to-end cell delay of a connecton s the summaton of the worst case delays at the leaky bucket and at all the network servers. Suppose the burst vector s ~. The end-to-end cell delay of connecton M can be expressed as the followng (gven n Secton 3): d ~ ˆd const ~ d fcfs ;1 ~ d fcfs ;2 ~... d fcfs ;K ~ ; 43 where ~ and d fcfs ;j ~ are the worst case queung delays at the leaky bucket and jth FCFS server, and K s the total number of servers on connecton M. In ths secton, we dscuss the computaton of ~ and d fcfs ;j. ~ The delay analyss at a server (or a leaky bucket) nvolves two steps: 1) the computaton of the worst case cell delay at the server; 2) the determnaton of the traffc descrptons at the output of the server. The output traffc of a server s the nput traffc of the next server. A.1 Delay Analyss of a Leaky Bucket The followng addtonal notatons, assumptons and defntons are frst ntroduced: ;lb I denotes the number of cells of M that can arrve at the nput of the leaky bucket n an nterval of length I: $ % I ;lb I P ;lb mn C C;lb ;lb $ % I ;I P ;lb P;lb! : 44 Q ;lb denotes the maxmal queue length at the leaky bucket of M. For each of the leaky buckets n the system we make the followng assumptons:. To ensure the stablty of the system, P server Cn ;lb ;lb : 45. The burst parameter of the leaky bucket s at least one,.e., 1: 46 Next, to facltate the understandng of a leaky bucket regulator, we ntroduce the concept of regeneraton pont. Defnton A.1. For a connecton regulated by a leaky bucket wth parameters ;,aregeneraton pont s a tme nstant t when the number of cells queued at the leaky bucket server s zero and the number of tokens n the leaky bucket s. The followng lemmas help us to prove the man theorem n ths subsecton. Lemma A.1. For connecton M whose traffc s descrbed by C;lb ;P;lb ;C;lb <P;lb, and regulated by a leaky bucket ( ; ), f t s a regeneraton pont, then there s another regeneraton pont n nterval t; t P;lb Š. Proof. We prove the lemma by contradcton. Let us assume that, for connecton M, regulated by a leaky bucket ( ; ), f t s a regeneraton pont, then there s no other ;lb Š. regeneraton pont n the nterval t; t P Snce t s a regeneraton pont, by Defnton A.1, at tme t, the number of cells n the leaky bucket queue s zero and the number of tokens n the leaky bucket s. Therefore, the total number of tokens avalable at the leaky bucket by tme t 0, t<t 0 t P b t 0 t c: ;lb,s 47 Snce there s no other regeneraton pont n nterval t; t P;lb Š, usng the defnton of the leaky bucket we get b t 0 t c ;lb t 0 t < : By algebrac manpulaton of (48), we get Hence, for t 0 ˆ t P b t 0 t c < ;lb t 0 t : b P ;lb ;lb, we have c < ;lb P;lb : 50 Usng (45), substtute the mnmum value of n (50). Then, we have

