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(UPDATED)homework2_solution

(UPDATED)homework2_solution
(UPDATED)homework2_solution

更正:2.5题(d)

Homework2:

Note :Submit your homework on next Saturday, Oct. 31, 2009

Problems from Data networks:

3.1 Customers arrive at a fast-food restaurant at a rate of 5/minute and wait to receive their order for an average of 5 minutes. Customers eat in the restaurant with probability 0.5 and carry out their order without eating with probability 0.5. A meal requires an average of 20 minutes. What is the average number of customers in the restaurant? (Answer: 75)

Solution :

顾客在餐馆中平均停留时间为5+20*0.5+0*0.5=15分钟。由Little ’s 定理,餐馆内的平均顾客数

N=λT=5*15=75人

3.2 Two communication nodes 1 and 2 send files to another node 3. Files from 1 and 2 require on the average R1 and R2 time units for transmission, respectively. Node 3 processes a file of node i (i =1, 2) in an average of P i time units and then requests another file from either node 1 or node 2 (the rule of choice is left unspecified). If λi is the throughput of node i in files sent per unit time, what is the region of all feasible throughput pairs (λ1, λ2) for this system?

Solution :待定。参考例题3.7,与例题3.7的不同之处在于该题目没有指定是分时系统。

3.3 A machine shop consists of N machines that occasionally fail and get repaired by one of the shop ’s m repair persons. A machine will fail after an average of R time units following its previous repair and requires an average of P time units to get repaired. Obtain upper and lower bounds (functions of R , N , P , and m ) on the number of machine failures per unit time and on the average time between repairs of the same machine.

如图,设B 、C 之间的延迟为D ,则: 1、最好情况下D=P ;

2、最坏情况下m 个工人都在维修,且剩余机器也都在B 处排队,此N-m 台机器每台的平均排队等待接受其前一个接受服务完成的时间为

m

P ,因此()

P m P m N D +-=。

λ

可得A 、C 之间的平均延迟T 满足:

()

P m P m N R T P R +-+≤≤+

m

NP R T P R +

≤≤+

根据Little ’s 定理,系统吞吐率λ满足:

P R N

m

NP R N +≤

≤+

λ 对修理工人处用Little ’s 定理有: P

m ≤

λ

最终得到

??? ??+≤≤+P R N P m

m

NP R N ,min λ 由Little ’s 定理,得到T 的范围:

m NP R T P R m PN +

≤≤??

?

??+,max 3.4 The average time T a car spends in a certain traffic system is related to the average number of

cars N in the system by a relation of the form T = α+βN 2, where α > 0, β > 0 are given scalars. (a) What is the maximal car arrival rate λ* that the system can sustain?

(b) When the car arrival rate is less than λ*, what is the average time a car spends in the system assuming that the system reaches a statistical steady state? Is there a unique answer? Try to argue against the validity of the statistical steady-state assumption.

3.4 (a) 由Little ’s 定理

N

N

N

N

T N βαβαλ+=

+=

=1

2

当βα=

N 时,上式取得最大值,即

αβ

λ21

=

*

(b) 当*

<λλ时,有

T

T T N λβ

α

λ=-?

=

可解出两个T 值,对应系统较为拥挤和较为空闲的两种情况。当系统较为拥塞时,车辆数目微小的增长会导致系统时间的增长,进而又导致车辆数目的继续增长,系统不会趋于稳定状态,T 和N 会趋近0和无穷大。

3.7 A communication line is divided in two identical channels each of which will serve a packet traffic stream where all packets have equal transmission time T and equal interarrival time R > T. Consider, alternatively, statistical multiplexing of the two traffic streams by combining the two channels into a single channel with transmission time T/2 for each packet . Show that the average system time of a packet will be decreased from T to something between T/2 and 3T/4, while the variance of waiting time in queue will be increased from 0 to as much as T 2/16. Solution :

由于两个信道各自的到达间隔R>T ,因此合并后,最好的情况下: 所有数据包都不需要排队等待。 最坏的情况下:

每两个数据包中有一个不需要排队等待,另一个排队等待时间为T/2。 设平均排队等待时间为W ,则W ∈[0,T/4]。 因此平均系统时间t ∈[T/2+0,T/2+T/4]

即t ∈[T/2,3T/4]

最坏情况下,平均排队等待时间的方差为1/2(0-T/4)2+1/2(T/2-T/4)2=T 2/16 因此平均排队等待时间的方差∈[0,T 2/16]。

3.11 Packets arrive at a transmission facility according to a Poisson process with rate λ. Each packet is independently routed with probability p to one of two transmission lines and with probability (1-p) to the other.

(a) Show that the arrival processes at the two transmission lines are Poisson with rates λp and λ(1-p), respectively. Furthermore, the two processes are independent. Hint: Let N 1(t) and N 2(t) be the number of arrivals in [0,t] in lines 1 and 2, respectively. V erify the correctness of the following calculation:

()(){}()()(){}

()()()()()()()

()()!

1!

!1!

|,,12121m p t e

n tp e

m n t e

p p n m n m n t e

m n t N m t N n t N P m t N n t N P m

p t n

tp

m n t

m n m

n t

-=

+-???

? ??+=++====

==---+-+-λλλλλλλλ

(){}()(){}()∑∞

=-=

===

=0

2

1

1!

