Stochastic Process and Queueing theory

. The arrival of trucks carrying tomatoes to a cannery is Poisson at rate A. Arriving trucks are weighted at scales #1, unloaded into a bin, weighted at scales #2, and they depart. A load of tomatoes arrive at the bin when the truck they were on completes unloading. Unloaded tomatoes are fed immediately to a canning line: Weighing, unloading, and canning times (of a load) are independent, where weighing times are N Emma), unloading times are N Emp(oz), and canning times are N Emmy). (a) (b) (g) Under what conditions is the entire system stable? Assume these conditions hold below. Notice that the ow of trucks is not affected by unloaded tomatoes. Use this observation to write down the productform solution for the distribution of the number of trucks at each of the scales and the bin. Notice the empty trucks have no effect on the ow of tomatoes. Use this observation to write down the productform solution for the number of trucks at the rst scales and the bin, and the number of loads of tomatoes at the canning line. Would you expect the entire fourstation system to have a productform solution? If trucks are all the same size and are full on arrival, we would expect that the time to can a load of tomatoes would be constant. Analyzing the canning line under this assumption. It has been proposed to combine the two singlechannel scales into one twochannel station that would serve both empty and full trucks in their order of arrival. What is the productform solution corresponding to (b)? Compares this proposal with the original mode of operation, from the point of view of trucks. [Hint We are pooling. (Why is the truck process no longer a Jackson network? How come we can still solve it?)] How would the proposal in (f) complicate the analysis of the canning line