Posted: March 29th, 2022

3. Suppose that N light bulbs are used in lighting a particular auditorium on campus. However, the university employee responsible for the lighting of the auditorium does not keep spare light bulbs on hand when bulbs burn out and need to be replaced. Instead, the employee s, at the beginning of each week, new light bulbs to replace the ones that burned out during the preceding week. These ed light bulbs are delivered to the university and are ready for use at the start of the following week. For ne Z, let X, be the number of light bulbs in operation at the beginning of the nth week, and let Y, be the number of light bulbs that burn out during the nth week. For ne Zt, assume that Y,(X) DU(0,r) for x=0,1,…,N At the beginning of the first week, suppose that all N light bulbs in the auditorium are properly functioning.

(a) Explain why {Xn Z) is a DTMC and specify the initial probability row vector for this DTMC.(b) Specify the elements P, of the TPM associated with the DTMC {Xn € 2¹).(c) Find the limiting probabilities of the DTMC (X,Z} if they exist. Justify your response.(d) A decision on whether to have a spare inventory of light bulbs on campus is being considered. The decision to support having such an inventory will be based on whether, in the long run, the expected proportion of operational light bulbs at the beginning of a week is less than 50%. Will the university proceed with plans to have a spare inventory of light bulbs on campus? Justify your response.

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