Question: A facility management company has recently acquired a number of new commercial properties within the city of Greenfield. They are faced with the problem of hiring a number of on-call janitors, who are responsible for fixing emergency problems that arise at these different facilities. The question is how many locations to assign to each janitor. The janitors work independently on the properties that they are responsible for serving. When the janitor gets a call from a location other than the one he or she is at, the request is placed in a queue and served on a FIFO basis. When visiting a certain location, the janitor will attend to all matters at this facility. This means that a location that has called for the janitor will not call again even if new matters arise. We can therefore consider a facility that has placed a call to the janitor as "nonoperational." If there are no matters that need janitorial attention, the facility is considered fully operational. For each fully operational facility, the times between emergency calls to the janitor are exponentially distributed with a mean of 2.5 h. The time it takes for the janitor to fix the problems at a location is exponentially distributed with a mean of 20 min, including travel time. The facility management company has determined service-level criteria, specifying that at least 75% of the time the facilities should receive immediate emergency service from a janitor.

a. Identify a queuing model that adequately describes this process of emergency janitorial assistance. Use the corresponding steady-state formulas to determine the largest number of facilities that can be assigned to a janitor without violating the specified service requirement.

b. Given the maximum number of facilities assigned to each janitor in (a), what is the expected fraction of time that a janitor is busy with emergency calls? What is the expected number of facilities not fully operational? What is the expected waiting time before a facility gets emergency assistance?