1. Customers arrive at a watch repair shop according to a Poisson distribution at a rate of one per every 10 minutes and the service time is an exponential random variable with mean 8 minutes. Find Ls, Lq, Ws, Wq. Ls = 4; Lq=3.2, Ws = 40 min., Wq = 32 min. 2. In a case manufacturing plant, a loading crane takes exactly 10 minutes to load a car into a wagon and again comes back to the position to load another car. If the arrival of cars is a Poisson stream at an average rate of one after every 20 minutes, calculate the average time of a car in a stationary state. Wq = 10 min. 3. Arrivals at a telephone booth are considered to be Poisson with an average of 12 minutes between two arrivals. The length of phone call is assumed to be distributed exponentially with mean of three minutes. (a) What is the probability that a person arriving at the booth will have to wait? (b) The telephone department will install a second booth when convinced that an arrival would expect waiting for at least 3 minutes for a phone call. By how much should the flow of arrivals increase in order to justify a second booth? (c) What is the average length of the queue? (d) What is the probability that it will take more than 10 minutes altogether to wait for the phone and complete the call? (a) 0.25 (b) 0.1667 (c) 0.815 (d) 0.08 4. Customers arrive at a one-man shop according to a Poisson process with a mean inter arrival time of 12min. Customers spend an average of 10min in the shop. (a) What is the expected number of customers in the shop and in the queue? (b) Calculate the percentage of time an arrival can walk straight into the shop without having to wait. (c) How much time can a customer expect to spend in the shop? (d) What is the average time customer spends in the queue? (e) Calculate the percentage of customers who have to wait prior to getting into shop. (a) 5 & 4.17 (b) 16.7% (c) 60 min. (d) 50 min. (e) 83.3%