The number of people who enter an elevator on the ground floor is a Poisson random variable with mean 10. If there are N floors above the ground floor, and if each person is equally likely to get off at any one of the N floors, independently of where the others get off, compute the expected number of stops that the elevator will make before dis-charging all of its passengers.

2. If electricity power failures occur according to a Poisson distribution with an average of 5 failures every 20 weeks, calculate the probability that there will not be more than one failure during a particular week.

A large ski mountain has a fleet of 4 snowmobiles it uses during the winter season. It believes that the time between consecutive repair operations for a single snowmobile is an exponential random variable with a mean of .75 weeks. There are two dedicated technicians that are capable of repairing the snowmobiles. A broken down snowmobile will be repaired by one of these technicians (when one becomes available) and the repair time of a snowmobile can be modeled as an exponential random variable with a mean of .4 weeks. (a) Describe a queueing system model of the repair operations of the snowmobiles by answering the following questions: What are the customers in the queueing system? What are the servers and what service is being provided? b) Provide the rate transition diagram of the birth and death process that captures the queueing system. Be sure to provide λn and µn for each relevant value of n and your logic behind any calculations used to obtain them (c) Provide the…

Data from 14 cities were combined for a​ 20-year period, and the total 280 ​city-years included a total of 107 homicides. After finding the mean number of homicides per​ city-year, find the probability that a randomly selected​ city-year has the following numbers of​ homicides, then compare the actual results to those expected by using the Poisson​ probabilities: