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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