Problem 1: Two kinds of raw materials arrive at two manufacturing stations following Poisson processes with rates and in, which have a single machine producing components following exponential times with parameters Hy and ly, respectively. Produced components in station 1 are sent to assembling station (station 3). A produced component in station 2 will be matched with a component at station 3 if there is a component waiting there for assembling. The assembling time in station 3 is assumed to be zero, that is, one component in station 3 disappears immediately in matching a component from station 2. However, if a component is produced in station 2 while there are no components in station 3, it is sent to station 4 for producing different products, which has a single machine producing the products following exponential time with parameter 14. Suppose all stations have infinite capacity. Station 1 Exp(ui) Station 3 PPD) Departure Station 2 Exp(uz) PP2) 1 Station 4 Exp(us) a) Show that the departure process from station 1 is a Poisson process. b) Analyze the stable condition of the queueing network. c) Analyze the limiting distribution of station 3, given that the stable condition is satisfied. Problem 1: Two kinds of raw materials arrive at two manufacturing stations following Poisson processes with rates and in, which have a single machine producing components following exponential times with parameters Hy and ly, respectively. Produced components in station 1 are sent to assembling station (station 3). A produced component in station 2 will be matched with a component at station 3 if there is a component waiting there for assembling. The assembling time in station 3 is assumed to be zero, that is, one component in station 3 disappears immediately in matching a component from station 2. However, if a component is produced in station 2 while there are no components in station 3, it is sent to station 4 for producing different products, which has a single machine producing the products following exponential time with parameter 14. Suppose all stations have infinite capacity. Station 1 Exp(ui) Station 3 PPD) Departure Station 2 Exp(uz) PP2) 1 Station 4 Exp(us) a) Show that the departure process from station 1 is a Poisson process. b) Analyze the stable condition of the queueing network. c) Analyze the limiting distribution of station 3, given that the stable condition is satisfied