16 932 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 C;lb < ;lb P;lb : 51 But, (51) contradcts (44), the defnton of the perodc descrptor. Therefore, our assumpton that there s no other regeneraton pont n the nterval t; t P;lb Š leads to a contradcton. Hence, the lemma holds. tu Before dervng the worst case delay at a leaky bucket, we ntroduce the followng lemma whch gves the maxmal queue length at the leaky bucket. Lemma A.2. For a connecton M regulated by a leaky bucket wth parameters ;, the maxmal queue length Q ;lb s gven by, Q ;lb ˆ C;lb : 52 Proof. We know the maxmal queue length at the leaky bucket wll occur between two regeneraton ponts. Further, by Lemma A.1, the maxmal nterval between any two consecutve regeneraton ponts s at most P;lb. Let t r be a regeneraton pont. Q ;lb, the maxmal queue length at the leaky bucket, occurs n the nterval t r ;t r P Q ;lb ˆ ;lb Š: max t r tt r P ;lb ;lb t t r b t t r c : 53 Further, from (44), we have $ % ;lb t t r t tr C;lb P;lb mn C;lb ; t t r $ % t tr P;lb : 54 P ;lb Now, we consder (54) n two cases. Case 1: t t r <P;lb. That s, b t tr P cˆ0. Hence, ;lb (54) becomes ;lb t t r mn C;lb ; t t r C;lb : 55 Case 2: t t r ˆ P;lb. That s b t tr P cˆ1. Hence, ;lb (54) becomes ;lb t t r C;lb mn C;lb ; 0 ˆC Therefore, n ether case, we have ;lb : 56 ;lb t t r C;lb : 57 Substtutng (57) n (53), we get Q ;lb ˆ max t r tt r P ;lb C;lb b t t r c : 58 When t t r ˆ 0, ª C;lb b t t r cº reaches the maxmum. Hence, we get Q ;lb ˆ C;lb : 59 tu Next, we use the results of Lemma A.2 to derve a bound on the maxmal cell delay, ~, experenced by a cell of connecton M, at the leaky bucket regulator, ;. Theorem A.1. For a connecton M, regulated by a leaky bucket parameterzed by ; ~ Q ;lb ; 60 where Q ;lb s gven by (52). Proof. The proof s obvous from the defnton of a leaky bucket. The maxmal number of cells buffered at the leaky bucket s Q ;lb (.e., the maxmal queue length) and cells pass through the leaky bucket at least at the rate of. Thus, n the worst case, t takes Q;lb to let all cells pass through the leaky bucket, whch s the worst case tme durng whch a cell s buffered before t s transmtted.tu Next, we derve the descrpton of traffc at the output of the leaky bucket. Theorem A.2. For a connecton M, regulated by a leaky bucket ;, the upper bound of the output traffc, denoted by out server I, s ;lb out server ;lb I I ˆ : 61 I Proof. The proof of ths theorem follows drectly from the defnton of the leaky bucket. tu A.2 Delay Analyss of an FCFS Server We now analyze an FCFS server j, 1 j K. For any connecton M, let ;j I be the maxmal number of cells from connecton M that can arrve at the nput of server j n any nterval of length I. Suppose the traffc of M at the nput of server j s descrbed by ;j I (the nput traffc to the frst FCFS server s the output of the leaky bucket regulator gven n (61)). Therefore, at server j we have ;j I ˆI ;j I : 62 Gven the nput traffc descrpton at FCFS server j, the followng theorem s used to fnd an upper bound on the delay experenced by a cell of M at server j. Theorem A.3. For an FCFS server j, d fcfs ;j, ~ the worst case delay experenced by a cell of connecton M at the server s gven by &!' d fcfs ;j ~ ˆ max IL j X N ˆ1 ;j I I ; 63 where L j, the length of the longest busy nterval at server j, s gven by! L j ˆ mn I j XN ;j I I 64 ˆ1 and ;j I s gven by (62).

