a. The general continuous-time random walk is defined by g_(ij)={[mu_(i)",",j=i-1","],[-(lambda_(i)+mu_(i))",",j=i","],[lambda_(i)",",j=i+1","],[0","," otherwise "]:} Write out the forward and backward equations.

b. The continuous-time queue with infinite buffer can be obtained by modifying the general random walk in the preceding problem to include a barrier at the origin. Put g_(0j)={[-lambda_(0)",",j=0","],[lambda_(0)",",j=1","],[0","," otherwise ".]:} Find the stationary distribution assuming sum_(j=1)^(oo)((lambda_(0)cdotslambda_(j-1))/(mu_(1)cdotsmu_(j))) < oo. If lambda_(i)=lambda and mu_(i)=mu for all i, simplify the above condition to one involving only the relative values of lambda and mu.

c. Repeat a and b assuming there is a boundary at j=N. Comments. need explicit calculation process, step by step