NTIA REPORT 79-21

Statistical Methods for

Estimating Time and Rate Parameters of

Digital Communication Systems

Edwin L. Crow

u.s. DEPARTMENT OF COMMERCE

Juanita M. Kreps, Secretary

Henry Geller, Assistant Secretary

for Communications and Information

June 1979

TABLE OF CONTENTS

ABSTRACT

1. INTRODUCTION

2. CHOICE OF CRITERION

3. DISTRIBUTION-FREE VERSUS DISTRIBUTION-DEPENDENT

ESTIMATION

4. EXAMINATION OF BELL SYSTEM AND ARPANET HISTOGRAMS

OF TIMES FOR APPROXIMATE FORM

5. ADVANTAGES AND DISADVANTAGES OF THE SAMPLE MEAN

AS AN ESTIMATOR OF THE POPULATION MEAN

6. CONFIDENCE LIMITS FOR TIME PARAMETERS

7. CONFIDENCE LIMITS FOR BIT AND BLOCK TRANSFER RATES

8. DESIGN OF EXPERIMENT TO ESTIMATE TIME AND RATE

PARAMETERS

9. SUMMARY OF PROCEDURE

10. ACKNOWLEDGEMENTS

11. REFERENCES

APPENDIX A. DESIGN OF EXPERIMENTS WHEN THERE ARE

WITHIN-DAY AND BETWEEN-DAY COMPONENTS

OF VARIATION

APPENDIX B. THE EFFECT OF TRUNCATING A DISTRIBUTION

AT THREE TIMES ITS MEAN

iii

Page

1

1

3

3

5

6

8

10

14

18

20

21

23

32
LIST OF FIGURES

Page

Figure 1. Minimum cost sample sizes for b=O.l , c=0.02. 29

Figure 2. Minimum cost sample sizes for b=.O.l , c=O.l. 30

Figure 3. Minimum cost sample sizes for b=O.l , c=l. 31

LIST OF TABLES

Page

Table 1. Probability of a Gamma Variable Exceeding Three Times

its Mean and the Mean of a Gamma Variable Truncated

at Three Times its Mean

iv

33
STATISTICAL METHODS FOR ESTIMATING TIME AND RATE

PARAMETERS OF DIGITAL COMMUNICATION SYSTEMS

Edwin L. Crow*

Statistical methods are provided for estimating time and rate

parameters of digital communication systems according to the

specifications of the Proposed Federal Standard 1033. The methods

may be applied to delay, disengagement, transfer and service times,

and to transfer rates. Some ARPANET and Bell System data are

examined for the form of distributions of times. The properties

of various types of estimators are discussed, and the sample mean

is recommended. Approximate confidence limits are given for time

and rate parameters. A step-by-step procedure is provided for

designing an experiment to estimate time and rate parameters with

prescribed accuracy.

Key words: ARPANET; autocorrelation; confidence limits; design

of experiments; distribution-free estimation; robust

estimation

1. INTRODUCT ION

The transmission of data over digital communication systems can be

characterized in part by various time intervals: delay times, disengagement

times, service times between outages~ outage durations, and bit, block, and

message transfer times. Transfer rates are derived from the latter times.

Times of a particular type may depend on various factors or conditions (e.g.,

a busy hour on weekdays versus Sunday morning), but within specified conditions they vary in an apparently random fashion and hence can be represented

by a random (stochastic) process or even a single theoretical probability

distribution. However, the theoretical or long-term distribution can be

estimated only from a finite amount of data. Criteria for specifying or

judging systems should include methods for measuring all time intervals of

interest and analyzing the data, and the methods should yield sufficiently

accurate results to conclude satisfactory or unsatisfactory performance.

This report considers various possible approaches to the analysis of

time interval data and recommends one that amounts in essence simply to

*The author is with the Institute for Telecommunication Sciences, National

Telecommunications and Information Administration, U.S. Dept. of Commerce,

Boulder, CO 80303.
calculating the sample mean and confidence limits for it. The recommended

procedure is outlined in the Summary (Sec. 9). It is a procedure based on

practi cal ity as well as theory. All time intervals (or times, as we shall

often state for brevity) have the common property of being nonnegative and

hence of having more or less asymmetrical or skewed distributions. In

Section 4 we examine some data on their shapes but find no information to

distinguish the distributions of the various times mentioned above with

respect to shape. Our discussions and results therefore apply equally well

to all of the time and rate performance parameters, but the required amount

of measurement (sample sizes) to achieve the specified accuracy may differ

from one parameter to another.

The Proposed Federal Standard 1033 takes the average (or mean) time to

characterize each time distribution. That is probably the most readily

understood and calculated criterion but conceivably could be improved upon

in theory. The question is discussed briefly in Section 2.

The classical methods of data analysis are based upon the assumption of

particular forms of probability distributions, suth as normal (Gaussian),

lognormal, and gamma, and they tend to be optimum only for a particular

form. Ideally methods should have properties independent of the form of

distribution. The problem of whether to use such "distribution-free" methods

or distribution-dependent methods is discussed in Section 3. Some of the

data available on time interval distributions of communication systems are

examined in Section 4.

We end up recommending use of the sample mean as the estimator of the

population or theoretical distribution mean. The sample mean has been

severely criticized by mathematical statisticians, primarily beCause of its

sensitivity to wild or contaminating outlying observations, that is, observations not belonging to the distribution of interest. This disadvantage is

disposed of for delay times, disengagement times, and block and message

transfer times by the definition in the proposed Federal Standard 1033 that

the maximum time is three times the nominal (presumably a specification)

time; any trial resulting in a larger time is counted as a failure and

enters into the performance assessment through the estimation of a probability. (The estimation of probabilities in digital communication systems,

especially confidence limits, is discussed by Crow and Miles, 1977.) The

2
statistical properties of the sample mean as an estimator of the population

mean are discussed more fully in Section 5.

The more concrete results are given in Sections 6, 7, and 8: confidence limits for time parameters, that is, for mean times, given the data,

are discussed in Section 6; confidence limits for transfer rates derived

from mean times are shown in Section 7; and the way to go about designing a

test or experiment to estimate such times or rates with a specified accuracy

is discussed in Section 8. It would be more logical to give the design

material first, but the design depends on the formulas for confidence limits.

2. CHOICE OF CRITERION

Is the mean, median, or 90% point of the distribution of times (such as

access times) of primary interest? Since it is often desirable that such

times be short, one might wish to require that almost all times are shorter

than a specified tolerance and therefore take the 90%, 95%, or 99% point as

his parameter. The proposed Federal Standard partially blunts such a desire

by cutting off most time distributions at three times the n6minal value.

Recent work on data communication systems seems to have emphasized means

(Bell System, cf. Duffy and Mercer~ 1978; ARPANET, cf. Kleinrock, 1976, pp.

