[試題] 105-1 蔡志宏 排隊理論 期末考
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︰電信所/電機所
考試日期（年月日）︰2017/1/11
考試時限（分鐘）：140 (mins)
試題 :
1. Consider a FCFS M/M/1 non-preemptive priority queue with 3 classes of
customers. The service rate for class-i is μ_i, the arrival rate is λ_i
for class-i, and arrival processes for all classes are Poisson. Please
answer the following questions, with λ_1/μ_1 + λ_2/μ_2 < 1, but
λ_1/μ_1 + λ_2/μ_2 + λ_3/μ_3 > 1.
(i) What is the expected residual (remaining) service time of the current
customer in service upon the arrival instant of a new customer? (5%)
(ii) Please derive the formula for L_q,2 (the average number of customer
waiting in queue for class-2 customers) (5%)
(iii) If λ_1 is almost zero, what is the mean waiting time for a class-1
customer (W_q,1)? (5%)
2. Consider a 2-class preemptive-resume priority queue. Let (n_1, n_2)
represents the system state, where n_i is the number of class-i customers
in the system. λ_1/μ_1 + λ_2/μ_2 < 1. Suppose the buffer for class-1
and class-2 are separated, and the buffer limit for these 2 classes are 1
(i.e. 1 for each class).
(i) Please draw the system transition diagram. (Hint: there are only 4
states: (0,0), (0,1), (1,0), (1,1)) (5%)
(ii) Please write down all the global balance equations. (5%)
(iii) Will detailed balance equations hold for this queue? Please explain.
(5%)
(iv) What is the expected residual service time of the current customer in
service upon the arrival instant of a new customer? (5%)
3. Consider 4 single server queues: Q1 is M/D/1, Q2 is M/E_2/1, Q3 is M/E_4/1,
Q4 is M/M/1. All these 4 queues have the same arrival rate (λ) and the
same mean service time (1/μ)
(i) Please compare their W, Wq and L (12%) Derivation required.
(ii) Please compare their server utilization (U) and p_0 (4%)
(iii) Please explain why the busy periods in these 4 queues have the same
expected length. (4%)
(iv) Please determine whether π_n = p_n in these queues. (π_n is the
steady state probability that system size is n upon departures, and
p_n is the steady state probability that system size is n at arbitary
point) (5%)
4. Let X_n be the system size observed by the n-th departing customer in an
M/M/1 queue. Suppose the service rate is μ and the arrival rate is λ.
(i) Please write the transition prob. Matrix for p_ij =
Pr{X_(n+1) = j | X_n = i} (9%)
(ii) If k_n represents the probability that n customers arriving during a
service time in this queue, what is K(z)?
( K(z) = Σ_{n = 0 to ∞} {k_n * z^n} )
5. (i) Please draw the state transition diagram and write all global balance
equations for an M/E_3/1/2 queue. (10%)
(ii) Will the Little's formula hold for this M/E_3/1/2 queue? If yes,
please write the formula for this queue to derive W from L. (5%)
6. Consider a closed Jackson queueing network as shown in the following. There
are 2 single server queues all with service rate μ and the third node is
with service rate 4μ. Suppose there are 2 customers circulating in the
network. Please use the Mean Value Analysis to derive (i) the mean system
size of each queueing node and (ii) mean cycle time. (iii) the customer
arrival rate of each node (20%)
Node 1 Node 2
μ μ
┌───→■■■■●─────→■■■■●──┐
｜ ｜
｜ ｜
｜ ｜
｜ Node 3 ｜
｜ ｜
└─●■■■■←───────────────┘
4μ
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※ 編輯: WhyThe (140.112.18.208), 01/11/2017 18:27:12
※ 編輯: WhyThe (140.112.18.208), 01/11/2017 18:27:54
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