Question $4(25$ marks$)$

Consider an $\frac{M}{M}\frac{?}{1}$ queue where the number of packets stored in the queue is the state of the system. The queue is in state $S_1$ when there is $1$ packet in the queue. Assume that the probability of packet arrival $"$a$"$ is $0\mathrm{.}1$ and probability of packets leaving the queue $c$ is $0\mathrm{.}5.$ The state of occupancy of the queue depends on the statistics of the packet arrivals as follows:

State of the queue will be empty with probability of $1-$ad

State of the queue will stay at the same state with probability ac $+$bd

State of queue will increase in size by one with probability ad

State of the queue will decrease in size by one with probability bc

a$.$ Draw the state transition diagram for the discrete time queue, include all the probability of in and out of the state.

b$.$ Construct the first four rows and columns of the transition matrix, P$.$

c$.$ Find the values of the first four components of the equilibrium distribution vector.

d$.$ Determine the wait time in time steps.