RELIABIHTY OF HOLDING TIME MEASUREMENTS 383 



' crossing will be simply .. That is, if the origination point falls within / 



seconds to the left of a switch count, the call will be marked up; if not it 

 will not be counted. If it is marked up there will occur for that call a plus 

 estimation error of x — i — t seconds since we shall eventually assume that 

 every switch count infers / call seconds of use on the trunk group. Likewise 



there is a probability of — : — that the call will not be counted with the 



resultant negative error of x = —t seconds. In summary the total proba- 

 bility of a positive error of .v = i — / is 



t ./•(/) dt -. ; 



and for a negative error of x = —I, 



At)dt '^. 

 I 



Case 2. I lies between i and 2i 



vSuch a call may be included either once or twice in the switch counts. 



The probability that it is counted twice is — ; — with a resultant plus error 



/ — i 

 of .V — 2i — /. The probability that it is counted but once is 1 — -^- = 



2i — t 



I . , with the corresponding negative error of .v = / — /. The overall 



[■ i 



probabilities of course will be formed by weighting these as in the first 



case with the probability, fit)dt, that the holding time of length t to t -\- dt 



actually occurs. 



General Case, t lies between qi and (q + l)i 



It will readily be seen that by extending the reasoning of the two cases 

 above to the case of / lying between qi and (q -\- \)i we shall have a plus 

 error (due to the call being marked up q -\- 1 times) of .v = (^ + 1)/ — /, 



with a probability of occurrence of ^ , and a negative error (the call 



marked u]) q times) of .v = qi — I with a probabilit}' of : . 



Summarizing the above cases, a negative error of size x can occur in a 

 great number of ways, due to / taking the values — .v, i — .v, 2i — .v, • • • 

 qi — .V, • • • with the corresponding probabilities of occurrence of the call 

 lengths, /( — .v),/(i — x),f{2i — x), •■■ f{qi — x) ••• , respectively. In 



