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 : Performance Evaluation 2003 Fall Midterm 1. Consider a normalized floating-point number in base (or radix) β so that the mantissa, X, satisfies the condition 1/β ≦ X ＜ 1. Experience shows that X has the reciprocal density: f_x(x) = k/x, k > 0. Determine: a. The value k b. The distribution function of X c. The probability that the leading digit of X is i as 1 ≦ i ＜ β 2. The CPU time requirement X of a typical job can be modeled by the following hyperexponential distribution: P(X≦t) = α(1 - e^(-tλ_1)) + (1-α)(1 - e^(-tλ_2)) 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]. 3. Consider a single-server queue with Poisson arrivals exponential service time with the following variation: Whenever a service is completed, a departure occurs only with probability α. With the probability 1-α, the customer, instead of leaving, joins the end of the queue. Note that a customer may be serviced more than once. a. Set up the balance equations and solve for the steady-state probabilities, stating conditions for it to exist. b. Find the expected waiting time of the customer from the time he arrivals until he enters service for the first time. c. What is the probability that a customer enters service exactly n times, for n = 1,2,3,…? d. What is the expected amount of time that a customer spends in service (which does not include the time he spends waiting in line)? e. What is the distribution of the total length of time a customer spends being served? 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 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. Memory Interference in Multiprocessor Systems: Consider a system with 2 memory modules and 3 processors. Assume that the time to complete a memory access is a constant (i.e. a memory cycle) and that all modules are synchronized. Processors are assumed to be fast enough to generate a new request as soon as their current request is satisfied. A processor cannot generate a new request when it is waiting for the current request to be completed. Assume that a processor-generated request is directed at memory module is q_1 = 0.4, and q_2 = 0.6. Let B be the number of memory requests completed per memory cycle. Find the average number of E[B]. 6. 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. The expected total traffic offered by the M sources per holding time. b. The expected total traffic carried by the switching system per holding time. c. The call congestion probability or the probability that the call is lost. --

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