Asymptotics for Open-loop Window Flow Control

An open-loop window flow-control scheme regulates the flow into a system by allowing at most a specified window size W of flow in any interval of length L. The sliding window considers all subintervals of length L, while the jumping window considers consecutive disjoint intervals of length L. To better understand how these window control schemes perform for stationary sources, we describe for a large class of stochastic input processes the asymptotic behavior of the maximum flow in such window intervals over a time interval [0, T] as T and L get large, with T substantially bigger than L. We use strong approximations to show that when T >> L > log T an invariance principle holds, so that the asymptotic behavior depends on the stochastic input process only via its rate and asymptotic variability parameters. In considerable generality, the sliding and jumping windows arc asymptotically equivalent. We also develop an approximate relation between the two maximum window sizes. We apply the asympto-tic results to develop approximations for the means and standard deviations of the two maximum window contents. We apply computer simulation to evaluate and refine these approximations.


Introduction
The Hungarian influence on probability theory, and mathematics more generally, seems to be greater than can be accounted for solely by chance.In this paper we pay tribute to an outstanding example (who we also claim as an American), Lajos Takcs, on the occasion of his 70th birthday, by applying some Hungarian probability theory to a problem of interest in the design of emerging high-speed communication networks.In particular, we apply strong approxi- mations as in Koml6s, Major and Tusndy [9], [12], Cs6rg5 and Rvsz [6] and Cshrgh, Horvth  and Steinebach [5] to study open-loop window flow-control schemes.We also refer to relate work by Erdhs and Rnyi [7].
The most familiar window flow-control scheme is a closed-loop control mechanism.Sources regulate their flow into the system (e.g., packets into a network) by keeping track of the flow that has been transmitted but not yet received, typically using acknowledgements sent back from the receiver to the source.The source stops sending when the unacknowledged flow reaches a speci- fied window size; e.g., see pp. 97 and 429 of Bertsekas and Gallager [2].
In contrast, open-loop window flow-control schemes use only local information.For high- speed communication networks, there is growing interest in open-loop control schemes, because end-to-end delays over long distances can become very large relative to the time significant applications, in the remaining sections we discuss asymptotics.We start in 2 by reviewing the central limit theorem and strong approximations, which justify an approximation by Brownian motion.In 3 we consider extreme-value asymptotics for Brownian motion as T-cx.In 4 we provide conditions for the Brownian approximation to be asymptotically correct and establish limits for S and J as both T and L get large.In particular, there we develop support for the following relatively simple approximations for the random variables (and their means) provided that log T << L << T, where << means much smaller than: (1) where ,3(T,L) = V/21og(T/L) + 2log log T 1.28 V/21og(T/L) J EJ AL + Cj(T,L)V/Ac2L, where Cj(T,L) V/21og(T/L)-log log(T/L)+ 1.38   2V/21og(T/L S-----ES-L E._ L " s (T, L)   for (I) S and (I)j defined above.In (1)-(3), A is the rate and c 2 is the asymptotic variability parameter of I (defined in (6) below).In (1)-( 3) and throughout the paper, log means the natural logarithm, i.e., to the base e.Without loss of generality (by the choice of the measuring units), we can let A-1.Hence, there are really only three parameters in (1)-(3)" L, T, and c2.
The approximations for S and J in (1) and (2) have a simple interpretation: The first term AL is the approximate mean for a single interval of length L; the term is the approximate standard deviation for a single interval of length L; and the remaining factors (s(T, L) for S in (1) and (j(T,L) for g in (2) represent the increase due to the maximization.It is significant that the factors (I) S and (I)j depend only on T and L, and not upon A and c2.
Note that, by (3), the ratio (J-AL)/(S-L)approaches 1 as T---,cx, but (3) provides an approximation for the ratio for finite T. This ratio estimate as a function of L and T is displayed in Table 1.We can also use ( 1) and (2) to develop an approximate expansion directly for the ratio J/S in powers of 1/.In particular, by (1) and ( 2), However, we will focus on (3).
The maximum window contents S and J are of course random variables.rough estimates for the standard deviations, namely, / Ac2L SOY(S) , SOY(J) 1.281Olo-27-7=,L V l We also derive (4) see Section 3.Moreover, we would approximate the distributions of S and J, after appropriate normalization, by the type-/ extreme-value distribution, i.e., the cdf (cumulative distribution function) exp(-e-x), -o < x < c.; see p. 4 of Leadbetter, Lindgren and RootzSn [11] and (10) -( 13) below.
We can combine (1), (2), and (4) to obtain estimates of the coefficients of variation (CV, standard deviation divided by the mean) of S-SL and J-SL.Keeping only the dominant V/21og(T/L) term in (I) and (I), we obtain CV(S-)tL) ,, CV(J-)tL) ,, l o g O ( ' 6 T L ) ._ (5) Hence, if T/L-10 k, then CV(S-$L) 0.278/k.If k > 3, then the standard deviation of S-SL should be less than 10% of the mean.
In 5 we briefly discuss discriminating windows with L O(log T), for which the asymptotics depend upon more than two parameters $ and c2, and thus have more potential for providing information about the input process.In 6 we make numerical comparisons using computer simulations.In particular, we consider 10 replications each of several renewal processes with -1, T 106 and various values of L.
These simulation experiments provide additional support for approximations (1)-( 5).Indeed, the last term in (I, s in (1) was suggested by the simulations.In 7 we state our conclusions.
We were motivated to look for approximations (1) and ( 2), because we want to compare the open-loop window flow-control schemes to other open-loop flow-control schemes.In particular, in [1] we compare the sliding window to a leaky-bucket-based flow-control scheme.There we conclude that the sliding window admits larger bursts than the leaky bucket for given peak rate and given sustainable rate.To draw this conclusion, we carry out a specific construction: We generate a sample path of a stationary point process I and specify a window length L. The reciprocal of the observed minimum distance between consecutive points is the realized peak rate, and the ratio S/L is the maximum possible sustainable rate, when S S(L,T,I) is the maximum sliding window content.With this peak rate and sustainable rate, the observed sample path just passes through the sliding window control.We then let the leaky bucket have drain rate v equal to the sustainable rate for the sliding window, and let the bucket capacity be the smallest level such that the observed stochastic sample path just passes through the leaky bucket.With this construction, both controls have the same peak and sustainable rates, and the given sample path just passes through both controls.Finally, with the control parameters so determined, we ask which control scheme allows larger bursts, where a burst is defined as the maximum number of consecutive arrivals at the peak rate.We find that the sliding window allows larger bursts, and we use (1) to estimate the difference.See [1] for more details.Reibman and Berger [14] reach the same conclusions for sample video teleconferencing sequences.