456, 480-482; and Payne, 1978), although histograms of entire distributions

are commendably also presented. The histograms give full information

(except possible time dependence), but that is probably not needed for a

specification, and it is desirable to have a specification no more complicated

than necessary. Since the total time wasted over the life of a communication system is simply a multiple of the mean time and longer delays may

not be individually crucial, it may well be justified to take the easily

understood mean time as the distribution parameter to be estimated. However, to do this with error bars or confidence limits usually requires

estimating one or more other parameters of the distribution also.

3. DISTRIBUTION-FREE VERSUS DISTRIBUTION-DEPENDENT ESTIMATION

If one knows the analytic form of a population distribution, he may be

able to estimate the (lltrue") mean of the population more accurately (e.g.,

3
with less variance) than if he doesn't know it. The sample (arithmetic)

mean is a minimum variance unbiased estimator of the population mean if the

distribution is normal Cor gamma also for large sample sizes) but not if the

distribution is lognormal. An alternative optimal estimator is available in

the lognormal case. Since one does not know the distribution form with

certainty, what should One do?

There are distribution-free, or nonparametric, estimation methods. The

sample mean is always an unbiased estimator of the population mean but not

always efficient (i .e., of minimum variance) and hence is not considered a

nonparametric estimator. The sample median is a median-unbiased estimator

of the population median, and distribution-free confidence limits for the

population median are easily obtained in the form of particular ordered

observat;ons~ but they do not apply to the population mean unless it

coincides with the median. Furthermore, the median is rather inefficient if

the population is normal Conly 64% efficiency as measured by ratio of

variances). (This raises the question of its efficiency for the lognormal

and gamma distributions; it is doubtful that this question has been answered.)

If individual measurements are not expensive - and they may not be, the

set-up time perhaps being so much larger than the individual measurement

time - then the statistical inefficiency of the mean for some distributions

(or of the median either for that matter) is not practically important.

Plenty of measurements can be taken, and the Central Limit Theorem can be

invoked to state that 95% (say) confidence limits for the population mean

are (approximately, but closely enough) x~s/In, where x is the sample mean,

s is the sample standard deviation, and n is the sample size. (It has been

tacitly ,assumed that essentially independent observations can be taken

within the specified conditions and that the communication system and its

environment have been compartmentalized for characterization and consequent

separate testing so that this essential independence holds. The procedure

for departures from independence is discussed in Sections 6 and 8.)

A type of estimation method intermediate to the nonparametric and

parametric types is the "robust" type. The properties of these methods are

not independent of the form of distribution but are relatively insensitive

to it, especially to outlying observations that may not belong to the population ofinterest~ For example, the "midmean,1I or 50% trimmed mean, defined

4
as the mean of the middle 50% of the ordered measurements, has efficiency of

80% over quite a range of symmetrical distributions.

If there is a substantial background of data and experience with the

type of time measurements /to which the Federal Standard is to be applied and

all of the data are consistent with one form of distribution, then it is

reasonable to consider that the form of distribution is known and proceed to

use the optimal estimation methods for that form. An interesting discussion

by Cox (1958) on the comparative merits of parametric and nonparametric

inference favors this sort of conclusion but suggests the auxiliary calculation of the nonparametric statistic as a check. The next section will

therefore be devoted to examining the immediately available data for the

form or forms of distribution.

4_ EXAMINATION OF BELL SYSTEM AND ARPANET HISTOGRAMS OF TIMES

FOR APPROXIMATE FORM

Duffy and Mercer's study (1978) of direct-distance-dialing call attempts

by telephone customers presents histograms of distributions of several

different random tjmes:

(a) from end of dialing to abandonment without a system response

(their Fig. 4),

(b) from end of dialing to the first system response after dialing a

valid number (Fig. 5),

(c) call attempt time for incomplete attempts (Fig. 6),

(d) call attempt time for successfully completed attempts (Fig. 7).

The order of decreasing asymmetry or skewness is a, c, b, and d; (a) is

quite J-shaped, probably approximately exponential, judged from visual

inspection, (b) and (c) are distinctly asymmetrical but with a single

central mode aside from sampling fluctuations, and (d) is almost normal in

shape, with a slightly longer tail on th~ right.

Duffy and Mercer also present means, medians, standard deviations, and

10% and 90% points of 16 to 22 time distributions (Table II). One can judge

the degree of asymmetry of these in three different ways: comparing mean

and median, comparing mean and standard deviation, and comparing the 10%-to50% and 50%-to-90% intervals. Based on these three criteria-jointly, one

5
would roughly classify three of Duffy and Mercerls distributions as symmetrical, 15 as asymmetrical, and four as of doubtful classification.

Kleinrock (1976, Fig. 6.16) shows three histograms of round-trip delay

time in ARPANET, for 1 hop, 5 hops, and 9 hops. The latter two are distinctly asymmetrical while the l-hop distribution is approximately symmetrical despite having much shorter times.

Payne (1978) presents a histogram of 200 successful access times which

appears roughly symmetrical except for one large outlying time.

The general conclusion is that most of the group of communications time

distributions examined have the asymmetrical shape expected of a random

variable bounded to the left by zero and unbounded on the right, but an

appreciable proportion are symmetrical or almost so. The form is generally

simple, with a single maximum, and would seem from visual inspection to be

satisfactorily represented by a gamma or lognormal model with two, or

occasionally three, parameters, some even by the normal model. It does not

seem worthwhile at this point to pursue the precise fitting of models further.

An important influence on the effective distribution of several time

variables is the decision in the proposed Federal Standard 1033 to regard

all times greater than three times the IInominal value ll as unacceptable and

to declare a failure on any trial resulting in such a time (Sec. A.3.3.1 for

access time, Sec. A.3.3.2 for block transfer time, and Sec. A.3.3.3 for

disengagement time). ItfollQw5 that there are no Ilout lyi ng observations. tI

Hence there is no substantial need to consider robust estimators.

5. ADVANTAGES AND DISADVANTAGES OF THE SAMPLE MEAN AS AN

ESTIMATOR OF THE POPULATION MEAN

The sample mean is now proposed as the estimator of mean time parameters

of communications systems regardless of the shape of the distribution. It

has the following advantages:

(a) It is an unbiased estimator and a cqnsistent estimator (i.e.,

converging in p~obability to the population mean as the sample

size increases), regardless of the distribution, under the

assumptions that the variance of the distribution is finite, -which

6

-----------_.-------_._------
is satisfied if the distribution is bounded, as it surely must be

in practice, and that successive observations are not completely

correlated.

(b) It is an efficient estimator (i.e., contains all the sample

information) if the distribution is either normal or gamma, which

may be sufficient to describe the data satisfactorily.

(c) It is very well known and simple.

(d) Its standard deviation is a/In if the observations are uncorrelated, regardless of the distribution, where n is the sample

size and a is the population standard deviation, which can be

estimated by the sample standard deviation s.

(e) It is asymptotically (as n becomes infinite) - and hence approximately for finite sample size - normally distributed. (This is,

in simple form, the Central Limit Theorem of probability theory

and mathematical statistics.) Approximate confidence limits for

the population mean can therefore be given as mentioned earlier

and are stated more fully in Section 6.

With respect to advantage (a) it should be noted that the sample

median, the best-known nonparametric estimator of location, is not an

unbiased estimator of the distribution mean if the distribution is asymmetric. Neither are the various "robust" esttmators proposed, such as the

trimmed mean. In fact, robust estimation of location has been developed

very little for asymmetric distributions except for small deviations from

asymmetry that are regarded as contamination of a symmetric distribution the

mean of which is the parameter of interest.

The sample mean has the following disadvantages:

(i) It is sensitive to outlying observations. If such outlying

observations are not part of the population of interest, then this

is a disadvantage but not necessarily otherwise.

(ii) It is not an efficient estimator for any distribution other than

the normal or gamma, though the loss is probably moderate for

moderate departures from those shapes. (Bondesson (1977) showed

recently that if x has high efficiency then the distribution

function must be "closell to the normal or to the gamma distribution function.) Bury (1975, p. 290, Fig. 8.4) shows the

7
efficiency of i as a function of the shape parameter 0 2 of the

lognormal distribution; the efficiency decreases from 100% to 63%

as 0 2 increases from 0 to 2.

(iii)Except for the normal distribution the sample standard deviation

is not an efficient estimator of the population standard deviation, so the confidence limits on the population mean are then not

estimated as well as might be if the distribution shape were

known. For example, Bury·s Fig. 8.4 shows the efficiency of the

sample standard deviation decreases rapidly as 0

2 increases in the

lognormal case.

These disadvantages are not considered sufficient to preclude use of the

sample mean in light of present knowledge.

6. CONFIDENCE LIMITS FOR TIME PARAMETERS

It is assumed that a sample of n uncorrelated measurements wl,w2, ... ,wn

of the time intervals of interest, say access time, is available aDd that

the long-term system or population mean or average time, W, is to be

estimated. For the reasons discussed in the preceding sections Wis

estimated by the sample mean

- 1 w = - L. w.. n · "1 1 1=

(6. 1)

The scatter of the sample is measured by the sample standard deviation,

which is the square root of the sample variance,

2 1 n - 2

s = -- I '». -w) n-l . 1 - 1 1=

(6.2)

This is immediately available simultaneously with won many pocket calculators. The sample standard deviation is an estimate of the corresponding

population standard deviation, o.

The variability of Vi is often measured by its "standard error",

s- = s/In. (6.3) w

8
The corresponding population characteristic, a- = a/In, is the standard

w

deviation of the theoretical sampling distribution of all possible sample

means of which the above wis the one observed. That theoretical distribution of sample means is approximately normal (arbitrarily closely for n

sufficiently large) even if the distribution of measurements w. is not.

1

It is desired to provide limits that contain Wwith high probability in

repeated application. Such limits are called confidence limits. If a is

known and the distribution of measurements w. is known to be normal, then 1

100(1-2a) confidence limits for Ware given by

W+ Z a/In, (6.4) - a

where z is the upper lOOa percent point of the standard normal distribution.

a

For

50% confidence, a = .25, z = 0.6745

a

80% .10 1-.282

90% .05 i.645

95% .025 1.960

99% .005 2.576

The 95% "confidence level" (or confidence coefficient) is most often used

(though it is an arbitrary choice).

If a is not known but the distribution of measurements is known to be

normal, then 100(1-2a) percent confidence limits for Ware given by

w + t 1 s/ffi, - n- ,a (6.5)

where t 1 is the upper 100a percent point of the Student t distribution

n- ,a

with n-1 degrees of freedom (d.f.). A table of t 1 is given in most n- ,a

statistics books and is abstracted here:

n 90% confidence -

2 6.314

3 2.920

4 2.353

5 2. 132

10 1.833

20 1.729

40 1.685

00 1.645

9

95% confidence

12.706

4.303

3.182

2.776

2.262

2.093

2.023

1.960
In practice the distribution of measurements cannot be known to be

normal, and time intervals, being nonnegative, cannot be normally distributed. However, since the distribution of sample means is approximately

normal for large n whatever the distribution of measurements, the confidence

limits (6.5) hold approximately. Because of the approximation, the three

decimal places listed above are unnecessary or even inappropriate, and "two

sigma" limits are often close enough for 95% limits, though they are on the

optimistic side for small samples. Examples of the use of (6.4) and (6.5)

are given at the end of Section 7.

If the observations are correlated, the situation is much more complicated. The one simple solution offered is outlined a little more fully in

Section 8 and is based on equally spaced time separations of the measured

time intervals, which are assumed to be short relative to the separations

between measurements. Then the half-length of the confidence interval in

(6.4) and (6.5) must be multiplied by the factor (l+Pl)/(l-Pl), where Pl is

the autocorrelation between successive measurements and is estimated by

(8 ..10) .

7. CONFIDENCE LIMITS FOR BIT AND BLOCK TRANSFER RATES

"Block transfer rate" is defined in Section A.3.3.2 of Proposed Federal

Standard 1033 as the total number, B, of successful block transfers counted

during a performance measurement period divided by the duration, w, of the

period. (A similar definition applies to bit transfer rate.) The rate

calculated from a particular performance period will be denoted by the lower

case letter r, while the time rate of the system, which would be obtained

from an infinitely long performance measurement period, will be denoted by

R.. Thus

whereas

r = B/w,

R = lim (B/w)

w-+oo

(7 .. 1)

Both numerator and denominator of this fraction could be random variables in

general, but a performance measurement period has been defined as the time

10
interval to transfer a IImessagefl (Seitz and McManamon, 1978, page 137),

and a message is a specified set of blocks. Hence the numerator would seem

to be a specified constant, but there may be slight variations because of

inability to end the period instantaneously, i.e., at a specified number of

blocks; this reason would seem to apply more strongly to bits. In any case

the random variation of the denominator is so much greater than that of the

numerator that the latter can be neglected.

In any case since the estimate of block transfer rate is a nonlinear

function of the duration of the period, it is a biase~ estimate of the true

rate, R, of the system. A heuristic demonstration of this may be worthwhile. In (7.1) we take a constant number Bl of block transfers and

consider the theoretical distribution of all possible times wl to transfer

8

1 blocks. Let Wl be the mean, or expected value, of that distribution;

that is, W=E(w). Since R is the true transfer rate, we assume that Wl=B1R.

Anyone random time wl is an unbiased estimate of W1 by definition. More

generally, the mean of any finite sample of wl 's, say wl' is also an unbiased estimate of W; that is, E(Wl )=Wl. What follows is true for any

unbiased estimate of Wl' but we shall use the notation wl. Let wl=Wl+d.

Then E(d)=O and E(d2) is the variance of wl' denoted by Vl. We can write

81 Bl 81

rl = W = Wl+d = Wl(l+d/Wl) 1

81 d d2

=-W(l--w +--2 - ... ).

1 1 W1

Then

E(rl) = R(l

R

Vl 0+-2- ... )

w1

if it is accepted that the infinite series does not equal 1, and that seems

plausible since the second-order approximation is not equal to 1. Hence rl

is not an unbiased estimate of R.

