948 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



Z = f N(t) dt 

 Jo 



M = Z/T = average of A^(^) over (0, T) 



\p„\ = the (discrete) probability distribution of n, the number ot 

 trunks found busy at the start of observation 



An estimator for a parameter is denoted t)y adding a cap (*) and a 

 subscript. The subscripts differentiate among various estimators for the 

 same parameter. We use d,. = ^■^/T, jc = H/Z, di = K/2T, 

 yi = K/2Z, and 72 = A/Z. 



Also, it is convenient to use the following abbreviations: r for yT, and 

 C for (1 — e~'')/r, where r is the dimensionless ratio of observation-time 

 to mean holding-time. The symbol E is used throughout to mean mathe- 

 matical expectation. 



V THE AVERAGE TRAFFIC 



We have adopted a model which depends on two parameters, the 

 calling rate a, and the mean holding-time h, or its reciprocal 7. Before 

 searching for a set of statistics that is sufficient for the estimation of 

 these parameters, let us consider the product ah = b. This product is 

 important because, as we saw in Section II, the equilibrium distribution 

 of the number of trunks in use depends only on 6, and not on a and h 

 individually. Indeed, the equilibrium probability that n trunks are busy 

 is 



— bin 



e 



n! ' 



and the average number of busy trunks in equilibrium is just b. 

 The average number of trunks busy during a time interval T is 



M 



- f r -^'<" "'' 



i.e., the integral of the random function Nit) over the interval T, divided 

 by T. This suggests that for large time intervals T, M will come close 

 to the value of b, and can be used as an estimator of b. Since M is a ran- 

 dom variable, the question arises, what are the statistical properties of 

 M? This question has been considered in the literature, the principal 

 references being to F. W. Rabe^° and to J. Riordan." Riordan's paper is 

 a determination of the first four semi-invariants of the distribution of M 

 during a period of statistical equilibrium, but without restriction on the 



