课程名称︰排队理论 课程性质︰选修 课程教师︰蔡志宏 开课学院：电资学院 开课系所︰工业工程/电信/电机 考试日期（年月日）︰2009/01/18 考试时限（分钟）：9:30 ~ 12:10 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Queueing Theory Final Exam 2009 1. Consider an M^[x]/D/1 queue with constant batch size c=2 , constant service time b , and arrival rate λ.Please answer the following questions. (i) What is average length of its busy period and what is average length of its idle period ? (10%) (ii) What is the transition probability matrix P = {p_ij} of its imbedded Markov Chain if the system is observed upon departure or an idle period is ended ? Hint : p_ij is the probability that give the current system size is i , and a transition occurs ,the next state is state j. (10%) 2. Consider an M/E_k/infinite queue with arrival rate λ customer/min., ave. service time 1/u min. (i) If one randomly selects a customer in the queue when the queue is in stationary ,what is its average residual service time ? (5%) (ii) If there is no customer at time 0 , what is the probability that there are n customers in the system at time t ? (10%) (iii) Please calculate the average system size L in stationary,(5%) 3. Comparing M/M/1 queue and M/D/1 queue with the same arrival rate λ and mean service tome b , please answer true or false for the following questions. Explanation and derivation is required for all questions. (i) M/D/1 always has longer average waiting time. (ii) M/D/1 has high server utilization. (iii) M/D/1 has longer mean busy period. (iv) M/D/1 has larger variance of busy period. (20%) 4. Consider a closed Jackson queueing network as shown in the following. There are 4 single server queues in the system , all with service rate u and 3 customers in the network. Please use Mean Value Analysis to derive the mean system size of each queue and mean cycle time.(20%) ───── ───── ─→│ │○──→│ │○─→ ↑ ───── ───── │ │ │ │ ───── ───── │ ←─○│ │←───○│ │─ ───── ───── 5. Consider an open Jackson queueing network , in which there are only 2 queueing nodes , node 1 and node 2 , and there is only 1 external traffic source γ for node 1 . Assume that the routing probability r11 = p1 , r21 = p2 , and service rate is u for both nodes. (i) Please write its state balance equation , assumeing the state is (n1,n2) (ii) Please derive its joint system size distribution.(20%) ───── ───── ──→│ │○──→│ │○───→ ↑ ↑ ───── │ ───── │ │ │ │ │ │ ───────── │ ────────────────────── 6. Please explain why 2-class FCFS M/M/1 queue, with arrival rate λ_i customer/min. , ave. service time 1/u min. for class-i , its mean system size is larger than single class FCFS M/M/1 queue with arrival rate λ=λ_1+λ_2 and mean service time (λ_1/λ)/u_1 + (λ_2/λ)u_2. (10%) --

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