A* a . ' Whichever of the methods are to be used,n t e16~~~ccuracy o 0 ing ime~~~the basic rules of sampling, as particu-

larly adapted for switch count methods,

Measurements apyFrom the preceding it will be under-

stood that the switch count process is a

IMRE MOLNAR particular method of time measurementNONMEMBER AIEE by sampling, and its accuracy can be ap-

praised by comparison with direct meas-urement of time, for instance, by a clock,

T WILL be helpful for the understand- apparent that any measurement of traf- where the latter term is used in its broad-ing of traffic problems to realize that, fic can be reduced in the last analysis to est possible interpretation. We also have

fundamentally, telephone traffic is the the measurement of time, more specifi- seen that the amount of traffic is equal tosum of accumulated holding times of cally to the measurement of accumu- the accumulated holding times of individ-switching channels. For instance, if one lated holding times. There are several ual calls; therefore, if in addition tochannel is loaded in a certain hour with established methods for the measurement measuring the traffic, the number of calls15 calls, each of 3 minutes or 0.05-hour of telephone traffic, the most widely used carried by the group of channels underduration, and since these calls must of these being the so-called switch count observation are simultaneously counted,necessarily be carried by the channel method; although under this generic the average holding time of a call can beconsecutively, the total duration of oc- term we shall also include other ones with determined by dividing the traffic by thecupancy will be 0.75 hour; which is then little resemblance in procedure though all total number of calls. Conversely, if byequivalent to the statement that the based on the same fundamental prin- some means we have obtained the averagetraffic is 0.75 traffic unit or that the oc- ciple. As it is well known, in the switch holding time of a call and the total num-cupancy of the channel is 75 per cent. count method a group of channels is ber of calls, the product of the two willFrom this simple statement two impor- sampled at regular scanning cycles to de- give us the total amount of traffic; thoughtant conclusions can immediately be termine the number of channels simul- the former procedure is the one moredrawn. First, that telephone traffic is taneously occupied at a certain instant. commonly occurring in practice, that is,completely equivalent to the length of oc- If these simultaneous occupancies are where the traffic and the number of callscupancy of the switching channels; that totalized for, say, one hour and the total are measured, and the holding time then isit may be expressed in any convenient divided by the number of scans, the directly determined. If the laws govern-unit of time; and that the same traffic or average number of simultaneously oc- ing the accuracy of peg count methodsoccupancy can be made up by many calls cupied channels during that hour will be and holding time measurements are deter-of short holding times each, or a few calls obtained. It can be demonstrated that mined, we have the necessary tools avail-of long holding times each, or in any con- this average number of occupied chan- able by which the accuracy of trafficceivable mixture of these as long as the nels, neglecting some fine points, is nu- measurements can be appraised. Thesum of the individual holding times add merically equal to the traffic carried by the former was recently studied and pub-up to the total length of occupancy. group during that hour (or traffic den- lished by the author,2 who found that theSecond, since a channel naturally cannot sity), expressed in traffic units. The commonly used peg counting methodsbe occupied for more than one hour in any agreement can be made very close if cer- provide an exactness well within that re-single hour, the traffic carried per channel tain rules in the technique of switch quired for traffic measurements.in one hour cannot exceed one traffic unit. counts and in the method of sampling are In 1941 Roger I. Wilkinson of the BellThese considerations apply equally to a observed, and the present paper will deal Telephone Laboratories published an out-single channel, or to any one channel in with some of the refinements in the ac- standing paper3 using methods of math-any arbitrary grouping of channels, or to curacy of traffic observations. ematical statistics, on the reliability ofthe average traffic per channel in a group. The scanning during the switch count holding time measurements, which can beFrom what we have said up to now, it is process can be carried out either by visual considered as the fundamental work for

inspection of the number of cord circuits all problems related to switch count-- ~~~~~~~~~pluggedin, or of the number of switches methods. This in conjunction with the

Puiaper51-51 ricommnedyed by the AIEE Corn- in use, or of the number of circuit buy paper qutdin the last paarp,pro-approved by the AIEE Technical Program Corn- lamps lit, or it can be entirely mechanized vides sufficient material to appraise themittee for presentation at the AIEE Fall GeneralMeeting, Cleveland, Ohio, October 22-26, 1951. by an automatic scanning device such as accuracy obtainable with switch countManuscript submitted July 11, 1951; made avail- one of the several types of Trafficorders.' programs occurring in practice. Mr.able for printing September 21, 1951. Figure 1 shows a picture of the Automatic Wilkinson reduced his theory into aIMRE MOLNAR is with tbe Automatic Electric ElcrcCmaystpCTrfcodr seisfeqton,ndhndvlpdaCompany, Chicago, Ill.ElcrcCmaystpCTrfiodr seisoeqain,ndhndvlpda

