At Joe's Pool Hall they rent tables by the hour and also rent Balabushka pool cues for purists that can't afford their own. The rental process for these cues is exhaustive and takes an hour and a half for each application. There are three leasing agents that conduct these interviews, and on average, it is one hour between customer arrivals.
-What percentage of the time is the system completely empty?
A) 9%
B) 13%
C) 17%
D) 21%

Question

Quizplus · Accepted Answer

The system is modeled as an M/M/3 queue (a queueing model with exponential service times, exponential interarrival times, and three servers) with an arrival rate ($\lambda$) of 1 customer per hour and a service rate ($\mu$) of 2/3 customer per hour (since it takes 1.5 hours to serve one customer). The traffic intensity ($\rho$) is given by $\lambda / (s\mu)$, where $s$ is the number of servers, which is 3 in this case. Thus, $\rho = 1 / (3 * 2/3) = 1 / 2 = 0.5$. The probability that the system is empty (P0) can be calculated using the formula for an M/M/s queue:$$P_0 = \left[ \sum_{k=0}^{s-1} \frac{(\lambda/\mu)^k}{k!} + \frac{(\lambda/\mu)^s}{s!(1-\rho)} \right]^{-1}$$Substituting $\lambda/\mu = 1.5$, $s = 3$, and $\rho = 0.5$ into the formula gives:$$P_0 = \left[ \sum_{k=0}^{2} \frac{1.5^k}{k!} + \frac{1.5^3}{3!(1-0.5)} \right]^{-1}$$$$P_0 = \left[ 1 + 1.5 + \frac{1.5^2}{2} + \frac{1.5^3}{6(0.5)} \right]^{-1}$$$$P_0 = \left[ 1 + 1.5 + 1.125 + 1.125 \right]^{-1}$$$$P_0 = \left[ 4.75 \right]^{-1} = 0.21$$Therefore, the system is completely empty 21% of the time.

At Joe's Pool Hall They Rent Tables by the Hour