Exam 20: Waiting-Line Models

A crew of mechanics at the Highway Department garage repair vehicles that break down at an average of λ = 10 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of μ = 12 vehicles per day with a repair time distribution that approximates a negative exponential distribution. (a) What is the probability that the system is empty? (b) What is the probability that there is precisely one vehicle in the system? (c) What is the probability that there is more than one vehicle in the system? (d) What is the probability of 6 or more vehicles in the system?

In the analysis of queuing models, the negative exponential distribution often describes arrival rates, while service times are often described by the Poisson distribution.

A waiting line has a(n) ________ population if, as arrivals take place, the likelihood of additional arrivals decreases.

A dental clinic at which only one dentist works is open only two days a week. During those two days, the traffic arrivals follow a Poisson distribution with patients arriving at the rate of three per hour. The doctor serves patients at the rate of one every 15 minutes. (a) What is the probability that the clinic is empty (except for the dentist and staff)? (b) What is the probability that there are one or more patients in the system? (c) What is the probability that there are four patients in the system? (d) What is the probability that there are four or more patients in the system?

Four of the most widely used waiting line models-M/M/1 or A, M/M/S or B, M/D/1 or C, and Finite population or D-all share three characteristics: Poisson arrivals, FIFO discipline, and exponential service times.

A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of 12 per hour. What is the average time a unit spends in the system and the average time a unit spends waiting (in hours)?

Why must the service rate be greater than the arrival rate in a single-server system?

In the finite population waiting line model, the average length of the queue is calculated from the probability the system is empty.

Which one of the following is NOT a characteristic of a Model A or M/M/1 system?

Suppose that 2 customers arrive on average each minute in a Poisson distribution. Is it more likely that 1 customer or 0 customers will arrive each minute?

A waiting line, or queuing, system has three parts, which are:

Arrivals or inputs to a waiting-line system have characteristics that include:

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). Which is cheaper, the existing system with one service crew, or a revised system with two service crews?

A single-phase waiting-line system meets the assumptions of constant service time or M/D/1. Units arrive at this system every 10 minutes on average. Service takes a constant 6 minutes. What is the average length of the queue, L_q, in units?

A waiting line model meeting the assumptions of M/M/1 has an arrival rate of 2 per hour and a service rate of 4 per hour. What is the utilization factor for the system?

The ________ probability distribution is a continuous probability distribution often used to describe the service time in a queuing system.

A crew of mechanics at the Highway Department Garage repair vehicles that break down at an average of λ = 5 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of μ = 8 vehicles per day with a repair time distribution that approximates a negative exponential distribution. (a) What is the probability that the system is empty? (b) What is the probability that there is precisely one vehicle in the system? (c) What is the probability that there are more than two vehicles in the system? (d) What is the probability of 5 or more vehicles in the system?

A finite population waiting line model is also called M/M/1 with finite source.

Which of the following is most likely to be served in a last-in, first-served (LIFS) queue discipline?

Describe the difference between FIFO and LIFO queue disciplines.