1912 Mfolnar-Accuracy of Holding Time Measurements AIEE TRANSACTIONS

Figure 1 (left). r-- TRAF'F1COF OERType C Traffi I I

corder ICOUNTERCHANNEL

Figure 2 (right).Simplified dia- cluded so that representative sampling ofgram of Traffi- all known or suspected major variationscorder connec- is accomplished. Second, if a choice is

tions available, a short scanning cycle will pro-duce more reliable results than a long one.As a result, for the holding times of usualtelephone conversations a fair accuracycan be obtained if the scanning cycle isone-half or less than the average holding

number of graphs to assist the traffic engi- ticularly when one considers the me- time. Third, the first and last countsneer in devising working schedules for the chanical limitations of the sampling pro- should coincide closely with the beginningsampling of holding times. cedure, the purpose for which the knowl- and end, respectively, of the observationAs in all sampling processes of homo- edge of the holding time is required, and period, and the intermediate counts, that

geneous populations the accuracy im- the persistence in the homogeneity of the is the scanning cycle, should be uni-proves with an increasing number of population. There are a few simple rules formly spaced. Fourth, each countsamples taken. However, a point of di- which can help to improve the accuracy of should be taken as quickly as possible, soniinishing returns will be reached, be- the program. First, other things being that a substantially instantaneous readingyond which the improvement in accuracy equal, the longest observation period of calls in progress is obtained. Fifth, thebecomes so slight that it would not justify should be selected, consistent with the graphs computed for this purpose are,the expense of further sampling; par- view that enough periods must be in- strictly speaking, valid only if the dis-

tribution of the holding times around the

Table I. Accuracy oF Holding Time Sampling average follow the exponential law; how-ever, they are not greatly dependent on

Number of Confidence (P) the form of holding time distribution as

Samples 0.50 0.80 0.90 0.95 0.99 0.999 long as the average call length covers(n) Per Cent Error (e) several scanning cycles, and the graphs

can probably be used with a slight allow-30. 12.315..... 23.407.... 30.034.... 35.786.. 47.033.... 60.08735. 11.401... 21.670.. 27.805.. 33.130.. 43.542.. 55.627 ance. For relatively constant holding40. 10.66. .. ..20.26. ..26.008. ..30.988. 40.727 ........52.031 times the accuracy can be further im-45. 10.055..... 19.111.... 24.523.... 29.218. 38.401.... 49.06050. 9.529... 18.111.. 23.240.. 27.690. 36.392.. 46.494 proved by choosing the scanning cycle so55 9.095 17.287. 22.181 26.429 . 34.733.5 . 44768 that it will be contained in the average60......8.708.....16.551.....21.237.....25.304.....33.257.....42.48865. 8.366... 15.901.. 20.403... 24.310.. 31.951... 40.819 holding time approximately a whole70......8.061.....15.321.....19.660.....23.424.....30.786.....39.331 nubrfme.A gh cra cne80... . 7.541. 14.333. 18.391. 21.913. 28.800. 36.79490. 7.110... 13.514...,.. 17.340... 20.660.. 27.154... 34.691 obtained with relatively few observations100. 6.745... 12.820... 16.450. 19.600.... 25.760. 32.910125 6.033. 11.467.. 14.714. 17.531.. 23.041. 29.436 if the scanning cycle is exactly equal to150. 5.507.. 10.467.. 13.431.... 16.003.. 21.032.. 26.870 the constant holding time.175 ., 5.099.... 9.692... 12.436 ....... 14.817... 19.474.... 24.879200. 4.769.... 9.064.... 11.631....... 13.858.... 18.213.... 23.269 Accordingto Mr. Wilkinson, neglecting230......4.447.....8.452.....10.846.....12.022.....16.984.....21.698 a few minor causes, there are three prim-260. 4.183 7.950 10.202. 12.155. 15.975. 20.410300 3.894.. 7.401.. 9.497.... 11.315.. 14.872.. 18.999 cipal sources affecting the accuracy of the350. 3.605.... 6.852... 8.792 ....... 10.476... 13.768.... 17.589400. 3.373.... 6.410.... 8.225....... 9.800.... 12.880.... 16.455 measurement.450......3.180.....6.044.....7.756.....9.241.....12.145.....15.516 1. The homogeneity of sampling processes500. .3.016. 5.732. 7.356. 8.764. 11.518. 14.716550. 2.876. 5.466. 7.014. 8.357.. 10.984.. 14.032 tn general, that is, the sampies which have600. 2.754 ........ 5.234...... 6.717...... 8.003. 10.518.. 13.437 been drawn at random represent the charac-650. 2.646 5.029 6.453 7.689. 10.105. 12.910 teristics of the population. This accuracy,700 ......2.549.....4.845.....6.217.....7.407.....9.735.....12.437800....... 2.385. 4.533....... 5.817. 6.930. 9.109.. 11.637 of course, can be improved by increasing the900.......... 2.248... 4.273........ 5.483... 6.532... 8.585... 10.968 number of samples.

