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/电机/电信 考试日期（年月日）︰2008/01/18 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 2008 1. Consider a non-preemptive priority single queueing system with infinite buffer for each class. There are 3 classes : class-1,class-2 and class-3, where class-1 has highest priority and class-3 has lowest.Suppose the service time is exponential and the service rate u is the same for all classes. And the arrival rate is λ_1 ,λ_2 and λ_3. Suppose (λ_1+λ_2)/u < 1 but (λ_1+λ_2+λ_3)/u > 1. (i) Please derive the residual service time of the customer in service , when a class-2 customer arrives at this system.(6%) (ii) Please derive the average waiting time (W) in the system for class-1 and class-2 ,i.e., W_1 and W_2. (14%) 2. Please use Mean Value Analysis to derive the average cycle time and average waiting time at each node,if there are K customers circulating in a circular queueing system consisting of 2 M/M/1 nodes , and ezch node is with service rate u. (15%) 3. Please compare the following two M/G/1 queueing systems , with service rate equal to 1 .(a) an M/D/1 system with service time equal to 2. (b) an M/G/1 system with 1/2 , P_r {A=3} = 1/2 .When the arrival rate is the same , please argue : (i) which system have larger system size ? (ii) which system have a longer mean busy period length ? Derivation required.(20%) 4. (i) Please explain why the output process of an queueing node in an open Jackson queueing network is still a Poisson process , if all routing are forward only (i.e.,without any feedback type routing ). Please give an example.(10%) (ii) Please explain how a finite source server queue with fixed population M can be solved as a closed loop Jsckson queueing network. Please give an example to explain.(10%) 5. Consider an M^[x]/D/1 queue with service time equal to 1/u , arrival rate equal to λ and batch size distribution p1=0.5 and p2=0.5 , where p_i is the probability that batch size equal to X. (i) Please derive the moment generating function Π(z) and average of system size observed by departing customers. (14%) (ii) Please derive P_0,i.e. the server idle probability.(6%) 6. Assume arrival rate λ and mean service time 1/u , (i) please derive the steady state probability that a customer is in phase-i for M/E_k/1/1 queue,i.e. an Erlang sevice node where no additional waiting queue is allowed to form. (ii) What is the blocking probability ? (15%) --

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