推 yushenlin :好有趣的課名 01/14 01:49
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/電信/電機
考試日期（年月日）︰2009/01/18
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 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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