1,000 .......... 2.133... 4.054........ 5.202... 6.198... 8.146.... 10.4071,250....... 1.908. 3.626....... 4.653. 5.544. 7.287.. 9.309 2. Errors at the beginning and at the end1,500....... 1.742. 3.311....... 4.248. 5.062. 6.653.. 8.500 of the observation period, due to observa-1,750.......... 1.612... 3.064........ 3.931... 4.684... 6.156........ 7.865 tions of calls being included at the beginning2,000.......... 1.508.. 2.866.. 3.678... 4.382. 5.759........ 7.3582,300....... 1.406. 2.672. 3.429. 4.086. 5.370........ 6.860 from the preceding period, and at the end2,600....... 1.323. 2.515. 3.227. 3.844. 5.053........ 6.455 which will project into the next period.3,000 .......... 1.231 .. 2.340........ 3.002 ........ 3.577 ........ 4.701 ........ 6.0113,500....... 1.140. 2.167. 2.780. 3.313. 4.354........ 5.562 3. Errors due to the nature of the method4,000....... 1.066. 2.026. 2.600. 3.098. 4.071........ 5.201 itself, that is by counting switches in definite4,500....... 1.005. 1.910. 2.451. 2.920. 3.838........ 4.904 Cycls; therefore an exact measurement of5,000.......... 0.954 ........ 1.813 ....,.2.327 ........ 2.772. 3.643........ 4.655 c

5.500. 0.909....... 1.728. 2.217. 2.641. 3.472. 4.435 the holding time of any one call will prac-6,000.. 0.871........ 1.655.. 2.124... 2.531... 3.326.. 4.250 ticallynever be made.6,500.. 0.837........ 1.591.. 2.041... 2.432... 3.197.. 4.0847,000. 0.806....... 1.532. 1.966. 2.342. 3.078. 3.933 When the average holding time is evalu-8,000.. 0.754........ 1.433.. 1.839... 2.191... 2.880.. 3.6799.0.00. 0.711....... 1.351. 1.734. 2.066. 2.715. 3.469 ated from the results of the switch count10,000.. 0.675. 1.282. 1.645S.- -----1.960-....... 2.576-....... 3.291 process, the best estimate for the aver-

1951, VOLUME 70 Molnar-Accuracy of Holding Time Measurements 1913

Table 11. Scanning Interval Corrections 5 per cent accuracy, Table III tells usthat two observation periods would have

Scanning Average Holding Time in Seconds sufficed.Interval 10 25 50 75 100 125 150 200

in Seconds Multiplier (k) In his paper Mr. Wilkinson compareshis theoretical conclusions with a series of

5. 1.035. 1.019. 1.031 . 1.047. 1.060. 1.075. 1.086. 1.112 observations taken in New Jersey be-10. 1.089... 1.032 ... 1.034... 1.049 ... 1.061... 01.076... 1.088... 1.11415.1.170. 1.050. 1.039. 1.051. 1.063. 1.077. 1.089. 1.115 tween the two. The tables of this paper20. 1.280. 1.075. 1.046. 1.052. 1.065. 1.078. 1.090..l.1.116 were applied in several traffic studies30 . .. 1.140... 1.065... 1.062 ... 1.071 ... 1.081 ... 1.092... 1.11840. . 1.220. 1.091. 1.076. 1.078. 1.086. 1.098. 1.121 madebyW. A. Alden andD. E. Klang of50. . 1.310. 1.122.. 1.091 .-. 1.088. 1.092.-. 1.102..... 1.124 the Automatic Electric Company Labora-60 . .. 1.420... 1.160... 1.111 ... 1.100... 1.102... 1.110... 1.13070.. . 1.202. 1.132. 1.115. 1.112. 1.117. 1.135 tories with Trafficorder equipment and80 ...1.246. 1.157. 1.131 1.123 1.125 1.140 the resulting accuracy was likewise con-.90.................. 1.293....1.181....1.146....1.135....1.134....1.147100.. . 1.340..1.1.210. 1.163. 1.147. 1.143. 1.152 firmedaslongas theholdingtimeswere in110 ...1.390... 1.239..... 1.182. 1.161. 1.154. 1.159120.. .. 1.270..... 1.204. 1.176. 1.164. 1.166 the usual telephone conversation range.130 .....1.301 .1.225 .1.191. 1.178. 1.175 However, the convenience, ease, and ac-140 ... 1.338... 1.248... 1.209. 1.190. 1.182150.. .. 1.371. 1.271. 1.225. 1.202 .1.192 curacy of Trafficorder measurement en-160 .. ..1.405.- 1.297.---.1.242.*. 1.208.--1.201 couraged us to measure several special170............................................. . . . 1.321 ... 262 . 1.232. 1.211180 ..... 1.348. 1.282. 1.249. 1.221 types of equipment which had holding190.---....-..-----........1.373. 1.302.-. 1.263..1.231 times much below that of telephone con-200 ... . 1.401. 1.323. 1.280. 1.242210 ......1.345. 1.299. 1.255 versations, and there some discrepancy220 ......1.367.1.315.1.268230 ......1.389. 1.332 .1.280 was noticeable. For instance, we at-240 .......1.350. 1.292 tempted to determine with a Trafficorder

(1) Obtain per cent error e from Table I for number of calls n included in one observation period and for having a 10-second scanning cycle theselected confidence P.* average holding time of translators used(2) Obtain multiplier k to e from this table for length of scanning interval and anticipated average holding in a Director system. By preliminarytime.