17 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 933 Fg. 14. Pseudocode of procedure to computer worst case delay at every varable server. In (63), P N ˆ1 ;j I s the maxmal number of cells from all the connectons that can arrve at server j durng an nterval of length I, and I s the number of cells that server j can transmt durng the nterval I (recall that, n ths paper, tme unt s normalzed by the tme for the transmsson of one cell). Therefore, P N ˆ1 ;j I I s the maxmal queue length at the nput of server j. Maxmzng P N ˆ1 ;j I I over all possble values of I, we have the worst case delay of a cell. The formal proof of Theorem A.3 can be obtaned by applyng Theorem 4.1 gven n [3]. The next theorem gves the output traffc descrpton of the connecton M at FCFS server j. Theorem A.4. For connecton M, f the nput traffc at FCFS server j s descrbed by ;j I, then the output traffc out server parameter I s gven by out server ;j ;j I ˆ 0 0 mn@ 1 d fcfs ;j ~ I ;j I d fcfs ;j 11 ~ AA: 65 Theorem A.4 follows from Theorem 4.2 n [19], whch s also mpled by Theorem 2.1 n [3]. A.3 Procedure for Computng d ~ In ths subsecton, we outlne the procedure to compute the worst case end-to-end cell delays for every hard real-tme connecton. Recall that we model an HRT connecton as a sequence of varable servers. In such a model, the worst case end-to-end delay experenced by a cell of a connecton s computed by frst determnng the worst case cell delay at each server and then summng up the delays at all servers. In the prevous subsectons, we presented the expressons for the worst case cell delay at a leaky bucket and an FCFS server. The computaton of delays at a server requres the nput traffc of all the connectons at the server to be known. It s qute possble that a traffc dependency loop exsts n the connecton-server graph,.e., the paths of some connectons form a loop. When such a loop exsts, none of the servers on the loop has ts nput traffc completely known. In [19], an on-lne admsson control algorthm was proposed, whch admts a new connecton nto the system under the condton that the delay requrements of all connectons are stll met. Ths on-lne algorthm specally consdered the stuatons where such dependency loops exst. We wll use ths on-lne method to compute d ~ ~ n the presence of traffc dependency loops. In our effcent algorthm of searchng for the ~ presented n Secton 4.4, each tme only one element of ~ s changed (ncremented by 1). For each change of, ~ the delays at all the servers whch are affected by ths change need to be recomputed. In order to compute the delays at all affected servers, the followng varables are ntroduced: H s the sequence of servers on the connecton path of M. Impact svr s an ordered lst of servers whch are affected by the change of a. ~ The ntal value of Impact svr for the change of s H. Next s s a functon whch returns the server next to server s on the connecton path of M. FCFS d s a vector of delays, one component for each FCFS server n the system. FCFS d sš s the delay at FCFS server s. In traffc s an array used to record the nput traffc of connectons at all FCFS servers. In traffc ; jš s the nput traffc of connecton M to server j.

18 934 IEEE TRANSACTIONS ON COMPUTERS, VOL. 48, NO. 9, SEPTEMBER 1999 Fg. 14 s the procedure for computng the worst case delays at all the servers. The procedure frst computes the delays and traffc outputs of connectons at all servers for the ntal value of ~ ˆ 1; 1;...; 1. The on-lne admsson control algorthm, presented n [19], s used to add the N connectons nto the system one after another. Every tme a new connecton s added nto the system, the nput traffc of the new connecton and the delays and the nput traffc of exstng connectons at all FCFS servers are known. The delays and nput traffc at all servers after addng the new connecton can thus be computed. The order of addng the N connectons nto the system