課程名稱︰系統效能評估 課程性質︰選修 課程教師︰周承復 開課學院：電資學院 開課系所︰資工所、網媒所 考試日期（年月日）︰ 考試時限（分鐘）： 試題 : Mid-term exam. Perf. 2014 1. (10%) Please show the inter-arrival time dist. of a Poisson process with rate λ is an exponential dist. with parameter λ. 2. (15%) 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 when the service time is exponential with mean 1/μ for job a and the service time is exponential with mean 1/λ for job b? 3. (25%) Consider the failure of a link in a communication network. Failures occur according to a Poisson process with rate 9.6 per day. Find a) P[time between failures ≦ 20 days] b) P[5 failures in 10 days] c) Expected time between 2 consecutive failures d) P[0 failures in next day] e) Suppose 12 hours have elapsed since last failure, find the expected time to next failure 4. (15%) Please show that, for a Poisson process with rate λ, its renewal equation m(t) = λt by the following definition t m(t) = F(t) + ∫ m(t-x) f(x) dx 0 5. (15%) The CPU time requirement X of a typical job can be modeled by the following hyerexponential distribution: -tλ_1 -tλ_2 P(X ≦ t) = α(1 - e ) + (1 - α)(1 - e ) where α = 0.6, λ_1 = 10, and λ_2 = 1. Compute (a) the probability density function of X, (b) the mean service time E[X], and (c) the variance of service time Var[x]. 1 - 3s 6. (20%) Find the function f(t) that satisfies F*(s) = ──────── (1-2s)(1-4s)^2 7. (10%) Customers arrive at a fast-food restaurant at a rate of 5 per minutes and wait to receive their order for an average 3 minutes. Customers eat in the restaurant with prob. 0.5 and carry out their order without eating with prob. 0.5. A meal requires an average of 20 minutes. What is the average number of customers in the restaurant? 8. (10%) For Geometric dist., please show that it has markov property or it is memoryless: P[x = i+n│x > n] = P[x = i], where P(i) = P(x=i) = p(1-p)^(i-1) --

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