Georgia Institute Of Technology ISYE 6644 hw1f17solns-QUESTIONS WITH VERIFIED ANSWERS

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. (Deterministic Model.) Suppose you throw a rock off a cliff having height h0 = 500 feet. You’re a strong bloke, so the initial downward velocity is v0 = −100 feet/sec (slightly under 70 miles/... hr). Further, in this neck of the woods, it turns out there is no friction in the atmosphere | amazing! Now you remember from your Baby Physics class that the height after time t is h(t) = h0 + v0t − 16t2: When does the rock hit the ground? Solution: Set 0 = h(t) = 500 − 100t − 16t2; and solve for t. Quadratics are easy: t = −b ± pb2 − 4ac 2a = 100 ± p1002 + 4(16)(500) 2(−16) = −100 ± 204:9 32 = 3:279; which we take as the answer since the negative answer doesn’t make practical sense.  2. (Stochastic Model.) Consider a single-server queueing system where the times between customer arrivals are independent, identically distributed Exp(λ = 2/hr) random variables; and the service times are i.i.d. Exp(µ = 3/hr). Unfortunately, if a potential arriving customer sees that the server is occupied, he gets mad and leaves the system. Thus, the system can have either 0 or 1 customer in it at any time. This is what’s known as an M/M/1/1 queue. If P(t) denotes the probability that a customer is being served at time t, trust me that it can be shown that P(t) = λ λ + µ + P(0) − λ +λ µe−(λ+µ)t: If the system is empty at time 0, i.e., P(0) = 0, what is the probability that there will be no people in the system at time 20 minutes? [Show More]

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