About the Work
“Time-Measurements” is a technical study of how telephone traffic engineers can reliably estimate the average duration of subscribers’ calls when they cannot measure each call individually. The work examines the statistical foundations of the switch-count method—a technique in which a group of circuits is scanned at regular intervals and the number found busy is recorded—and develops a mathematical apparatus for quantifying the errors that this indirect approach introduces. It draws its empirical evidence from pen-register observations made on an inter-office trunk group in Newark, New Jersey, and compares theoretical predictions against those observed data throughout.
“ONE of the fundamental quantities in traffic engineering is the average duration of subscribers’ calls.”
The central problem the work addresses is practical rather than abstract. Traffic engineering depends on knowing the average holding time of calls, because that figure, multiplied by the expected number of calls per hour and divided by 3600, yields the traffic load expressed in average simultaneous calls. From that load, tables and curves allow the engineer to determine how many paths must be provided so that no more than a small percentage of calls encounter an all-paths-busy condition. If the average holding time is estimated inaccurately, the provisioning of circuits will be inaccurate in turn. The work therefore asks: when holding times are inferred from switch counts rather than measured directly, how large are the resulting errors, and how can those errors be kept within acceptable bounds?
The study is quantitative and formula-driven. It derives probability distributions for three distinct sources of error, checks each against Newark data, combines them into an overall error estimate, and presents the result in a form an engineer can use when designing a switch-count schedule. The exposition moves from the general problem, through a preliminary analysis of the Newark data, into the detailed mathematics of end-of-period effects and individual-call measurement error, and finally to the total error on a sample of calls and the practical question of how long an observation period should be.
Context
The work belongs to the field of telephone traffic engineering as it stood in the early 1940s, a period when the Bell System was systematically mathematizing the behavior of its plant. The notes reference A. K. Erlang’s observation, published in Denmark in 1909, that local subscriber call holding times follow an exponential frequency distribution—a finding that had become foundational for the field. They also cite an unpublished memorandum by E. C. Molina dated September 7, 1920, on the analytics of a related case in which errors at the ends of a call were assumed to fall equally likely between plus and minus half an interval. The empirical data come from Newark, New Jersey, with dates in the tables running through 1918, indicating that the underlying observations were made some years before the analysis was written up.
The immediate context is the choice among three methods of obtaining holding-time data. Direct measurement with a stopwatch is simplest in principle but seldom used, because accumulating observations is slow and the many observers required—each watching one call at a time—are expensive. Mechanical or electrical recording of each call’s length over a group of switches or trunks has been used more or less extensively in the Bell System, but it demands considerable time and labor to summarize results and poses problems of maintaining the measuring equipment. The third method, switch or plug counts, scans the group of paths at regular intervals mechanically, electrically, photographically, or by eye, recording each time the number found busy. This method immediately yields estimates of average load carried and, by relatively simple analysis, a measure of the reliability of that estimate; and if a call- or peg-count meter records the number of calls passing over the group during the same period, estimates of average call holding time and of the reliability of that average follow as well. The work is essentially a rigorous accounting of what the switch-count method costs in accuracy.
The notes also situate the inquiry within a broader methodological concern: deciding how long a period to study requires some knowledge of the persistence of the same holding-time universe, obtainable through small holding-time samples taken in the hours of interest every day for several weeks in the busy season. If spottiness or lack of control is not apparent, the problem simplifies. If trends exist—for instance, if some day of the week shows high holding times—their nature must be investigated and the main sampling procedure apportioned to give these peculiarities proper weighting. Direct measurement or switch counts should be made on groups during periods presumably typical of those toward which engineering is ultimately directed, usually, though not always, the busy or busiest hours of the day during the busy season.
Structure of the Book
The work proceeds in a logical sequence from problem statement to practical recommendation. It opens by identifying the average duration of subscribers’ calls as a fundamental quantity in traffic engineering and by laying out the three available methods for obtaining holding-time data, with their respective costs and drawbacks. It then proposes to examine limited data taken by the pen register method some years earlier on an inter-office trunk group in Newark, New Jersey, as an indication of procedure applicable to other, less-known groups.
The preliminary study of the Newark data establishes the empirical baseline. It confirms that local subscriber call holding times follow remarkably closely the simple exponential frequency distribution, and it presents hourly holding-time data across a period of months, revealing an apparent division between forenoon and afternoon hours. This leads into a test of whether the afternoon averages could reasonably come from the same universe as the forenoon hours.