(3) Obtain number of observation periods N required from Table III for ke and for selected over-all per cent calculations from the individual operatingerror E. times of the circuit components, the

holding time was estimated to be aboutage holding time of the population will be As an example for the use of this table 400 milliseconds; the results, however, ofthis result; however, there is a certain let us assume that we devise a switch the Trafficorder observation showed aamount of uncertainty inherent to this count schedule in which approximately greater discrepancy than could have beenvalue due to the above three causes. As a 1,000 samples are to be observed in one reasonably expected from the accuracy ofresult of Mr. Wilkinson's work this un- period, that the scanning cycle is 10 the preliminary estimate. The samecertainty can be estimated, and by ex- seconds, and that from some preliminary translator holding time was then meas-tending the pregram to include many ob- information we know that the average ured with the aid of the delay register4servation periods any specified degree of holding time is in the order of 125 seconds. to a fairly high degree of precision, andaccuracy is obtainable. We wish to know from this program the found to be approximately 100 milli-

Tables I, II, and III were calculated average holding time, so that we may have seconds less than the holding time ob-from Mr. Wilkinson's equations. Table a confidence P of 0.95 that the result will tained from the Trafficorder processI represents the limits e of the percentage be accurate within =i= 1 per cent. From Furthermore, by making comparison ofof error in the average holding time which Table I we find opposite 1,000 in the the two methods on various types ofcan be expected with a given confidence column for 0.95 an error e of 6.198. In equipment and in a holding time rangeP if n samples are drawn in one observa- Table II we find opposite 10 seconds from 300 to 600 milliseconds, the sametion period from an exponentially dis- scanning cycle and in the column for 125- discrepancy, that is about 100 milli-tributed holding time population, and is second holding time the multiplier 1.076, seconds, was obtained regardless of theequally valid whether the holding times (6.198X1.076=6.67). From Table III length of the holding time or the functionof the samples are exactly measured by a opposite 6.5 and in the column for 1 per of the circuit. By a mathematical anal-clock method, or obtained by a switch cent, we find that 42.3 observation periods ysis and subsequent experimental coni-count process. Table II gives the mul- are required. Interpolating between 6.5 firmation, it was discovered that there istiplier to the per cent error e obtained and 7.0 single period error we conclude an additional source of error inherent tofrom Table I, due to causes (1) and (2) that about 45 or 46 observation periods switch count methods not disclosed byabove and which are peculiar to switch would provide the desired degree of ac- previous studies, and in order to under-count methods. Table III specifies the curacy. Had we been satisfied with, say, stand this more clearly we must firstnumber of observation periods requiredfor any desired limit in the percentage Yover-all error if the single period error e is tobtained from Table I, and multiplied by CHANNEL lk of Table II. The basis of Table III is 1the central limit law, according to which(provided that certain conditions are u''i- Usatisfied as they are in this problem) the TRAFFICORDER U ,h 1_ -averages derived from groups of n sampleshave a tendency to be normally dis- b lbtributed with increasing numberNofsuch A M B A' L' EB-groups. Figure 3. Generalized sequence diagram for Trafficorder measurements

1914 Molnar Accuracy of Holding Time Measurements . ATEE TRANSACTIONS

Table 1ll. Number of Observation Periods Required

% Error Single Per Cent Over-All Error (E)Observation 0.5 1 2 3 4 5 6 8 10 15 25Period (ke) Number of Observation Periods Required (N)

