Exam 11: Waiting Line Models

The equations provided in the textbook for computing operating characteristics apply to a waiting line operating

The Arctic Flyers minor league hockey team has one box office clerk.On average,each customer that comes to see a game can be sold a ticket at the rate of eight per minute.For normal games,customers arrive at the rate of five per minute.Assume arrivals follow the Poisson distribution and service times follow the exponential distribution. a.What is the average number of customers waiting in line? b.What is the average time a customer spends in the waiting line? c.What is the average number of customers in the system? d.What is a customer's average time in the system? e.What is the probability that someone will be buying tickets when an arrival occurs? The Flyers are playing in the league playoffs and anticipate more fans,estimating that the arrival rate will increase to 12 per minute.Output is supplied for two- and three-cashier systems. ​ ​ ​

The postmaster at the Oak Hill Post Office expects that the mean arrival rate of people to her customer counter will soon increase by 50% due to a large apartment complex being built.Currently,the mean arrival rate is 15 people per hour.The postmaster can serve an average of 25 people per hour.By what percentage must the postmaster's mean service rate increase when the apartment complex is completed in order that the average time spent at the post office remains at its current value?

Two new checkout scanning systems are under consideration by a retail store.Arrivals to the checkout stand follow the Poisson distribution with λ = 2 per minute.The cost for waiting is $18 per hour.The first system has an exponential service rate of five per minute and costs $10 per hour to operate.The second system has an exponential service rate of eight per minute and costs $20 per hour to operate.Which system should be chosen?

A variation of the waiting line models involves a system in which no waiting is allowed,and arriving units are denied access to the system.

For a single-server queuing system,the average number of customers in the waiting line is one less than the average number in the system.

In a waiting line situation,arrivals occur,on average,every 10 minutes,and 10 units can be received every hour.Which of the following represents λ and μ in this situation?

Use of the Poisson probability distribution assumes that arrivals are not random.

The insurance department at Shear's has two agents,each working at a mean speed of eight customers per hour.Customers arrive at the insurance desk at a mean rate of one every six minutes and form a single queue.Management feels that some customers are going to find the wait at the desk too long and take their business to Word's,Shear's competitor.​ In order to reduce the time required by an agent to serve a customer,Shear's is contemplating installing one of two minicomputer systems: System A that leases for $18 per day and will increase an agent's efficiency by 25% or System B that leases for $23 per day and will increase an agent's efficiency by 50%.Agents work eight-hour days.​ If Shear's estimates its cost of having a customer in the system at $3 per hour,determine if it should install a new minicomputer system,and if so,which one?

Arrivals at a box office in the hour before the show follow the Poisson distribution with λ = 7 per minute.Service times are constant at 7.5 seconds.Find the average length of the waiting line.

Waiting line models describe the transient-period operating characteristics of a waiting line.

Waiting line models consist of mathematical formulas and relationships that can be used to determine the operating characteristics (also known as performance measures)for a waiting line.

The manner in which units receive their service,such as FCFS,is the

Adding more servers always improves the operating characteristics of the waiting line and reduces the waiting cost.

A waiting line situation with a single server is referred to as an M/M/1 model with a finite calling population.

The body of knowledge dealing with waiting lines is known as queueing theory.

For an M/M/1 queuing system,if the service rate,µ,is doubled,the average wait in the system,W,is cut in half.

During summer weekdays,boats arrive at the inlet drawbridge according to the Poisson distribution at a rate of three per hour.In a two-hour period, a.what is the probability that no boats arrive? b.what is the probability that two boats arrive? c.what is the probability that eight boats arrive?

The time to process a registration at the Sea View Resort follows the exponential distribution and has a mean of six minutes. a.What is the probability of a registration time shorter than three minutes? b.What is the probability of a registration time shorter than six minutes? c.What is the probability of a registration time between three and six minutes?

The Quick Snap photo machine at the Lemon County bus station takes four snapshots in exactly 75 seconds.Customers arrive at the machine according to a Poisson distribution at the mean rate of 20 per hour.On the basis of this information,determine a.the average number of customers waiting to use the photo machine. b.the average time a customer spends using the system. c.the probability an arriving customer must wait for service. ​