The analysis then turns to the three distinct sources of holding-time error in switch-count studies: errors at the start of the observation period, errors at the end, and errors at the beginning and end of each call. Each source is treated in turn, with theoretical probability distributions derived and then compared against actual Newark switch counts. The treatment of individual-call error branches into three cases according to the assumed distribution of holding times: constant holding times, an equally likely distribution between adjacent multiples of the counting interval, and exponentially distributed holding times.
The final movement of the work combines the separate error components into a total error on a sample of n calls. It derives the best estimate of switch counts after correction for carry-over and end-of-period effects, expresses the standard deviation of that estimate, adds the contribution from imprecise measurement of individual calls, and arrives at an overall standard error for the switch-count method. The concluding practical question is how the overall error behaves as the observation period is lengthened, and what that implies for the design of switch-count schedules.
Inside the Book
The opening establishes the stakes. The average duration of subscribers’ calls, expressed in seconds, multiplied by the average number of calls expected over a given route in an hour and divided by 3600, gives the traffic load submitted in average simultaneous calls—commonly called “the average.” Tables and curves then determine how many paths must be provided so that no more than a desired small percentage of calls find all paths busy. A footnote observes that the number of all-paths-busy cases is highly variable, and that even on properly engineered groups, few or no such cases need cause concern, since such peaks are short and easily missed except over a very long series of counts.
Against this background, the three methods of obtaining holding-time data are weighed. Direct stopwatch measurement is simplest in principle but seldom used, owing to slowness in accumulating observations and the considerable expense of the many observers required, each watching one call at a time. Mechanical or electrical recording of each call’s length over a group of switches or trunks during an interval has been used more or less extensively in the Bell System through various holding-time recorders or cabinets, but its chief disadvantages are the considerable time and labor needed to summarize results, plus problems of maintaining the measuring equipment. Switch or plug counts—scanning the group of paths at regular intervals mechanically, electrically, photographically, or by eye, and recording each time the number found busy—immediately give estimates of average load carried and, by relatively simple analysis, a measure of the reliability of that estimate. If a call- or peg-count meter records the number of calls passing over the group for the same period, estimates of average call holding time and of the reliability of that average can also be obtained. The work then proposes examining limited data taken by the pen register method some years earlier on an inter-office trunk group in Newark, New Jersey, as an indication of procedure applicable to other, less-known groups.
The preliminary study of Newark data begins from the long-known fact that local subscriber call holding times, t, follow remarkably closely the simple exponential frequency distribution f(t)dt = ke^(−kt)dt, where 1/k is the average holding time. A footnote credits A. K. Erlang as apparently the first to notice this holding-time distribution, in a Danish mathematical journal in 1909. This was found substantially true of the Newark inter-office trunk group data, shown for 7385 calls observed in 19 hours having loads in the range 15.0–16.0 average simultaneous calls. The fit of the exponential curve with average equal to the observed average of 2.380 minutes is quite good. In the exponential distribution the standard deviation equals the mean; in practice the standard deviation is usually found slightly larger than the mean, though not markedly so. This information may be used to test the homogeneity of the holding-time universe if all hours of the days are grouped indiscriminately.
Hourly holding-time data for Newark are then presented, covering hours from 9–10 a.m. through 4–5 p.m. across dates from Monday 7-8-18 through Thursday 10-11-18, with a summary giving 139 hours and averages by hour ranging from about 141.63 to 174.3 seconds. Holding-time averages for 135 hours observed at various times of day over a period of 3 months appear at first glance to fall into two rather distinct groups: those before noon and those after noon. Treating the 115 hours before noon as a homogeneous group, the question is whether the afternoon holding-time averages could reasonably come from the same universe. The average holding time of the 115 forenoon hours is found to be 143.5 seconds. Since these hours averaged about 390 calls each, the standard deviation of the means should by theory be closely σ/√n = 7.29, with σ taken as the mean. The observed standard deviation is 9.26, some 27 per cent higher, which agrees with the earlier observation that the standard deviation is usually slightly larger than the mean. On the hypothesis that the universe of 115 early hours has parameters 143.5 and 7.29, the observations for each of the three clock hours could readily have occurred: their deviations from 143.5 are 1.9, 0.17, and 2.5 seconds, with corresponding standard errors.