Of course, the longer the period is the less the bias is. In practice,

periods from several different days should be used, and a combined period

11
can be obtained by adding together the several numbers, Bi, of block

transfers and by adding together the several durations, wi. Then the

estimate of transfer rate ;s

"'-

R =

m

I B. · .1 1 1=

m

I w.

1=· 1 1

rnB B.

= _.. =----- mw w

(7.2)

Under the assumption that all Bi are equal, confidence limits for the true

rate R can be derived from the confidence limits WL and Wu for the population

mean duration W, which are given in Section 6. They are simply

(7.3)

"'-

In the same way it follows that R is a IImedian-unbiased ll estimate of R;

i.e., it will be larger than R in 50% of the cases it is calculated

(infinitely many cases being assumed), and less than R in 50% of such cases,

even though it is biased in the usual sense that its mean over such cases

will not be R.

IIBlock rate efficiency" is defined in Section A.3.3.2 as the ratio of

th~ product of block transfer rate and average block length, denoted by L

here, to the signalling rate of the communication service, R(max):

Q = RL/R(max). (7.4)

As implied by the notation, Q, like R, is a characteristic of the communication s,ystem (or service), and is merely estimated by a measurement from a

single performance period. If Land R(max} are known constants of the

system, then confidence limits for Q follow directly from those for R in

(7.3):

(7.5)

Example 1. Consider the nonswitched private line system in Table 1.1

of Kimmett and Seitz (1978), for which the block transfer rate is given as 5

blocks per second. Suppose that a message is specified as 10 blocks and

that ina single performance measurement period the time to transfer the

message is measured to be 2.13 seconds. The estimated block transfer rate "'-

from this measurement would be R=r=lOj2.13=4.69 blocks per second. We Can not

12
say how accurate an estimate this is without further information. If it"

were known that the standard deviation of such transfer times is 0.4 seconds

and a normal distribution is assumed, then 95% confidence limits for the

mean transfer time Wfrom the single measurement would be, by (6.4),

2.13! 1.96(0.4) = {~:~~} seconds,

and by (7.3) the corresponding limits on R would be 3.43 and 7.43 blocks per

second. If in fact individual transfer times are gamma distributed with 50

degrees of freedom (d.f.) (which corresponds to the combination of a time

mean of 2 seconds and 0=0.4 seconds) rather than normally distributed, then

the correct 95% confidence limits for Wbased on that knowledge (with 2-1/2%

chance of being wrong in each direction) are 1.49 and 3.29 seconds (derived

from a table of chi-squared percentage points, details not being shown),

and those for Rare 3.04 and 6.71 blocks per second. The change from the

previous limits shows the effect of even a small amount of asymmetry of the

distribution of times. (Neither one of these calculations takes into account

the convention that trials that result in transfer times greater than three

times the nominal are counted as failures in the Proposed Federal Standard

1033. )

Example 2. Same as Example 1 except that measurements are taken in 9

independent performance measurement periods. The total time to transfer the

90 blocks is found to be 19.17 seconds (taken artificially as 9x2.13 so that

the results will be centered the same as in Example 1 for comparison). Then A

by (7.2) R=90/19.17=4.69 blocks per second, but now the 95% confidence

limits on Wfrom the normal assumption are

2.13 1. 96 (0. 4)/19 = {~: ~~} seconds.

and the corresponding limits on Rare 4.18 and 5.35 blocks per second.

Making use of knowledge of the gamma distribution with 50 d.f. would now

change the limits on Wonly to 1.90 and 2.47 seconds and those on R only to

4.04 and 5.26 blocks per second. Not only are the limits narrower than in

Example 1, but they are less dependent on knowledge of the shape of the

distribution, a result of the Central Limit Theorem.

13
Example 3. Same as Example 2 but instead of relying on a given prior

value for the standard deviation 0 of transfer times, it is calculated from

the 9 observed times. For the purpose of comparison assume it turns out to

be 0.400 seconds. Then the 95% confidence limits on Ware somewhat wider

than before because of the uncertainty in the estimate of o. Based on the

assumption of the normal distribution of transfer times we need to use the

Student t percentage point with 8 d.f. rather than the normal percentage

point, that is, 2.306 rather than 1.96. Hence the limits on Ware, by

(6.5),

2.13 2.306(0.400)//9 = {~:ri~} seconds

and those on Rare 4.10 and 5.49 blocks per second.

8. DESIGN OF EXPERIMENT TO ESTIMATE TIME AND RATE PARAMETERS

To be specific, we shall consider a distribution of access times, 'but

the discussion will apply equally well to any other time parameters. The

purpose of the experiment is taken to be the estimation of the mean access

time with specified accuracy and confidence for a specified time period of

the day, week, and year and a specified class of source-destination user

pairs of the communication system. All user pairs may be of interest, but

if they can be classified into subsets with predictably different access

times, such as close neighbors and distant ~sers, it would be poor design to

lump them together. Likewise, traffic and hence the distribution of access

times may be different for different hours of the day~ week days and weekends, and perhaps months of the year. Of most interest may be the busiest

hour of a week day. If the busiest hour is not known, a preliminary experiment to determine that should be conducted. Such an experiment may be

necessary anyway to estimate the standard deviation 0 of access times,

because design of the final experiment to achieve specified accuracy

requires prior knowledge of 0.

The preliminary experiment may be taken to consist of nl (say about 10,

more if they are not expensive) access time measurements wl,w2, ... ,wnat

equal intervals within each hour of a day. For each hour the sample ~ean

and standard deviation are calculated:

14
nl

w = _1 \'

l L w.,

nl · 1 1 1=

(8.1)

The hour (or hours) of interest may be selected as that one with the longest

w1' or the largest sl' or the most messages transmitted if that is available, or by some other criterion. Such selection introduces some bias,

since the random variable used may be larger than average for the hour

picked just by chance. Hence it is desirable to have prior knowledge of the

time period of interest.

We wish to estimate the true mean access time Wwith error less than or

equal to a specified value A with confidence level 1-2a. A rigorous twostage sampling plan for doing this was given by Stein (1945) if the distribution sampled is normal and the observations are independent. We assume that

the distribution is sufficiently close to normal to apply the plan, and we

can rely on the Central Limit Theorem to claim a good approximation even if

the sample distribution is not normal. We will consider alternatives to the

independence assumption later. We follow Seelbinder's (1953) statement of

Stein's method. A confidence interval for Wcan be calculated from the

above initial sample. The half-width of this interval is t

as1

/ ln1, where t

a

is the Student t lOOa upper percentage point with nl-l degrees of freedom

(d.f.). If

(8.2)

then no further observations need be taken, as the desired accuracy has

already been attained with the initial n1 observations. If t

as1

/ ln1>A,

then n2 additional observations wi are taken, where n2 is the smallest

integer satisfying

> 2 2

n2 = tasl fA -nl.

The true mean Wis estimated by

- = 1 w W., n = nl + n2 , n i =1 1

15

(8.3)

(8.4)
and 100(1-20,) percent confidence limits on lJ are w+A (or negligibly narrower

by virtue of the inequality in (8.3)).

The second sample size n

2 may turn out to be impractically large,

especially if n

l is chosen too small. Seelbinder discusses the choice of nl

so as to minimize n on the average, but such choice depends on guessing what

the true standard deviation a is and is not discussed further in this report.

If a is known, then no preliminary sample is needed. From a sample of

size n, a 100(1-20,) percent confidence interval for Wis given by

W+ Z a/In (8.5) - a

where z is the upper 1000, percentage point of the normal distribution (1.96 a

for 95% confidence level). Hence we can achieve accuracy A by taking

(8.6)

If the observations are not independent, it is necessary to know or

estimate from a preliminary sample the degree of dependence in order to

determine the sample size needed to attain a prescribed accuracy. Since

only an approximate solution can be hoped for in practice, only a simple

first order approximation is proposed. We assume that the process can be

modeled, within the extent of our likely knowledge, as a stationary first

order Markov process. Then the autocorrelation function P

t between access

time measurements made at a time separation t is equal to p

l

t , where t=l is

a convenient unit time separ~tion, large relative to the largest access

time. If a sample of size n is taken at unit time separations, then its

mean x has variance

2 2 'V a

a- = -. x n (8.7)

If Pl as well as a is known, then a 100(1-20,) confidence interval for lJ is

given approximately by

a l+Pl 1/2

x+z -( ) _r'Il -lp' \.N In - 1

16

(8.8)
Hence, we can achieve accuracy A by taking sample size

n = (8.9)

It is unlikely that Pl and a are known, but they can be estimated from

a preliminary sample at the same unit intervals. We estimate a from (8.1)

and P

l by

where

cov

r l = srs"

n -1 _ 1 1\ ( -. ) ( -II )

COy - n:T .L xi-x xi+l-x , 1 1=1

(8.10)

1 nl-l Xl = -- L x. nl-l i=l l' [

1 nl-l - 2Jl /2

s' = n:T .l (xi-x') , 1 1=1

(8.11)

1 nl-l XII = -- I x nl-l i=l i+l' [

1 nl-l 2]1 /2

sII = -.- L (x. . -x") . nl-l i=l 1+1

Substitutions of sl and r1 for 0 and P1 in (8.9) results in further approximation, part of which might be avoided by substituting instead in Steinls

two-stage formula (6.3) (adjoining the factor (l+r,)/(l-r1) to s1 2), but to

no great point since several other approximations are already incorporated.

Still another model seems possible for the measurement of access times

up to a prescribed accuracy for their mean value. There may be a common

component of variation for all measurements taken on the same day, as well

as a component of variation that varies randomly from one measurement to

another on the same day. Thus a biased picture would be obtained by taking

measurements on only one day. The problem of economically taking n measurements on each of m days is solved in Appendix A. It is necessary to have

prior knowledge of the cost of adding another day of measurement relative

to another measurement on each day and of the relative sizes of the two

component standard deviations. Charts are given for reading off m and n

for a wide range of values of these inputs, and examples of their use are

given. Since the required inputs are probably not known and cannot be

17
estimated from preliminary samples accurately enough unless those samples

are quite large, Appendix A may have more an instructional than an operational value. Nevertheless, the possibility of differences between days

seems substantial and should not be neglected. An example of the use of

the components-of-variance model is given in Appendix A itself.

Example 1. Standard deviation of access times is known. We consider

the example in Proposed Federal Standard 1033, Vol. II, Table 1.1, of a

nonswitched private line. The (mean) access time is given as 5.2 seconds.

We assume that the standard deviation 0 of individual access times is 1.0

second, that they are independent, that we don't know the mean, and that we

want to sample enough times to estimate it within +0.2 second with 95%

confidence. We apply (8.6) to obtain

n = (1. 96) 2( 1) 2/ (O. 2) 2 = 96.

Example 2. Same as Example 1 but we don't know o. We take a preliminary sample of 10 observations (using a table of normally distributed

random numbers) and calculate )(1=5.413 seconds, sl=1.195 seconds. From a

table of Student t percentage points we find t. 025=2.262 for 9 d.f.

Substituting in the left-hand side of (8.2) gives

2.262(1.195)//10 = 0.8546 second.

Hence we must make n

2 further observations, where, by (8.3),

n

222

2 (2.262) (1.195) /(0.2) - 10 = 172.6.

Thus in these examples at least, ignorance of the standard deviation leads

to needing almost twice as many observations.

Example 3. Same as Examp l e 1 except that we assume there is autocorrelation between equally spaced observations obeying the Markov model

with Pl=0.5. Then by (8.9) the required number of observations is three

times what it was in Example 1, that is, 288.

9. SUMMARY OF PROCEDURE

It has been the purpose of this report to provide methods of designing

tests or experiments on a digital communication system and of analyzing the

resulting data-so that the time and rate parameters are estimated with

18
prescribed accuracy. An outline of procedure is given here, with necessary

references to parts of the report.

1. If nothing is known about the various time parameters and measurements

on them, it is necessary to carry out a preliminary experiment to

determine the classes of source-destination user pairs and the hour or

hours of the day to be considered. There may be only one class of

user pairs, but it should be reasonably homogeneous. The preliminary

experiment may be taken as 10 or more measurements at equal intervals

within each hour of the day. On the basis of the hourly sample means

w1 it should be decided which hour or hours is of interest. See the

first part of Section 8.

2. It must also be decided whether there are significant differences

between time measurements from day to day. Hence the preliminary

experiment needs to be expanded to include several days. The decision

is made on the basis of an !IF test" as stated briefly after equation

(A.9) in Appendix A. If there is a significant between-day component

of variance, then Figures 1-3 may be used to design an experiment to

estimate the mean time parameter with prescribed accuracy, as illustrated in the Appendix A example; the design requires inputs that

may be estimated from the preliminary experiment, but it must be

realized that these estimates may have poor accuracy.

3. In the main body of the report the between-day variance is ignored.

It is assumed either that the standard deviation 0 of the time measurements is known beforehand or that it is estimated as sl from (8.1) and

a preliminary experiment. If the measurements are uncorrelated and 0

is known, the sample size necessary to achieve accuracy A is determined from equation (8.6), where z is given after (6.4); after the

a

data are obtained, the mean time parameter Wis estimated by w as in

(6.1), and confidence limits are calculated by (6.4) or (8.5).

4. If the measurements are uncorrelated and 0 is estimated from sl' then

a further sample of measurements may be needed in accordance with

(8.2) and (8.3). The mean time parameter Wis then estimated by the

combined sample mean was in (8.4), and confidence limits are w+A

where A is the prescribed accuracy A.
5. If an experiment has not been designed to achieve prescribed accuracy,

and one simply has a sample of data to analyze, then confidence limits

for the mean time parameter Wcan be calculated from (6.4) if the true

population a is known, or, in the more likely situation that a is

estimated by the sample standard deviation s as in (6.2), from (6.5).

