There are three markets, M1, M2, and M3, owned by a single firm. Depending on their previous experience, customers will either shop at the same market that they shopped at during their last visit or they will switch to one of the other markets. If a customer currently shops at M1, the probability that they will continue shopping at M1 on the next trip is 0.7, while the probability of switching to M2 is 0.2, and the probability of switching to M3 is 0.1. If they currently shop at M2, then the probability of shopping at M2 on the next trip is 0.6, while the probability of switching to M1 is 0.3, and the probability of switching to M3 is 0.1. Finally, customers who currently shop at M3 will return to M3 on their next trip with a probability of 0.8, switch to M1 with a probability of 0.1, and switch to M2 with a probability of 0.1. The Operations Manager is concerned with the time that customers must spend waiting in line at each of the three markets. We expect a total of 100 customers per hour to arrive at the markets. These 100 customers per hour are sent to M1, M2, and M3 based on the steady-state probabilities of M1, M2, and M3. For example, if the steady state probabilities for M1, M2, and M3 are 0.6, 0.3, and 0.1, respectively, then the arrival rates to M1, M2, and M3 are 60, 30, and 10 customers per hour, respectively. Dividing by 60, we can translate these rates to arrivals per minute, giving 1, ½, and 1/6 customers per minute. Each market has a single server, but the actual probability distribution of service times at the markets is unknown. We do know that M1 serves an average of 20 customers per minute with a standard deviation of 5, while M2 serves an average of 15 customers per minute with a standard deviation of 5, and M3 serves an average of 20 customers per minute with a standard deviation of 3.
Find the average time spent in line for each of the three markets