STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 961 



Appendix A 



PROOF THAT {u, A, H, Z} IS SUFFICIENT. 



We observe the system during the interval (0, T), and gather the in- 

 formation specified in Section I, and summarized in Table I. From this 

 information we can extract four sets of numbers, described as follows: 

 Sa the set of complete observed inter-arrival times, not counting 



the interval from the last arrival until T 

 Sh the set of complete observed holding times 

 Si the set of hang-up times for calls of category (i) 

 Si the set of calling-times for calls of category (iv) 

 In addition, our data enable us to determine the following numbers: 

 n the number N{0) of calls found at the start of observation 

 k the number of calls of category (iii); i.e., of calls which last through- 

 out the interval (0, T) 

 X the length of the time-interval between the last observed arrival 



and T 

 In view of the negative exponential distributions which have been 

 assumed for the inter-arrival times and the holding-times, and in view 

 of the assumptions of independence, we can write the likelihood of an 

 observed sequence of events as 



utSa ~tSh wlSi V^Si 



SO that 



hi L = —ykT — ax + In p„ + A In o — ^ au 



u(Sa 



+ H \n y - Y, yz - Y, yw - ^ y{T - ij) 

 ziSh wtSi ytSi 



It is easily seen that the summations and the two initial terms can be 

 combined into a single term, so that we obtain 



In L = In pn + .1 In a -{- H hi y - yZ - aT. 



This shows that L depends only on the statistics n, .4, H, and Z; it 

 follows that the information we have assumed can be replaced by the 

 set of statistics {n, .4, H, Z\, and that these are sufficient for estimation 

 based on that information. 



The likelihood is sometimes defined without reference to the initial 

 state, by leaving the factor p„ out of the expression for L. Strictly speak- 

 ing, this omission defines the conditional likelihood for the observed 



