STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 949 



assumed frequency distribution of holding-time. It follows from Rior- 

 dan's results that M converges to 6 in the mean, which is to say that 



lim^ {[3/ - 6 1'} = 0. 



r->oo 



It also follows that M is an unbiased estimator of b; i.e., that E{M\ = h, 

 and that M is a consistent estimator of b, which means that 



lim pr{\ M - b \ > £} = 



T-»QO 



for each f > 0. 



VI MAXIMUM CONDITIONAL LIKELIHOOD ESTIMATORS 



As shown in Appendix A, the likelihood Lc of an observed sequence, 

 conditional on N{0), is defined by 



In L, = A In a -{- Hhiy - yZ - aT. 



According to the method of maximum likelihood, we should select, as 

 estimators of a and y respectively, cjuantities dc and -y^. which maximize 

 the likelihood Lc . Now a maximum of L,. is also one of In Lc , and vice 

 versa. Therefore Oc and 7^ are determined as roots of the following two 

 ecjuations, called the likelihood e(iuations: 



I- In L. = 0; ^ In L, = 0. 

 da oy 



The solutions to the likelihood equations are 



. _ A '^ ^H 



These are the maximum conditional likelihood estimators of a and 7. 

 The estimator dc is the number of requests for service in T diA'ided by T; 

 this is intuitively satisfactory, since d,. estimates a calling rate. 



Since maximum likelihood estimators of functions of parameters are 

 generally the same functions of maximum likelihood estimators of the 

 parameters, we see that AZ HT is a maximmn likelihood estimator of b. 



VII PRACTICAL ESTIMATORS SUCJtiESTED BY MAXIMIZING THE LIKELIHOOD 



L, DEFINED IX APPENDIX A 



We obtain as likelihood ecjuations 



— In L = 0, — In L = 0. 

 da oy 