,m n

tp

n tp e

m t N n t N P n t N P λλ

(b) Use the result of part (a) to show that the probability distribution of the customer delay in a (first-come first-serve) M/M/1 queue with arrival rate λ and service rate μ is exponential, that is, in steady-state we have

{}()τ

λμτ--=≥e

T P i

where T i is the delay of the i th customer. Hint: Consider a Poisson process A with arrival rate μ, which is split into two processes, A 1 and A 2, by randomization according to a probability ρ=λ/μ; that is, each arrival of A is an arrival of A 1 with probability ρ and an arrival of A 2 with probability (1-ρ), independently of other arrivals. Show that the interarrival times of A 2 have the same distribution as T i . Solution : (a ) 证明:略

(b ) 方法1:参见课件;

方法2:

设A 2的时间间隔为t 的概率为P 2(t),M/M/1系统顾客延迟时间为t 的概率为P(t)。

则有

P 2(t) =

∑∞

=0

(n P{t 时间内A 中总共到达了n+1个顾客} * P{前n 个顾客进

入A 1,最后一个顾客进入A 2} )

P(t) = ∑∞

=0

(n P{顾客到达时发现前面有n 个顾客} * P{t 时间内服务完了n+1个顾

客} )

其中

P{前n 个顾客进入A 1,最后一个顾客进入A 2} = ????

??-???? ??μλμλ1n

P{顾客到达时发现前面有n 个顾客} = ???

?

??-???? ??μλμλ1n

而A 的到达过程与M/M/1系统的服务过程同分布,因此

P{t 时间内A 中总共到达了n+1个顾客} = P{t 时间内服务完了n+1个顾客}

综上,P 2(t) = P(t)

因此A 2的时间间隔与M/M/1系统顾客延迟时间是同分布的。又由(a)中结论,A 2的时间间隔为参数λμ-的指数分布。因此M/M/1系统顾客延迟时间也为参数

λμ-的指数分布。

3.41 Little ’s Theorem for Arbitrary Order of Service; Analytical Proof [Sti74]. Consider the

analysis of Little ’s Theorem in Section 3.2 and the notation introduced there. We allow the possibility that the initial number in the system is positive [i.e., N(0) > 0]. Assume that the time-average arrival and departure rates exist and are equal:

()()

t

t t

t t t βαλ∞

→∞

→==lim

lim and that the following limit defining the time-average system time exists:

()()()

()()?

??

?

??

-++=∑∑∈∈∞→t D i i t D i i k t t T t N T α01

lim

where ()t D is the set of customers departed by time t and ()t D is the set of customers that are in the system at time t. (For all customers that are initially in the system, the time T i is counted starting at time 0.) Show that regardless of the order in which customers are served, Little ’s

Theorem (N = λT) holds with

()?∞

→=t

t d N t

N 0

1

lim

ττ Show also that

Hint: Take ∞→t below: ()

()()()

∑?

∑?∈∈≤

t D t D i i

t

t D i i T

t

d N t T t

11

10

ττ

Proof :对任意时刻t ,有

()()

()()

∑∑?

∈∈-+

=

t D i i

t D i i

t

t t T

d N 0

ττ

()

()()()

∑?∑?∈∈≤

?

t D t D i i

t

t D i i T

d N T 0τ

τ

()()()

()()()

()()

()()

001

N t T

t

N t d N t

t T

t

t t D t D i i

t

t D i i

++≤

?

∑?∑?∈∈ααττββ

其中|)()(|)(t D t D t +=α 因此当∞→t 时,有 T N T λλ≤≤ T N λ=?

Problems from Kleinrock ’s book (Queueing system V ol.1)

2.5 Consider the homogeneous Markov chain whose state diagram is

∑=∞

→=k

i i

k T k

T 1

1lim

(a) Find P , the probability transition matrix.

(b) Under what conditions (if any) will the chain be irreducible and aperiodic? (c) Solve for the equilibrium probability vector π. (d) What is the mean recurrence time for state E 2 ?

(e) For which values of α and p will we have π1=π2=π3 ? (Give a physical interpretation of this case.) Solution: (a) ?????

???

??--=αα

10

00

1

10p

p P

(b) 10≤

()()???

?

??

?

+=

+==????????=++-+=+-==?=++=απααπππ

π

ππαππαππππ

ππ

π

πππ221

111

,321

3

2

1

3

13312

2

1

3

2

1p p p p p P

1-α

1-p

因此平稳分布????

??+++=αα

α

α

α

π2,

2,

2p p

p p

(d) ()α

α

π21

2

22+=

=p T E

(e) p =α

2.6 Consider the discrete-state, discrete-time Markov chain whose transition probability matrix is given by

?????

?

???

??

?=41432121P (a) Find the stationary state probability vector π. (c) Find the general form for P n .

Solution : (a)

???

???

?==?=+=52531,2121ππππππP

(c)

?

??

?

??????

?

?????? ??-+??? ??--??? ??--??? ??-+=?????

??

?????-????

???

?-??????

?

?-=n n n

n

n n

P 415352415353415252

415253525

2525341

00123111

2.7 Consider a Markov chain with states E 0, E 1, E 2, … and with transition probabilities

()∑=----???

? ??=j

n n

j n

i n ij n j q p n i e

p 0!

λ

λ

Where p+q=1 (0

(a) Is this chain irreducible? Periodic? Explain. (b) We wish to find

πi = equilibrium probability of E i

write πi in terms of p ij and πj for j= 0, 1, 2, ….

注意:此处题目有误,应该是write πi in terms of p ji and πj for j= 0, 1, 2, ….

Solution :(a )给定的马氏链是不可约,非周期的,因为任意0,>j i p ,并且任意0>ii p 。

(b )ji j j

i p ∑∞

==

π

π

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