Question

3. For a single-server, single-line, single-phase waiting line system, where l represents the mean arrival rate of customers and m represents the mean service rate, what is the formula for the average utilization of the system?

a) l / m

b) l / (m-l)

c) l² / m(m-l)

d) 1 / (m-l)

e) l / m(m-l)

4. For a single-server, single-line, single-phase waiting line system, where l represents the mean arrival rate of customers and m represents the mean service rate, what is the formula for the average number of customers in the system?

a) l / m

b) l / (m-l)

c) l² / m(m-l)

d) 1 / (m-l)

e) l / m(m-l)

5. An information desk at a rest stop receives requests for assistance (from one server). Assume that a Poisson probability distribution with a mean rate of 5 requests per hour can be used to describe the arrival pattern and that service times follow the exponential probability distribution, with a mean service rate of 1 request per minute. What is the probability that there are no requests for assistance in the system?

a) 0%

b) 8.3%

c) -4%

d) -400%

e) 91.7%

6. Consider a single-server queuing system (with Poisson arrivals and exponential service times). If the arrival rate is 30 units per hour, and each customer takes 30 seconds on average to be served, what is the average length of the line?

a)   1.03

b) -1.03

c)   0.166

d)   0.333

e)   0.083

7. Consider a single-line, single-server waiting line system. The arrival rate λ is 80 people per hour, and the service rate μ is 120 people per hour. What is the probability of having 3 or more units in the system?

a) 29.63%

b) 33.33%

c)   0.00%

d)   9.88%

e) 66.67%

Answer 1

5. An information desk at a rest stop receives requests for assistance (from one server). Assume that a Poisson probability distribution with a mean rate of 5 requests per hour can be used to describe the arrival pattern and that service times follow the exponential probability distribution, with a mean service rate of 1 request per minute. What is the probability that there are no requests for assistance in the system?

Hence option (e) is the right answer.

6. Consider a single-server queuing system (with Poisson arrivals and exponential service times). If the arrival rate is 30 units per hour, and each customer takes 30 seconds on average to be served, what is the average length of the line?

Hence the required option is (e)

7. Consider a single-line, single-server waiting line system. The arrival rate λ is 80 people per hour, and the service rate μ is 120 people per hour. What is the probability of having 3 or more units in the system?

Hence the required answer is (a)

4. For a single-server, single-line, single-phase waiting line system, where l represents the mean arrival rate of customers and m represents the mean service rate, what is the formula for the average number of customers in the system?

a) l / m

b) l / (m-l)

c) l² / m(m-l)

d) 1 / (m-l)

e) l / m(m-l)

Hence the required answer is (b), which is the required formula.

Answer 2

3. For the average utilization of the system in a single-server, single-line, single-phase waiting line system, the correct formula is:

a) l / m

Average utilization represents the proportion of time the server is busy serving customers. In this formula, "l" represents the mean arrival rate of customers, and "m" represents the mean service rate.

4. For the average number of customers in the system in a single-server, single-line, single-phase waiting line system, the correct formula is:

c) l^2 / m(m-l)

Average number of customers in the system refers to the average number of customers waiting in line and being served. In this formula, "l" represents the mean arrival rate of customers, and "m" represents the mean service rate.

5. To find the probability that there are no requests for assistance in the system, we can use the exponential distribution for arrival times.

a) 0%

Since we are looking for the probability that there are no requests (zero arrivals) in the system, it means the service desk is idle, and the probability of that happening is zero.

6. To calculate the average length of the line in a single-server queuing system with Poisson arrivals and exponential service times, we can use Little's Law.

d) 0.333

Average length of the line = λ * W where λ is the arrival rate and W is the average time a customer spends in the system.

Average length of the line = (30 units/hour) * (30 seconds/unit) / (3600 seconds/hour) ≈ 0.333 units

7. To find the probability of having 3 or more units in the system in a single-line, single-server waiting line system, we can use the formula for the probability of having n or more customers in the system, which follows the Poisson distribution.

b) 33.33%

The formula for the probability of having n or more units in the system is:

P(n or more units) = 1 - Σ[P(i)], where the sum is from i = 0 to n-1, and P(i) represents the probability of having i units in the system.

P(3 or more units) = 1 - [P(0) + P(1) + P(2)] P(3 or more units) ≈ 1 - (0.449 + 0.269 + 0.081) ≈ 0.201

To convert this probability to a percentage, multiply by 100: P(3 or more units) ≈ 0.201 * 100 ≈ 20.1% (rounded to 2 decimal places) Therefore, the probability of having 3 or more units in the system is approximately 20.1% or 20.1%.