The Central Limit Theorem and Strong Approximations
First, as T gets large, we can apply extreme value theory, as in Leadbetter, Lindgren and Rootzn [11].In general, though, the limiting behavior depends quite delicately upon the input process I.However, if we can assume that L also is suitably large, then we can hope for significant simplification.As L gets large, in great generality, each increment I(t + L)-I(t) becomes approximately normally distributed by the central limit theorem (CLT), i.e., I(t+L)-I(t)-AL g(0,1)as L-oc (6) V/Ac2L for all t, where =: denotes convergence in distribution, N(0,1) denotes a standard (zero mean, unit variance) normal random variable, A is a positive constant called the input rate and c 2 is a positive constant called the asymptotic variability parameter.
To have the CLT (6), we need to have the second moment EI(t) 2 finite and we need to have the dependence between the increments I(t + h)-I(t) and I(s)-I(s-h) for s <t and h > 0 decrease suitably as t-s increases.We assume that we do not have infinite second moments or the long-range dependence that would invalidate (6).
If I is a stationary stochastic point process (i.e., has stationary increments) then c 2 is typically the limiting value of the index of dispersion for counts (IDC), i.e., c 2-I c(oc -lira Vat I(t) t-o EI(t------ see p. 71 of Cox and Lewis [4].(The role of the second moment condition is clear in (7).)If I is a renewal counting process with i.i.d interrenewal times X k, then c 2 is the squared coefficient of variation of (SCV) of X1, i.e., (It is important that a square appears in the denominator of (8), but not in (7).) In fact, we need more than (6), and we often have it.In great generality, a properly normalized input process such as I can be approximated by a Brownian motion.Indeed, our key assumption will be that I can be approximated strongly by a Brownian motion, i.e., there exists a standard (zero drift, unit variance) Brownian motion (BM) B {B(t) _> 0} such that 0 <_ t<_T}-O(logT) w.p.lasToe; (9) see CsSrg6 and R6v4sz [6] and Csarg6, Horvth and Steinebach [5].These references show that (9) holds in great generality, so that this seems to be a reasonable starting point for us.Note that (9) can be regarded as a form of functional central limit theorem (FELT) generalizing the CLT in (6).To convert (9) into the FELT, replace in (9) by nt, T by n, and then divide by X/.This yields V/Ac2n "0 <_ t<_l -0 V/-, w.p. 1 as n----oe and note that tbr each n, {B(nt)/v/-" t>_O} is distributed the same as {B(t)" t>_O}.Hence, from (9) we get I(nt)-Ant =V B(t) as with the convergence being in the Nnction space D[0, 1]; see pp. 11-15 and 88-95 of [6] for %rther discussion.
It is also significant that the error in (9) can be regarded as best possible; see [5], [6].For a renewal process with interrenewal times X k having mean Iand SCV c 2, (9) holds when e sX1 < for some s > 0; see Corollary 3.1 on p. 1450 of CsSrg5 ttorvth and Steinebach [5].
(Other strong approximations hold with bigger errors when only EX < Nr some p > 2; see [5], [6].)Theorem 3.1 of [5] also provides a means to treat a large class of counting processes with dependent intervals.Moreover, I(t) need not be a counting process; e.g., it could be the input in a fluid model.
In applications, e.g., to communications networks, we are often interested in large L and large T. A key issue in the asymptotics is the way these two parameters grow.We consider this issue in 4.

