Scenario B.4
Customers arrive at an airline ticket counter at the rate of 50 customers per hour, according to a Poisson distribution. There are three ticket agents. Customers select the first available agent from one line. Each agent can process 20 customers per hour with exponential service times.
-Use the information in Scenario B.4. What is the average time spent in the system?
A) fewer than or equal to 6.5 minutes
B) greater than 6.5 minutes but fewer than or equal to 7.5 minutes
C) greater than 7.5 minutes but fewer than or equal to 8.5 minutes
D) greater than 8.5 minutes

Question

Quizplus · Accepted Answer

Given the arrival rate ($\lambda$) is 50 customers per hour and each of the three agents can service 20 customers per hour ($\mu$), the total service rate is $3 \times 20 = 60$ customers per hour. The system can be analyzed as an M/M/3 queue. The traffic intensity, $\rho$, is the arrival rate divided by the total service rate, which is $\frac{50}{60} = \frac{5}{6}$. To find the average time spent in the system ($T$), we use the formula $T = \frac{1}{\mu - \frac{\lambda}{c}}$, where $c$ is the number of servers. However, this direct formula applies to M/M/1 queues. For M/M/c queues, the average time in the system is more complex to calculate directly without the detailed formulas or tables specific to M/M/c queues, but it involves considering both the service rate and the probability of having to wait due to all servers being busy. Given the system is not overly congested ($\rho < 1$), and considering the efficiency of having multiple servers, the average time in the system is expected to increase compared to a single-server scenario but not excessively. The correct estimation, without performing the exact M/M/3 calculations, would consider the balance between the arrival rate and the total service capacity, hinting that customers spend more time in the system than the minimum service time (due to waiting in line) but not an excessive amount of time, given the system's capacity to handle the arrival rate efficiently. Thus, the average time spent in the system is likely to be greater than 6.5 minutes but fewer than or equal to 7.5 minutes, considering the efficiency of three servers handling the flow of customers, which aligns with choice B.

Scenario B.4 Customers Arrive at an Airline Ticket Counter at the Rate