As a model for an ATM switch we consider the overow frequency of a queue that

Transcription

1 BUFFER OVERFLOW ASYMPTOTICS FOR A BUFFER HANDLING MANY TRAFFIC SOURCES COSTAS COURCOUBETIS, University of Crete RICHARD WEBER, University of Cabridge Abstract As a odel for an ATM switch we consider the overow frequency of a queue that is served at a constant rate and in which the arrival process is the superposition of N trac streas. We consider an asyptotic as N! in which the service rate N c and buer size N b also increase linearly in N. In this regie, the frequency of buer overow is approxiately exp(?n I(c; b)), where I(c; b) is given by the solution to an optiization proble posed in ters of tie-dependent logarithic oent generating functions. Experiental results for Gaussian and Markov odulated uid source odels show that this asyptotic provides a better estiate of the frequency of buer overow than ones based on large buer asyptotics. ATM SWITCHES; BUFFER OVERFLOW ASYMPTOTICS; EFFECTIVE BANDWIDTHS; LARGE DEVIATIONS; MARKOV MODULATED FLUID AMS 99 SUBJECT CLASSIFICATION: PRIMARY 60K30, SECONDARY 60F0, 60K25, 68M20, 90B0, 90B22. Switches handling any bursty sources In a high speed data counications network, such as one operating according to the ideas of ATM (asynchronous transfer ode), data is packaged into cells of xed size which are Received 9 Deceber 994; revision received 3 May, 995 Postal address: Departent of Coputer Science, University of Crete, PO Box 470 Heraklion, 70 Greece Postal address: Statistical Laboratory, 6 Mill Lane, Cabridge CB2 SB, UK

2 transitted between switches along high capacity links. The trac sources are of various types, such as voice, video and le transfer, and they are bursty, in the sense that the rate at which cells are produced by a source is not constant, but uctuates around a ean rate. Switches are buered, as a safeguard against those occasions when the total rate at which cells enter the buer fro all sources that are routed through that switch exceeds the rate at which they can be served by the output links. Occasionally, the buer will not be large enough and cell loss will occur. This should happen rarely. Since the nuber of sources that are routed through a switch is large, there is a statistical ultiplexing which reduces the probability of buer overow. The idea is that when soe sources are producing cells at above their average rates other sources will be producing calls at below their average rates. A easure of Quality of Service (QoS) is the cell loss rate; this rate should be very sall, typically of the order of 0?6 to 0?0. The cell loss rate is dicult to easure, but is iportant to estiate, both to decide questions of call acceptance and to identify paths in the network that are heavily or lightly loaded for the purpose of call routing. Except for very siple odels it is ipossible for a queueing theory analysis to evaluate the savings in bandwidth due to statistical ultiplexing or to estiate the cell loss rate. For this reason researchers have focused their attention on analysis based on asyptotics and upon on-line easureents of the cell loss rate. One such asyptotic has been developed for systes with large buers. In this paper we study an asyptotic for a large nubers of sources, and show that it can be used ore successfully to estiate the cell loss rate. The odel The behavior of a switch in an ATM network can be odelled as queue with a constant service rate and a buer for B cells. We shall suppose that tie is discretised into epochs, n = ; 2; : : :; and that during epoch n the cell service rate and arrival rate are both constant, and equal to C and X n cells per epoch respectively. The workload at the start of epoch n is denoted W n and W n+ = axf0; in[(w n + X n? C); B]g: The iniization in the above reects the fact that cells are lost when the buer is full, and the axiization insures that the the queue cannot becoe negative. We shall assue that fx n g is a ergodic process, and that E[X n ] < C. This condition iplies that fw n g is ergodic. 2

