Consider the steady state $M / D / 1$ queueing system. Prove the following:
(a)
$$
W_q[t]=\sum_{n=0}^{k-1} p_n+p_k\left(\frac{t-(k-1) W_s}{W_s}\right)
$$
where
$$
(k-1) W_s \leq t<k W_s, \quad k=1,2, \ldots
$$
(b)
$$
W[t]= \begin{cases}0 & \text { for } t<W_s \\ \sum_{n=0}^{k-1} p_n+p_k\left(\frac{t-k W_s}{W_s}\right) & \text { for } t \geq W_s,\end{cases}
$$
where
$$
k W_s \leq t<(k+1) W_s, k=1,2, \ldots .
$$
(In Example 3.4.6 we show how to compute the values of $p_n$.)
(c) Consider the $\mathrm{M} / \mathrm{D} / 1$ queueing system of Example 5.3.1. Calculate $W_q[2], W_q[5], W[8], W[14.4]$, and $W[14.43]$.