STATISTICS FOR A SIMPLE TELEPHONE EXCHANGE MODEL 959 



of statistical equilibrium; however, if the sequence {pn} does form the 

 stationary distribution discussed in Section II, then 



Y^x'pn = exp {b{x - 1)}, (10.6) 



and 



* = ... r^^cTTf^ - ^} ,« 



is the Laplace transform of the distribution of Z for an interval (0, T) 

 of statistical equilibrium. 

 The Laplace transform is a moment generating function expressible as 



t^o n! 



where 7n„ is the 71'' ordinary moment of Z. Differentiation of 10.7 then 

 gives a recurrence relation for the moments upon equating powers of 



(-r). Thus, 



= V'-{2ar(l - e-''^'^') + (f + y)[yaT + 6nV^^+^^1}, 

 and 

 ynin+i — 87 mn„ + 37w(n — l)7n„-i — n{n — 1)(« — 2)m„_2 



= ay'^Tyrin - (2a + ayT)?i7nn-i + 2ane~''''{m + T)"~' + n (10.8) 

 '{n-l)aTe-^''{m + T)'"' - n{n - l){n - 2)bTc-'\7n + T)""', 

 where (w + T) " is the usual symbolic abbreviation of 



From the recurrence (10.8) it is easily verified that 



im = bT, 



m, = (bTY + ^ [1 - C], 

 7 



from which it follows that the variance of Z is 



var {Z} = ^^ [1 - CI 



7 



