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）︰2012/01/13 考試時限（分鐘）： 9:30~12:10 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Queueing Theory Final Exam 2012 1. Consider a FCFS M/M/1 queue with 2 classes of customers. The service rate for class-i is u_i , the arrival rate is λ_i for class i , and arrival processes for all clasddes are Poisson . Please answer the following questions if class-1 has non-preemptive priority over class-2 , with λ_1 / u_1 ＜ 1 ,but λ_1 / u_1 + λ_2 / u_2 ＞ 1. (i) What is the expected residual service time of the current customer in service upon the arrival instant of a new customer ? (5%) (ii) What is mean waiting time in queue W_q,1 for class-1 customers ? (5%) (iii) Can the Little's formula be applied to derivethe the mean number of class-1 customers of this queue (L_1)? Please explain (3%) (iv) What is the average number of class-2 customers served between two class-1 customer busy periods ? (Here a class-1 busy period is a period in which class-1 customers are served continuously) (5%) 2. Repeat the problem 1 , but the non-preemptive priority queue only allows at most 1 class-2 customers and the number of class-1 customers is still unlimited . Let (n_1 ,n_2) represents the system state,where n_i is the number of class-i customers in the system. λ_1/u_1 + λ_2/u_2 ＜1. (i) Please draw the system transition diagram. (6%) (ii) Please write down the global balance equations. (6%) (iii) What is the expected residual service time of the current customers in service upon the arrival instant of a new customer ? (5%) (iv) Please derive directly W_q,1 for an arbitrary class-1 customer. (5%) 3. Consider an M/G/1 queue with service time LST B^* (s) and arrival rate λ. Suppose a busy period is initiated by a special customer with service time H^* (s). The LST of the length distribution of this busy period is given by F^* (s) = H^* [s＋λ－λG^*(s)] , where G^*(s)=B^*[s＋λ－λG^*(s)]. Please us these equations to derive the expected busy period length E(F) , under such an initial customer.(12%) (Hint:take derivatives) 4. If you are a customer in M^[x]/D/1 queue ,with group arrival rate λ , mean service time b ,and the group size is equal to i with equal probability , for i=1,2,3,4,...L_max , where L_max id the maximum group size. (i) Please derive the expected group size that you are associated with. (6%) (ii) Please determine the moment generating function and expected vales of the number of arrived customers during your service time. (12%) 5. Consider an open Jackson queueing network with only 2 single server queues (node 1 and node 2) in series , with exponential service rate u_1 and u_2 for node 2 respectively. Suppose the external arrival rate for node i is γ_i. And all output form node 1 goes to node 2. Node 1 has unlimited buffer but node 2 has only K buffers. (including the one in service). If node 2 is full, all arriving customers are lost . Please dertermine joint steady state probability p_n1,n2 where n_i is the number of customers in node i. (12%) γ_1 γ_2 ↘ ↘ ───── ───── │ │○─────→ │ │○──→ ───── ───── 6. Consider a closed Jackson queueing network as shown in the following. There are 2 single server queues all with service rate u and 1 infinite server queue with mean delay d. Suppose there are 3 customers in the network. Please use the Mean Value Analysis to derive (i) the mean system size of each queueing node and (ii) mean cycle time. (iii) the customer arrival rate of each node (18%) Node 1 u Node 2 u ───── ───── →│ │○──────→│ │○───→ ↑ ───── ───── │ │ │ │ infinite server node │ │ ───── ↓ ← ○│ │←──────────────── ───── Mean service time = d --

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