The analysis then turns to the errors introduced by the switch-count method itself. If each call’s holding time is not measured with considerable accuracy, additional calls must be observed to compensate. This is precisely the situation in the method of switch counts, which merely notes at regular intervals whether a particular call exists; thus no calls are closely measured for individual length. Further errors arise because at the start of a period some switch counts inevitably include calls from the preceding hour, and at the end some calls registered on the peg count meter end beyond the period, losing part of their proper switch counts. Three distinct sources of holding-time error are named: errors at the start of the observation period, errors at the end, and errors at the beginning and end of each call.
For the start-of-period error, if observation period T is divided into r equal intervals of length i, with switch counts at the beginning and end of each interval, there are r + 1 observations. The first count gives the number of calls extending into the period from preceding hours, but their contributions to subsequent counts cannot be segregated, so a theoretical estimate is required. Assuming average holding time and an exponential distribution, the probability that a single call is included only in switch count number 1 is 1 − e^(−i/t), with probabilities for exactly two, three, and more counts following similarly. If m such calls are observed on count number 1, m variables of this type must be added; exact addition becomes complex when m exceeds about 3 or 4, so moments are combined instead. Theory is compared with actual 1-minute switch counts contributed by carry-over calls in 30 hourly periods on the Newark interoffice trunk group; average m is about 15, normality was assumed, and control lines were drawn. One point falls on the outermost line, 16 of 30 within the tighter lines, corroborating the theory.
For the end-of-period error, if counts are made at intervals i so that the final count occurs at the exact end of the period, the situation resembles the start. The last observation tells how many calls continue into the next hour. Probabilities of losing exactly zero, one, and more switch counts are derived, with average and standard deviation, and the sum distribution for w variables and its parameters follow. These are checked against 30 busy-hour periods; two hours fall outside the three-sigma limits, but about half the points fall within the tighter lines. The work stresses that the first and last counts should coincide closely with period beginning and end, intermediate counts should be uniformly spaced, and each count should be taken as quickly as possible for a substantially instantaneous reading; observer-made counts may carry highly variable errors. Newark data showed the last switch count often failed to coincide with period end, falling from a little after the close to nearly an interval ahead. Uncorrected, this would omit about a third of switch counts properly included, lowering the average holding-time estimate by approximately one per cent. Formulas give expected lost counts and standard deviation when elapsed time is known, assuming Poisson-distributed originations. For small elapsed times the distribution is very skew; for larger ones it becomes unimodal. Agreement is fair; each incorrectly estimated lost count causes about .2 second error in average holding time.
For the error at the beginning and end of each call, exact measurement of any call length is seldom made. In a field of switch counts, some calls receive about one count per interval, others are undercounted, and others overcounted. Three cases are distinguished. If the call length falls between 0 and i, the probability of crossing a count point is t/i, giving a plus error of i − t, or a negative error of −t with probability (i − t)/i. If the call length falls between i and 2i, it is counted twice with probability (t − i)/i, giving a plus error of 2i − t, or once with probability (2i − t)/i, giving a negative error of i − t. In the general case, where the call length falls between qi and (q + 1)i, the plus error is (q + 1)i − t with probability (t − qi)/i, and the negative error is qi − t with probability ((q + 1)i − t)/i. Total probabilities of negative and positive errors are given. For constant holding times falling between qi and (q + 1)i, only one negative error and one positive error are possible; their probabilities are given, with mean zero and a standard deviation equal to the square root of the product of the two errors, maximized at i/2 when the positive error equals i/2. An example follows: an hour with 372 calls of constant holding time 131.8 seconds under a 60-second switch count study; 284 calls, or 76.3 per cent, gave counts of “2” with error 120 − 131.8 = −11.8 seconds; 88 calls, or 23.7 per cent, gave counts of “3” with error 180 − 131.
The theoretical values check satisfactorily with observations. The observed average holding time is 134.2 seconds against a true value of 131.8 seconds; the error of 2.4 seconds is compatible with a standard deviation of about 23.85 seconds and 372 calls observed. The analysis then considers an equally likely distribution of holding times between adjacent multiples of the counting interval. Such a distribution would probably occur but rarely, if ever, in practice; however, if intervals are short compared to the average holding time, the assumption may not introduce serious discrepancy. Upon inspection, the sum of terms in the relevant equations is constant for all values of the error and the two are equal. Hence the error distribution on individual calls is isosceles triangular, with average error zero and standard deviation .408i. A footnote states that the analytics of the allied case, in which errors at the ends of a call were assumed to fall equally likely between plus and minus i/2, were discussed by E. C. Molina in an unpublished memorandum dated September 7, 1920.