0.8 ........ 2.56.... 0.640.... 0.160.... 0.071 .... 0.040.... 0.026.... 0.018.... 0.010.... 0.006.... 0.003... .0.0010.9 ........ 3.24.... 0.810.... 0.203 .... 0.090.... 0.051 .... 0.032 .... 0.023.... 0.013 .... 0.008.... 0.004.... .0.0011.0 ........ 4.00.... 1.00 .... 0.250.... 0.111 .... 0.063.... 0.040.... 0.028.... 0.016.... 0.010. .. . 0.004.... .0.0021.2 .... ..... 4.84... 1.44 .... 0.360.... 0.160.... 0.090.... 0.048.... 0.040.... 0.023.... 0.014.... 0.006.....0.0021.4......... 7.84... 1.96 .... 0.490.... 0.218.... 0.123 .... 0.078.... 0.054.... 0.031 .... 0.020.... 0.009.... .0.0031.6......... 10.2... 2.56 .... 0.640.... 0.284 .... 0.160.... 0.102 .... 0.071 .... 0.040.... 0.026.... 0.011.... .0.0041.8 ........ 13.0 . 3.24 .... 0.810.... 0.360.... 0.203.... 0.130.... 0.090.... 0.051 .... 0.032.... 0.014.... .0.0052.0 ........ 16.0.... 4.00 .... 1.00 .... 0.444.... 0.250.... 0.160.... 0.111 .... 0.063 .... 0.040.... 0.018.... .0.0062.2......... 19.4... 4.84 .... 1.21 .... 0.538.... 0.303.... 0.194.... 0.134.... 0.076.... 0.048.... 0.022.... .0.0082.4......... 23.0 5.76 .... 1.44 .... 0.640.... 0.360.... 0.230.... 0.160.... 0.090.... 0.058.... 0.026.... .0.0092.6......... 27.0... 6.76 .... 1.69 .... 0.751 .... 0.423 .... 0.270.... 0.188.... 0.106.... 0.068.... 0.030.... .0.0112.8......... 31.4 7.84 .... 1.96 .... 0.871 .... 0.490.... 0.314.... 0.218.... 0.123 .... 0.078.... 0.035.... .0.0133.0......... 36.0... 9.00 .... 2.25 .... 1.00 .... 0.563.... 0.360.... 0.250.... 0.141 .... 0.090.... 0.040.... .0.0143.3......... 43.6... 10.9 .... 2.72 .... 1.21 .... 0.681 .... 0.436.... 0.303 .... 0.170.... 0.109 .... 0.048.... .0.0173.6......... 51.8 13.0 .... 3.24 .... 1.44 .... 0.810.... 0.518.... 0.360.... 0.203.... 0.130.... 0.058.... .0.0214.0......... 64.0 16.0 .... 4.00 .... 1.78 .... 1.00 .... 0.640.... 0.444.... 0.250.... 0.160.... 0.071.... .0.0264.5......... 81.0 20.3 .. 5.06 .... 2.25 .... 1.27 .... 0.810.... 0.563.... 0.316.... 0.203.... 0.090.... .0.0325.0 ... ...... 100 25.0 .... 6.25 .... 2.78 .... 1.56 .... 1.00 .... 0.694.... 0.391 .... 0.250.... 0.111... .0.0405.5 ... ...... 121 30.3 .... 7.56 .... 3.36 .... 1.89 .... 1.21 .... 0.840.... 0.473 .... 0.303.... 0.134.... .0.0486.0... ...... 144 36.0 .... 9.00 .... 4.00 .... 2.25 .... 1.44 .... 1.00 .... 0.563.... 0.360.... 0.160.....0.0586.5... ...... 169 42.3 .... 10.6 .... 4.69 .... 2.64 .... 1.69 .... 1.17 .... 0.660.... 0.423 .... 0.188... .0.0687.0 ... ...... 196 49.0 .... 12.3 ... 5.44 .... 3.06 ... 1.96 .... 1.36 .... 0.766 .... 0.490.... 0.218.....0.0787.5 ... ...... 225 56.3 .... 14.1 .... 6.25 .... 3.52 .... 2.25 .... 1.56 .... 0.879.... 0.563 .... 0.250... .0.0908.0 ... ...... 256 64.0 .... 16.0 ... 7.11 .... 4.00 ... 2.56 .... 1.78 .... 1.00 ... 0.640.... 0.284.... .0.1028.5 ... ...... 289 72.3 .... 18.1 .... 8.03 .... 4.52 .... 2.89 .... 2.01 .... 1.13 .... 0.723 .... 0.321.... .0.1169.0 ... ...... 324 81.0 .... 20.3 .... 9.00 .... 5.06 .... 3.24 .... 2.25 .... 1.27 .... 0.810.... 0.360.... .0.13010.0 ... ...... 400 100 .... 25.0 .... 11.1 .... 6.25 .... 4.00 .... 2.78 .... 1.56 .... 1.00 .... 0.444.... .0.16012.0 ... ...... 484 144 .... 36.0 .... 16.0 .... 9.00 .... 4.84 .... 4.00 .... 2.25 .... 1.44 .... 0.640.... .0.23014.0 ... ...... 784 196 .... 49.0 .... 21.8 .... 12.3 .... 7.84 .... 5.44 .... 3.06 .... 1.96 .... 0.871.... .0.31416.0.........1,024 256 .... 64.0 .... 28.4 .... 16.0 .... 10.2 .... 7.11 .... 4.00 .... 2.56 .... 1.14 .... .0.41018.0.........1,296 324 .... 81.0 .... 36.0 .... 20.3 .... 13.0 .... 9.00 .... 5.06 .. 3.24 .... 1.44 .... .0.51820.0.........1,600 400 .... 100 .... 44.4 .... 25.0 .... 16.0 .... 11.1 .... 6.25 .. 4.00 .... 1.78 .... .0.64022.0.........1,936 484 ... 121 .... 53.8 .... 30.3 .... 19.4 .... 13.4 .... 7.56 .... 4.84 .... 2.15 .... 0.77424.0.........2,304 576 .... 144 .... 64.0 ... 36.0 ... 23.0 .... 16.0 ... 9.00 .... 5.76 .... 2.56 .... 0.92226.0.........2,704 676 .... 169 .... 75.1 .... 42.3 .... 27.0 .... 18.8 .... 10.6 .... 6.76 .... 3.00 ... .1.0828.0.........3,136 784 .... 196 .... 87.1 .... 49.0 .... 31.4 .... 21.8 .... .12.3 .... 7.84 .... 3.48 .... .1.2530.0.........3,600 900 .... 225 .... .100 .... 56.3 .... 36.0 .... 25.0 .... .14.1 .... 9.00 .... 4.00 .... .1.4433.0.........4,356 ...1,089 .... 272 .... .121 .... 68.1 .... 43.6 .... 30.3 .... .17.0 .... .10.9 .... 4.84 .... .1.7436.0........ ...5,184 ...1,296 .... 324 .... .144 .... 81.0 .... 51.8 .... 36.0 ...20.3 ... .13.0 .... 5.76 ... .2.0740.0.........6,400 1,600 .... 400 .... .178 .... .100 .... 64.0 .... 44.4 .... .25.0 ... .16.0 .... 7.11 .... .2.5645.0.........8,100 ...2,025 .... 506 .... 225 .... .127 .... 81.0 .... 56.3 ... .31.6 ... .20.3 .... 9.00 .... .3.2450.0.........10,000 ...2,500 .... 625 .... .278 .... .156 .... .100 .... 69.4 ... .39.1 .... .25.0 .....11.1 .... .4.0055.0.........12,100 ...3,025 .... 756 .... .336 .... 189 .... .121 .... 84.0 ... .47.3 .....30.3 ....13.4 .... .4.8460.0.........14,400 ...3,600 .... 900 .... .400 .... .225 .... .144 ... .100 .... .56.3 ....36.0 .....16.0 .... .5.7665.0.........16,900 ... 4,225 .... .1,056 .... .469 .... .264 .... .169 ... .117 .... .66.0 . ....42.3 .... .18.8 .....6.7670.0.........19,600 ...4,900 .... .1,225 ... .544 ... .306 .... .196 .... .136 ... .76.6 .....49.0 .... .21.8 .... .7.84