6. Whether the time measurements are uncorrelated within an hour (or

longer period of the day) can be tested by taking the measurements at

equal time separations and calculating the autocorrelation function as

in (8.10). Even if it is not zero, it may not be significantly

different from zero. This can be tested approximately by using a

table of percentage points of the ordinary correlation coefficient to

be found in many statistics books. The correlation problem can be

avoided by taking essentially independent observations only, i.e.,

by taking them sufficiently separated. If it is decided to use

observations with correlation included, then the sample size for a

prescribed accuracy must be larger in accordance with (8.9),' and the

confidence limits on Ware calculated by an approximation as in (8.8) r

or as described at the end of Section 6.

7. Confidence limits for transfer rates and for block rate efficiency are

derived from those for transfer times as stated in equations (7.3) and

(7 . 5) .

10. ACKNOWLEDGMENTS

The work has been conducted as part of the data communications project

of the National Telecommunications and Information Administration under the

supervision of Neal B. Seitz, project leader, whose problem statement,

encouragement, and discussions are gratefully acknowledged. The report was

typed by Patricia Sanchez.

20
11. REFERENCES

Bondesson, L. (1977), When has the sample mean as an estimator of a location

or scale parameter high efficiency? Zeitschrift fUr

Wahrscheinlichkeitstheorie und verwandte Gebiete, Vol. 38, pp. 127-146.

Bury, K.V. (1975), Statistical Methods in Applied Science (John Wiley and

Sons, New York, NY).

Cox, D.R. (1958), Some problems connected with statistical inference,

Annals of Mathematical Statistics, Vol. 29, pp. 357-372.

Crow, E.l., and M.J. Miles (1977), Confidence limits for digital error rates

from dependent transmissions, OT Report 77-118, National Telecommunications and Information Administration, (Available thru NTIS, 5285 Port

Royal Road, Springfield, VA 22161, Accession No. PB267777).

Duffy, F.P., and R.A. Mercer (1978), A study of network performance and

customer behavior during direct-distance-dialing call attempts in the

U~SeAe, Bell System Technical Journal, Vol. 57, pp. 1-33.

Guenther, W.C. (1964), Analysis of Variance, Prentice-Hall, Inc., Englewood

Cliffs, N.J.

Kimmett, F.G., and N.B. Seitz (1978), Digital communication performance

parameters for proposed Federal Standard 1033, Volume II: Application

examples, NTIA Report 78-4, National Telecommunications and Information

Administration (Available from NTIS, 5285 Port Royal Road, Springfield,

VA 22161, Accession No. PB 284235/AS).

Kleinrock, L. (1976), Queueing Systems, Vol. II: Computer Applications

(John Wiley and Sons, New York, NY).

Payne, J.A. (1978), ARPANET host to host access and disengagement measurements,

NTIA Report 783, National Telecommunications and Information Administration (Available from NTIS, 5285 Port Royal Road, Springfield, VA

22161, Accession No. PB 283554/AS).

Pearson, K., ed. (1965), Tables of the Incomplete r-Function (Cambridge

University Press, New York, NY).

Proposed Federal Standard 1033 (1978), Telecommunications: Digital

communication performance parameters, Version 10, September 15.

21
Seitz, N.B., and P.M. McManamon (1978), Digital communication performance

parameters for proposed Federal Standard 1033, Volume I~ Standard

parameters, NTIA Report 78-4, National Telecommunications and Information Administration (Available from NTIS, 5285 Port Royal Road,

Springfield, VA 22161, Accession No. PB 283580jAS).

Seelbinder, B.M. (1953), On Stein's two-stage sampling scheme, Annals of

Mathematical Statistics, Vol. 24, pp. 640-649.

Stein, C. (1945), A two-sample test for a linear hypothesis whose power

is independent of the variance, Annals of Mathematical Statistics,

Vol. 16, pp. 243-258.

22
APPENDIX A. DESIGN OF EXPERIMENTS WHEN THERE ARE WITHIN-DAY AND

BETWEEN-DAY COMPONENTS OF VARIATION

The design of a test or experiment is at least as important as its

analysis. Indeed, the complete design will include specifications of the

method of analysis, so that in theory the actual analysis will involve only

routine substitution in formulas or running of a computer program. In

practice, unforeseen difficulties often arise that require some adjustment.

The difficulties can be cl~ssified as various nonconformances with the

assumed mathematical statistical model, and of course the data should be

used to test the model as well as to make the minimal calculations needed

if the system meets the primary specification.

The design of the experiment will follow from the precise statement of

its purpose. The purpose will be taken as the following:

To determine with minimum cost the mean delay (or other) time for

the busiest hour and busiest pair of terminals of a specified

data communications system within specified accuracy with 95%

confidence.

It may not be known what the busiest hour and the busiest pair of terminals

are, and other hours and pairs may be of interest. However, it is assumed

that the time period of interest is known and that time measurements of the

type of interest can be made within this period. It is assumed in this

Appendix that the measurements are mutually independent within each day but

may contain a daily component of error. The modifications necessitated by

autocorrelations between measurements on the same day are discussed briefly

in Secs. 6 and 8. In practice, the measurements should be taken at equal

time intervals each day such that the interval is not a period or quasiperiod of the system. Subsets of the state-time space of the system other

than days, busiest hours, and busiest terminal pairs may be of interest and

may be substituted for those terms, especially days, in the sequel.

Since it is recommended that the true mean delay time be estimated by

the sample mean, the accuracy will be specified in terms of its standard

deviation 0-. (It is assumed that the measurements are unbiased, so that

x

we can speak of accuracy and precision interchangeably.) Thus more

specifically the experiment should be designed so as:

to determine with minimum cost the number of measurements per day,

n, and the number of days of measurements, m, to make 20x < B,

where B is specified.

23
It is assumed that there are random fluctuations within each day~ to

be denoted by e .. which may include measurement error as well as system 1J

variations, and that there may be additional random variations from day to

day, to be denoted by a .. Thus the measurements x.. are assumed to be of 1 1J

the form

x. · = 11 + a. + e .. ; i = 1,2, ... ,m; lJ 1 1J

j 1,2, ... ,n; (A.l )

where the ai and ei j are mutually independent tuncorrelated is sufficient

for most purposes) with standard deviations aaeand a .

Before discussing the determination of m and n it is desirable to

present the method of analysis to be used after the data have been obtained.

Ideally the same number of measurements should be made each day, but

inevitably some variations will occur~ so we let j vary from 1 to n.. We

1

estimate the true mean 11 by

where

1 m n . - I I

1

X = N x·.

i =1 j=i lJ

m

N = Ln .. 1=· 1 1

Approximate 95% confidence limits for 1J are given by

x + 2sx

where

(A.2)

(A.3)

(A.4)

(A.5)

2 _ 1 m

se - N-m l: i =1

m-l

N-l:n;2/N

ni 2 L (x ..-x.) , ·1 . lJ I > " J= .

