THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 507 



ABRIDGED BIBLIOGRAPHY OF ARTICLES ON TOLL ALTERNATE ROUTING 



Clark, A. B., and Osborne, H. S., Automatic Switching for Nationwide Telephone 

 Service, A.I.E.E., Trans., 71, Part I, p. 245, 1952. (Also B.S.T.J., 31, p. 823, 

 Sept., 1952.) 



Pilliod, J. J., Fundamental Plans for Toll Telephone Plant, A.I.E.E. Trans., 71, 

 Part I, p. 248, 1952. (Also B.S.T.J., 31, p. 832, Sept., 1952.) 



Nunn, W. H., Nationwide Numbering Plan, A.I.E.E. Trans., 71, Part I, p. 257, 

 1952. (Also B.S.T.J., 31, p. 851, Sept., 1952.) 



Clark, A. B., The Development of Telephony in the United States, A.I.E.E. 

 Trans., 71, Part I, p. 348, 1952. 



Shiplev, F. F., Automatic Toll Switching Systems, A.I.E.E. Trans., 71, Part I, 

 p. '261, 1952. (Also B.S.T.J., 31, p. 860, Sept., 1952.) 



Myers, O., The 4A Crossbar Toll System for Nationwide Dialing, Bell Lab. 

 Record, 31, p. 369, Oct., 1953. 



Clos, C, Automatic Alternate Routing of Telephone Traffic, Bell Lab. Record, 

 32, p. 51, Feb., 1954. 



Truitt, C. J., Traffic Engineering Techniques for Determining Trunk Require- 

 ments in Alternate Routing Trunk Networks, B.S.T.J., 33, p. 277, March, 

 1954. 



Molnar, I., Some Recent Advances in the Economy of Routing Calls in Nation- 

 wide Dialing, A.E. Tech. Jl., 4, p. 1, Dec, 1954. 



Jacobitti, E., Automatic Alternate Routing in the 4A Crossbar System, Bell Lab. 

 Record, 33, p. 141, April, 1955. 



Appendix I* 



DERIVATION OF MOMENTS OF OVERFLOW TRAFFIC 



This appendix gives a derivation of certain factorial moments of the 

 c(iuilibrium probabilities of congestion in a di^dded full-access multiple 

 used as a basis for the calculations in the text. These moments were de- 

 rived independently in unpublished memoranda (1941) by E. C. Molina 

 (the first four) and by H. Nyquist; curiously, the method of derivation 

 here, which uses factorial moment generating functions, employs auxili- 

 ary relations from both Molina and Nyquist. Although these factorial 

 moments may be obtained at a glance from the probability expressions 

 given by Kosten in 1937, if it is remembered that 



pw = |:(-i)'-'(';)^, (1.1) 



where p{x) is a discrete probability and M (k) is the A;th factorial moment 

 of its distribution, Kosten does not so identify the moments and it may 

 1)0 interesting to have a direct derivation. 



Starting from the equilibrium formulas of the text for f(;ni, n), the 

 l)robability of m trunks busy in the specific group of x trunks, and n in 



Prepared by J. Riordan. 



