Question

Students arrive randomly at the help desk of the computer lab. There is only one service​ agent, and the time required for inquiry varies from student to student. Arrival rates have been found to follow the Poisson​ distribution, and the service times follow the negative exponential distribution. The average arrival rate is 12 students per​ hour, and the average service rate is 20 students per hour. On​ average, how long does it take to service each​ student?

Answer 1

Answer:

It takes on average 3 minutes to service each​ student.

Step-by-step explanation:

In queueing problem, an M/M/1 queue model is an arrangement with a single queue, where arrivals at the queue are approximated by a Poisson distribution with mean λ and the service times follows an exponential distribution with mean μ.  

The mean number of arrivals in this system is given by:  

$\text{Avearge}=\frac{\rho}{1-\rho}$  

Here the variable ρ is defined as:

 $\rho=\frac{\lambda}{\mu}$

From the average number of arrivals in the system, we can calculate the average number of arrivals in the queue by subtracting the average number of arrivals in service.

That is,

$\text{Average arrivals in queue}=\frac{\rho}{1-\rho}-\rho=\frac{\rho^{2}}{1-\rho}$  

The information provided is:

The average number of arrivals of customers is, λ = 12 students/hour.

The average service rate of a single server is, μ = 20 students/hour.

The average of an exponential distribution is:

$E(X)=\frac{1}{\mu}$

Compute the average time it takes to service one student as follows:

$E(X)=\frac{1}{\mu}$

         $=\frac{1}{20}\\\\=0.05\ \text{hour}\\\\=3\ \text{minutes}$

Thus, it takes on average 3 minutes to service each​ student.