Calculation of the limit in Exercise 4 of Section 6.11 in G&S

Clear[G,mu,lambda,s,t];G[mu_,lambda_,s_,t_]:=(mu*(1-s)-(mu-lambda*s)*Exp[-t*(lambda-mu)])/(lambda*(1-s)-(mu-lambda*s)*Exp[-t*(lambda-mu)]);

Limit[(G[mu,lambda,s,t]-G[mu,lambda,0,t])/(1-G[mu,lambda,0,t]),t->Infinity,Assumptions->mu>lambda]

Note that this is equal to

(1-rho)*s/(1-rho*s)

Simplify[Sum[s^n*(1-p)^(n-1)*p,{n,1,Infinity}]]

Hence we have a geometric distribution with parameter 1-rho