3 Suppose that fx n g is the superposition of sources of M dierent types, with there being N i sources of type i, i = ; : : :; M. Thus the trac is dened by the paraeter = ( ; : : :; M ) and the scaling paraeter N. Let b and c denote respectively the aounts of buer space and bandwidth per source, so that B = Nb and C = Nc. The cell loss rate can be expressed as L(c; b; N) = E[(W n + X n? C? B; 0) + ], when W n and X n have their stationary values. A related easure is the proportion of tie the buer is full, which we denote (c; b; N) = P (W n = B). Alternatively, we ight suppose the buer is innite and study the proportion of tie that the content it is above level B, say Q(c; b; N). Note that L(c; b; N), (c; b; N) and Q(c; b; N) are all functions of, but to lighten the notation this dependence is not shown. The large N asyptotic The large N asyptotic with which we are concerned takes the following for. (c; b; N) = exp(?ni(c; b) + g (c; b; N)); () where li N! g (c; b; N)=N = 0. Both I(c; b) and g (c; b; N) depend on but as we have done above this is suppressed to lighten the notation. The rate I(c; b) is found as the solution to an optiization proble posed in ters of tie dependent logarithic oent generating functions. Asyptotic () can be copared with a well-known asyptotic in B, (c; b; N) = exp(?nbh(c) + g 2 (c; b; N)); (2) where li b! g 2 (c; b; N)=b = 0. This is an asyptotic that has been studied in a long series of papers, by Courcoubetis and Weber (995), de Veciana and Walrand (994), Elwalid and Mitra (993), Kelly (99), Kesidis, Walrand and Chang (993) and Whitt (993). Either () or (2) ay be used to estiate, as exp(?ni(c; b)) or exp(?nbh(c)) respectively. In section 2 we show that I(c; b)=b! H(c) as b! and so if the buer space per source is large then either of these estiates ight be used. However, we nd that it is usually better to base an estiate of on () rather than (2). For one thing, to base an estiate upon the asyptotic for large N takes ore account of what really happens in practice. In the output link of an actual ATM switch, the nuber of sources is of the order 0 3 {0 4 and so it is correct to apply a results that assues N is large. 3

4 One cannot say the sae for the approach that assues the buer is large; in any ATM switch designs, the total buer space ay be oderate, say 00? 200 cells. This eans that in actuality there is a uch saller aount of buering per source than is supposed by use of the estiate based on the large b asyptotic. The utilization of the switch by the bursty trac also have an ipact on the relative erits of the approaches. In an ATM network, soe of the real-tie trac, such as voice and video, will be assigned high priority and soe, such as le transfer and interactive trac, can be delayed and subject to ow control. The real-tie trac will only utilize a proportion of the bandwidth. If we are concerned to odel the high-priority trac fx n g can be considered to be a odel for this trac alone. Siulations show that when the utilization of the switch is low, say X n = (2=3)C, the estiation of the overow rate using the large N asyptotic reains accurate, whereas the estiate based on the large b asyptotic is very poor, overestiating the overow rate by as uch as 0 5. Section 2 begins with a proof of the large N asyptotic. We discuss the for of I(c; b) in cases when b is sall or large, and the iplications of expressing a quality of service constraint as the requireent that I(c; b) should exceed soe target value. In Section 3 we calculate I(c; b) for a Markov odulated uid odel of a source and observe fro soe calculations with typical values that the large N asyptotic does a uch better job of estiating the cell loss rate for this odel than does the large b asyptotic. Our approach reproduces ore easily soe of the results of Weiss that were obtained by a ore rened analysis. In Section 4 we consider a odel of a source as an autoregressive Gaussian process. Again the large N asyptotic does a better job of estiating the cell loss rate than does the large b asyptotic. In the special case where the paraeters of the autoregressive process are chosen so that the source odel becoes relatively unbursty then the large b asyptotic underestiates the cell loss rate, whereas the large N asyptotic continues to overestiate it and therefore lead to conservative decisions. The analysis in this section conrs and explains observations of Choudhury, Lucantoni and Whitt (994a), (994b). Section 5 concludes with a discussion of on-line estiation and of asyptotics for trac whose coposition is unknown. 4

5 2. The large N asyptotic Both the large N and large b asyptotics are expressed in ters of logarithic oent generating functions. Suppose fy i ; Y i 2 ; : : :g denotes cells generated in epochs ; 2; : : :; by a trac source of type i, and dene ' i () = log E "exp X n=!# Yn i : It is well-known that ' i () is a convex increasing function of. We shall also suppose that the asyptotic logarithic oent generating function exists, dened as ' i () = li! 'i (): We write ' () = P i i ' i () and '() = li! ' (). The large b asyptotic of (2) has its rate given by, H(c) = sup[ : c '()=]: (3) Note that H(c) does not depend on N, but only on proportions of the dierent types of trac. Since the sources are independent, and ' i ()= is increasing in, H(c) () c X i ' i () i ; (4) and the quantity ' i ()= is identied as the eective bandwidth of source type i. Exactly the sae asyptotic holds for L(c; b; N), (c; b; N) and Q(c; b; N). In eect, these quantities dier by approxiately constant ultiplicative factors, and these factors are absorbed by g 2. Siulation results have shown that an approxiation of L(c; b; N) by exp(?nbh(c)) can be good for large values of Nb. However, as entioned above, the approxiation is good only if b, the buer per source, is quite large. For realistic specications of an ATM switch, b ay be very sall, and the approxiation can overestiate L(c; b; N) by several orders of agnitude. Equivalently, it overestiates the eective bandwidths. The large N asyptotic result is expressed in the following theore. This result has also recently been proved independently by Dueld (994) and by Sionian and Guilbert (994) for the case of on-o Markov uid sources. Theore For the odel above and under appropriate assuptions, li N! log (c; b; N) =?I(c; b); N 5

