STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 



955 



E{dc} = a, 



and 



var lad = 



a 



T' 



so that dc is an unbiased and consistent estimator of a. We now compare 

 the variances of estimators dc and di. From Table IV we have 



var {di} = |f 1 



2" ) < jT = var {ttcl, 



so that di is a better estimator of a for any T > 0, in the sense that its 

 variance is less. 



X THE DISTRIBUTIONS of Z AND M 



Since we have defined 



= [ N{t) dt, 



we can regard Z as the result of growth whose rate is given by the ran- 

 dom step-function Nit) ; when N{t) = n, Z is growing at rate n. An idea 

 similar to this is used by Kosten, Manning, and Garwood , and by Kos- 

 ten alone. Now the Z{T) process by itself is not Markovian, but it can 

 be seen that the two-dimensional variable \N{t), Z{t)\ itself is Marko- 

 vian. Let Fn{z, t) be the probability that N{t) = n and Z{t) ^ z. Since 

 the two-dimensional process is Markovian, we can derive infinitesimal 

 relations for Fn{z, t) by considering the possible changes in the system 

 during a small interval of time A^ 



Table V — p\X ,Y) 



