Markov Process question

Question

There are two tennis courts. Pairs of players arrive at a rate of 3 per hour and play for an exponentially distributed amount of time with mean 1 hour. If there are already two pairs of players waiting, new arrivals will leave. Find the stationary distribution for thenumber of courts occupied.

Answer 1

The model has $(P_0(t), P_1(t), P_2(t))$ as the probabilities, at time $t$, that there are $0$, $1$, or $2$ courts occupied, with $P_0(t)+P_1(t)+P_2(t) = 1$. In the steady state the time derivatives of each of those probabiliites will be zero.

So we need to solve the system $$ 0 = \frac{dP_0}{dt} = -3P_0 + 1 P_1\\ 0 = \frac{dP_1}{dt} = -1P_1 -3 P_1 +3 P_0 + 1 P_2\\ 0 = \frac{dP_2}{dt} = 3P_1 - 1 P_2 \\ P_0+P_1+ P_2 = 1 $$ This is four equations in three unknowns, but since the first 3 preserve probability ($\frac{d(P_0+P_1+P_2)}{dt}=0$) we can drop one of the first three; say the middle one. Then $$ -3P_0 + 1 P_1 = 0\\ 3P_1 - 1 P_2 = 0\\ P_0+P_1+ P_2 = 1 $$ and the solution is $$ P_1 = 3P_0\\ P_2 = 3P_1 = 9P_0\\ P_0 + 3P_0 + 9P_0 = 1 \\ P_0 = \frac1{13}, P_1 = \frac3{13},P_2 = \frac9{13}, $$ WHich should jibe with experience: When more people want to use the courts than the courts have capacity, most of the time both courts will be full.