6 where I(c; b) = inf sup [(b + c)? ' ()]: (5) For a xed the supreu over in (5) calculates a large deviation coecient (appropriate to an asyptotic in N) for the probability that the buer overows in periods; the way in which this overow occurs is by each of the N sources producing c + b cells, and so contributing an equal share to the total of C + B cells that causes the buer to ll. The inu xes upon the for which this probability is greatest. Proof of Theore. Recall that the switch is odelled as a queue with a constant service rate of Nc. If the buer is overowing at epoch 0 then there ust have been soe epoch,?, at which the buer was last epty and since which at least N(c + b) cells have been received. Thus (c; b; N) P (S > Nc + Nb for soe ), where S = P n= X n. Pick a such that c? '( ) > 2. Then since ' ( )! '( ), there ust be an such that for all >, both and A Cherno bound is that for all > 0, c? ' ( ) > b + > sup[c? ' ()]: P (S > Na) E exp((s? Na)) = exp(?n[a? ' ()]): Using this, with a = c + b, we have Hence P (S > Nc + Nb for soe ) li N! X = X? = X? = P (S > N(b + c)) e?n sup [(b+c)?' ()] + e?n [ b+fc?' ( )g] = X X e?n sup [(b+c)?' ()] + = e?n [ b+] (? )e?n in< sup [(b+c)?' ()] + e?n [ b+]? e?n : log (c; b; N)? in N < sup [(b + c)? ' ()] 6

7 ? inf sup [(b + c)? ' ()]: In the reverse direction, let us consider consecutive epochs. The expected nuber of these epochs in which overow occurs is at least as large as the probability that there is overow during at least one of the epochs. Thus we have that for all, (c; b; N) P (S > Nc + Nb) : Therefore, using Craer's lower bound for the su of N i.i.d. rando variables li N! N As this holds for all, the proof is coplete. log (c; b; N)? sup[(b + c)? ' ()]: By observing fro (5) that for any given, NI(c; b) Nb if and only if for each there exists a such that (b + c)? ' () b, we have the following equivalent expression for bandwidth requireent. Corollary. b I(c; b) b () c sup inf? + ' () : (6) The far right hand side of Equation (6) species the least value that ay be taken by c if the large deviation coecient I is to exceed b (a nuber related to axiu allowable cell loss rate). Alternatively, for xed values of c and, it expresses in ters of the quantity ' () the constraint on the trac that ay be carried by the switch. Notice that there is no notion of eective bandwidths siilar to that which occurring in Equation (4) for the large b asyptotic. The large N asyptotic ay be interpreted in another way: it is as if in a syste, indexed by N, a single source is replaced by the average of N identically distributed sources. The asyptotic is taken as N! while the total buer and total bandwidth are held constant. This asyptotic has been considered by Weiss (983), for identical sources that are Markov odulated uids. His approach is ore rened in that he uses a large deviation result for saple paths to nd the ost likely path by which a buer will ll. We consider now soe issues raised by the large N asyptotic. 7

8 Discretization of tie We have assued a discrete tie odel and it is illuinating to consider how this aects the results. Suppose the length of an epoch is doubled. Then c ust be replaced by 2c, and ' () by ' 2 (). We get I(c; b) = inf sup [(b + 2c)? 2' 2 ()]: So it is as if we are now only taking inf only over the even values of. The answer will be greater. This is to be expected since by taking an epoch to be longer there is soe averaging of the trac within each epoch. This dependence on the discretization is not seen in the analysis of the large b asyptotic. The case of no buer If there is no buer then we have the following. Theore 2 Proof. We have and I(c; 0) = sup[c? ' ()]: I(c; 0) = inf sup [c? ' ()]: (7) ' () = log E log E " exp " = ' (); X n=!# X X n n= exp (X n ) # where the rst inequality follows by convexity. Thus I(c; 0) inf sup [c? ' ()] = sup[ 0 c? ' ( 0 )]: (8) 0 The nal equality follows by letting 0 =. On the other hand, by choosing = in the iniization of (7), the right hand side in (8) attained. Thus I(c; 0) = sup [c? ' ()]. Note that (7) is siply the large deviation asyptotic for the probability that the buer should receive ore than C cells in a single epoch. 8