does not affect the fnal results because the ntal values of for all connectons are the same. After the ntal values of delays and nput traffc at all servers have been determned, each tme when a value s changed, ts mpact to the exstng system can be computed by startng from the server where the value s changed. Suppose j s changed. The delay analyss starts from the frst server of M j. Snce the delay and the nput traffc at each server before the change of j has been already computed and recorded n the system, the delay analyss at the servers of M j can be done one after another along M j 's connecton path. Procedure compute_delay(s) (Lne 12) computes the worst case cell delays at server s by usng the results n (60) and (63). Procedure compute_output(s) (Lne 13) computes the output traffc to the succeedng server by usng the results n (61) and (65). Snce the change of j wll also change the delays and nput traffc of other connectons, the algorthm appends the servers whch are affected to lst Impact svr (Lne 14-17). Thus, the algorthm systematcally traces all the servers affected by the change of j. Lst Impact svr wll eventually become empty and, therefore, the algorthm termnates, because the mpact caused by the change of j converges. The formal proof can be found n [17]. ACKNOWLEDGMENTS Ths work was supported n part by Strategc Research Grant no from Cty Unversty of Hong Kong. REFERENCES [1] A.W. Berger, ªPerformance Analyss of a Rate-Control Throttle where Tokens and Jobs Queue,º IEEE J. Selected Areas n Comm., vol. 9, no. 2, pp , Feb [2] D.D. Clark, S. Shenker, and L. Zhang, ªSupportng Real-Tme Applcatons n an Integrated Servces Packet Network: Archtecture and Mechansm,º Proc. ACM SIGCOMM '92, pp , Aug [3] R.L. Cruz, ªA Calculus for Network Delay,º IEEE Trans. Informaton Theory, vol. 37, no. 1, pp , Jan [4] M. de Prycker, Asynchronous Transfer Mode: Soluton for Broadband ISDN. Ells Horwood, [5] A. Demers, S. Keshav, and S. Shenker, ªAnalyss and Smulaton of a Far Queueng Algorthm,º Proc. ACM SIGCOMM '89, pp. 1-12, Sept [6] D. Ferrar and D.C. Verma, ªA Scheme for Real-Tme Channel Establshment n Wde-Area Networks,º IEEE J. Selected Areas n Comm., vol. 8, no. 3, pp , Apr [7] M. Garrett, ªContrbutons Toward Real-Tme Servces on Packet Swtched Networks,º PhD thess, Columba Unv., [8] S.J. Golestan, ªA Framng Strategy for Congeston Management,º IEEE J. Selected Areas n Comm., vol. 9, no. 7, pp. 1,064-1,077, Sept [9] Int'l Telecomm. Unon, ITU-T Recommendaton I.311ÐB-ISDN General Network Aspects, [10] C.R. Kalmenek, H. Kanaka, and S. Keshav, ªRate Controlled Servers for Very Hgh-Speed Networks,º Proc. IEEE Global Telecomm. Conf., pp , Dec [11] D.D. Kandlur, K.G. Shn, and D. Ferrar, ªReal-Tme Communcaton n Mult-Hop Networks,º Proc. 11th Int'l Conf. Dstrbuted Computng Systems, pp , May [12] S. Kweon and K.G. Shn, ªTraffc-Controlled Rate-Monotonc Prorty Schedulng of ATM Cells,º Proc. IEEE Infocom '96, pp , Mar [13] T. Lng and N. Shroff, ªSchedulng Real-Tme Traffc n ATM Networks,º Proc. IEEE Infocom '96, pp , Mar [14] C.L. Lu and J.W. Layland, ªSchedulng Algorthms for Multprogrammng n a Hard-Real-Tme Envronment,º J. ACM, vol. 20, no. 1, pp , Jan [15] N. Malcolm and W. Zhao, ªHard Real-Tme Communcaton n Multple-Access Networks,º J. Real-Tme Systems, Jan [16] A.K.J. Parekh, ªA Generalzed Processor Sharng Approach to Flow Control n Integrated Servces Networks,º PhD thess, Dept. of Electrcal Eng. and Computer Scence, Massachusetts Inst. of Technology, [17] A. Raha, ªReal Tme Communcaton n ATM Networks,º PhD thess, Dept. of Computer Scence, Texas A&M Unv., [18] A. Raha, S. Kamat, and W. Zhao, ªGuaranteeng End-to-End Deadlnes n ATM Networks,º Proc. 15th IEEE Int'l Conf. Dstrbuted Computng Systems, June [19] A. Raha, S. Kamat, and W. Zhao, ªAdmsson Control for Hard Real-Tme Connectons n ATM LAN's,º Proc. IEEE Infocom '96, Mar [20] S.S. Sathaye, W.S. Ksh, and J.K. Strosnder, ªResponsve Aperodc Servces n Hgh-Speed Networks,º Proc. IEEE Int'l Conf. Dstrbuted Computng Systems, pp , May [21] L. Trajkovc and S.J. Golestan, ªCongeston Control for Multmeda Servces,º IEEE Network, vol. 6, no. 5, pp , Sept [22] R.J. Vetter, ªATM Concepts, Archtectures, and Protocols,º Comm. ACM, vol. 38, no. 2, Feb [23] H. Zhang and D. Ferrar, ªRate-Controlled Statc Prorty Queueng,º Proc. IEEE Infocom '93, pp , Mar [24] H. Zhang and S. Keshav, ªComparson of Rate-Based Servce Dscplnes,º Proc. ACM SIGCOMM '91, pp , Sept [25] L. Zhang, ªVrtual Clock: A New Traffc Control Algorthm for Packet Swtchng Networks,º Proc. ACM SIGCOMM '90, pp , Sept Amtava Raha receved hs PhD n computer scence from Texas A&M Unversty, College Staton, Texas, n He s currently the senor desgner at Fujtsu, where he s nvolved n the research and development of a management platform to support end-to-end (applcatons, systems, and network) qualty of servce management. In addton, he s also nvolved n the desgn and development of QoS-based transport for supportng multcastng applcatons n dstance learnng. Hs nterests nclude QoS provsonng n areas of networks and e-commerce, network protocols, mddleware, and desgn patterns. Sanjay Kamat receved the PhD degree n computer scence from Texas A&M Unversty, College Staton, Texas, n He receved the BTech degree n mechancal engneerng n 1985 and the MTech degree n computer scence n 1987 from the Indan Insttute of Technology, Bombay. Dr. Kamat s currently a member of the techncal staff of the Hgh Speed Networks Research Group at Bell Laboratores, Lucent Technologes. He was prevously a research staff member at the IBM T.J. Watson Research Center. Hs research nterests nclude hgh speed networks, constrant-based routng, polcy-based networkng, real-tme communcaton, fault tolerance, and dstrbuted systems.

19 RAHA ET AL.: USING TRAFFIC REGULATION TO MEET END-TO-END DEADLINES IN ATM NETWORKS 935 Xaohua Ja receved hs ScD n nformaton scence from the Unversty of Tokyo n He s currently an assocate professor n the Department of Computer Scence at the Cty Unversty of Hong Kong. Dr. Ja's research nterests nclude operatng systems, dstrbuted systems, network systems, computer communcatons, and real-tme communcatons. He s a member of the IEEE Computer Socety. We Zhao receved hs BSc n physcs from Shaanx Normal Unversty, Xan, Chna, and hs MSc and phd n computer and nformaton scence from the Unversty of Massachusetts at Amherst n 1983 and 1986, respectvely. In 1990, he joned the Department of Computer Scence at Texas A&M Unversty, College Staton, Texas, where he s currently a full professor and department head. Hs current research nterests nclude secured real-tme computng and communcaton, dstrbuted operatng systems, database systems, and fault-tolerant systems. Dr. Zhao was a member of the edtoral board of the IEEE Transactons on Computers. He served as guest edtor of a specal ssue on real-tme operatng systems for the ACM Operatng Systems Revew and for a specal ssue on real-tme mddleware for the Internatonal Journal of Real-Tme Systems. He was the general and program char of the IEEE Real-Tme Technology and Applcatons Symposum n 1995 and 1996, respectvely. he s the general and program char of the IEEE Real-Tme Systems Symposum n 1999 and 2000, respectvely. He wll also serve as the general char of the IEEE Internatonal Conference on Dstrbuted Computng Systems n He s an edtor for the Internatonal Journal of Real-Tme Systems and a member of the executve commttee of the IEEE Techncal Commttee on Real-Tme Systems. Dr. Zhao and hs students receved the Outstandng Paper Award from the IEEE Internatonal Conference on Dstrbuted Computng Systems and the Best Student Paper Award from the IEEE Natonal Aerospace and Electroncs Conference. He has publshed more than 100 papers n journal, conferences, and book chapters. He s a member of the IEEE.