Extreme Values for Brownian Motion
In this section we simply sur ose that I can be approximated by a BM.Consistent with (9), we assume that I(t) t / /c 2 B(t), where B is a standard BM.In this case, we can apply classical results to describe J and S as T--,cx for any fixed L, omitting I from the notation.In particular, by the extreme-value theorem for i.i.d, normal random variables in Theorem 1.5.3 on p. 14 of [11], a(L,T)(J(L,T)-)L-b(L,T)):::Z as Toc, (10 where Z is the classical type-/extreme-value distribution with cdf and Note that EZ-0.5772 (Euler's constant), VarZ-r2/6-1.645,SCV(Z)-4.94,median(Z) 0.3667 and mod(Z)= 0.9624 for the type-/ extreme-value random variable Z in (10); see Chapter 21 of Johnson and Kotz [8].
We use (10) to develop approximation (2) for the mean of J and thus J itself.
we let In particular, J(L,T) L + b(L,T) + EZ a(L,T) which reduces to (2).We also use (10) to obtain an estimate of the variability of J. (10), we have the approximation (14) which yields (4).

Based on
We point out that it is not obvious that approximations ( 14) and ( 15) (( 2) and (4)) will be good.First, we need to approximate I by Brownian motion and, second, we need the extreme- value asymptotics for Brownian motion to perform suitably well for realistic values of T. It is well known that the CLT (6) often yields remarkably good approximations for moderate values of the limiting variable (when there is weak dependence).In contrast, the extreme-value limit SDV(Z) SDV(J), a(L,T)' (15) theorems do not yield such good approximations.In part, this is due to the rate of convergence in the extreme-value theorems being remarkably slow; see Section 2.4 of Leadbetter, Lindgren and Rootzn [11].For example, for the limit (10) the rate of convergence of the normalized cdf's is slower than l/log(T/L); see (2.4.8) and (2.4.10) of [11].Hence, even if T 106 and L 103, good approximations based on (10) are not automatic.Thus, we should be interested in the computer simulations in 6.
As a corollary to (10), we can divide by a(L,T) and obtain the simpler result J(L,T)-AL-2c2Llog(T/L) = 0 as T--c. (16) When we divide by v/log(T/L)in ( 16), we obtain v/log(T/L) for fixed L. In ( 16) and ( 17) convergence in distribution is equivalent to convergence in probability since the limits are deterministic.
(19) J(L,T)-AL Hence, using the Brownian approximation, by (19) S and J are asymptotically equivalent as T---,cx for any fixed L.Moreover, (17) and ( 18) are consistent with S and J being asymptotically approximated by (1) and ( 2), but we need additional refinement to get the approximations in (1) and (2).

