hw7-sol-f08 - Hw#? “5/2 S‘X’w’h'WS MI7. smkms A“...

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Unformatted text preview: Hw#? “5/2 S‘X’w’h'WS MI7. smkms A“ $3th- rfls owe Wu‘a’fifaud. Axum.» N even. Le} 7; be ’Hfl erto‘ whfia _ N M “(11: 2x +%‘2+”"+~E—w{)z=~§—(i+3-). ‘8~5. .p—vo‘ N210, sFoflA 50—0 Mues wfm’r . Lmé quack C 7“ 43°G'bt’g- PL‘H'H OMS—$153!- 4‘% his [011% . “ks =7- 30 M39" ' ?W\’“fia‘kw Adda: 2mg? {01: mil“); R011 -— Ca“ Foitnua : 7:120 msec - fngégv? :10 msec. U) confide/v a} m and a? «km s-k-v‘mgz 1/: E—(H—N) =6éo msecj L: Ni? + N‘cs-i- 2" a ‘160 wsec‘ ('2) Can+YOHW a} WdIt ‘7'? We sjnmg. ./::(:_ +£ ~—.. -. I ‘ z 20 2)-—36o/m>ec; L= N+F+Nts+T’=6éomsea Hub FO‘Ufla; Tké PasiHon :7? H42 WH‘TOHWV is Mo .an “I: ill? ‘vusec. La“, ‘32?” MFMML+ . a: NTS + z" 3349 (msec). §;‘i_._ a: Nym .— Mmofix numbw 4*answxssims Lw M sec. 3 = N Am m my»? mama w% SuaSsM "cram; mgsws LA, m sec, G: s + vangx MMMbWO+MrWMLSS(0us w, m‘ sec. 2: S + Q ~ Prohiai‘ Jud" mm. uiLisSDn ocwavs Ln. 2 m inwmfl Bid? ProbEll}. 9145*“ m Colustm 0Com: 6w 2w. inkyqu] =1_e’~—2MANI\’: i‘efzfi Hana. Cx=$+€TU-e"“) M S=&e‘2‘*. (X—u). 2:39: m=2izz =0.083 see}. )u: I WSSW/mo sac . 240-0 bias channel . Fine. Moha- : NMM; AM *4 SW.” ‘—‘= 043 —» Nva=1 257 Shared Atom: SMM := 9.35 --a- NM}; 513. Fm 5w~ioz+ missagil 2 = W a 0.208 sec. 2400 Pure Aloha ; wax 5105 Jr Shirk—0‘ MOM: NW 2": 207; FW 209.49% massage. bait k: 1 MSW/3 Min: Ema. ARM: “new 5 333} Std-kt; ALoM: NM”: $7773 344’. Mimerhg MWWSM = H (Wt "'P)“ = “PU—P)“ .~ -~ , $1.: ~ ‘ W .P 7%” m 0. WW .7; Pm Z, . Terminal A is transmitting messages over a common channel shared with a second source. The data rate of the common channel is C, The second source generates messages according to a Poisson process with rate A2. Each message is of fixed length m2. Once a message is generated at the second source, it is immediately transmitted on the channel, regardless whether any other messages are being transmitted. The message from the second source is NOT retransmitted even if a collision occurs. Messages of length m1 > m2 arrive at Terminal A according to a Poisson process with rate A1. Assume that Terminal A has infinite buffer space, and it serves the messages in a first-come—first-serve manner. Terminal A uses the following pure Aloha scheme. Once a message is ready for transmission (i.e., when it is the first message in the queue), the message is first immediately transmitted to the common channel. If there is no collision with messages from the second source, the message is considered successfully transmitted, and Terminal A will immediately serve the next message in the queue. If there is a collision, Terminal A will wait for a constant (non-random) time equal to R with R > m1 / 0. Terminal A will then retransmit the message. This retransmission procedure repeats until the message is transmitted successfully. See the illustration in Fig. 3 for an example of this process at Terminal A. the previous message first second is successfully retransmission retransmission trasnmitted collision and collision and no collision l l l l ml/Cl mI/C ml/C ml/C ml/C > T <—————>l l<—> A time the first message R R transmission of in the buffer is retransmission the next message immediately delay begins transmitted Figure 3: Terminal A uses pure Aloha to serve messages from the buffer You goal is to determine the average delay from the time that a message arrives at Terminal A, to the time when it is successfully transmitted. (a) (10 points) Derive the probability p that a message transmission by Terminal A is successful (i.e., no collision occurs)? (b) (10 points) Let tv denote the delay from the time a typical message is ready for transmission at Terminal A (i.e., it is the first message in the queue), to the time the message is successfully transmitted. Find the distribution of the random variable tv. (c) (10 points) Find the expected value E(tv) of the random variable t1). You may use formulas of some frequently-used summations at the end of the exam booklet. (d) (10 points) From the distribution obtained in part (b), we could also calculate the second moment of the random variable 22,. Assume that you have already obtained the value of E (you do NOT need to calculate it). Find the expected delay that a typical message experiences at Terminal A, Le, determine the expected delay from the time the message arrives at Terminal A, to the time when it is successfully transmitted on the common channel. Your final expression may contain E02,) and/or S 19 MW‘W t Ca) Cow-C AM" a, pike/i imam «kw/m m‘ilteéi in) m‘md off “New: “i’. l3 { “Fr/‘3 1113 I. 7) ho WWWMEJCM L96) IECH): §§~Gm+nfr)‘0wafl figflmmi.4.wW&afla A M/é/! “(LL «3/20 fax/f“ng )W-W‘\Lg ‘7€”’(“% LS M ' A, ‘ F ( 64w”,. :w ‘ F i what" 710?: €[+vj , 61 "7" )* HM a; Cm “E: (D) ‘ f“ A EMU] j, r‘affb’M MUJ PUNK R rill fi‘wXSAs‘d‘m Vwmfi _ ...
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This note was uploaded on 01/27/2010 for the course ECE 547 taught by Professor Xiaojunlin during the Fall '09 term at Purdue University.

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hw7-sol-f08 - Hw#? “5/2 S‘X’w’h'WS MI7. smkms A“...

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