Exam 12: Waiting Lines and Queuing Theory Models

Customers enter the waiting line to pay for food as they leave a cafeteria on a first-come,first-served basis.The arrival rate follows a Poisson distribution,while service times follow an exponential distribution.If the average number of arrivals is four per minute and the average service rate of a single server is seven per minute,what proportion of the time is the server busy?

Arrivals are random when they are dependent on one another and can be predicted.

Table 12-5 -According to the information provided in Table 12-5,which presents the solution for a queuing problem with a constant service rate,what percentage of available service time is actually used?

Table 12-3 -According to the information provided in Table 12-3,which presents a queuing problem solution,what proportion of time is the system totally empty?

Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school.Sam can vaccinate a dog every 3 minutes.It is estimated that the dogs will arrive independently and randomly throughout the day at a rate of 1 dog every 4 minutes,according to a Poisson distribution.Also assume that Sam's vaccinating times are exponentially distributed. (a)Find the probability that Sam is idle. (b)Find the proportion of time that Sam is busy. (c)Find the average number of dogs receiving or waiting to be vaccinated. (d)Find the average number of dogs waiting to be vaccinated. (e)Find the average time a dog waits before getting vaccinated. (f)Find the average amount (mean)of time a dog spends between waiting in line and getting vaccinated.

The school of business has 3 fax machines.The toner in each machine needs to be changed after about 5 hours of use.There is one unit secretary who is responsible for the fax machine maintenance.It takes him 15 minutes to replace the toner cartridge.What is the probability the system is empty?

Most systems use a queue discipline known as

At an automatic car wash,cars arrive randomly at a rate of 7 cars every 30 minutes.The car wash takes exactly 4 minutes (this is constant).On average,what would be the length of the line?

At an automatic car wash,cars arrive randomly at a rate of 7 every 30 minutes.The car wash takes exactly 4 minutes (this is constant).On average,how long would each car spend at the car wash?

With regard to queuing theory,define what is meant by balking.

Table 12-6 Customers arrive at coffee shop at a rate of 40 per hour.There are 2 servers available and it takes an average of 1 minute to serve each customer. -Using Table 12-6,what is the probability of no customers in the system?

Which of the following is not an assumption for the M/M/1 model with finite population source?

At the start of football season,the ticket office gets busy the day before the first game.Customers arrive at the rate of four every ten minutes.A ticket seller can service a customer in four minutes.Traditionally,there are two ticket sellers working.The university is considering an automated ticket machine similar to the airlines' e-ticket system.The automated ticket machine can service a customer in 2 minutes. (a)What is the average length of the queue for the in-person model? (b)What is the average length of the queue for the automated system model? (c)What is the average time in the system for the in-person model? (d)What is the average time in the system for the automated system model? (e)Assume the ticket sellers earn $8 per hour and the machine costs $20 per hour (amortized over 5 years).The wait time is only $4 per hour because students are patient.What is the total cost of each model?

Cars arrive at a parking lot entrance at the rate of 20 per hour.The average time to get a ticket and proceed to a parking space is two minutes.There are two lot attendants at the current time.The Poisson and exponential distribution appear to be relevant in this situation. (a)What is the probability that an approaching auto must wait? (b)What is the average waiting time? (c)What is the average number of autos waiting to enter the garage?

Table 12-6 Customers arrive at coffee shop at a rate of 40 per hour.There are 2 servers available and it takes an average of 1 minute to serve each customer. -Using Table 12-6,what proportion of time are the servers busy?

Bank Boston has a branch at Bryant College.The branch is busiest at the beginning of the college year when freshmen and transfer students open accounts.This year,freshmen arrived at the office at a rate of 40 per day (8-hour day).On average,it takes the Bank Boston staff person about ten minutes to process each account application.The bank is considering having one or two tellers.Each teller is paid $12 per hour and the cost of waiting in line is assumed to be $8 per hour. (a)What is the total daily waiting cost for the single teller model? (b)What is the total daily waiting cost for the two-teller model? (c)What is the total daily service cost for the single teller model? (d)What is the total daily service cost for the two-teller model? (e)Which model is preferred?

What is meant by a single-channel queuing system?

The three major characteristics of the input source that generates arrivals or customers for the service system are

A suburban specialty restaurant has developed a single drive-thru window.Customers order,pay,and pick up their food at the same window.Arrivals follow a Poisson distribution,while service times follow an exponential distribution.If the average number of arrivals is 6 per hour and the service rate is 2 every 15 minutes,what is the average number of customers in the system?

With regard to queuing theory,define what is meant by reneging.