The Role of L in the Brownian Motion Approximation
The asymptotics in the previous section apply if we do indeed have a BM approximation.
However, the quality of the Brownian approximation for the increments of another process I depends on L being large.Indeed, for the CLT in (6), we required that Lc.Moreover, in order to have the Brownian extreme-value theory apply to the process I, it is evident that L must go o infinity as T goes to infinity.In particular, we need LT/logT---c, where the notation L T indicates that L depends on T. When L T O(log T), the asymptotics for I depend on more than two parameters and c2.The case in which I(t) is a partial sum of i.i.d.random variables is discussed in Sections 2.4 and 3.1 of [6]; also see 5 below.
When we stipulate that L T grows suitably fast as Tc, we can obtain a positive result.
However, (LT, T in (21) arises naturally in the proof.Moreover, a partial result holds even when log(TILT)/log log T-,r, 0 < r < oc; see Remark 1.2.2 on page 35 of [6].Hence, there is a basis for (21) in its current form.
Note that, except for the last term, the approximations for S in (1) comes directly from (20); i.e., we use the approximation S , ,LT+ (LT, T)vc2.The last term in ( 1) is added based on our numerical experience with simulations; see Section 6.It evidently remains to establish a theorem of the form (10) for the sliding window applied to Brownian motion.
To have a more concrete form for approximations, we suppose that L T xT p for fixed x and 0<p<l.Note that the conditions on L T in Theorem 1 are satisfied if L T=xT p for any p, 0 < p < 1, but not for p 1, no matter how small x is. Corollary: 1, then where If I admits the strong approximation (9) and if L T xTp, for x > 0 and 0 < p < S(zTV,T,1)-zT T)V/ c We now obtain a related result for the jumping window J and relate the asymptotics for J and S. In particular, we show that they are asymptotically equivalent in this asymptotic regime and provide support for (2).
Proof: Combine the strong approximation (9) with the extreme-value result (10).The limit (10) holds as Toc for fixed L, but we need a limit as Toc and L--,oc.By the properties of Brownian motion, (10) holds uniformly in L provided that T/L oc; i.e., given that T is an integer multiple of L, [J(L,T)-,L]/x/--for Brownian motion with drift , and variance coefficient 1 is distributed the same as J(1,T/L) for Brownian motion with drift 0 and variance coefficient 1.For the case in which I in J(LT, T,I is Brownian motion, (22) follows from (10)  by dividing by a(LT, T)b(LT, T ).For the case of a general I in J(L T, T,I), we apply (9).Under the assumptions of Theorem 1, Lr/log T--,oc, so that (log T)/b(LT, T)O as T--,oc.For (23), we use the additional assumption to obtain a(LT, T)log T-O as T--oc. [-1 In Theorems 1 and 2 we have provided asymptotics that can serve as the basis for approximations.If better numerical results are desired, especially for smaller values of T and L, then the more involved approximation method of Naus [12], [13] is useful.

Discriminating Windows
The appeal of the asymptotics in 4 is the simple formulas depending on the,process I only via the two parameters ,k and c2.However in some circumstances we may want to learn more about the process I from window processes.For this purpose, it is natural to be guided by 2.4 of (24) For the case of partial sums of i.i.d, random variables, i.  26) uniquely determines the distribution function of X 1.More generally, the limit (25) regarded as a function of x seems promising for characterizing the entire process I.This seems to be a promising direction for research.

NumericM Comparisons
In this section we use computer simulation to investigate the approximations in (1)-( 5).
all cases we let ,-1.We start by letting T-106.