For exponentially distributed holding times, the sum of the bracketed terms depends on the particular magnitudes of errors assumed. Substituting exponential expressions into the relevant equation yields an integral expression and a resulting constant. The mean of this unusual-shaped distribution is zero, and its standard deviation is given. The distribution of individual errors found by 60-second switch counts on 746 varying holding time calls—2 hours on the Newark group—is shown. Their true average holding time was 131.45 seconds; the mean error was +1.84 seconds and the standard deviation 25.55 seconds. The corresponding theoretical distribution has a standard deviation of 24.56 seconds with a mean of zero, and gives quite a good fit to the data.
It is interesting that the theoretical average error for each of these three widely dissimilar holding time distributions should be zero, while their standard deviations and analytical forms assume quite different characteristics. A footnote observes that the average error will be zero for any assumed holding time distribution, since each length of call may be segregated and considered as a constant holding time. Comparing the standard deviations for an average holding time of 145 seconds and an interval of 60 seconds gives: constant holding time, 29.58 seconds; equally likely holding time, 24.48 seconds; exponential holding time, 24.03 seconds. The constant holding time standard deviation is largely a function of whether the average is closely a whole multiple of the interval; comparing the other two indicates it is slightly advantageous that most variable holding time calls switch counted in practice are of roughly exponential form.
The work then combines errors from the three sources to formulate general conclusions for the switch count method. Suppose r + 1 switch counts are spaced uniformly by interval i, covering a period ri units long, with the call count covering exactly the same total interval. Suppose the first switch count showed m calls up, the last showed w calls up, and the sum of all r + 1 counts totalled s. The average correction in s for calls from the previous period is m/(1 − e^(−i/ī)), with a standard deviation involving the square root of m. At the end of the hour, the average correction is we^(−i/ī)/(1 − e^(−i/ī)), with a standard deviation involving the square root of w. These are independent corrections if the observation period is several times the average holding time. The best estimate of switch counts is s minus the first correction plus the second, with a standard deviation equal to the square root of the sum of the two variances. A preliminary figure for average holding time is the corrected switch count times the interval divided by n, with a standard deviation given. This must be modified by added fluctuation from switch counting at finite intervals; these cause no change in expected value but the standard deviations for individual calls must be included. Since the individual-call standard deviation is for individual calls, the standard error of the average is that value divided by the square root of n. This error is practically independent of end-of-hour errors; joint effects are additive, giving a final result.
The work notes that switch counts occur at finite intervals, and presents the overall error distribution for the illustrative example alongside the distributions of its two elements. The method of combining standard deviations explains why the two error distributions do not add directly to yield the overall error curve. Comparisons of observed and theoretical overall errors are presented, with one axis in multiples of the theoretical standard deviation and the other in per cent of observed values. A typical overall distribution of errors in estimating average holding times distinguishes the overall error, the end-effect error, and the call-measurement error.
A key conclusion is that errors due to uncertainties at the ends of the observation period may in certain cases be considerably greater than those due to imprecise measurement of each individual call’s length. The relative prominence of these two errors does not change rapidly with different observation-period sizes, since the end-effects standard deviation varies inversely as n, while the standard deviation from measuring individual calls varies inversely as the square root of n. Doubling the observation period would halve the first sigma and reduce the second to .707 of its former value.
The overall error in estimating average holding time is then expressed as a function of the number of calls, the average holding time, the counting interval, and the observation period. Since the numbers of calls at the start and end are unknown before switch counts, the average value of their sum can probably be substituted without seriously disturbing the average value of the standard deviation. This makes the standard deviation a function of only four variables: n, the average holding time, the interval, and the observation period. The errors predicted by this formula are compared with those found by theory for the particular observations in each hour’s observations at Newark. An average of 332 calls per hour was observed with an average holding time on the order of 145 seconds. The switch count interval was approximately 60 seconds, and the observation period was 3600 seconds. Taking a probability of .90, the per cent error is about 3.05 per cent, or about 4.42 seconds. This point bears about the same relationship to observed errors as the theoretical curve which accounts for actual calls carried beyond the start and end of each hour.
