490 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



tables). The respective sums of the overflow a's and v^s, give A' and V 

 by (28) and (29); they provide the necessary statistical description of 

 traffic offered to the alternate route. 



According to the Equivalent Random method for estimating the alter- 

 nate route trunks required to provide a specified grade of service to the 

 overflow traffic A', one next determines a random load A which when 

 submitted to S trunks will yield an overflow with the same character 

 {A', V) as that derived from the complex system's high usage groups. 

 An alternate route of C trunks beyond these S trunks is then imagined. 

 The erlang overflow a', with random offer A, to S + C trunks is found 

 from standard i^i-formula tables or curves (such as Fig. 12). 



The ratio R2 = a! I A' is a first estimate of the grade of service given to 

 each parcel of traffic offered to the alternate route. As discussed in Sec- 

 tion 7.5, this service estimate, under certain conditions of load and 

 trunk arrangement, may be significantly pessimistic when applied to a 

 first routed parcel of traffic offered directly to the alternate route. An 

 improved estimate of the overflow probability for such first routed 

 traffic was found to be R\ as given by (30). 



8,1 Determination of Final Group Size with First Routed Traffic Offered 

 Directly to the Final Group 



When first routed traffic is offered directly to the final group, its 

 service Ri will nearly always be poorer than the overall service given to 

 those other traffic parcels enjoying high usage groups. The first routed 

 traffic's service will then be controlling in determining the final group 

 size. Since Ri is a function of *S, C and A in the Equivalent Random 

 solution (30), and there is a one-to-one correspondence of pairs of A and 

 S values with A' and V values, engineering charts can be constructed at 

 selected service levels Ri which shoAv the final route trunks C required, 

 for any given values of A' and V. Figs. 42 to 45 show this relation at 

 service levels of Ri = 0.01, 0.03, 0.05 and 0.10, respectively.* 



* On Fig. 42 (and also Figs. 46-49) the low numbered curves assume, atjfirst 

 sight, surprising shapes, indicating that a load with given average and variance 

 would require fewer trunks if the average were increased. This arises from the 

 sensitivity of the tails of the distribution of offered calls, to the V'/A' peaked- 

 ness ratio which, of course, decreases with increases in A'. For example, with C 

 = 4 trunks and fixed V = 0.52, the loss rapidly decreases with increasing A': 



