944 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



functions, that of the intervals between demands for service, and that 

 of the lengths of conversations. We have supposed that these distribu- 

 tions are both of negative exponential type, each depending on a single 

 parameter. Thus we know the functional form of each distribution, and 

 each such form has one unknown constant in it. Since the mathematical 

 structure of the model is fully specified except for the values of the two 

 unknown constants, we can assign a likelihood or a probability density 

 to any sequence S of events in the model during the interval (0, T). 

 This likelihood will depend on the parameters, on 2, and on the number 

 of calls in existence at the start of the interval. If the likelihood L(2) 

 can be factored into the form L = FH, where F depends on the param- 

 eters and on statistics from the set S only, and H is independent of the 

 parameters, then the set S of statistics may be said to summarize all the 

 information (in a secjuence 2) relevant to the parameters. If L can be 

 so factored, then S is sufficient for the estimation of the parameters. 



The mathematical model to be used in this paper is described and 

 discussed in Sections II and III, respectively. Section IV contains a 

 summary of notations and abbreviations which have been used to sim- 

 plify formulas. 



In Appendix A we show that the original data we have allowed our- 

 selves can be replaced by four statistics, which are sufficient for estima- 

 tion. In Appendix B and Sections Y-VIII we discuss various estimators 

 (for parameters of the model) based on these four statistics. To determine 

 the anticipated accuracy of these methods of measurement, we consider 

 the statistics themselves as random variables whose distributions are 

 to be deduced from the structure of the model. 



A primary task is the determination of the joint distribution of the 

 sufficient statistics. In view of the sufficiency, this joint distribution tells 

 us, in principle, just what it is possible to learn from a sample of length 

 T in this simple model. By analyzing this distribution we can derive 

 results about the anticipated accuracy of measurements in the model. 



The joint distribution of the sufficient statistics is obtainable in prin- 

 ciple from a generating function computed in Appendix C, using methods 

 exemplified in Section X. This generating function is the basic result of 

 this paper. The implications of this result are summarized in Section 

 IX, which quotes the generating function itself, and presents some 

 statistical properties of the sufficient statistics in the form of four tables: 

 (i) a table of generating functions obtainable from the basic one; (ii) a 

 table of mean values; (iii) a table of variances and covariances; and 

 (iv), a table of squared correlation coefficients. (The coefficients are 

 all non-negative.) 



