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）︰2008/01/18
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 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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