In
We first consider the Poisson process, which has c 2-1. Tables 2 and 3 display results for S and J based on two experiments, each with 10 independent replications.We let L range from 20 to 5000.Since T-106 and log T-13.8, in this range for L we clearly have L << T, but we only have log T << L for larger values of L, say L _> 100.
Frown the first experiment, we saw that the approximation for S provided by Theorem 1 performs reasonably well, but that it can be improved by subtracting the standard deviation in (4_).This yields the approximation for S or ES in (1).This refinement was found to help in other experiments as well, so we include it in (1).However, it still remains to develop theoretical justification for this refinement.Note that the standard deviation in (4) is of the same order as 1/a(L,T) in ( 10)-( 14), so that it is natural to anticipate a refinement for S of this form.To confirm this refinement, we conducted the second experiment with different random seeds.
In Table 2 we display the error in the mean of S for each simulation experiment.This is the approximation value (1) minus the simulation estimate.Notice that the approximation for the mean performs well for the entire range of window lengths.Also notice that the estimated errors in the mean are quite a bit smaller than the standard deviation when L is not too small (e.g., for L > 100), both predicted by (4) and estimated by simulation.Hence, in the region log T << L << T, the approximation is accurate to within the noise, e.g., the maximum of the strong approximation error O(log T)-O(13.8) and one standard deviation.
Table 3 describes similar results for the jumping window, based on the same simulation experiments.Again, the error in the mean is the approximate value minus the simulation estimate.Here no additional refinement was deemed necessary, so approximation (2) for J follows directly from Theorem 2 and (10)-( 14).As with the sliding window, the approximation for the mean performs quite well for a Poisson process over the entire range of window lengths.
Again the error in the mean is less than one standard deviation for L > 100.
From Tables 2 and 3, we see that there is considerable variability in the standard deviation estimates.However, (4) is evidently a reasonable estimate of the standard deviation of both S and J.
To test the invariance principle, we also performed the identical simulation experiment for two rate-1 renewal processes with common SCV c 2-15.4.One renewal process is an on-off process with a deterministic distance of 0.1 between arrivals during an on period.The on period is geometrically distributed with mean 1, so that the mean number of arrivals during an on period is 10.As usual, the off period is exponentially distributed.An interarrival time is the mixture of 0.1 and 0.1 plus the exponential off period.The mean of the exponential random variable is chosen so that the mean interarrival time is 1.0.This makes the SCV c 2 15.4.
The other renewal process has interarrival times equal to a constant 0.1 plus a hyperexponential (H2) random variable, which has a distribution which is a mixture of two exponential distributions; i.e., the H 2 distribution has density It The H 2 distribution is given balanced means, i.e., P$1-1_ (1-P)'2-1.The remaining two parameters are chosen so that the overall distribution has mean 1 and SCV c 2 15.4.Tables 4 and 5 display the results for these two renewal processes with common SCV c 2-15.4.Table 4 displays results for the sliding window, while Table 5 displays results for the jumping window.As in Tables 2 and 3, the simulation estimates are based on ten independent replications for each renewal process.A second experiment of ten independent replications yielded similar results.
From Tables 4 and 5, we see that there are significant differences between these two renewal process.However, for larger values of L, these differences are relatively small compared to the standard deviation.For L < 500, though, the differences are dramatic.For larger values of L, say L > 500, the results are roughly consistent with the invariance principle.The approximate mean overestimates the (D + H2) mean values by about the same amount it underestimates the on-off mean values.The error in the mean relative to the standard deviation decreases significantly as L increases.At about L > 700, the errors are no more than one standard deviation.
For the smaller values of L, we see that the invariance principle is not appropriate.
Consistent with 5, we see that for L < 100 the two renewal processes behave very differently.
Both the means and the standard deviations differ by a factor of two to five in this range.It is interesting that the ratio of the standard deviation estimates of the D + H 2 renewal process to the on-off renewal process increases from 0.22 at L 20 to 1.8 for L 5000.As with the mean, the approximation falls nicely between the two standard deviation estimates.
We also performed similar experiments with less bursty variants of these same two renewal processes.We found that the approximations performed better when the SCV is decreased from 15.4 to 4.0.As might be anticipated, this case falls in between the cases c 2-1 and c 2-15.4 that we have described in detail.
Next, Table 6 compares the approximation for the ratio (J-L)/(S-.L) in (3) with simulation estimates for the three renewal processes considered above.
Note that the approximation is the same for all three processes.The estimates are the sample means and sample standard deviations of the observed ratio.The average errors over the 14 values with L _> 500 are -0.015 for the Poisson process, -0.001 for the on-off renewal process and -0.027 for the (D / H2)-renewal process.Notice that the estimated standard deviations are about 0.09 for the larger L values and about 0.05 for the smaller L values.The estimated errors in the mean are quite small compared to these standard deviation estimates.
Finally, it is interesting to see how the approximations perform for smaller values of T than 106.To give an idea, Tables 7 and 8 display results for the sliding and jumping windows, respectively, applied to the Poisson process for T-103 and T-104.Again, the simulation estimates are based on ten independent replications and the error in the mean is the approximation minus the sample mean.Consistent with Tables 2 and 3, the approximation for the means tend to be slightly low for small values of L. Table 7 shows that for T-104 and L >_ 1000, where T/L < 10, the approximation degrades, but it still is within one standard deviation.Table 8 shows that the approximation for the jumping window degrades more for large L. The ratio estimates are also highly variable in this region.For example, for T-104 and L-2.5 x 103, the sample mean and variance of (J-,L)/(S-L) were 1.19 and 1.83, respectively.This should not be surprising, since T/L-4 in this case.Contrary to the conditions of Theorems 1 and 2, log(T/L) is not large compared to log log T in this range, so this degradation in performance can be anticipated.
However, for L neither very small nor very large, Tables 7 and 8 show that the approximations still perform well.

