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）︰2010/01/08 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 2010 1. Consider a FCFS M/M/1 queue with 2 classes of customers , and the service rate for class-i is u_i , the arrival rate is λ_i for class i , and arrival process for all classes are Poisson. Please answer the following questions if all there is no priority and the sum of λ_i / u_i =1,2 is smaller than 1. (Hint , you apply either formulas for M/G/1 or multi-class Markovian queue) (i) What is the probability of observing an idle server upon an arrival if the system is stationary ? (4%) (ii) What is mean waiting time in queue W_q for an arbitrary customer ? (4%) (iii) What is mean waiting time W for a customer of class-i ? i=1,2 (6%) (iv) What is the probability that a class-1 arriving customer is followed by a class-2 customer ? (4%) (v) What is the remaining service time of the current customer upon an arriving instant of a new customer ? (6%) 2. Consider an M/M/c/c queue with arrival rate λ=0.4 customer/min , service rate u=1 customer/min (per server), (i) If a server is randomly selected in an arbitrary time instant , what is the probability that this server is idle ? (6%) (ii) Please compare the blocking probability of this queue with an M/G/c/c queue with the same mean arrival rate, mean service time and the same c.(4%) (iii) Is the probability of observing n customer in this system upon a departure point equal to the probability of observing n customer in an arbitrary point in time ? (10%) 3. Consider a preemptive resume priority single server queue with 2 classes of Poisoon arriving customers. Class-1 is with rate λ_1 and class-2 is with arrival rate λ_2. Class-1 is with high priority. Suppose the service time is always exponetial and both classes are with the same service rate u=1. Assume the buffer is infinite. (i) Please derive the average length of its busy period and the average number of customers served in a busy period. (8%) (ii) Will the probability of observing n customers in the system upon arrivals change if the service discipline is changed from preemptive-resume to non-preemptive ? Please explain.(4%) (iii) What is the probability of observing a class-2 customers in service upon arrival ? Will this probability changes if the service discipline is changed to preemptive repeat ? (8%) 4. Compare an M/D/1 queue and an M/S/1 queue with the same arrival rate λ and the same mean service time 1/u. Please determine (i) Which queue has higher mean system size (L) ? (4%) (ii) Which queue has higher mean waiting time in queue (W_q) ? (4%) (iii) Which queue has longer residual service time for the current customer in service if the system is observed in rendom ? (4%) (iv) Which queue has higher probability of observing an empty queue upon arrival ? (4%) (v) Can the Little's formula be applied in both queues ? (4%) (vi) Will the departure processes of these two queue be Poisson ? (4%) (vii) Will their second moments of busy periods be the same ? Please determine this from the equation G^*(s)=H^*[s+λ-λG^*(s)] (10%) (Discussion or derivation is required) 5. Consider a closed Jackson queueing network as shown in the following. There are 3 single server queues in the system , all with service rate u and 3 customers in the network. Please use the Mean Valut Analysis derive (i) the mean system size of each queue and (ii) mean cycle time. (iii) the customers arrival rate of each node (20%) ───── ───── →│ │○─────→│ │○─→ ↑ ───── ───── │ │ │ │ │ │ ───── ↓ ────←○│ │←────────── ───── --

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