Probability distributions and their applications

Copying Statistics

Probability Distributions and Copier Repair: An Analysis and Presentation of a Pending Acquisition

When the number of copiers that this company maintains increases, it only makes sense that the number of technicians employed by the company to make service calls also increases. Due to the random and essentially unpredictable nature of these service calls, a balance needs to be achieved between hiring enough technicians to ensure speedy (i.e. same-day) service, and not hiring so many that a large number of them sit idle for long periods, requiring the company to pay them when they aren't performing any work. There is no "right" answer to this problem; it is fairly simple to determine what the expected average number of repairs per day will be, and thus to determine the average number of necessary technicians, but hiring only this number will leave the company without enough technicians to handle anything but an average day. Obviously, more technicians should be hired, but the more technicians over the average requirement the company employs, the more likely they are to end up sitting around unneeded. Ultimately, determining the correct balance is up to you, but I can certainly assist.

A quick look at the expected averages of repairs calls and needed technicians now and after the acquisition helps to frame the issue. With 2105 office-use copiers that break approximately once every fifty days, the average day will see forty-two service calls (2105/50=42.1). Dividing this by the eight repairs an office-use copier technician can typically repair in a single day means that six office-use copier technicians are needed on the typical day, with some off-time already present (42/8=5.25). Changing the average rate in which a copier needs servicing to forty or sixty days, as has been suggested, yields a need for seven and five technicians, which still produces a mean of six, but an average of forty-three daily repairs, which necessitates seven technicians.

For the 386 production-use copiers, which break down more often (approximately every twenty days) and take longer to service (technicians can complete four service calls in a typical day), the average number of repairs and technicians a day ranges from four technicians and sixteen repairs to six technicians and twenty-four repairs. Assuming that the company wishes to err on the side of providing excellent service rather than cost saving, a certain number of technicians over the average amounted of expected necessary technicians should be hired. Assuming that the number of service calls follows the normal distribution -- which almost all truly random probability distributions do -- the standard deviation can be calculated from the numbers that produced the range of expected repairs above. This, in turn, can tell us how likely it will be to experience higher than average volumes of service calls on any given day.

The standard deviation for office-use copier repairs is nearly nine, while for professional-use copiers it is about four. According to the normal distribution, there is an 84% chance that an outcome will be lower than one standard deviation higher than the average. For example, adding the standard deviation nine to the average of forty-three repairs gives us fifty-two repairs; on only 16% of working days would we expect to see a higher rate of service calls. Therefore, seven technicians can handle the work 84% of the time -- an adequate number. Applying this same method to the professional copier situation reveals that twenty-four service calls or less will come in 84% of the time, and six technicians can cover this. So a total of thirteen technicians seven for office use and six for professional-use copiers, ought to be plenty to ensure speedy service to customer without employing too many idle technicians.