Conclusions
Our purpose in this paper was to develop simple approximations for the maximum sliding and jumping window content associated with a general input process I. Since our intended application is high-speed communication networks, we are interested in relatively long sample paths (very large values of T) and large window length L, with T much larger than L. Hence, it is natural to consider applications based on asymptotics.
We proposed approximating the general input process I by a Brownian motion.This approximation depends on the general input process I through only two parameters" its rate A and the limiting value of its index of dispersion, c 2 in (7).Strong approximations allow us to do this for the entire process yielding an error of O(log T); see (9).
We then consider extreme-value asymptotics for Brownian motion in order to obtain the basic approximations in (1) and (2).These approximations were supported by Theorems 1 and 2, which established limits as both Toc and Loc with L/log Toc.
We developed the final form of the approximations for ES in (1) and provided additional support for the other approximations by making numerical comparisons with computer simulations.(However, a refinement of Theorem 1 supporting approximation (1) is still needed.)When T 106 and L 103, the basic approximations (1)-(4) perform well.However, when T-106 and L 50, and L is no longer much bigger than log T, the simulations confirm that the invariance principle can break down, as predicted by the theory (see 5).This break down was illustrated by the example of the two renewal processes with SCV 15.4.4. A comparison of approximations with simulation estimates for the mean and standard deviation of the maximum sliding window content of two rate-1 renewal processes with SCV c 2= 15.4 for a total time period of length T--106 as a function of the window length L. A comparison of approximations with simulation estimates for the mean and standard deviation of the maximum jumping window content of two rate-renewal processes with SCV c 2= 15.4 for a total time period of length T 10 6 as a function of the window length L. The simulation estimates are based on ten independent replications. .A comparison ofthe approximation for the normalized ratio (J-;L)/(S-,L) with simulation estimates for the Poisson process, the on-off renewal process, and the D+H 2 renewal process for a time period T-106 as a function of the window length L. The simulation estimates are based on ten independent replications.8.A comparison of approximations with simulation estimates for the mean of the maximum jumping window content of a rate-Poisson process for total time periods T-103 and T 104 as a function of the window length L. The simulation estimates are based on ten independent replications.

Table 1 "
The estimated ratio (J-L)/(S-.L) in (3) as a function of L and T.

Table 2 .
A comparison of approximations with simulation estimates for the mean and standard deviation of the maximum sliding window content of a rate-1 Poisson pro- cess for a total time period T 106 as a function of the window length L. The simula- tion estimates are based on two experiments, each of ten independent replications.

Table 3 .
A comparison of approximations with simulation estimate for the mean and standard deviation of the maximum jumping window content of a rate-1 Poisson process for a total time period T 106 as a function of the window length L. The simulation estimates are based on two experiments, each of ten independent replications.
The simula- tion estimates are based on ten independent replications.

Table 6 .
The error is the approximation minus the estimated mean.