9 Large buer asyptotics The following theore connects the asyptotics for large b and large N. Theore 3 li b! I(c; b) = H(c): b Proof. Recall that = H(c) satises '( )= = c. Let us take b = (' 0 ( )? c). Note that since ' 0 ( )! ' 0 ( ), as!, we also have b! as!. Furtherore, the growth in b is approxiately linear for large. Thus for any suciently large that b b, b I(c; b) b I(c; b ) b b b b sup [ + (=b )(c? ' ())] b b sup [ + (=b )(c? ' ( )? (? )' 0 ( ))] = b b '0 ( )? ' ( )= ' 0 ( )? '( )= ; where the third inequality follows by convexity of ' (). But by the fact that b grows linearly in for large, we can let! as b! in such a anner that b =b!. So fro the above it follows that li b! b I(c; b) : To establish an inequality in the reverse direction, pick <. Recall that '()= is increasing in and li! ' () = '(). Then there exists 0 such that for all 0, c?' () > 0. So for all 0 we have [b + (c? ' ())] > b and I(c; b) = inf sup [(c + b)? ' ()] inf [b + (c? ' ())] inf in <0 [b + (c? ' ())]; bg = b + in <0 [(c? ' ())? ] The second ter on the right hand side of the nal expression is a constant and hence li b! I(c; b) : b 9

10 Since this holds for all <, the proof is coplete. Notice that in the case that fx n g is a sequence of i.i.d. variables, ' () = '() and so we can take 0 = 0. Thus I(c; b) bh(c) for all b. Moreover, for those b for which b = (' 0 (H(c))? c), = ; 2; : : :, we have I(c; b) = bh(c). Also, fro the rst part of the proof we have at upper bound, and so in the i.i.d. case, H(c) I(c; b) b b b = ('0 (H(c))? c) H(c) b The following corollaries are worth noting. + '0 (H(c))? c b H(c) Corollary 3. As b! the value of that is optial on the right hand side of (5) also tends to innity. Proof. Suppose the result is not true and that for arbitrarily large b, the optial is less than soe 0. Choose any. Then for arbitrarily large values of b, I(c; b) > b inf <0 [ + (=b)(c? ' ())]: This iplies li b! (=b)i(c; b) >. For > H(c) this contradicts Theore 2. Corollary 3.2 g (c; b; N)=b! 0 as b! : Proof. This follows fro g (c; b; N) = log (c; b; N) + NI(c; b). For xed N we have that as b! both log (c; b; N)=b!?H(c) by de Veciana and Walrand (994), and I(c; b)=b! H(c) by Theore 3. Thus g (c; b; N)=b! 0. The shape of the acceptance region Recall that there are N i sources of type i. Thus the trac is dened by the paraeters N and = ( ; : : :; M ). Given a desired loss probability of less than exp(?nb), we ight dene as acceptable those possible ixes of sources 2 R, where R = f : Nb < NI(c; b; )g; 0

11 or equivalently, " b c sup inf? + MX i= # ' i () i : (9) Now we can see that in the case that the cells produced in each period are i.i.d. rando variables, then odulo discreteness eects, the acceptance region is linear. For in this case, ' () = '(), and I(c; b) = inf sup [(b + c)? '()] inf sup t>0 [(b + tc)? t'()] = sup[b : '()= c] = sup " b : X i i ' i ()= c # : So I(c; b) b if and only if '()= c; this is the sae as (4) that holds for the large b asyptotic and shows that in this case the boundary to the acceptance region is linear in the i. However, if we do not disregard discreteness eects, or the cells produced in successive periods are not i.i.d., the shape of the acceptance region can be ore coplicated. To show the eect introduced by the discreteness of we can consider two sources in which the nuber of cells in successive epochs are i.i.d. Gaussian variables, with eans = 5=3, 2 = and variances 2 = =3, 2 2 =. For a source of type i, ' i () = i + 2 i 2 =2. Hence the acceptance region dened in (9) becoes c MX i= i i + sup 2s 4 2b P i i 2 i 3? b 5 : The rst ter is linear in, but the ter within the supreu is concave in. For the values above, and = 2, b =, c = 2, soe values of ; 2 that lie on the boundary of the acceptance region are These show that the boundary of the acceptance region is neither locally concave nor convex. We believe that it is not only because of discreteness of that the shape of the acceptance boundary is not always either concave or convex. However, we have yet to give an exaple of this.