The section then turns to the overall error in estimating average holding time. The engineer devising a switch count schedule wants to estimate roughly the accuracy obtained in average holding time from data across a number of observation periods. Up to this point, the work has concerned only errors inherent in measuring the average length of a particular n calls of exponential type in an observation period of length T. Even when such an average length is accurately known for the n calls, it may not exactly or even closely coincide with the true average of the universe of calls from which the n are presumed to form a random sample. The errors just studied and those from the earlier section must now be combined to measure the overall accuracy of the switch count method.
Applied to exponential holding times, the standard error of the average in a sample is given. This error is independent of the error represented by the standard deviation of the switch-count estimate, so the overall standard error is determined by combining them. Of particular interest is how much the switch-count error contributes to the overall standard deviation, conveniently expressed by a ratio q. From this ratio, q depends on the interval, the average holding time, and the observation period. If the observation period is held constant, curves between q, the interval, and the average holding time may be plotted. If the observation period is varied, say increased by a factor k, the ratio shows that if the interval and the average holding time are also increased by the same factor, the values of q may still be read directly from the same curves.
An example follows: if approximately 100-second exponential calls are switch counted for an hour with observation intervals of 120 seconds, the curves show the overall standard error in estimating true average holding time is q = 1.134 times the basic standard error from taking a random sample of n calls from a very large universe. That is, the residual sampling error in a stopwatch measurement of the n calls is increased by 13.4 per cent due to resorting to switch count methods. If a continuous period of 2 hours is switch counted the same way, one reads on the 50-second curve opposite 60 seconds, giving an increase over basic sampling error of 12.4 per cent. This meets common-sense conclusions: when the observation period is increased by a factor k, the sampling error decreases as 1/√k since k times as many calls are observed; the error of measurement on individual calls also decreases as 1/√k for the same reason; and the end effect errors are unchanged in actual magnitude but prorated over k times as many calls, so this element decreases as 1/k. The overall effect as k increases is for the overall standard deviation to decrease faster than the sampling standard deviation. Thus not only is sampling error decreased by lengthening the observation period, but overall error decreases even faster. An observation period as long as is consistent with a controlled universe of holding times is to be preferred.
Main Arguments
The work’s central argument is that the switch-count method, despite its practical advantages over stopwatch measurement and mechanical recording, introduces measurable and predictable errors that can be quantified and controlled. The argument proceeds by decomposing the total error into its constituent parts, deriving a probability distribution for each, checking each against empirical data, and then recombining them into an overall error estimate.
The first substantive argument is that local subscriber call holding times follow an exponential frequency distribution closely enough that the exponential form can serve as the basis for theoretical analysis. This is established both by reference to Erlang’s earlier observation and by the Newark data, where the fit of the exponential curve with average equal to the observed average of 2.380 minutes is quite good. The practical qualification is that the standard deviation is usually found slightly larger than the mean, though not markedly so—a fact that recurs in the analysis of the forenoon hours, where the observed standard deviation of the means is some 27 per cent higher than theory would predict.
The second argument concerns the homogeneity of the holding-time universe. The Newark data appear at first glance to fall into two rather distinct groups, those before noon and those after noon. The work treats the 115 forenoon hours as a homogeneous group and asks whether the afternoon averages could reasonably come from the same universe. The deviations of the three clock hours from the forenoon average are small enough that the observations could readily have occurred under the hypothesis of a common universe. This matters because the reliability of any estimate depends on whether the sample is drawn from a controlled universe; if trends or day-of-week peculiarities exist, the sampling procedure must be apportioned to give them proper weighting.
The third argument is that the three sources of error—start-of-period carry-over, end-of-period truncation, and individual-call measurement—can each be modeled probabilistically and that the models agree with observation. For the start-of-period error, the theoretical distribution of carry-over calls is corroborated by 30 hourly periods on the Newark group, with one point on the outermost control line and 16 of 30 within the tighter lines. For the end-of-period error, two hours fall outside the three-sigma limits, but about half the points fall within the tighter lines. For individual-call error, the observed average holding time of 134.2 seconds against a true value of 131.8 seconds is compatible with the theoretical standard deviation, and the distribution of errors on 746 varying holding time calls fits the theoretical exponential-error distribution quite well.
The fourth argument is that the average error is zero for any assumed holding time distribution, even though the standard deviations and analytical forms differ. This is because each length of call may be segregated and considered as a constant holding time. The practical consequence is that the switch-count method does not systematically bias the estimate of average holding time; it only adds variance. The comparison of standard deviations for constant, equally likely, and exponential holding times shows that the exponential form—which is the form actually encountered in practice—produces a slightly smaller standard deviation than the equally likely form, and both are smaller than the constant form when the average is not closely a whole multiple of the interval.
