University:
Massachusetts Institute of Technology
Course:
15.501/516 | Accounting
Academic year:

2023
Views:

105

Pages:

2
Author:

Rigoberto Hale

Take home ﬁnal exam Note: The work must be done individually. Problem 1 A device consists of n main units, all of which must be operational for the device to be operational. Successive failure times of the main units are exponentially distributed with rate λ. There are m + k additional units, m of which are active, that is their failure times have the same distribution as the main units, while the remaining k are passive and cannot fail. Failed units are sent for repair. The service time distribution is exponential with rate µ. If some of the main units fail, they are replaced by active units, and these in turn are replaced by passive units. Find the probability that the unit is operational. Problem 2 Consider a closed single class queueing network with N jobs, and let π + (x − e1 ) be the probability that the system is in state x − e1 at the departure epoch of a job from node 1. Note that we do not count the departing job. Prove that π + (x − e1 ) = πN −1 (x − e1 ), where πN −1 (x − e1 ) is the probability that the state is in x − e1 for a closed queueing network with N − 1 jobs. Problem 3 Consider a ﬂuid model (α, µ, P, C). Establish that every server σj , 1 ≤ j ≤ J empties eventually in ﬁnite time. Namely, establish that for every time t and j = 1, 2, . . . , J � there exists a time τ > t such that k∈σj lk (τ ) = 0. Why does not this imply that the ﬂuid model is stable? HINT. Consider the workload Wj (t) corresponding to a given server σj . Problem 4 Consider a ﬂuid model (α, µ, P, C) with initial ﬂuid level l(0) = (l1 (0), . . . , lN (0)). Find a ﬂuid solution (l(t), u(t)), not necessarily workconserving, which empties in shortest time, and express the emptying time τ ∗ in terms of α, µ, P and l(0). The emptying time of a ﬂuid solution (l, u) is � τ (l, u) � inf {t : �l(t)� = lk (t) = 0}. 1≤k≤N Thus you need to ﬁnd inf τ (l, u) over all (not necessarily workconserving) ﬂuid solutions (l, u). HINT: First obtain a lower bound on the shortest emptying time and then show that there exists an optimal solution with constant u which achieves this lower bound. 2 Take home ﬁnal exam Problem 5 (Extra credit) Consider single server multiclass ﬂuid model (α, µ, P ) (J = 1). Suppose there is a cost c = (c1 , . . . , cN ) ≥ 0 associated with the ﬂuid level l(t). Given a solution (l, u) the associated cost is � ∞ c� l(t)dt. 0 Note that the cost is ﬁnite provided that the ﬂuid solution is stable. Find a ﬂuid solution (l∗ , u∗ ) which minimizes the cost.