take a closer look at the conditions under ent example the call extends beyond the counted for by Mr. Wilkinson in sectionwhich switch counts, either by visual or point A', and it should therefore be in- IV-c of his paper,' and our present prob-mechanical means, are performed. cluded in the scanning interval A' -B'. lem represents an error in excess and

Figure 3 shows the occupancy of a The point Y indicates the termination of practically independent of those causedchannel in parallel with the scanning proc- the call, which will be properly registered by the inaccuracy in measuring each call,ess of the Trafficorder. In this example if .the point Y falls anywhere to the right since the latter will always be present re-

the holding time of a call extends ap- of the point L'. If, however, Y falls in gardless whether the scanning interval u isproximately over two scanning cycles and the interval A '-L' it will not register, finite or infinitely short. (u representsmay, of course, start and end at any time. because the call will not be long enough the time during which the counter isFor our problem we assume that it starts for the counter to respond, and, there- connected to the circuit to be measured,at the point X which is somewhere within fore, would not be included in the switch and b represents the time required by thethe scanning interval A-B, but it did not count, causing a negative error in the counter to respond.) By extending theexist at the instant A , which we will as- average holding time. switch count program, the magnitude ofsume for our present discussion to be the It should be noted that these two er- such error can be reduced, and by takingreference point: that is, only calls in ex- rors are over and above any others oc- sufficient number of observations we can

istence at A should be counted. curring in sampling or because of other arrive at the true average holding time as

We now see that calls originating be- peculiarities of switch count methods, closely as we wish to. On the other hand,

Xl bct-U+bY negative error b recorded. Since thesetwo cases are mutually exclusive the total

CHANNEL error will be u-2b.I 1 Step 3: t is greater than i-u+b but less

£i b than i-u+2b. This case is illustrated ineut u r Figure 5. If the point X falls beyond

TRAFFICORDER point A, a positive error will be obtainedlI l I . Ias long as the point Yof the same call is to

AMb WA'Lb B' the left side of point A'. The probabilityof such an event is (i-t)/i. A negative

Figure 4. Sequence diagram for Step 2 error will be recorded if the point X fallsbetween M-B and the point Y of thesame call falls between points A '-L',

work, by assuming infinitely short scan- scanning cycle i, and is completely deter- with a probability of (i+b-t-u+b)/i.ning interval; and applying independ- mined by the length of the scanning in- For any other point of origin there will beently for the other a correction according terval u and the counter response b. no error recorded, since the positive errorsto our presently discussed method. Furthermore, it shows that it will com- occurring will compensate for the nega-We now shall make the assumption that pletely disappear if the scanning interval tive one of the same call. Since the prob-

