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）︰2006.11.14 考試時限（分鐘）：120 試題 : Mid-term Exam. 11/14/2006 (120min) 1. Let 12 f_n - 11 f_(n-1) + 2 f_(n-2) = (1/2)^n, n=2,3,4,…. And f_0 = 0, f_1 = 4. Please find out fn, n=0,1,2,…. 2. Please prove the PASTA, i.e,. Poisson arrivals see the system as it is (averaged over all time). 3. Please prove that for an exponential distribution, it has Markov property or it is memoryless. 4. Consider a Poisson process {N(t), t ≧ 0} having rate λ, and suppose that each time an event occurs it is classified as either a type I or a type II. Suppose further that each event is classified as a type I event with probability p and a type II event with probability (1-p) independently of all other events. Prove type I process, {N_1(t), t ≧ 0 } and type II process {N_2(t), t ≧ 0} are both Poisson processes having rates λp and λ(1-p) respectively. 5. Consider a model of telephone switching system consisting of n trunks with a finite caller population of M callers and n < M. The average call rate of an idle caller is λ calls per unit time, and the average holding time of a call is 1/μ. If an arriving call finds all trunks busy, it is lost. Assuming that call holding times and the inter-call times of each caller are exponentially distributed. Find a. A: the expected total traffic offered by the M sources per holding time. b. C: the expected total traffic carried by the switching system per holding time. c. B: the call congestion probability or the probability that a call is lost. 6. Consider a single processor with an infinite waiting room. Customer arrivals are assumed to be Poisson with rate λ and service times are exponentially distributed with expectation 1/μ. Suppose that the processor fails at rate γ and, when failed, all of the customers in the system are assumed to be lost. The failed processor is fixed at an exponential rate of α. Please draw the state transition diagram and find the probability that there are i customers in the system. 7. Suppose we are told that exactly one job of a Poisson process with rate λ has submitted to our computer system by time t, please determine the distribution of the time at which the job is submitted. 8. 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? 9. Consider the failures of a link in a communication network. Failures occur according to a Poisson process with rate 4.8 per day. Find (i) P[time between failures ≦ 10 days] (ii) P[5 failures in 20 days] (iii) Expected time between two consecutive failures. (iv) P [0 failures in next day] (v) Suppose 12 hours have elapsed since the last failure. Then, find the expected time to next failure. --

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