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 : Midterm Exam 2005 1. (20%) Derive the stationary distribution of an M/M/2 system where the two servers have different service rates μ_l and μ_h respectively. The arrival process is Poisson arrival with rate λ. A customer that arrives when the system empty is routed to the faster server, i.e., the server with service rate μ_h. 2. (20%) A telephone company establishes a direct connection between two cities expecting Poisson traffic with rate 30 calls/min. The duration of calls are independent and exponentially distributed with mean 3 min. Inter-arrival times are independent of call durations. How many circuits should the company provide to ensure that an attempted call is blocked (because all circuits are busy) with probability less than 0.01? It is assumed that blocked calls are lost (i.e., a blocked call is not attempted again). 3. (20%) In our system, there are k machines and 2 repairmen. Each machine breaks down after a time that is exponentially distributed with parameter α. When a breakdown occurs, a request is sent to the repairman for fixing it. Requests are buffered. It takes an exponentially distributed amount of time with parameter μ for the repairman to repair a machine. What is the probability p(i) that i machines are up? What is the overall failure rate? 4. (15%) Packets arrive at a transmission facility according to a Poisson process with rate λ. 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 λp and λ(1 - p), respectively. 5. (25%) Consider a system with 2 components. We observe the state of the system every hour. A given component operating at time n has prob. 0.2 of failing before the next observation at time n+1. A component that was in a failed condition at time n has a probability 0.6 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. 6. (10%) Customers arrive at a bank at a Poisson rate λ. Suppose 3 customers arrived during the first hour. What is the probability that (a) All customers arrived during the first 30 mins? (b) At least one arrived during the first 30 mins? --

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