Section 3.6, Exercise 17, introduced the geometric distribution with parameter 0
implying that E[H] = 1/γ and Var[H] = (1 – γ)/γ2
. Suppose that a Markov chain enters state i, a state for which 0
ii
ii.
Exercise 17
An automated guided vehicle (AGV) transports parts between four locations: a release station A, machining station B, machining station C, and an output buffer D. The movement of the AGV can be described as making trips from location to location based on requests to move parts. More specifically:
• If the AGV is at the output buffer, it is equally likely to move next to any of the other three locations, …, …, C.
• If the AGV is at the release station, it is equally likely to move next to machining station B or C.
• If the AGV is waiting at either of the machining stations, it is equally likely to move to any of the other three locations.
(a) Develop a model of this system that is capable of answering the questions below. Be sure to define the states, time index, and one-step transition matrix.
(b) If the AGV is currently at machining station B, what is the probability that it will be at the output buffer after five trips?
(c) What is the long-run fraction of time the AGV spends traveling to the output buffer?