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University of Illinois Spring 2017 ECE 313 (Section G) In-Class Activity 5 Solution - Wednesday, Apr 5th Write your name and NetID here: ___________________________________ ___________________________________ Question 1 (75 points): Consider the following storage system composed of two subsystems S1 and S2. S1 is the primary system composed of three disks A, B, and C, where the failure of any of them will cause the failure of the subsystem S1. S2 is composed of a disk D which acts as a backup (cold standby) of the subsystem S1 and will be powered on only after the primary subsystem fails. Assume that A, B, and C are identical disks and the switching circuit S is perfect. We model the lifetime of the disks A, B, C, and D with three independent random variables X1, X2, X3 and X4. Assume X1, X2 and X3 are exponentially distributed with parameter š. Part A (25 points) ā Find the reliability function and failure rate (š1) of the primary subsystem (S1). Reliability function for each of disk A, B, and C: eāšt From reliability point of view, A, B and C are in series: R S1 = R A R B R C = eā3št (15 points) Therefore, failure rate š1 = 3š (10 points) Part B (25 points) ā If X4 is exponentially distributed with parameter 4š . What distribution best models the time to failure of the whole system? Use the results of part A to derive the reliability function and instantaneous failure rate of the system in terms of š. The time to failure is sum of two independent sequential phases with different parameters. Therefore, a 2-stage Hypo-Exponential with š1 = 3š and š2 = 4š (10 points) š (š”) = 1 ā š¹(š”) = š2 š1 š āš1 š” ā š āš2 š” = 4eā3šš” ā 3š ā4šš” š2 ā š1 š2 ā š1 (7 points) š(š”) Hence, the instantaneous failure rate ā(š”) = š (š”) = 12š(eā3šš” āš ā4šš” ) 4eā3šš” ā3š ā4šš” (8 points) Part C (25 points) ā If X4 is exponentially distributed with parameter 3š . What distribution best models the time to failure of the whole system? Derive the reliability function and instantaneous failure rate of the system in terms of š. With š2 = 3š, we have a 2-stage Erlang Distribution with š = 3š Therefore, š (š”) = 1 ā š¹(š”) = eā3št (1 + 3šš”) for t > 0 (9š2 š”)š ā3šš” Then, h(t) = (1+3šš”)š ā3šš” = 9š2 š” 1+3šš” for t > 0 (10 points) (7 points) (8 points) Question 2 (25 points): Consider a communication network in which packets are being sent from node N1 to node N2. The packet can take one of two paths that exist between N1 and N2. The time to transfer is exponentially distributed with the parameter dependent on the path i.e. š1 for path 1 and š2 for path 2. The path to be taken is decided by tossing a coin with probability of heads š. The packet takes path 1 if heads turns up and path 2 if tails turns up. What is the distribution of the time of transfer of a packet from N1 to N2? Derive the density function of the distribution. In this problem, the process consists of alternate phases (alternate paths between N1 and N2), i.e. during any single experiment the process experiences one and only one of the many alternate phases (a packet can take only one of the two paths). Both these paths have an exponential distribution. Therefore, it is a two-phase hyper-exponential distribution. (15 points) The pdf is given by: š(š”) = š š1 š āš1 š” + (1 ā š) š2 š āš2 š” (10 points) 2