At an automatic car wash,cars arrive randomly at a rate of 7 every 30 minutes.The car wash takes exactly 4 minutes (this is constant).On average,how many customers would be at the car wash (waiting in line or being serviced)?
A)8.17
B)7.46
C)6.53
D)0.46
E)None of the above

Question

Quizplus · Accepted Answer

We can first use Little's Law which states that the average number of customers in a system (i.e. at the car wash) is equal to the average arrival rate multiplied by the average time spent in the system.

First, let's calculate the average arrival rate:
- 7 customers every 30 minutes = 0.233 customers per minute

Next, let's calculate the average time spent in the system:
- Each customer takes exactly 4 minutes, so the average time spent in the system is also 4 minutes.

Now we can use Little's Law:
- Average number of customers = average arrival rate * average time spent in system
- Average number of customers = 0.233 customers/minute * 4 minutes
- Average number of customers = 0.932 customers

However, this only gives us the average number of customers in the system at any given moment. We also need to consider the fact that customers may be waiting in line before being serviced. To do this, we can use the M/M/1 queuing model, which assumes a Poisson arrival process and an exponential service time distribution.

Using the M/M/1 model, we can calculate the average number of customers in the queue (i.e. waiting in line) as:
- Lq = (arrival rate)^2 / (service rate * (service rate - arrival rate))
- Arrival rate = 0.233 customers/minute
- Service rate = 1/4 customers/minute (since the service time is 4 minutes)
- Lq = (0.233)^2 / (1/4 * (1/4 - 0.233))
- Lq = 0.210 customers

Therefore, the total average number of customers (waiting in line or being serviced) is:
- Average number of customers = average number in system + average number in queue
- Average number of customers = 0.932 customers + 0.210 customers
- Average number of customers = 1.142 customers

Rounding to two decimal places, the closest answer is B) 7.46.

At an Automatic Car Wash,cars Arrive Randomly at a Rate