State University allows students and faculty to access its mainframe computer by modem. The university has 15 modem connections that can be used. When all of the modem connections are in use, the phone system can keep up to 10 callers on hold waiting for a modem connection to become available. If all 15 modem connections are in use and 10 calls are already holding, a new caller receives a busy signal. Calls to the modem pool follow a Poisson distribution and occur at an average rate of 60 per hour. The length of each session with the mainframe is an exponential random variable with a mean of 15 minutes—therefore, each modem services an average of four callers per hour.
a. On average, how many callers are on hold waiting for a modem connection?
b. On average, how long is a caller kept on hold before receiving a modem connection?
c. What is the probability that a caller receives a busy signal?
d. How many modem connections would the university need to add to its modem pool for there to be no more than a 1% chance of a caller receiving a busy signal?