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Uses of Waiting-Line Theory Uses of Waiting-Line Theory
Applies to many service or manufacturing situations Relating arrival and service-system processing
characteristics to output
Service is the act of processing a customer Hair cutting in a hair salon Satisfying customer complaints Processing production orders Theatergoers waiting to purchase tickets Trucks waiting to be unloaded at a warehouse Patients waiting to be examined by a physician
First-come, first-served (FCFS)—used by most service systems
Preemptive discipline—allows a higher priority customer to interrupt the service of another customer or be served ahead of another who would have been served first
Other rules Earliest due date (EDD) Shortest processing time (SPT)
Probability of Customer ArrivalsProbability of Customer Arrivals
EXAMPLE C.1
Management is redesigning the customer service process in a large department store. Accommodating four customers is important. Customers arrive at the desk at the rate of two customers per hour. What is the probability that four customers will arrive during any hour?
SOLUTION
In this case customers per hour, T = 1 hour, and n = 4 customers. The probability that four customers will arrive in any hour is
The management of the large department store in Example C.1 must determine whether more training is needed for the customer service clerk. The clerk at the customer service desk can serve an average of three customers per hour. What is the probability that a customer will require less than 10 minutes of service?
SOLUTION
We must have all the data in the same time units. Because = 3 customers per hour, we convert minutes of time to hours, or T = 10 minutes = 10/60 hour = 0.167 hour. Then
The manager of a grocery store in the retirement community of Sunnyville is interested in providing good service to the senior citizens who shop in her store. Currently, the store has a separate checkout counter for senior citizens. On average, 30 senior citizens per hour arrive at the counter, according to a Poisson distribution, and are served at an average rate of 35 customers per hour, with exponential service times. Find the following operating characteristics:
The checkout counter can be modeled as a single-channel, single-phase system. Figure C.4 shows the results from the Waiting-Lines Solver from OM Explorer.
Figure C.4 – Waiting-Lines Solver for Single-Channel, Single-Phase System
Both the average waiting time in the system (W) and the average time spent waiting in line (Wq) are expressed in hours. To convert the results to minutes, simply multiply by 60 minutes/ hour. For example, W = 0.20(60) minutes, and Wq = 0.1714(60) = 10.28 minutes.
Customers arrive at a checkout counter at an average 20 per hour, according to a Poisson distribution. They are served at an average rate of 25 per hour, with exponential service times. Use the single-server model to estimate the operating characteristics of this system.
c. We use the same logic as in part (b), except that is now a decision variable. The easiest way to proceed is to find the correct average utilization first, and then solve for the service rate.
Estimating Idle Time and CostsEstimating Idle Time and Costs
EXAMPLE C.5
The management of the American Parcel Service terminal in Verona, Wisconsin, is concerned about the amount of time the company’s trucks are idle (not delivering on the road), which the company defines as waiting to be unloaded and being unloaded at the terminal. The terminal operates with four unloading bays. Each bay requires a crew of two employees, and each crew costs $30 per hour. The estimated cost of an idle truck is $50 per hour. Trucks arrive at an average rate of three per hour, according to a Poisson distribution. On average, a crew can unload a semitrailer rig in one hour, with exponential service times. What is the total hourly cost of operating the system?
Estimating Idle Time and CostsEstimating Idle Time and Costs
SOLUTION
The multiple-server model is appropriate. To find the total cost of labor and idle trucks, we must calculate the average number of trucks in the system.
