M2: Waiting Line Models

What is another term for a waiting line?

A queue

What is the body of knowledge dealing with waiting lines called?

Queueing theory

What are the average time characteristics of a waiting line?

Average time a unit spends in the waiting line and in the system.

What is the probability that no units are in the system?

It is one of the operating characteristics of a waiting line.

What is the average number of units in the waiting line?

It is a performance measure in waiting line models.

What does the Poisson probability distribution describe?

The arrival pattern of units in a waiting line.

What is the mean number of arrivals per time period in Poisson distribution denoted as?

λ (lambda)

What does the service time in a waiting line refer to?

The time a customer spends at the service facility once the service has started.

What type of probability distribution is used for service time when arrivals follow a Poisson distribution?

Exponential probability distribution.

What is the service rate in waiting line models?

The mean number of units that can be served per time period.

What does FCFS stand for in queue disciplines?

First-Come, First-Served.

What is the transient period in a waiting line system?

The startup period before the system reaches steady-state operation.

What happens if the arrival rate (λ) exceeds the service rate (μ)?

The waiting line will continue to grow without limit.

What is the average number of customers waiting in line at Burger Dome?

2.25 customers.

What is the average waiting time for customers at Burger Dome?

Three minutes.

What is the significance of the service time varying in a waiting line?

It affects the efficiency and speed of service.

What is the probability that an arriving unit has to wait for service?

It is one of the key performance measures in waiting line models.

What is the formula for factorial denoted as n!?

n! = n(n−1)(n−2)⋯(2)(1), with 0! defined as 1.

What is the average number of services per time period denoted as?

μ (mu)

What is the average time a unit spends in the system?

The sum of the average time spent in the waiting line and the service time.

What does the probability function in Poisson distribution calculate?

The probability of x arrivals in a specific time period.

What is the impact of adding more serving stations in a waiting line?

It can reduce waiting time but may increase costs and idle staff.

What is the average arrival rate at Burger Dome?

0.75 customers per minute.

What is the relationship between mean and variance in a Poisson distribution?

They are the same.

What does the equation λ < μ indicate?

The service facility has sufficient capacity to handle the arriving units.

What is the average time a unit spends in the waiting line at Burger Dome?

It is part of the performance measures calculated in the waiting line model.

What is the goal of waiting line models?

To keep waiting times within tolerable limits while optimizing service.

What is the probability that there are seven or more customers in the restaurant at one time?

0.1335

What is one method to improve the waiting line operation?

Increase the service rate by design changes or adding servers.

What design change did the restaurant management implement to improve service?

Customers fill out and submit a paper order form directly to the kitchen while waiting.

What is the estimated service rate after the design change?

75 customers per hour or 1.25 customers per minute.

What is the probability of no customers in the system after the new system is implemented?

0.400

What is the average time a customer spends in the waiting line after improvements?

1.200 minutes

What is the average time a customer spends in the system after improvements?

2.000 minutes

What is the probability that an arriving customer has to wait after improvements?

0.600

What is the average time a customer spent in the waiting line before improvements?

3 minutes

What is the average time a customer spent in the system before improvements?

4 minutes

What is a multiple-server waiting line?

A system with two or more identical servers where customers can wait in a single or dedicated queue.

What are the two typical queueing possibilities in multiple-server systems?

1) Single waiting line for all servers, 2) Dedicated queues for each server.

What are the conditions for applying formulas for multiple-server waiting lines?

1) Arrivals follow a Poisson distribution, 2) Service time follows an exponential distribution, 3) Same service rate for each server, 4) Single waiting line.

What is Little's flow equation for the average number of units in the system?

L = λ × W (arrival rate times average time in the system).

What is the economic analysis formula for waiting lines?

TC = cwL + csk (where L is the number of units, k is the number of servers, and c are respective costs).

What does the Kendall notation A/B/k represent?

A denotes the arrival probability distribution, B denotes the service time distribution, k denotes the number of servers.

What does the letter 'M' signify in Kendall notation?

A Poisson probability distribution for arrivals or an exponential distribution for service time.

What does the letter 'D' signify in Kendall notation?

Deterministic or constant arrivals or service times.

What does the letter 'G' signify in Kendall notation?

General probability distribution with known mean and variance.

How is the single-server waiting line model classified in Kendall notation?

M/M/1 (Poisson arrivals and exponential service times).

How is the two-server waiting line model classified in Kendall notation?

M/M/2 (Poisson arrivals and exponential service times).

What is the average time a customer spends in the system for a two-server system?

Reduced from 4 minutes to 1.1636 minutes.

What is the average number of customers in the waiting line for a two-server system?

Reduced from 2.25 customers to 0.1227 customers.

What is the probability that a customer has to wait for service in a two-server system?

Reduced from 0.75 to 0.2045.

What is the average number of units in the system denoted as?

L

What is the average time a unit spends in the waiting line denoted as?

Wq

What is the average time a unit spends in the system denoted as?

W