12 Of course we know that I(b; c)=b! H(c), as b! and so the acceptance region dened as Nb NI has a boundary that is asyptotically the linear one dened in Equation (4) by c P i i ' i ()=. 3. Markov odulated uid sources The Markov odulated uid odel of a source is one in which the rate of the source alternates between 0 and a peak rate a according to the state of a continuous tie Markov process. The o and on states have exponentially distributed holding ties with paraeters and respectively. The proportion of tie the source in on is =( + ), and so for sensible scenarios, we require a=( + ) < c < a. Theore was proved under the assuption of a discrete tie odel, but it also clearly holds for the continuous-tie setting here (cf. Dueld (994) for a proof). To nd I(c; b) for the a Markov odulated uid, we will copute the oent generating functions Z t z i;t () = E i exp x(s)ds ; i = 0; ; where x(t) is the uid rate at tie t and i = 0; as the source is o or on initially. Now 0 z 0;t+ = (? )z 0;t + z ;t + o() z ;t+ = e a [z 0;t + (? )z ;t ] + o(): So 0 _z 0 and _z 0 C A B a? 0 C A z 0 z C A z 0;t =! 2! 2?! e! t?!! 2?! e! 2t z ;t =! 2? a! 2?! e! t + a?!! 2?! e! 2t ; where! =?( +? a)? p ( +? a) 2 + 4a 2! 2 =?( +? a) + p ( +? a) 2 + 4a 2 2

13 are the two roots of! 2 + ( +? a)!? a = 0. Hence Z t z t () = E exp x(s)ds The proble which we wish to solve is = 0 + z 0;t() + + z ;t() =! 2 + (! 2? a) (! 2?! )( + ) e! t +?! + (a?! ) (! 2?! )( + ) e! 2t : I(c; b) = inf sup t>0 [(b + tc)? log(z t ())]: By Theore 3, we know that I(c; b)=b! H(c), and also that the t that is optial in the right hand side above tends to as b!. Note that '() = li t! t log z t() =! 2 (): The bandwidth requireent c! 2 ()= () H(c) is precisely that given in other papers, such as Gibbens and Hunt (99). An alternative expression of this constraint is H(c) = ( + )c? a c(a? c) : Nuerical results We have conducted a nuber of experients to deterine the eectiveness of estiating buer overow rates using the large N asyptotic. For a Markov odulated uid odel, Q(c; b; N) = P (W > Nb) can be obtained exactly using techniques of Anick, Mitra and Sondhi (982). To odel a voice source we use the paraeters = 650s, = 353s and a = 64kbps. The ean bandwidth is 22:48kbps. In Figure, we take c to be as :5 ties the ean rate, i.e., 33:72kbps. We plot as functions of b graphs of I(c; b), bh(c) and the exact value of (=N) log Q(c; b; N) as coputed using the forula of Anick, et. al. for N = 50; 00. Notice that the large N asyptotic is a uch better estiator of Q(c; b; N) than is the large b asyptotic, and that it is ore exact for larger N. For exaple, at b = 0:05,?(=N) log Q(c; b; N) is 0:0903 for N = 50, 0:09758 for N = 00; these are well-estiated by I(c; b) = 0:07989, but not by bh(c) = 0:0024. Both estiates are conservative in the sense that they both overestiate the true probability that the queue should exceed N b. 3

