The M/M/c/K queueing system can be modeled as a birthand-death process (see Sections 4.3 and 5.2).
(a) Draw the state-transition rate diagram and from it deduce that
$$
\lambda_n= \begin{cases}\lambda & \text { for } n=0,1, \ldots, K-1, \\ 0 & \text { for all other } n,\end{cases}
$$
while
$$
\mu_n= \begin{cases}n \mu & \text { for } n=1,2, \ldots, c \\ c \mu & \text { for } n=c+1, \ldots, K \\ 0 & \text { otherwise. }\end{cases}
$$
(b) Show that part (a) yields
$$
p_n= \begin{cases}\frac{a^n}{n !} p_0 & \text { for } n=1,2, \ldots, c \\ \frac{a^c}{c !}\left(\frac{a}{c}\right)^{n-c} p_0 & \text { for } n=c+1, \ldots, K,\end{cases}
$$
where
$$
p_0=\left[\sum_{n=0}^c \frac{a^n}{n !}+\frac{a^c}{c !} \sum_{n=1}^{K-c}\left(\frac{a}{c}\right)^n\right]^{-1},
$$
and $a=\lambda W_s=\lambda / \mu$.
(c) From the formula
$$
L_q=\sum_{n=c+1}^K(n-c) p_n
$$
show that
$$
L_q=\frac{a^c p_0 r\left[1-(K-c+1) r^{K-c}+(K-c) r^{K-c+1}\right]}{c !(1-r)^2},
$$
where $r=a / c$.
(d) Show that
$$
\begin{aligned}
L & =L_q+E\left[N_s\right] \\
& =L_q+\sum_{n=0}^{c-1} n p_n+c\left(1-\sum_{n=0}^{c-1} p_n\right) .
\end{aligned}
$$
(e) Let $q_n$ be the probability that an arriving customer finds $n$ customers in the service facility. Use the same argument as that in Exercise 12 to show that
$$
q_n=\frac{p_n}{1-p_K}, n=0,1,2, \ldots, K-1 .
$$
(f) Show that
$$
E[q \mid q>0]=\frac{W_q}{1-\sum_{n=0}^{c-1} q_n} .
$$
(g) Let $\lambda_a$ be the average arrival rate of customers who actually enter the system. Show that
$$
\lambda_a=\lambda\left(1-p_K\right) .
$$