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Elis [28]
3 years ago
12

Assume that the arrival of airplanes at a one-runway airport is a Poisson distribution with a mean rate of 8 planes per hour. Th

e landing time (service time, not service rate which would be planes/min) is an exponential distribution with a mean of 5 minutes per plane. What is the mean number of planes in a holding (planes circling around waiting to land) pattern (compute exact number)? a. 1 b. 1.33 c. 2 d. 1.67
Mathematics
1 answer:
Delvig [45]3 years ago
3 0

Answer:

Option b that is 1.33 is the right choice.

Step-by-step explanation:

Given:

Mean rate of arrival (\lambda) = 8 planes/hr

Service time = 5 minute/plane

Mean service rate  (\mu) = \frac{60}{5} = 12 planes/hr

Applying the concept Poisson-distributed arrival and service rates (exponential inter-arrival and service times)(M/M/1) process:

We have to find mean number of planes waiting to land that is mean number of customers in the queue .

Mean number of customers in queue (L_q) .

⇒ L_q=\frac{\rho \lambda}{\mu-\lambda}

   Considering, \rho=\frac{\lambda}{\mu} , \rho is also mean number of customers in service.

⇒ L_q=\frac{ \lambda^2}{\mu(\mu-\lambda)}

⇒ Plugging the values.

⇒ L_q=\frac{8^2}{12(12-8)}

⇒ L_q=\frac{64}{48}

⇒ L_q=1.33

So,

Mean number of planes in holding and waiting to land = 1.33

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