14 The shape of I(c; b) near b = 0 Our nuerical results agree with the calculations of Weiss (983) that for the Markov odulated uid the behavior for sall b is I(c; b) = C + C 2 p b. We have not been able to establish the p b behavior fro (5), but we can copute the constant C = I(c; 0). Following the ideas in Theore 2, suppose s is very large. Then I(c; 0) = inf sup s>t>0 inf sup s>t>0 [tc? t' t ()] [n(t=n)c? (t=n)' t=n (n)] = inf sup[tc? t' t ()] s=n>t>0 li [tc? t' t ()] sup t!0 h = li sup tc? log E[exp([tX(0) + o(t)])] t!0 "!# e = li sup c? log a t! o(t)=t = c 0 c 0 log + (? c 0? c 0 ) log ; p? p where p = =( + ) and c 0 = c=a. Clearly I(c; 0) li t!0 sup [tc? t' t ()]. So this is in fact the value of I(c; 0), and it is just the asyptotic rate for the probability that the nuber of on sources exceeds c=a. i 4. Gaussian sources Gaussian stationary sources Suppose fx n g is the superposition of N Gaussian sources, each distributed as fy n g with ean, variance 2 and autocovariance function (k). Then ' () = =2, where 2 is given by 2 = var X i= Y i! = 2 + 2[(? )() + (? 2)(2) + + (? )]: Notice that li! 2 =, where is the index of dispersion and equal to P? (k). Now I(c; b) = inf (b + c? ) 2 ; 2 2 with the optiu achieved by = (b + c? )= 2. 4

15 Also the bandwidth requireent (6) is b c sup inf? + ' () with the optiu achieved by = p 2b= 2. 2 = sup 4 s2b + 2 3? b 5 ; An autoregressive source A siple autoregression has soeties been used as a odel for a videotelephone source. Let us take Y n = Y n? + (? ) + n ; where f n g is Gaussian white noise with variance 2. Then 2 = 2 =(? 2 ), = ( + ) 2 =(? ) and 2 can be derived as follows. 2 = (? ) 2? ( ? ) 2 = (? ) 2? + 2 2??? 2 = (? ) + + (? 2 )? (? ) 2? =?(? 2) ? = +? 2? 2? + (? ) 2 2 =? 2 (? ) (? ) 2 2 : "?? 2? +? It turns out that 2 is a convex increasing function of when > 0. Notice 2 =? 2 (? ) (? ) 2 2 < if > 0. But 2 > for < 0. Now since '() is quadratic in, we can easily nd that sup[(b + c)? ' ()] = (b + c? )2 2? 4 (? ) (?) 2 2 : For large b this is iniized by large, and so assuing is sall this is iniized with respect to by ' b c? + =; # 5

16 where The optial value is I(c; b) ' 2b(c? ) + = 4 (? ) 2 2 : (c? )2 2b(c? ) 4(c? )2 = + 2 ( + ) 2 : 2 This can be copared with bh(c) = 2b(c? )=. One sees that bh(c) < I(c; b) when > 0, but that the inequality is reversed when < 0. The case > 0 corresponds to a relatively bursty source, while < 0 corresponds to relatively sooth source, because of the negative rst order autocorrelation. When > 0 the use of the large b asyptotic to estiate L(c; b; N) leads to a larger estiate of cell loss rate than does the large N asyptotic: the experiental evidence is that both asyptotics lead to over-estiates of the true cell loss rate but that the large N asyptotic is closer. When < 0 the large b asyptotic estiates a saller loss rate than the estiate based on a large N asyptotic: the experiental evidence is that the large b asyptotic under-estiates the true loss rate while the large N asyptotics over-estiates it and is closer. In this case the large N asyptotic is clearly preferable. Nuerical results For an autoregressive Gaussian process with =?0:5, = 8, 2 = 64 and c = 8:5, we plot exp(?n bi(b; c)) and exp(?bh(c)) against b, together with siulation results (over 2,000,000 epochs) for the probability Q(c; b; N) = P (W > Nb), for N = 000 and ve values of b, :0; :2; :4; :6; :8. The proportion of the tie that the buer is epty is 60.95%. Figure 2 shows that for these sall aounts of buer per source, the large N asyptotic over-estiates the tail probability by a factor of about 2{3, while the large N asyptotic under-estiates it by a factor of about 2{5. For exaple, at b = :2, the true value has a 95% condence interval (92; 04) 0?5, and ean 98 0?5 ; the large N asyptotic gives 263 0?5 and the large b asyptotic give 32 0?5. The sign of I(c; b)? bh(c) The above exaple is illuinating in considering the dierence between the large N and large b asyptotics. Our results are consistent with the work of Choudhury, Lucantoni and Whitt 6

