Show that the density function for queueing time in the $\mathrm{M} / \mathrm{M} / \mathrm{c} / \mathrm{K} / \mathrm{K}$ queueing system is given by
$$
f_q(t)=W_q[0] \delta(t)+\frac{c^c \mu p[K-c-1 ; c(t+z)]}{(c-1) ! \times p[K-1 ; c z]} p_0[K-1], \quad t \geq 0,
$$
where
$$
z=\frac{E[O]}{W_s} .
$$
Hint: Differentiate the formula (5.309) below for $W_q[t]$. Note that $q$ has a probability mass at the origin equal to $W_q[0]=q_0$. In differentiating (5.309), use the fact that
$$
\frac{\partial}{\partial y} Q[k ; y]=-p[k ; y],
$$
which follows from the formula
$$
Q[k ; y]=\int_y^{\infty} p[k ; x] d x
$$
of Exercise 33(b).
$$
W_q[t]=P[q \leq t]=1-\frac{c^c Q[K-c-1 ; c z] p_0[K-1]}{c ! \times p[K-1 ; c z]}, t \geq 0,
$$
where
$$
Q[k ; \alpha]=e^{-\alpha} \sum_{n=0}^k \frac{\alpha^n}{n !} .
$$