STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 951 



It seems reasonable, and can be shoAni rigorously (Appendix C), that 

 for an interval (0, T) of statistical equilibrium, the distribution of .4 

 and that of H are the same. Thus we can argue that, for long time inter- 

 vals, A and H will not be ^-ery different. This suggests using 



/ =±±£ = ^ 

 ' 2T 2T 



as an estimator of a. This estimator does not involve y, and it uses not 

 only information given by A, but also information supplied by arrivals 

 occurring possibly before the start of observation. 



Similarly, since b = a y, and .1/ is a consistent and unbiased estimator 

 of 6, we may use 



-^ = ^ = L 



to estimate 7, and its reciprocal to estimate h. Finally, since for long {0,T) 

 we have .4 ^ H, we may try 



A 1 

 as an estimator of 7, and its reciprocal as an estimator of h. 



IX THE JOINT DISTEIBUTION OF THE SUFFICIENT STATISTICS 



The basic result of this paper is a formula for the generating function 



E{z''x'''''w\"e:'''\ (9.1) 



for the joint distribution of the random variables n, N(T), A, H, and Z. 

 This formula is derived in Appendix C, by methods illustrated in Section 

 X. For an initial n distribution {p„ j , the generating function is 



V- n [ U-r + y-r - yu)e~^^^^^^ + 7^* 



""^ 1 (r + 7)' f + 7 /■ 



It is proved in Appendix A that the set of statistics [n, A, H, Z} is 

 sufficient for estimation on the basis of the information assumed, which 

 was described in Section I. Thus the generating function (9.2) .specifies, 

 at least in principle, what can be di.sco^•ered about the process from an 

 observation interval (0, 7"), for which N{0) has the distribution |p„}. 

 All the results summarized in this section are consequences of (9.2). 



T 