Figure C.5 shows the results for the American Parcel Service problem using the Waiting-Lines Solver from OM Explorer. Manual calculations using the equations for the multiple-server model are demonstrated in Solved Problem 2 at the end of this supplement. The results show that the four-bay design will be utilized 75 percent of the time and that the average number of trucks either being serviced or waiting in line is 4.53 trucks. That is, on average at any point in time, we have 4.53 idle trucks. We can now calculate the hourly costs of labor and idle trucks:
Suppose the manager of the checkout system decides to add another counter. The arrival rate is still 20 customers per hour, but now each checkout counter will be designed to service customers at the rate of 12.5 per hour. What is the waiting time in line of the new system?
s = 2, = 12.5 customers per hour, = 20 customers per hour
DBT Bank has 8 copy machines located in various offices throughout the building. Each machine is used continuously and has an average time between failures of 50 hours. Once failed, it takes 4 hours for the service company to send a repair person to have it fixed. What is the average number of copy machines in repair or waiting to be repaired?
The Worthington Gear Company installed a bank of 10 robots about 3 years ago. The robots greatly increased the firm’s labor productivity, but recently attention has focused on maintenance. The firm does no preventive maintenance on the robots because of the variability in the breakdown distribution. Each machine has an exponential breakdown (or interarrival) distribution with an average time between failures of 200 hours. Each machine hour lost to downtime costs $30, which means that the firm has to react quickly to machine failure. The firm employs one maintenance person, who needs 10 hours on average to fix a robot. Actual maintenance times are exponentially distributed. The wage rate is $10 per hour for the maintenance person, who can be put to work productively elsewhere when not fixing robots. Determine the daily cost of labor and robot downtime.
The finite-source model is appropriate for this analysis because the customer population consists of only 10 machines and the other assumptions are satisfied. Here, = 1/200, or 0.005 break-down per hour, and = 1/10 = 0.10 robot per hour. To calculate the cost of labor and robot downtime, we need to estimate the average utilization of the maintenance person and L, the average number of robots in the maintenance system at any time. Figure C.6 shows the results for the Worthington Gear Problem using the Waiting-Lines Solver from OM Explorer.
Manual computations using the equations for the finite-source model are demonstrated in Solved Problem 3 at the end of this supplement. The results show that the maintenance person is utilized only 46.2 percent of the time, and the average number of robots waiting in line or being repaired is 0.76 robot. However, a failed robot will spend an average of 16.43 hours in the repair system, of which 6.43 hours of that time is spent waiting for service. While an individual robot may spend more than two days with the maintenance person, the maintenance person has a lot of idle time with a utilization rate of only 42.6 percent. That is why there is only an average of 0.76 robot being maintained at any point of time.
The Hilltop Produce store is staffed by one checkout clerk. The average checkout time is exponentially distributed around an average of two minutes per customer. An average of 20 customers arrive per hour.
A photographer takes passport pictures at an average rate of 20 pictures per hour. The photographer must wait until the customer smiles, so the time to take a picture is exponentially distributed. Customers arrive at a Poisson-distributed average rate of 19 customers per hour.
a. What is the utilization of the photographer?
b. How much time will the average customer spend with the photographer?
SOLUTION
a. The assumptions in the problem statement are consistent with a single-server model. Utilization is
The Mega Multiplex Movie Theater has three concession clerks serving customers on a first come, first-served basis. The service time per customer is exponentially distributed with an average of 2 minutes per customer. Concession customers wait in a single line in a large lobby, and arrivals are Poisson distributed with an average of 81 customers per hour. Previews run for 10 minutes before the start of each show. If the average time in the concession area exceeds 10 minutes, customers become dissatisfied.
a. What is the average utilization of the concession clerks?
b. What is the average time spent in the concession area?
The Severance Coal Mine serves six trains having exponentially distributed interarrival times averaging 30 hours. The time required to fill a train with coal varies with the number of cars, weather-related delays, and equipment breakdowns. The time to fill a train can be approximated by an exponential distribution with a mean of 6 hours 40 minutes. The railroad requires the coal mine to pay large demurrage charges in the event that a train spends more than 24 hours at the mine. What is the average time a train will spend at the mine?
SOLUTION
The problem statement describes a finite-source model, with N = 6. The average time spent at the mine is W = L[(N – L)]–1, with 1/ = 30 hours/train, = 0.8 train/day, and = 3.6 trains/day. In this case,