Questions about queue theory

Question

A service center consists of one server, working at an exponential rate of four services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at most three customers, what fraction of potential customers enter the system?

Answer 1

This is a $M/M/1/3$ queue with arrival rate $\lambda=3$ and service rate $\mu=4$. Let $Z(t)$ be the number of customers in the system at time $t$, then $\{Z(t):t\geqslant0\}$ is a CTMC on $\{0,1,2,3\}$ with generator $$Q=\begin{bmatrix}-3 & 3 & 0 & 0\\ 4 & -7 & 3 & 0\\ 0 & 4 & -7& 3\\ 0 & 0 & 4& -4\end{bmatrix}.$$ The stationary distribution $\pi$ satisfies $\pi Q=0$ and $\sum_i \pi_i=1$, so we have the system of linear equations \begin{align} \pi_0 &= -3\pi_0 + 4\pi_1\\ \pi_1 &= 3\pi_0 -7\pi_1 + 4\pi_2\\ \pi_2 &= 3\pi_1 -7\pi_2 + 4\pi_3\\ 1 &= \pi_0 + \pi_1+\pi_2+\pi_3. \end{align} Solving these equations yields $$\pi_0=\frac15,\ \pi_1=\frac15,\ \pi_2=\frac14,\ \pi_3=\frac7{20}. $$ If the system is full, then arriving customers leave without entering the queue, so the fraction of potential customers entering the system is $$1-\pi_3=1-\frac7{20}=\frac{13}{20}.$$

Note that this means the effective arrival rate is $$\lambda_e = \lambda(1-\pi_3)=\frac{39}{20}.$$