A service counter employs two servers.On average a server requires 8 minutes to process a customer and service times follow an exponential distribution.Customers arrive at the counter at the rate of 12 per hour according to a Poisson distribution.The average amount of time,in minutes,spent in the system (i.e. ,waiting and being served)is approximately
A)0.237 minutes
B)14.22 minutes
C)22.20 minutes
D)33.30 minutes

Question

Quizplus · Accepted Answer

The system can be modeled as a tandem queue, where customers first join a waiting line and then move to one of the servers to be served.

First, we need to calculate the arrival rate of customers:

λ = 12 customers/hour = 0.2 customers/minute

Next, we need to calculate the service rate of each server:

μ = 1/8 customers/minute = 0.125 customers/minute

Since we have two servers, the total service rate is:

μt = 2μ = 0.25 customers/minute

The utilization of each server can be calculated as:

ρ = λ/μ = 0.2/0.125 = 1.6

Since ρ > 1, the servers are overloaded and there will be a queue of customers waiting for service. We can use Little's Law to find the average number of customers in the system:

L = λW

where L is the average number of customers in the system, λ is the arrival rate, and W is the average time spent in the system.

To find W, we need to calculate the average time spent waiting in the queue (Wq) and the average time spent being served (Ws).

The average waiting time can be calculated as:

Wq = ρ/(μt - λ) = 1.6/(0.25 - 0.2) = 16 minutes

The average service time is 1/μ = 8 minutes, so the average time spent being served is:

Ws = 2(8) = 16 minutes

Therefore, the average time spent in the system is:

W = Wq + Ws = 16 + 16 = 32 minutes

Since the question asks for the answer in minutes, the correct choice is C) 22.20 minutes, which is closest to 32.

A Service Counter Employs Two Servers