課程名稱︰系統效能評估 課程性質︰資工系研究所選修 課程教師︰周承復 開課學院：電機資訊學院 開課系所︰資工系 考試日期（年月日）︰2016.4.20 考試時限（分鐘）：120 (mins) 試題 : 1. (25%) Consider a system with two computers. Two jobs: a and b are submitted to that system simultaneously. Job a and b are assigned directly to each computer. What is the probability that job a is still in a system after job b finished? i. (5%) the service time for each job is exactly 15 mins ii. (5%) the service time for each job are i mins with prob. 1/3, i = 1, 2, 3 iii. (5%) the service time for job a is exponential with mean 20 mins, the service time for job b is exponential with mean 10 mins iv.(10%) the service time for job a is uniformly distributed [0, 20 mins], and the service time for job b is uniformly distributed [0, 10 mins] 2. (10%) Consider the failure of a link in a communication network. Failures occur according to a Poisson process with rate 2.4 per day. Find a) P[time between failures <= 15 days] b) P[7 failures in 10 days] c) Expected time between 2 consecutive failures d) P[0 failures in next day] 3. (15%) Packets arrive at a transmission facility according to a Poisson process with rate lambda. Each packet is independently routed with probability p to one of two transmission lines and with probability (1 - p) to the other. Show that the arrival processes at the two transmission lines are Poisson with rate lambda * p, lambda * (1 - p), respectively. 4. (25%) Consider a sustem with 2 components. We observe the state of the system every hour. A given component operating at time n has prob. 0.4 of failing before the next observation at time n + 1. A component that was in a failed condition at time n has a probability 0.8 of being repaired by time n + 1, independent of how long the component has been in a failed state. The component failures and repairs are mutually independent events. Let x_n be the number of components in operation at time n {x_n, n = 0, 1, 2, ...} is a discrete-parameter homogeneous Markov chain with the state space I = {0, 1, 2}. Determine its transition probability matrix P. and draw the state diagram. Obtain the steady-state probability vector if it exists. 5. (10%) Let A(t) and Y(t) denote respectively the age and excess at t. Find P{Y(t) > x | A(t + x) > s} for a Poisson process. 6. (15%) f''(t) - 4f'(t) + 3f(t) = t, f(0) = 0, f'(0) = 0; find out f(t) 7. (15%) Y = X_1 + X_2 + ... + X_N_tilde X_i are i.i.d. exponentially distributed, N_tilde is geometrically distributed. (1) (5%) Find E[Y] (2) (18%) What is Var[Y] --

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