Question

a call center has analyzed the calls received in the last month and found that, on average, 64 calls were received per hour. assume that call arrivals follow poisson distribution and come from an infinite population. service times follow an exponential distribution with an average of 45 seconds per call served. assume the queue length can be infinite with fcfs discipline. a) what is the average number of people in line? (2.5 points) b) what is the average number of people in the system? (2.5 points) c) what is the average amount of time that a person can expect to spend in line? (2.5 points) d) on average, how much time will a person spend in the system? (2.5 points) e) during the weekends, the call arrival rate can be expected to increase to over 80 per hour. what effect will this have on the average number of people waiting in line? (2 points)

Answer 1

To find the average number of people in line, we calculate the average arrival rate and service rate using Little's Law. The average number of people in line is 1.422. The average number of people in the system is 2.422. The average time a person spends in line is 0.0222 hours, and the average time a person spends in the system is 0.0378 hours.

Step-by-step explanation:

To find the average number of people in line, we need to calculate the average arrival rate and the average service rate. The average arrival rate is given as 64 calls per hour. The average service rate is the reciprocal of the average service time, which is 1/45 calls per second. Using Little's Law, we can calculate the average number of people in line.

The formula is:

L = λ * W

where L is the average number of people in line, λ is the average arrival rate, and W is the average time a person spends in the system.

Using the given values, we have:

L = 64 calls/hour * (1/45 calls/second) = 1.422 people

To find the average number of people in the system, we add the average number of people in line and the average number of people being served.

The formula is:

Ls = L + ρ

where Ls is the average number of people in the system and ρ is the traffic intensity.

Using the given values, we have:

Ls = 1.422 people + 1 = 2.422 people

To find the average amount of time a person can expect to spend in line, we divide the average number of people in line by the arrival rate.

The formula is:

Wq = L / λ

Using the given values, we have:

Wq = 1.422 people / 64 calls/hour = 0.0222 hours

To find the average time a person spends in the system, we divide the average number of people in the system by the arrival rate.

The formula is:

Ws = Ls / λ

Using the given values, we have:

Ws = 2.422 people / 64 calls/hour = 0.0378 hours

During the weekends, the call arrival rate increases to over 80 calls per hour. This will have a direct effect on the average number of people waiting in line because the arrival rate is a determining factor. With a higher arrival rate, more people will be in line on average.

Answer 2

a call center has analyzed the calls received in the last month and found that, on average, 64 calls were received per hour. The average number of people in line is 0.8 people. The average number of people in the system is 80.8 people.

Step-by-step explanation:

a) To find the average number of people in line, we need to use Little's Law. Little's Law states that the average number of people in a system is equal to the average arrival rate multiplied by the average time spent in the system. In this case, the average arrival rate is 64 calls per hour and the average service time is 45 seconds per call. However, we need to convert the service time into hours, so dividing 45 seconds by 3600 (the number of seconds in an hour), we get 0.0125 hours. So the average number of people in line is 64 calls per hour multiplied by 0.0125 hours, which equals 0.8 people.

b) The average number of people in the system is the sum of the average number of people in line and the average number of people being served. Since the queue length can be infinite with FCFS discipline, the average number of people being served is equal to the average service rate, which is 1 call per 0.0125 hours, or 80 calls per hour. Therefore, the average number of people in the system is 0.8 people in line plus 80 people being served, which equals 80.8 people.

c) To find the average amount of time a person can expect to spend in line, we can use Little's Law again. The average time in the system is the average number of people in the system divided by the average arrival rate. So the average time a person can expect to spend in line is 80.8 people divided by 64 calls per hour, which is approximately 1.26 hours.

d) Similarly, the average time a person spends in the system is the sum of the average time in line and the average service time. So the average time a person spends in the system is 1.26 hours plus 0.0125 hours, which is approximately 1.27 hours.

e) If the call arrival rate increases to over 80 per hour during the weekends, the average number of people waiting in line will increase. This is because the arrival rate is higher than the service rate, causing a backlog in the queue. The exact increase in the average number of people waiting in line will depend on the new arrival rate and the average service rate.