The fifth argument is that end-of-period errors may in certain cases be considerably greater than individual-call measurement errors, and that the two behave differently as the observation period lengthens. The end-effects standard deviation varies inversely as n, while the individual-call standard deviation varies inversely as the square root of n. Doubling the observation period halves the first and reduces the second to .707 of its former value. This asymmetry drives the practical recommendation that the observation period should be as long as is consistent with a controlled universe of holding times, because the overall error decreases faster than the sampling error alone.
The sixth argument is that the overall error of the switch-count method can be expressed as a ratio to the basic sampling error of a stopwatch measurement. For the example of 100-second exponential calls counted for an hour with 120-second intervals, the overall standard error is 1.134 times the basic sampling error—an increase of 13.4 per cent. For a 2-hour period counted the same way, the increase is 12.4 per cent. This gives the engineer a concrete sense of what is sacrificed by using switch counts rather than direct measurement, and it shows that the sacrifice diminishes as the observation period lengthens.
Key Concepts
The exponential frequency distribution of holding times is the foundational concept. Expressed as f(t)dt = ke^(−kt)dt, where 1/k is the average holding time, it describes the probability that a call has a given duration. In this distribution the standard deviation equals the mean, though in practice the standard deviation is usually found slightly larger. The exponential form is used throughout the analysis as the basis for deriving error distributions, and the Newark data confirm its applicability.
The switch-count method is the technique under examination. It scans a group of paths at regular intervals and records the number found busy, without measuring any individual call. Its advantages are speed and low cost; its disadvantage is that it measures calls only indirectly, through the number of counts each call happens to receive. The method yields estimates of average load carried directly, and with a call- or peg-count meter it also yields estimates of average holding time.
The three sources of error are the analytical backbone of the work. Start-of-period error arises because the first switch count includes calls that originated in the preceding hour and whose contributions to subsequent counts cannot be segregated. End-of-period error arises because some calls registered on the peg count meter end beyond the period, losing part of their proper switch counts. Individual-call error arises because a call’s length is not measured exactly but only inferred from how many count points it crosses, leading to undercounting or overcounting depending on where the call falls relative to the count points.
The carry-over correction and the end correction are the practical remedies for the first two error sources. The average correction for calls from the previous period is m/(1 − e^(−i/ī)), where m is the number of calls up at the first count. The average correction at the end is we^(−i/ī)/(1 − e^(−i/ī)), where w is the number up at the last count. These corrections are independent if the observation period is several times the average holding time, and their variances add.
The best estimate of switch counts is the observed total minus the start correction plus the end correction, with a standard deviation equal to the square root of the sum of the two variances. From this corrected total, a preliminary figure for average holding time is obtained by multiplying by the interval and dividing by the number of calls. This preliminary figure is then modified by the added fluctuation from switch counting at finite intervals.
The overall standard error combines the switch-count error with the basic sampling error of a random sample from a large universe. The ratio q expresses how much the switch-count error contributes to the overall standard deviation, and it depends on the interval, the average holding time, and the observation period. Curves of q allow the engineer to read off the overall error for various combinations of these variables.
The concept of a controlled universe of holding times is the practical constraint on observation-period length. If the holding-time universe is not controlled—if there are trends or day-of-week peculiarities—then lengthening the observation period does not simply reduce error, because the sample is no longer drawn from a single homogeneous population. The recommendation that the observation period be as long as is consistent with a controlled universe follows directly from this constraint.
Themes
The dominant theme is the tension between statistical precision and practical economy. Direct measurement with a stopwatch is the most accurate method in principle, but it is slow and expensive because it requires many observers, each watching one call at a time. Mechanical recording is more efficient but demands considerable time and labor to summarize results and poses maintenance problems. Switch counting is the cheapest and fastest method, but it sacrifices accuracy. The entire work is an attempt to quantify exactly how much accuracy is sacrificed, so that the engineer can make an informed trade-off.
A second theme is the power of probabilistic modeling to predict and correct for measurement error. The work does not simply report that switch counts are inaccurate; it derives the exact form of the inaccuracy for each of its sources, checks the derivations against data, and shows that the theoretical distributions fit the observations. The agreement between theory and data—whether in the carry-over counts, the end-of-period counts, or the individual-call errors—is presented as evidence that the method is sound and that the corrections are reliable.
