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Question 1.1 (10%) Next, the drone model chosen for each location will determine that location's effective 5G- covering area. Shown in the following figure, if drone model is chosen for location m , then Dm will equal radiusi ; but if a different drone model in dronesm  is chosen, Dm could have a different value. For each location m in the locations set, please develop an equation to determine the real value of Dm.

Question 1.2(10%) To make sure that SG signals will cover the entire area, we must enforce a specific overlap(特 定程度的重疊) on the two areas covered by any two nearby locations. Shown in the following figure, if m and n are two such locations, then the sum of Dmand Dn must be at least 1.5 times of the distance between the two locations (no matter the drone model used in each location to emit 5G signals). Please develop an equation to enforce that requirement in the model.

Question 1.3 (15%) Finally, please develop an objective function for the MIP model that will minimize the total costs to choose an appropriate drone model for every location in the locations set.

2. (15% in total) Consider the following linear system with its coefficient matrix A given right next to it. What is the basic feasible solution corresponding to the first four columns of A? That is, use the first four columns of A as the basis to find a basic feasible solution(求解A 的前四行所對應到 的基本可行解). **Note**: You must use linear algebra related mathematics to find your answer 求解必需使用與「線性代數相關」的數學計算,否則不予計分).

(a) What are the absorbing states of this Markov chains? (4 points)

(b) What is the trans sition probability matrix of this chain? (10 points)

(c) Let T = {1,... , c - 13} is a finite set of transient states and xj  is the probability that player 1 wins given T. Write down the systems of equations that xj need to satisfy. (10 points)

(a) (6 points) Consider two independent M/M/1 queues, the first one with arrival rate λ and service rate μ1; and the second one with arrival rate  λ and service rate μ2. What is the probability that there are n customers at server 1? What is the probability that there are m customers at server 2 ?

(b) Consider a two-server system in which customer inter-arrival times are exponentially distributed with rate λ at server 1. After being served by server 1, customers then join the queue for server 2. We assume that there is infinite waiting space at both servers. Each server serves one customer at a time with server i taking an exponential time with rate μi for a service, i = 1,2. To analyze this system, we need to keep track of the number of customers at server 1 and the number of customers at server 2. Let define the state by pair (n,m)- meaning that there are n customers at server 1 and m customers at server 2 and denote the probability of being in that state. (b-1) (10 points) What are the balance equations for this system? (b-2) (S points) If the number of customers at server 1 and server 2 were independent random variables, what would be the expression for  ? (b-3) (S points) Verify that your solution satisfies the balance equations from (b-1).

b)_________is the boundary around the mean of a set of replicate analytical results within which the population mean can be expected to lie within a certain probability.

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