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）︰2011/01/14 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 2011 1. Please show that for a 2-class FCFS M/M/1 queue , with arrival rate λ_i customer/min , average service time 1/u_i minfor class-i , its mean system size is large than a single class FCFS M/M/1 queue with arrival rate λ_1＋λ_2 ＝λ and mean service time (λ_1/λ)/u_1 ＋ (λ_2/λ)/u_2 . (14%)(detail proof is required) 2. Consider a non-preemptive 2-class priority M^[x]/M/1 queue with constant batch size c=2 , mean service time 1/u and arrival rate λ_1 and λ_2 for class-1 and class-2 respectively. Please answer the following questions. (i) What is average length of its busy period and what is average length of its idle period ? (8%) (ii) What is the utilization of the server ? (4%) (iii) What is the mean waiting time of a class-1 customer if it is a batch leader and the mean waiting time of a randomly selected class-1 customer ? (6%) (iv) Repeat (iii) for class-2. (6%) (Hint : modify the waiting ime formula of multi-class M/E_2/1 queue , which is similar to multi-class M/M/1) 3. Consider an M/M/infinit queue with arrival rate λ customer/min , ave. service time 1/u min.. (i) Please write down the transition matric of its embedded markove chain , i.e., The markov chain formed by the system sizes observed upon arrivals or departures. (10%) (ii) Is the output process of M/M/infinite queue a Poisson process? Proof or explanation is required.(10%) (iii) Please calculate the average system size L in stationary. (4%) 4. Comparing M/E_2/1 and M/D/1 queue with the same arrival rate λ and mean service time b, please answer true or false fot the following questions. Explanation and derivation is required for all questions. (i) M/D/1 always has longer average waiting time in queue(W_q). (ii) M/D/1 has higher server utilization. (iii) M/D/1 has longer mean busy period. (iv) M/D/1 has larger variance of system size. (v) M/E_2/1 queue has longer residual service time for the current customer in service if the system is observed in random. (vi) M/E_2/1 and M/D/1 have the same maximum throughput. (24%) 5. Consider a close Jackson queueing network, in which there are only 2 queueing nodes ,node 1 and node 2 , and there is no external traffic source. Assume that there are only 2 customers in the network and the routing probability r_11=p_1 r_21=1 , and service rate is u for both nodes. (i) Please writs its state balance equation, assuming the state is (n_1,n_2) (ii) Please derive the customer arrival rate at node 2 .(24%) ───── ───── ──→│ │○────────→│ │○──→ ↑↑ ───── ↑ ───── │ ││ │ │ │ ────────── │ │ P1 │ │ │ ──────────────────────────── --

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