96-1 THE BELL SYSTEM TECHXICAL JOURNAL, JULY 1957 



If, ill this last expression, we let x approach 1, z approach 1 , and u ap- 

 proach 1, we obtain 



exp 



^ _ ^i^Y ^rie---- - 1) _ ,,; 



(C) 



r + 7/\ r + 7 



as the generating function E\w^e^^^] for an interval of equiUbrium. 

 Alternately, if instead we let x approach 1, z approach 1, and w ap- 

 proach 1, we obtain (C) with u substituted for w; this implies the not- 

 surprising result that for an interval of equiUbrium, the two-dimen- 

 sional variables {A, Z\ and [U, Z\ have the same distribution. From 

 this and (C) it follows that for eciuilibrium (0, T), (i) .4 and E both 

 have a Poisson distribution with mean oT, and (ii) the estimators Ac and 

 Ao have the same distribution. 



ACKNOWLEDGEMENT 



The author would Uke to express his gratitude for the helpful com- 

 ments and suggestions of E. N. Gilbert, S. P. Lloyd, J. Riordan, J. 

 Tukey, and P. J. Burke. 



REFERENCES 



1. H. Cramer, ■Mathematical Methods of Statistics, Princeton, 1946. 



2. W. Feller, An Introduction to Probability Theory and its Applications, John 



Wilev and Sons, New York, 1950. 



3. W. Felier, On the Theory of Stochastic Processes with Particular Reference to 



Applications, Proceedings of the Berkeley Symposium on Math. Statistics 

 and Probability, Univ. of California Press, 1949. 



4. A. Jensen, An Elucidation of Erlang's Statistical Works Through the Theory 



of Stochastic Processes, in The Life and Works of A. K. Erlang, Copenhagen, 

 1948. 



5. L. Kosten, On the Accuracy of Measurements of Probabilities of Delay and of 



Expected Times of Delay in Telecommunication Systems, App. Sci. Res., 

 B2, pp. 108-130 and pp. 401-415, 1952. 



6. L. Kosten, J. R. Manning, F. Garwood, On the Accuracy of Measurements of 



Probabilities of Loss in Telephone Systems, Journal of the Royal Statistical 

 Society (B), 11, pp. 54-67, 1949. 



7 P A. P. Moran, Estimation Methods for Evolutive Processes, Journal of the 



Royal Statistical Society (B), 13, pp. 141-146, 1951. 



8 W. F. Newland, A Method of Approach and Solution to Some Fundamental 



Traffic Problems, P.O.E.E. Journal, 25, pp. 119-131, 1932-1933. 

 9. C. Palm, Intensitatsschwankungen im Fernsprechverkehr, Ericsson Tech- 

 nics, 44, 1943. 



10. F. W. Rabe, Variations of Telephone Traffic, Elec. Comm., 26, 243-248, 1949. 



11. J. Riordan, Telephone Traffic Time Averages, B.S.T.J., 30, 1129-1144, 1951. 



12. H. Stormer, Anwendung des Stichprobenverfahrens beim Beurteilen von 



Fernsprechverkehrsmessuugen, Archiv der Elektrischen Ubertragung, 8, 

 pp. 439-436, 1954. 



