STATISTICS FOE A SIMPLE TELEPHONE EXCHANGE MODEL 963 



fact gives them a practical ascendancy over the unconditional estimators, 

 even though the latter may be more efficient statistically by dint of 

 using all the information available in an observation. 



Appendix C 



the joint distribution of n{t), tl, a, h, and z 



By methods already used in Section X one can obtain a gen- 

 erating function for the joint distribution of all the random variables n, 

 N(t), A, H, and Z. Let 



^ = L[x w u e \. 



Then $ satisfies the differential equation 



^ + [f.i- + 7.i- - yu]— = aiwx - 1)^, 

 dt dx 



whose solution has the form 



/aw[^x -i- yx — yu][l — e"^^"^^^'] , aywut \ 



■^H o^^r + r+^ - "V- 



where the function R is determined by the initial distribution {p„} 

 through the relation 



^ + yu 



n^O L r + 7 J 



From these results it follows that the generating function 



E\z X w u e ^ \ 

 is given by 



2^ PnZ ( — r ) 



"go \ f + 7 / 



fawi^x + 7.r - yu)[l - e'^^^'^^] . aywuT „\ 



If {p„\ forms the stationary distribution, this reduces to 



2(fx + 7.T — 7tOe~^^^^'^ + yiiz \ 



R{^\ = E Pn 



exp 



r + 7 / 



awj^x + yx - yu)[l - e~^^+"^^] aywuT _ 

 ■^ (f + 7)^ "^ f + 7 "" 