calls originate at random, which is cus- is made equal to twice the response time abilities of the positive and negative errorstomary in most telephone traffic prob- of the counter. are mutually exclusive, the net error willlems, and also, for the time being, that the We may now extend our problem to the be equal u -2b as before.holding times t are exponentially dis- constant holding time case, where all calls Step 4: t is greater than i-u+2b buttributed about their average. The latter have the same length. Here calls still less than i. This case is illustrated incondition is sufficient to insure that the originate at random as before; how- Figure 6. If the point X is to the right ofprobability for a call to terminate is as- ever, their termination is no longer in- A and the point Y of the same call is toymptotically proportional to the length of dependent of the time already elapsed. the left side of point A' there will be athe time interval under consideration and On the contrary, by knowing that the positive error recorded with a probabilityis independent of the time when that call call has originated at the point X we defi- (i -t)/i. If the point X falls to the left ofhas originated; that is, the probability of nitely also know that it must end at the point M and the point Y of the same calltermination of a call is independent of the point Y, which is t (the constant holding to the right of point L' a positive errorlength of time it has already been in time) apart from X by hypothesis. will likewise be recorded with a prob-progress. These assumptions are con- The problem of constant holding times ability of (t+u-b-i-b)/i. In allsidered generally true for telephone con- can be broken down into five separate other cases, that is, if Y falls betweenversations. It might be mentioned here steps. points A'-L', the positive and negativethat some advanced studies indicate that Step 1: t is less than b. In this case the errors compensate each other. The totalthese are not necessary conditions for a Trafficorder will not respond at all but it error will therefore be u - 2b as in all thePoisson process to take place, but, be- is of no interest' , ' 1S o no lntrsisnce, naturally, there ...............othercases.cause they are usually satisfied in tele-phone traic,they are very helpfin te would be no attempt made to measure Step 5: t is greater than i. The previouss implify demonstrations. holding times less than the minimum re- considerations can readily be extended to

Since calls originate at random, the sponse time inherent to the measuring the case if t is between i and i+b, that is, aprobability that it originates between the apparatus. positive error u -2b will be registered onlypoints A and M, if it does originate within Step 2: t is greater than b but less than if a call begins to the left of point M andthe scanning cycle i, is simply (u-b)/i. i-u+b. This case is illustrated in Figure terminates to the right of L, with a prob-If this has taken place we have added i to 4. If X falls between points A -M there ability of (u-2b)/i. If t covers more thanthe true holding time of this particular will be a positive error u-b recorded and one scanning period i it can be consideredcall, therefore the average positive error there will be no negative error. If point as made up by two parts: one whichon all calls will be i(u-b)/i=(u-b). Y falls between A'- L' there will be a covers exactly one or several scanningSince according to our second assumptionthis call may terminate at any moment x iVU.b<t. y

independently of the time it has already <t i-U+ 2bbeen in progress, the probability for a call CHANNELto terminate in the interval A'-L', if it Idoes terminate during a scanning cycle i, ywill be b/i and the resulting negative er-Xror will be ib/i= b.lIThe occurrences of positive and nega- CHNE I

tive errors are represented by two randomIvariables, therefore the expectation of anhierror will be the sum of their expectation, U -8 U-Iwith due regard to the sign. Combining TRAFFICORDER [L . LI. Ithe positive and negative errors will give . Iusa total error, u-2b. -. bkl -bAs we now see, this error is independent A M B AL' B'

of the holding time t or the length of the Figure 5. Sequence diagram for Step 3

1916 MolnarAccuracy of Ho-lding Time Mealsurements AIEE TRANSACTIONS

X , Y sidering the magnitude of u and b fori-U+2b < t < Trafficorders, the value for crb is about

CHANNEL 0.23. Since, because of the other sourcesor errors, holding time measurement al-

X Y ways includes at least several hundredt - u + 2 b < t < samples, the accuracy in the holding titrme

CHANNLIwill be within 10 milliseconds due to using

CHANNEL the expectation u-2b for the constanterror term and neglecting its own fluctua-tion in sampling.

tU u -j It now remains to verify our theoreticalTRAFFICORDER cconsiderations by factual evidence. The

type C Trafficorder, which was used in4bK -b these investigations, has a scanning in-