A third theme is the importance of understanding the underlying distribution of the phenomenon being measured. The exponential distribution of holding times is not merely a convenient assumption; it is an empirical fact that shapes the error analysis. The comparison of constant, equally likely, and exponential holding times shows that the form of the distribution affects the standard deviation of the error, even though the average error is zero in all three cases. The practical conclusion that it is slightly advantageous that most variable holding time calls are of roughly exponential form follows from this comparison.
A fourth theme is the asymmetry between different sources of error and its implications for experimental design. The end-of-period error and the individual-call error behave differently as the observation period lengthens, and this difference means that the overall error decreases faster than the sampling error alone. The recommendation to make the observation period as long as possible—subject to the constraint of a controlled universe—is a direct consequence of this asymmetry. It is a theme that connects the abstract mathematics to the practical business of designing a switch-count schedule.
A fifth theme, present throughout but never stated as a slogan, is the value of checking theory against data. The work repeatedly derives a theoretical distribution and then compares it with actual Newark observations. Sometimes the agreement is good, sometimes only fair, and sometimes a point falls outside the control lines. The work does not hide these discrepancies; it discusses them and accounts for them where possible. This gives the analysis a cumulative, self-correcting quality: each theoretical result is tested, and the tests inform the next stage of the argument.
The Author’s Conclusions
The work concludes that the switch-count method is a reliable way to estimate average holding time, provided that its errors are properly understood and corrected. The average error is zero for any assumed holding time distribution, so the method does not introduce systematic bias. The variance it introduces can be calculated from the interval, the average holding time, the number of calls, and the observation period, and the corrections for start-of-period and end-of-period effects can be applied using the observed numbers of calls at the first and last counts.
The most important practical conclusion is that the observation period should be as long as is consistent with a controlled universe of holding times. When the observation period is increased by a factor k, the sampling error decreases as 1/√k, the error of measurement on individual calls also decreases as 1/√k, and the end effect errors decrease as 1/k. The overall effect is that the overall standard deviation decreases faster than the sampling standard deviation alone. Lengthening the observation period therefore improves accuracy more than proportionally, and the only limit is the requirement that the holding-time universe remain controlled.
A second conclusion is that end-of-period errors may in certain cases be considerably greater than individual-call measurement errors. This is a caution against assuming that the main source of inaccuracy is the imprecision of individual call measurement. The relative prominence of the two errors does not change rapidly with different observation-period sizes, but the end-effect error diminishes faster as the period lengthens, so the balance shifts in favor of the longer period.
A third conclusion is that the switch-count method increases the standard error over a stopwatch measurement by a modest amount. In the example of 100-second exponential calls counted for an hour with 120-second intervals, the increase is 13.4 per cent; for a 2-hour period, 12.4 per cent. This is the price of using switch counts instead of direct measurement, and it is a price that can be reduced by lengthening the observation period.
A fourth conclusion concerns the practical details of switch counting. The first and last counts should coincide closely with the beginning and end of the period, intermediate counts should be uniformly spaced, and each count should be taken as quickly as possible for a substantially instantaneous reading. Observer-made counts may carry highly variable errors, and the Newark data showed that the last switch count often failed to coincide with period end, falling from a little after the close to nearly an interval ahead. Uncorrected, this would omit about a third of switch counts properly included, lowering the average holding-time estimate by approximately one per cent. Each incorrectly estimated lost count causes about .2 second error in average holding time.
A fifth conclusion is that the exponential form of the holding-time distribution is slightly advantageous for switch counting. Comparing the standard deviations for constant, equally likely, and exponential holding times at an average of 145 seconds and an interval of 60 seconds gives 29.58, 24.48, and 24.03 seconds respectively. The constant holding time standard deviation is largely a function of whether the average is closely a whole multiple of the interval; the exponential form produces the smallest standard deviation of the three.
Significance
The work’s significance lies in its demonstration that a cheap and convenient measurement method can be made rigorous. Switch counting had been used in the Bell System and elsewhere because it was faster and less expensive than direct measurement or mechanical recording, but its accuracy had not been systematically analyzed. By deriving the probability distributions of the errors it introduces, checking those distributions against real data, and combining them into an overall error estimate, the work turns switch counting from a rough-and-ready technique into a quantitatively understood instrument.
The analysis also provides a template for thinking about measurement error in any setting where a phenomenon is observed indirectly.
