Question: "Customers arrive at a sporting goods store at times of a Poisson process with rate 10 per hour. 60% of the customers are men and 40% are women. Women spend an amount of time shopping that is uniformly distributed on [0, 30] minutes, while men spend an exponentially distributed amount of time with mean 30 minutes. Suppose the store has been open for a long time. Find the joint distribution of X = the number of men and Y = the number of women shopping."
This doesn't make sense to me. If the rate of arrival of men is 6 per hour but the rate of departure for men is 2 per hour (mean 1/2 hour per visit = 2 visit/hour), wouldn't the long-run distribution of men explode?
Could someone please give me a hint on how to construct this joint distribution? Thanks a bunch!