DIALING HABITS OF TELEPHONE CUSTOMERS 43 



The probability of a call encountering congestion, which is equivalent 

 to the probability of a call having a delay greater than zero units of 

 time is: 



POO) = E/CaO (7) 



DELAY FORMULA FOR THE DIAL TONE TESTER 



The probability that a dial tone test call encounters congestion is 

 gi\-en by expression (7). Once a test call has encountered congestion it 

 will experience a delay depending upon a number of variables. The 

 assumptions underljdng the dial tone tester formula are: 



1. A dial tone test call when encountering a delay waits until served. 



2. A dial tone test call does not add to the load offered and carried 

 by the line finders. 



3. Upon encountering a delay, a dial tone test call is served in the 

 order of its arrival with respect to all other waiting calls. For example, 

 if the test call finds three other calls waiting, it waits fourth in line. 



Under the third assumption as calls drop out, due to conversations 

 terminating on the occupied line finders or due to waiting calls aban- 

 doning their requests for service, the test call advances from an initial 

 position of say fourth in line to third in line, then to second, then to 

 first in line, and finally is served. The overall delay distribution of the 

 test calls depends therefore upon the number of calls they find waiting 

 ahead of them. The delay distribution for each such number must be 

 weighted by the probability of its occurrence in order to obtain the 

 overall distribution. The delay distribution for a test call which finds 

 zero calls waiting is: 



PoOO = exp {-d/jH) (8) 



The probability is /(c) that a call made at random will find all line 

 finders busy with no calls waiting. Hence the weighted delay distribution 

 Po(>0,is: 



PoOO = Kc)po(>t) = m exp (-d/jH) 



(9) 



The delay distribution for a test call which finds one call waiting 

 ahead of it is : 



PiOO = [1 + c/j - (c/j) exp (-t/H)] exp (-d/jH) (10) 



The probability is /(c + 1) that a call made at random will find all 