A M B A' L' B' terval u of 167 milliseconds and a responsetime b of 25 milliseconds. Therefore wemay expect that the average holding time,as measured by the Trafficorder, will be

~~~~~~~~~~~ u2b= 17 milliseconds too long. As itcycles for which, of course, the correct = -iu-(u - 2b)w u b 11cmllied cton setoo tong s itnubr o cont wil be oband an V.U will be recalled, actual observations on

thesecond consistingof afraction ofa The standard deviation for the other working equipment indicated a discrep-seanninge

ycle which will cause the same steps can be estimated similarly; for ancy of about 100 milliseconds.

amount of errors asth ose enumerated instance for step 5 the probability of an For further experimental verificationin the preceding three cases. error -i is 0, and for +i is (u-2b)/i: the trafficorder was subjected to a series ofThe foregoing analysis indicates that thus 0Tb = /7(u - 2b) (i-u +2b). However tests, in which it was connected to a source

the magnitude of the error will be u -2b, considering the magnitudes of the various of artificial traffic of constant holdingregardless whether the holding times are intervals of Trafficorders, step 2 or its time. In each test about 1000 holdingexponentially distributed or constant. "harmonics" will be most frequently en- times weremeasuredand varied betweenThese two represent extreme cases of countered. tests from approximately 0.5 second uptoholding time distributions, therefore it For the exponential case the probabili- t15 seconds. Simultaneously, the holdingmay perhaps be justified to conclude ties for -i and +i are about the same as times ofthe same alls were measured towithout further analysis that we are fac- in step 2, except that allowance would the require ddegree of accuracy by a de-ing here an error term which is independ- have to be made for those calls where the lay register. Th eaverage differened be-ent of the scanning cycle of the traffic positive and negative errors cancel out tweenthe holding timesobtained by therecording process, of the magnitude of the each other. Such calls will start at a eTraffiorderand the delay register, re-holding time to be measured, and of their point x between 0 and u - b, and termi- speetively,118 milliseeonds. The milli-distribution about the average, since an natebetweeni-xandi-x+b,orbetween mumobserved difference was 101 milli-arbitrary distribution could be considered 2i-x and 2i-x+b, etc. later. The seconds and the maximum 140 milli-as a step function subdivided into ex- probability for such call lengths is: seconds. The standard deviation of thetremely narrow widths, and for each such Pr = (e (i-x)/t_e-(i(.-X+b)/t)+ average differences of each test was 12.8subdivision the holding time t is con- (e -(2i -x)/t-e- (2i -X+b)/t)+ milliseconds while the expected standardstant, with an error term independent of

ee' tl(1-e - N/)-)/(1e(i/t)) deviation of the same averages from the

t. - foregoing formula alone is 7.3 milliseconds

It is however, dependent on the length The probability for the errors to cancel and 9.0 milliseconds if the errors in meas-

of the scanning interval u and the re- out each other will thus be: uring each call as well as end effect errors

sponse time of the counting device b, and fU Prdxli = (t/i)(e(u -1b)I are also included. The average standardcan be minimized by choosing the scan- deviation derived from subsets of 100ning interval to be approximuately twice (1-e-

b )/(ef/- 1) calls each was 39 milliseconds, while thethe counter response time. The error For the computation of the standard de- expectation forsuch subsets, includingu -2b should be deducted from the aver- viation this probability should be sub- the various sources of errors, is 29 milli-age holding time as obtained from the tracted from b/i, as well as from (u -b)/i. seconds.switch count process, and the net value However, b/t is usually a small fraction, To check further the validity of theshould be considered as the best estimate on the order of at least 1/20, and (u - b)/t theory, the scanning interval it was re-

for the average holding time. is likewise small: thus the above correc- duced to 100 milliseconds; thus a dif-This error also appears as a variable in tion can be disregarded for most applica- ference of 50 milliseconds was expected.

the sampling process, and the true amount tions, the positive and negative errors con- .From a similar series of tests the averageof error can only be obtained after a suf- sidered as essentially uncorrelated, and .difference was 51 mrilliseconds.ficiently large number of samples have the standard deviation will be about the .A series of tests were made, with 1,800been obser-ved. Actually, it can be rep- same as for step 2 in the constant holding .artificial random calls of exponential dis-resented by a three valued variable, timeecase. .tribution about an average holding timewhich is observed either as -i or 0 or +i. The correct procedure would be now to .of 3.76 seconds, using again 167 mii-Their respective probabilities for instance enter the above expressions for the stand- .seconds scanning interval. The averagein the constant holding time case, step 2, ard deviation into Mr. Wiilkinson's equa- .difference was 124 milliseconds comparedare: b/i, 1- (u/i), and (u -b)/i. The tions 40 or 42, and use the new values to .with 117 milliseconds expectation; andstandard deviation of each variable is: recalculate Table II. However, con- .the s;tandard deviation of the average

1951, VOLUME 70 Molnar-Accuratcy of Holding Time Measurements 1917

differences in each test was 10 milli- expectation. This average difference is Referencesseconds compared with 12 milliseconds not as close as the previous one, but con- 1. THE AUTOMATIC ELECTRIC TRAFFICORDER,expectation. sidering the magnitude of the standard J. E. Ostline. Automatic Electric Technical Journal

Finally, the holding times of 340 tele- deviation may still be considered as (Chicago, Ill.), April 1948.2. A STUDY OF TRAFFIC IN AUTOMATIC TOLL

phone calls, subdivided into five lots, satisfactory. TELEPHONE SYSTEMS, Imre Molnar. Northwest-were found to have a measured average Furthermore, the average of each lot ern University (Evanston, Ill.), part 2, 1950, pagesholding time of 60.5 seconds. The aver- was consistently higher than the true 2-27s

3. TsHE RIELIABILITY oF HOLDING TIME MEASURE-age difference was 96 milliseconds com- measured average holding time of the MENTS, Roger I. Wilkinson. Bell System Technicalpared with 117 milliseconds expectation; same lot, suggesting that there is a Journal (New York, N. Y.), October 1941.and the standard deviation 26 milli- systematic positive error present in con- 4. TOLL ANSWERING DELAYS AND THEIR

MEASURIEMERNTS, Imre Molnar. Automatic Electricseconds compared with 28 milliseconds formancewith the prediction of thetheory. Technical Journal (Chicago, III.), July 1951.

No Discussion

1918 M'olnar- Accuracy ofHolding Time Mealsurements AIBE TRANSACTIONS