Many systems in operations management can be formulated as waiting line models (also called queuing models). Most common situations are in service systems, such as banks, medical clinics, and supermarkets. But some waiting line situations are embedded in manufacturing systems, as when mechanics go to a tool crib to obtain the tools necessary for the next job or in the typical scheduling of a machine for a variety of jobs. In all these situations, someone or something must wait in line for service. The service is provided by a processing system, followed by a departure process our typical input–output processing model for a productive system.
In structuring waiting line models, the inputs are called “arrivals”, and arrival times are controlled by some probabilistic process. The time to process or service the arrivals is also controlled by a probabilistic process. The output rate of such systems depends on the interplay between the random arrivals and the variable service times, and waiting line models are used to predict these values. A common example is found in a bank, where the arrival of customers to do business at a teller window is random. Each individual deciding when to go to the bank. The length of the waiting line depends in part on these arrival times. Equally important is the time it takes for a teller to service each individual. These times vary because the time for the human performance of a task is variable and, more important, because each customer requires different services. For instance, the first customer may simply want to make a deposit, whereas the second may want to make a deposit obtain cash and pay the gas bill.
The following example of a university outpatient clinic provides a background of the nature of the data needed and the service system design problems involved.
The example outpatient clinic at the University of Massachusetts treated an average of 400 to 500 patients per day with a staff that included 12 full time physicians. Because the physicians had a variety of other duties, only 260 physician hours per week were available during regular clinic hours, about 22 hours per physician per week. Only about half of the patients were seen by a physician. The others were treated by nurses under a physician’s supervision or in specialized sub-clinics for tests or immunizations.
In aggregate terms, approximately 178 patients per day needed access to an average of 52 available physician hours. Thus, the average time with a physician was about 17.5 minutes.
Demand of Services:
Demand is not uniform through the week being approximately 20 percent above the average on Mondays, 84 to 88 percent of the average on Thursdays, and increasing slightly on Fridays.
Furthermore, the daily variation is significant. A figure showing arrival data from Monday and Thursday (the days with the heaviest and lightest loads, respectively), highlighting the great demand variation during the day, with peaks at 8 am, 10am and 2pm can be graphically represented. When the arrival data are placed on an inter-arrival time basis (the time between arrivals), they exhibit a negative exponential distribution.
Time for Service:
The amount of time physicians spent with patients was measured in three separate categories: walk in, appointment, and second service times. Histograms show the service times recorded for the three categories. The second service category represents a return of the patient to the physician following diagnostic tests or other intervening procedures. Although the three distributions are different, they share the common general properties of being skewed to the right and having relatively large standard deviations. Thus the average appointment service time is only 12.74 minutes, but the standard deviation is nearly 10 minutes and the maximum recorded time is 40 minutes. These typical service time distributions reflect the variety of tasks involved in a consultation, depending on the nature of the patient’s complaint.