17 (994a), (994b), who discuss the fact that their nuerical experience suggests that the large B asyptotic should be odied by prior ultiplication by an exponential factor in N, so Q(c; b; N) e?n e?b : This can now be explained theoretically by writing Q(c; b; N) = e?ni(c;b)+g (c;b;n ) = e?n [I(c;b)?bH(c)]?NbH(c)+g (c;b;n ) = e?n [I(c;b)?bH(c)?g (c;b;n )=N ] e?bh(c) : Of course = H(c) and g (c; b; N)=N! 0 as N! with b xed. It is therefore clear that has the sign of I(c; b)? bh(c)? g (c; b; N)=N which for xed b is deterined by the sign of I(c; b)? bh(c) for large N. We have seen in the discussion above that I(c; b) > bh(c) when > 0 and I(c; b) < bh(c) when < 0, and that these correspond to greater or less burstiness of the source, and give > 0 and < 0 respectively. Notice that I(c; b) > bh(c) if ' () > '(), for all and. Note that if for two rando variables, X and Y, E exp(x) > E exp(y ) then X is ore variable than Y. Siilarly, an ordering of oent generating functions corresponds to a ordering of burstiness. This eans that the quantity of cells arriving over a nuber of periods is actually less bursty than is iplicitly assued when ones uses with '() in (3) to calculate H(c). Therefore exp(?bh(c)) tends to overestiate Q(c; b; N). In the cases we have considered it appears that ' () is onotone in, and thus > 0 and ' () < '(), corresponds to the case in which ' () < ' (), and in which we would say that the source is `ore bursty than Poisson', in the sense that for a Poisson source, or any which is i.i.d., we have ' () = ' (). The sae coents apply, with inequalities reversed, so < 0 occurs for sources that are `less bursty than Poisson.' 5. Open issues and rearks On-line estiation An iportant issue in ATM networks is the accurate and tiely estiation of the cell-loss probabilities that occur at various switches in the network, so that correct decisions can be 7

18 ade about whether or not to accept ore trac or how it should be routed. A generic proble that the network anageent syste faces is that because the events of interest are rare, e.g., cells loss rates of the order of 0?6? 0?0, any brute-force on-line estiation procedure would fail because of the large tie that would be required to ake an accurate estiate. Providing a reasonable condence interval for the estiator requires tie of the order of agnitude of tens of inutes, which is ipractical since decisions about accepting new trac in the network ust be done iediately, and because the coposition of the trac can change in such a large tie interval. The large N asyptotic can be used to estiate in real-tie the cell loss rate in the following anner. Suppose N sources (coprising N i sources of type i) are active in a switch with total bandwidth C and total buer B. Suppose that we can ake an on-line easureent to estiate the cell loss rate that would occur if N=k sources were routed through a switch that has total bandwidth C=k and total buer B=k. This can be done by a special device at the switch level which siulates a `virtual' switch of =k the size of the actual one, operating on a representative saple of the actual input of size =k. The device provides an estiator of the buer overow rate that occurs in the virtual syste. The large N asyptotic suggests that the loss rate in the virtual syste should be about p = exp(?ni(c; b)=k), hence it can be estiated accurately in a tie that is orders of agnitude saller than the tie required to ake an estiate of the buer overow rate in the original syste, and we can extrapolate the actual loss rate to be p k. For exaple, suppose the channel is carrying 000 sources, with a loss rate of about 0?0. As we have already entioned, such a sall cell loss rate is dicult to easure directly. However, an on-line easureent of the loss rate for 200 sources in a virtual syste with one-fth the bandwidth and buer space should easure a loss rate p of about 0?2. At this loss rate cell losses will be observed in a relatively short tie and the loss rate can be easured satisfactorily. Then p 5 is an estiate of the actual cell loss rate. Of course there is the opportunity to ake ve independent estiates of p, using ve groups of 200 sources. This idea can be rened further to provide a ore accurate estiate of the cell-loss probability by coputing the o(n) ters in (). On can assue that (N) = AN exp(?ni) and estiate the values of A,, I by easuring in three virtual systes of size N=k ; N=k 2 ; N=k 3, where N=k < N=k 2 < N=k 3 << N. Then (N) is coputed by substituting the correspond- 8

