Exam 20: Waiting-Line Models

A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a service rate of 15 per hour. What is the average time a unit spends in the system and the average time a unit spends waiting (in hours)?

A waiting-line system with one waiting line and three sequential processing stages is a multiple-server, single-phase system.

In the basic queuing model (M/M/1), what probability distribution describes service times?

A crew of mechanics at the Highway Department Garage repair vehicles that break down at an average of λ = 7 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of μ = 11 vehicles per day with a repair time distribution that approximates an exponential distribution. (a) What is the utilization rate for this service system? (b) What is the average time before the facility can return a breakdown to service? (c) How much of that time is spent waiting for service? (d) How many vehicles are likely to be waiting for service at any one time?

A crew of mechanics at the Highway Department Garage repair vehicles that break down at an average of λ = 8 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of μ = 11 vehicles per day with a repair time distribution that approximates an exponential distribution. The crew cost is approximately $300 per day. The cost associated with lost productivity from the breakdown is estimated at $150 per vehicle per day (or any fraction thereof). What is the expected cost of this system?

Which of the following occurs as the level of service decreases?

Which of the following is NOT a measure of a queue's performance?

In queuing problems, which of the following probability distributions is typically used to describe the number of arrivals per unit of time?

A finite population single-server waiting line model with N = 5 has an average arrival rate of 1/2 customer per hour and a mean service rate of 4 customers per hour. What is the probability the system is empty?

Which of the following represents a customer who reneged due to the waiting line?

In a busy DMV office, a customer will have to wait in one line to complete the application (the first service stop), queue again to take the test, and finally go to a third counter to pay the applicable fee. In this case, it is a ________ system.

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 10 students per hour, and the average service rate is 20 students per hour. A student has just entered the system. How long is she expected to stay in the system?

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. What is the utilization factor?

Which of the following is an example of a finite arrival population?

At the order fulfillment center of a major mail-order firm, customer orders, already packaged for shipment, arrive at the sorting machine to be sorted for loading onto the appropriate truck for the parcel's address. The arrival rate at the sorting machine is at the rate of 140 per hour following a Poisson distribution. The machine sorts at the constant rate of 150 per hour. (a) What is the utilization rate of the system? (b) What is the average number of packages waiting to be sorted? (c) What is the average number of packages in the sorting system? (d) How long must the average package wait until it gets sorted?

A waiting-line system that meets the assumptions of M/M/1 has λ = 1, μ = 5. For this system, what is the probability of more than two units in the system?

Suppose that a service facility has an average line of 2 customers that must wait, on average, 5 minutes for service. How many customers are arriving per hour?

What are L_s and L_q, as used in waiting line terminology? Which is larger, L_s or L_q? Explain.

Study of waiting-line models helps operations managers better understand:

The cost of waiting decreases as the service level increases.