24

CA.6)

(A.7)
2 _ 1 - 2

sl - m-l L ni (xi.-X) , i=l

(A.8)

and

1 x. =-

1 • n.1

n.

1

2

j=l

x ... lJ CA.9)

(See, for example, Guenther, 1964, pp. 59-60.) Whether the true

between-day variance 0 2 is in fact positive or zero is tested by calculating 2 2 a ' the ratio F = sl /se and comparing it with the 5 percent (say) point of

the F distribution with m-l and N-m degrees of freedom, tables of which are

available in most statistics books.

With all n. = n the variance of x is

1

2

2 1 2 0 e 0- = -(0 +-=-). x man' (A .10)

It is helpful to have the bound B specified as a fraction b of the true

mean ~, so that B=b~. Also 0

a and 0

e

are represented as fractions of ~:

Then the accuracy requirement becomes

k 2

(k 2+__. e__)1/2 < b.

m- a nThen m could be specified as

k 2

m = !- (k 2~) b2 a n

(A. 11 )

(A . 12)

(A.13)

if k and k are known and n is given. In fact k and k have to be known

a e a e

or guessed at prior to testing if m and n are to be determined.

Even with knowledge of k

a

and ke the one equation (A.13) is not

sufficient to determine the two unknowns m and n. The further condition

needed will result from minimizing the total cost, but first let us simply

minimize the total number of observations N = mn. From (A.13)

25
(A. 14)

Since N increases linearly with n, N is minimized by taking n=l, i.e., only

one measurement per day. This results because increasing m decreases both

within-day and between-day components of variance while increasing n

decreases only the within-day component. If there is no between-day

component, k =0, and it does not matter whether the measurements are made

a

on Ndays or all on one day. However, the possible existence of a betweenday component should not be ignored.

The above optimal solution is not attractive in practice because it

surely is less costly to make N observations by taking many each day. If

all were taken on one day, the between-day component of variance could not

be estimated. Likewise, if n=l, the within-day component could not be

estimated. Hence m>2 and n~2 in practice. It will be assumed that the

total cost depends linearly on the number of days and also on the total

number of observations:

Total Cost = Co + Clm + C2mn.

However, this is minimized if we minimize the simpler expression

C = m + cmn.

(A.15)

(A .16)

Substituting for m from CA.13), differentiating with respect to n, setting

the derivative equal to zero, and solving for n gives the solution

n · = c-1/ 2k /k . (A.l7) mln e a

Substitution in (A.13) yields

Also

(A. 18)

m . mln

26

(A. 19)
Since m and nshould be integers, we must modify the above continuous

solution. We calculate C for the two integers nearest n

mi n and let

(A.20)

whichever yields the smaller C, using [yJ to denote the largest integer

less than or equal to y. Then

k 2

:::: [~k 2+_1 e_)+1] .

b

2 an.mln

(A. 21 )

These will surely achieve the accuracy requirement (A.12) but still will

not necessarily be the integers that minimize the cost; if mi. is small mln

(roughly smaller than nl • ), it may be possible to decrease C by allowing a mln

slight increase in m and obtaining a substantial decrease in n. However,

this gain is illusory in practice because the balance shifts rapidly with

change in k and k , and the integer pair (m,n) absolutely minimizing C

e a

with the assumed k

e and ka will not achieve the accuracy bound (A.12) for

some true k and k quite nearby. Hence (A.20) and (A.2l) yield the practical e a

solution in integers and form the theoretical solution also unles~ m~in is

small.

Since n ·n depends on k and k only through their ratio, it is convenient ml e a

to use ke and kalke as parameters in m~;n rather than ke and ka. We also

neglect taking the next larger integer than mmin in calculations. Hence we

let

II = 4ke

2

[(~) 2 + 1 ] (A•22) mmin b2 ke n~in·

Example. b=c=O.l, k Ik =1. Then n . =/10=3.162, m ,·n=526.49l k

e

2, 2 a e mln m

C

mi n=692.982 k

e. But the integral solution for n is 3 or 4. From (A.13)

and (A.16)

2 2

m3 = 533.333 ke ' C3 = 693.333 ke '

2 2

m4 = 500.000 ke ' C4 = 700.000 k

e ·

27
_.-..- __.- .._--_._~._ _.-_ __ ..__ _..-.._._-----_..__._----~--_._--_.-._----_._--_ •.._ .._-----_._--------_._------------_.._-,

Hence nl • =3, mil. =533.333 k 2. If k =0.1, then the most economical design mln . mln e e

is to take 3 measurements each day for 6 days. If k =1, then mil. =533 and the \ e Ill ] n

design is much less practical. Thus a criterion no doubt entering into the

design beyond accuracy and cost is the maximum number of days permissible

for the experiment.

Charts of the values of m and n that provide the most economical

experiments for determining mean delay time with specified accuracy are

given in Figures 1, 2, and 3. The results depend on the confidence level

(already limited to 95%), the relative accuracy desired (b), the relative

cost (c), and the relative standard deviations k

a

and ke. Since n~in (called

n on the charts for simplicity) depends on k and k only through their

a e

ratio, it is convenient to use k Ik as a parameter on curves (which are

a e

straight lines) giving mil. (called m .. on the charts) as a function of mln mln

k. The only relative accuracy value used is b=O.l, i.e., 10% of the true

e

mean. Three different relative costs of each measurement in a day~ relative

to the incremental cost of another day are considered, c=0.02, 0.1, 1.

Values of kalke from 1/128 to 4 by factors of 2 are offered. In a few

cases these correspond to n=l, just one measurement per day, an impractical

solution, and in any case at least two measurements should be taken per day

in order to estimate 0e by (A.6).

The ranges of values of m and n on the charts are enormous. It is

emphasized that in order to use the charts one must accept the relative

accuracy b=O.l as adequate, must decide which, if any, relative cost c fits

his situation, and must know or guess the relative standard deviations k

a

and k

e.

If Figures 1-3 do not cover the situation at hand, the values of

n=n l • and m . =m". can easily be calculated from (A.17), (A.20), (A.16), mln mln mln

and (A.22).

As an example of the use of the charts, we consider the above example.

We enter Figure 2 and find the line labeled ka/ke=l. It is also labeled

n=3, so 3 measurements are to be made each day irrespective of k

e. With k

e

given as 0.1, we find on the line the solution m . =5.3, so the experiment mln

should continue for 6 days to achieve the desired accuracy (on the safe

side) .

28
0.3 0.5 0.7 3 5 7

1'---_----It.--~_..L_J. _ ___I_~J..._..J_..L_.J~.....L._ ___L___J.._____L_ _L.__'__..1._..!..._L......

0.1 10'

7

5

3

70

10

30

50

100

700

300

500

Figure 1. Minimum Cost Sample Sizes for b=O.l, c=O.02

29
0.5 0.7 3 5 7

Ke

0.3

5

7

3

30

10

50

10

700

100

300

500

Figure 2. Minimum Cost Sample Sizes for b=O.l, c=O.l

30
0.3 0.5 0.7 3 5 7

5

1

3

10

30

50

10

700

500

100

300

Figure 3. Minimum Cost Sample Sizes for b=O.l, c=l.