19 ing values. This provides an alternative to the MINOS procedure described in Courcoubetis et al. (995). Nuerical work has validated the utility of this heuristic approach. Trac of rando coposition The class of trac ay not be known exactly. Suppose that a certain class of trac the sources are actually of k possible types, with each source having probability p i of being of type i. The probability of buer overow is axiized when the observed ix deviates fro the expected ix in an appropriate way and sources produce cells at above their ean rates. The probability that N sources of such rando ixture should have epirical distribution ; : : :; k is P () = exp?n kx i= and thus taking the product of probabilities, li N! log (c; b) =? inf N P inf sup M i : i= i= i log( i =p i ) + o(n) " (b + c)? kx i=! ; i f' i ()? log( i =p i )g Assuing the optiization over i occurs at an interior point, the stationarity condition iplies i is proportional to p i exp(' i ()), and hence after soe algebra, # : I (c; b) = li N! N =? inf sup =? inf sup log (c; b) " " (b + c)? log (b + c)? log kx i= kx i= h i!# p i E exp S i p i exp(' i ())!# ; (0) where S i has the stationary distribution of the nuber of cells produced by a source of type i over epochs. Note that the nal ter in (0) is, by convexity, log kx i= p i exp(' i ())! kx i= p i ' i ()) = ' (); and thus the asyptotic overow frequency, given by (0), is greater than when the ix is known exactly. Then as b! the optial value of tends to innity and the su in (0) is doinated by the axial ter, hence I (c; b) li b! b = H (c) = axf : ' i ()= < c; for all ig = in H i (c); i 9

20 where H i (c) is obtained for a source consisting only of type i. Siilarly, H (c) > if and only if ' i ()= < c for all i, and so this analysis concludes that the eective bandwidth of the rando ix is equal to the axiu of the eective bandwidths of the individual source types. This akes sense since the ost likely way that a single source of unknown type will ll a large buer is if the source type turns out to be the ost bursty of the k possible types. Open probles A nuber of further issues require study. These include: (a) the ecient nuerical solution of the double optiization proble dening I(c; b); it is not clear that the function always has a saddle point, though this is the case in the exaples we have studied; (b) further understanding of the properties of the solution and the iplied acceptance region, and (c) the on-line estiate of the ' (), or on-line estiation of I(c; b). Acknowledgent We are very grateful to Georgos Fouskas for coputing the nuerical result reported in Sections 4 and 5. References Anick, D., Mitra, D. and Sondhi, M. (982) Stochastic theory of a data-handling syste with ultiple sources. The Bell Syste Technical Journal 6, 872{894. Choudhury, G., Lucantoni, D. and Whitt, W. (994a) On the eectiveness of eective bandwidths for adission control in ATM networks. In J. Labetouille and J. Roberts, eds., ITC, volue 4, pp. 4{420. Elsevier Sciences B.V. Choudhury, G., Lucantoni, D. and Whitt, W. (994b) Squeezing the ost out of ATM. Preprint. Courcoubetis, C., Kesidis, G., Ridder, A., Walrand, J. and Weber, R. (995) Adission control and routing in ATM networks using inferences fro easured buer occupancy. IEEE Transactions on Counications 43, 778{

21 Courcoubetis, C. and Weber, R. (995) Eective bandwidths for stationary sources. Probability in the Engineering and Inforational Sciences 9, 285{296. to appear. de Veciana, G. and Walrand, J. (994) Eective bandwidths: call adission, trac policing and ltering for ATM networks. to appear in Queueing Systes. Duffield, N. (994) Econoies of scale in queues with sources having power-law large deviation scalings. Preprint. Elwalid, A. and Mitra, D. (993) Eective bandwidth of general Markovian trac sources and adission control of high speed networks. IEEE/ACM Trans. Networking, 329{343. Gibbens, R. and Hunt, P. (99) Eective bandwidths for the ulti-type UAS channel. Queueing Systes, 7{28. Kelly, F. (99) Eective bandwidths at ulti-class queues. Queueing Systes 9, 5{5. Kesidis, G., Walrand, J. and Chang, C. (993) Eective bandwidths for ulticlass Markov uids and other ATM sources. IEEE/ACM Transactions on Networking, 424{428. Sionian, A. and Guilbert, J. (994) Large deviations approxiation for uid queues fed by a large nuber of on-o sources. Preprint. Weiss, A. (983) The large deviation of a Markov process which odels trac generation. Technical report, AT&T Bell Laboratories. Whitt, W. (993) Tail probabilities with statistical ultiplexing and eective bandwidths in ulti-class queues. Telecounication Systes 2, 7{07. 2