課程名稱︰排隊理論 課程性質︰選修 課程教師︰蔡志宏 開課學院：電資學院 開課系所︰工業工程/電機/電信 考試日期（年月日）︰2011/11/11 考試時限（分鐘）：130分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1. Consider an M/M/1 queue with service rate u and two Poisson arrival processes. Suppose the arrival rate is λi from source i,i=1,2.Please answer the following questions. (i) If the previous arriving customer is from source 1 , what is the probability that the next arrivals are still both from source 1 ? (3%) (ii) If the previous arriving customer is from source1 and that event happened before time 0 , what is the probability that the next arrival is from source1 and it will occur within (0,t)? (3%) (iii) What the probability that only one customer is served in a busy period of this queue ,if the busy period is observed from an arbitrary arriving customer who initiated this busy period ? (3%) (iv) Can PASTA property be applied to customers from both sources and the same mean system size is observed by them? (3%) (v) If the server is observed at a random time epoch,what is the probability that the server is busy and the current customer is from source 2 ? (3%) 2. Consider an M/M/c/c queue representing a small telephone switch,with call arrival rate λ call/min.,mean call duration equal to m seconds,and c=4 (i.e. with 4 voice channels). (i) Please draw the state transition diagram.(5%) (ii) Please calculate the traffic intensity,L and W of this queue.(9%) (iii) If these c servers are indexed from 1 to c , and a call is always assigned to the idle channel with smallest index, what is the probability that a randomly selected channel is idle ? (3%,derivation required) (iv) What is its mean duration for staying in a blocking state?(3%) 3. For a finit source multi-server Markovian queue with 2 servers and 3 sources. Suppose the arrival rate from each source is λ and service rate is u for each server. (i) Please obtain the effective mean arrival rate λeff.(6%) (ii) Please derive draw the state transition diagram if the state represents the numbers of customers in the system? (5%) (iii) Please write the generator matrix of the corresponding continues time Markov Chain. (7%) (iv) Please derive the transition matrix of the embedded Markov chain, when observing the system only at arrival or departure events. (7%) 4. Comparing M/E2/1 and M/M/1 queue. Assume that the arrival rate is λ,and the mean service time is 1/u for both queue. Please compare them and answer the following questions: (i) Which queue has smaller L? (ii) Which queue has smaller W? (iii) Which queue has smaller Lq? (iv) Which queue has smaller Wq? (v) Which queue has longer mean busy period length? (20%, derivation required for all answers) 5. Consider an M^[x]/M/1/k queue and if an arriving group cannot be completely accepted,the whole group of customers are all rejected. Suppose the group size is either 1 or 2,with probability c1 and c2 respectively.Let the batch arrival rate be λ,and the service rate be u, and k=3. (i) Please draw the state transition diagram.(5%) (ii) Please write the global balance equations for all states(6%). (iii) Please determine whether the system size process is a birth-and-death process. (iv) Please calculate the formula for λeff , the effective customer arrival rate (6%). ------------------------------------------------------------------------------ --

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