31
APPENDIX B. THE EFFECT OF TRUNCATING A DISTRIBUTION AT

THREE TIMES ITS MEAN

Since the proposed Federal Sta~dard 1033 considers any trial that

results in a delay, disengagement, or block or message transfer time more

than three times the II nomi nal li time as a failure, it seems of interest to

consider the effect on an asymmetrical type of distribution of truncating

it at three times its mean. That is, the nominal time is assumed to be the

mean prior to truncation. A convenient type of asymmetrical distribution

to use is the gamma family of distributions, all members of which extend

from 0 to 00 with continuous probability density functions (pdfls) having a

single maximum and varying asymmetry. The particular effects of truncation

considered are: (1) the probability that a time that follows a gamma

distribution will be discarded by the truncation and (2) the change in mean

due to the truncation.

A well-known subfamily of the gamma distributions is the set of chisquared (x 2) distributions, which have the single parameter called the

"degrees of freedom," a positive integer, denoted here by v. It is used to

specify the particular pdf and thus its asymmetry, which ranges from very

large for v=l to 0 in the limit for V7OO. The mean of a chi-squared distribution is v. In order to maintain the mean value constant, at unity, we

consider the variable

z = x

2j V.

It is easy to calculate the probability that a gamma variable is three

times its mean value, that is, that Z>3, from Karl Pearson1s Tables of the

Incomplete f-Function (1965). After a change of variable we find (details

being omitted) that

1/2

P(Z>3) = l-I(3(v/2) ,v/2 - 1)

where I(u,p) is the incomplete f-function tabulated by Pearson. Table 1

gives this probability for v ranging from 1 to 32. The pdf1s of the

distributions are also plotted in Table 1. The probability of considering

a trial a failure is appreciable for v=l, about 8%, and v=2 (the exponential

distribution), about 5%, but decreases towards 0 as v increases. Two

further remarks are in order: (1) More extreme cases than v=l are quite

possible, from fractional values of v, yielding much larger values of

32
Table 1. probability of a gamma variable exeeding three

times its mean and the mean of a gamma variable

truncated at three times its mean

v

(degrees of

freedom)

1

2

3

4

P(Z>3)

.0833

.0498

.0293

.1074

.663

.843

.918

.9545

Probability density function

6

8

12

16

24

32

.0062

.0023

.00033

.000048

.0000011

.0000000

.9849

.9947

.99928

.999897

.9999977

.9999999

33

~~

~~::::::==-_-

IL~=b

I~

IL,

I~, o I.
P(Z>3); the table has a limited range. (2) Although the mean value of Z

remains constant, its standard deviation steadily decreases as v increases,

as suggested by the pdf plots.

It is also easy to calculate the mean of the truncated distribution

from Pearson's Tables. If denotes the mean of the gamma distribution and

~t the mean of the same distribution after truncation at 3~, then

1/2 ~t _ I(3v/(2v+4) , v/2) II - 1/2

I(3(v/2) ,v/2 - 1)

This truncated mean is also tabulated in Table 1. It is about 34% less

than the original mean for v=l and 16% less for v=2, but rapidly approaches

the original mean as v increases. (The last digit in the ratio of means,

as well as in the probability, may be in error by about one unit because of

linear interpolation in Pearson's Tables.)

It may be useful to note that the mode (abscissa of the maximum of the

pdf) of the gamma distribution with mean unity is 1-2/v for v~2 and that the

median is very close to 1-2/(3v) for v~2. Thus, for v=3 the mode is 1/3 and

the median is 0.789. For v=24 the mode is 11/12 and the median is 0.9725.

These illustrations using the gamma distribution do not imply any firm

conclusions for actual time distributions in digital communication systems

because the gamma distribution may not be the precise form and the degrees

of freedom parameter no doubt varies with the system and the type of time

interval. It is believed that the gamma distribution is a plausible model,

but the second uncertainty is a considerable one, as indicated by Table 1.

It is hoped that some intuitive feeling may be gained by associating the

probabilities and truncated means with the distribution shapes.

The examination of Bell System and ARPANET time interval distributions

in Section 4 revealed shapes varying over roughly the entire range shown in

Table 1. Despite this wide range of shapes and the corresponding probabilities and means shown in the table, there is no suggestion that truncation

should occur at any other point than at three times the mean.

34
FORM OT·29 (3-73)

u.s. DEPARTMENT OF COMMERCE

OFFICE OF TELECOMMUNICATIONS

BIBLIOGRAPHIC DATA SHEET

1. PUBLICATION OR REPORT NO.2. Gov't Accession No.3. Recipient's Accession No.

NTIA Report 79-21

4. TITLE AND SUBTITLE

STATISTICAL METHODS FOR ESTIMATING TIME AND RATE

PARAMETERS OF DIGITAL COMMUNICATION SYSTEMS

7. AUTHOR(S)

Edwin L. Crow

5. Publication Date

June, 1979

6. Performing Organization Code

NTIA/ITS

9. Project/Task/Work Unit No.

8. PERFORMING ORGANIZATION NAME AND ADDRESS

U.S. Dept. of Commerce

National Tel econmun lcattons and Infonnation Admin;stratipmo.contract/GrantNo.

Institute for Telecommunication Sciences

325 Broadway, Boulder, CO 80303

1J. Sponsoring Organization Name and "'Address 12. Type of Report and Period Covered

U.S. Dept. of Commerce

National Telecommunications and Information AdministratiDn

1325 G Street, NW 1-1

-

3.- - - - - - - - - - -1

Washington, DC 20005

14. SUPPLEMENTARY NOTES

i 5. ABSTRACT (A 200-word or les s factual summary of most significant information. If document includes a significant

bibliography ot literature survey, mention it here.)

Statistical methods are provided for estimating time and rate parameters of

digital communication systems according to the specifications of the Proposed

Federal Standard 1033. The methods may be applied to delay, disengagement,

transfer and service times, and to transfer rates. Some ARPANET and Bell System

data are examined for the form of distribution of times. The properties of various

types of estimators are discussed, and the sample mean is recommended. Approximate

confidence limits are given for time and rate parameters. A step-by-step

procedure is provided for designing an experiment to estimate time and rate

parameters with prescribed accuracy.

16. Key Words (Alphabetical order, separated by semicolons)

ARPANET; autocorrelation; confidence limits; design of experiments;

distribution-free estimation; robust estimation

17. AVAILABILITY STATEMENT

[X] UNLIMITED.

o FOR OFFICIAL DISTRIBUTION.

18. Securi ty Class (This report)

Unclassified

19. Security Class (This page)

Unclassified

20. Number of pages

41

21. Price:

tr u.s. GOVERNMENT PRINTING OFFICE: 1979-0-680-338/498 L'SCOMM-DC 29716-P73