There are some problems in your description:

Look: P[0](s) calls P[1](s)

P[1](s) again calls P[0](s) and so on. =>recursive

Therefore, assigning Ps with ":=" can not be used, you can assign each equation to a var named f[i]. like this: 

f1:= P[0](s) - (P[1](s)*mu[1]+P[2](s)*mu[2]+1)/(s+3*lambda+3*sigma);

f2:= P[1](s) - (3*P[0](s)*lambda+3*P[3](s)*mu[1]+P[4](s)*mu[2])/(s+mu[1]+2*lambda+2*sigma);

f3:= P[2](s) - (3*P[0](s)*sigma+P[4](s)*mu[2]+P[3](s)*mu[1])/(s+mu[2]+2*lambda+2*sigma);

f4:= P[3](s) - 2*lambda(P[1](s)+P[2](s))/(s+2*mu[1]+lambda);

f5:= P[4](s) - 2*sigma(P[1](s)+P[2](s))/(s+2*mu[2]+sigma);

f6:= P[5](s) - lambda(P[3](s)+P[4](s))/s;

f7:= P[6](s) - sigma(P[3](s)+P[4](s))/s;

 Then solve the set of 7 eqs, you will get P[i](s)