mcenroe thesis ch4 analysis

8/3/2019 McEnroe Thesis Ch4 Analysis

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CHAPTER IVANALYSIS

The analysis focuses on transient faults produced by alpha particle radiation.A single error correcting code minimizes the effects of the alpha particles faults. Twointeresting questions are examined. The first is the probability that a word is bad, i.e.has two or more bits flipped. The second question addresses the number of goodwords between bad words. The analysis will require several results from queueingtheory developed in the next section.

FIFO OperationThe FIFO (First-In First-Out) memory device stores a finite number of data

words K. We assume data words of width n arttve at the FIFO as part of a Poissonprocess with rate X. Unless the FIFO is full, the word takes its place in line, afterthe words already present. When there are words present in the FIFO, we assumedepartures take place as part of a Poisson process with rate L. The population in theFIFO at any time /, is a birth-death process. When there are i words in the FIFO,then after an exponential random interval the number of words present can transitionto either i*1 or i-l via a birth or death respectively.

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Steady State DistributionsThe MlMlllK is the well known queueing model for the FIFO behavior just

described. The first M stands for Poisson arrivals, the second M for Poisson servicetimes. The 1 refers to the single server. The K indicates the capacity of the queue,including the customer in service. In our model we will refer to 'customers' as'words.' The word at the head of the line is not actually being serviced, it is waitingto be called forth from the FIFO. The time between pr@essor calls is assumed to beexponentially distributed. Thus there is an equivalence between Possion service timesand Poisson departure times. We will use the term departure rather than the queueingterm service when referring to the death event in the birth-death process.

The birth-death process of the MlMlllK queue is the Markov chain in frg. 14.

Figure 14.--Markov Chain for the MlMlIlK QueueDefine: pit) = Pr{i words in the queue at time r}. The steady-state or limitingdistribution for the MlMlllK exists and is given by (Kleinrock 1975 vol. I, p. 104):

r,=limn,(r)=ffid 0


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Since words can only enter the queue when it is in states 0 to K-1, the steady-statedistribution seen by words that can enter the queue is:

"i =t#nj{D=#0,

p,=tpp,(r) = 1et-t

0


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one sample path. Figure 15 shows a sample path from i:3 to l:2 for the queuewith capacity 4 (K - 4).

-d ATRJVALARRTVAIJDPARTURES

.frArtw^LgsTWOWdDS r+tt AnRIVAL sEsONB WOTD

Figure l5.--Sample Path from i:3 to l:2.Poisson events in the birth-death process (rccur with rate \+p. When there are

words in the queue, but the queue is not full, the probability that an event is adeparture is pa = h Under the same conditions, the probability that the Poissonevent is an arrival is po = l-pa = * The birth-death process is a Markovchain, the probabilities pa nd po {a independent of the state and the word index m.Thus:

PI'^*t = P, (r2)

Irt the notation a^ d^-z dn-t an+t indicate the following sequence of events: thearrival of the nf word, departure of the m-P word, departure of the m-I't word,arrival of the m*L't word. This is the sample path from i:3 to l:2 as illustratedin fig. 15. The probability that event A is a departure ispr. Similarly the probabilitythat event B is a departure is also pr. The probability that event C is an arrivat is p,.

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After event C, we do not care if more arrivals occur and since the n{ departure isinevitable, there is no probability associated with its departure. So the probability ofthis particular sample path occurring is popopo.

Table 11.-- Sample Paths and Probabilities for the Case K:4Departuresil

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2t23

3t234

4I234

Sample Paths

ql d- a-+tql to*r d.ql d--r d- fo+ra-! d^-r a-*r d-tt.l a-*r d--r d-a-l d-" d--, d- o-+rfol d^., d--, a-*r d-a-l d--z a-*r d.-, d-a-l a-*r d--z d.-r d-fol d--, d._, d-., d- a.+ra-l d-" d-, d-_r fo*r d-a^ I d--, d--, a-*, d.-r d-a- I d--: fo*r d--, d--, d^

P;r, PathProbabilities

I d-*tI d-*tI d-*tI d-*tI d^*tI d-*tI d-*tI d-*tI d-*tI d-*tI d^*tI d-*tI d-*t

PaP"p]PdP"P.

tPa-Pdb"PaP"P"Pd3pizT."PaP"P.

Table 11 lists all the sample paths and path probabilities for the case K:4. Theseresults are generalized in table 12. Later we will use the results in table 12 tocompute how the distribution of A(l) changes from departure to departure.

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Table I2.-- Pib Path Probabilities for the General CaseDepartures:i1


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an exponential pdf and rate p. T, has the Erlang pdf:fr.6.(tlAn(o)) =

Using the law of total probability:

pe-lt(p11t-rCI-1)!

(15)

(16)

(t7)

Substituting eq. 15:

Kfr.(t) = Ef, -n.QlA -QD P(A.Q))

Kfr^(t) =Furrn^a)rffi

SummaryThis section developed three results used in solving the two central questions of

error event probability and error event interarrival distribution. The three resultsinclude the steady-state entry-to-exit departure distribution pr, the transitionprobabilities Pir of the number of departures A^, and the pdf of the system time forthe mh word /",(r)

Error Event ProbabilityWhen using a single error correcting code, an error event is defined as two or

more bits in a word flipped from a data value of 1 to a value of 0 (1 -, 0). Thus wedefine the probability of an error event, an uncorrectable word, for word m.

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Define:P(E^)

Find:

Where:

t Pr{mh word written has at least 2 bits flipped when read out}= Pr{mh word is in error}

P(E) = I:"t l

P(8.) = f- r1z^lTn=t) fr_{t) dt

(1 8)

(1e)

P(B*lTn = t) is Pr{mth word is in error I mth word spent time f in queue}For single error correcting codes the probability of a word error is one less the

probability of either zero or one bit flips from 1 * 0.P(E^lTn=t) = 1 - P(0 "flips l-}"lTn=t) - P(l "flip l*O"lTn=t) Q0)

In general, for a codeword n bits wide:

P(g 'flips l-O'lT^=t) = if te 'flips l-O'lTn=t, w(c)=j) .P(w(c) =jf")i=oWhere w(c), the weight of the codeword, is the number of l's in the codeword.

P(w(c) =i) -- (;) ",*=1)i (p(bit=0))'-i (22)For nontrivial linear codes, the number of 1's is equal to the number of 0's.


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Therefore it is reasonable to model codeword bits as i.i.d with P@it : 1) : p@i1 :0) : Vr.

P(w(c) =i) = (;)(i)' Q3)Atpha particles strike as part of a Poisson process where \ is the arrival rate for

a single cell. Alpha particles always cause the cell data value to change from a 1 toa 0 as explained in the review of literature {B. l2).

Pr(t) I Pr{a single cell is flipped 1 * 0 in time /}Pr(/) is one less the probability that the cell is never hit during [0, r], i.e.,

(24)

So:

P(4 "flips l-0'lT^=t, w(c)=j) =

Substituting q,. 25 for appropriate values of 4 into eq. 20:

p(E.-lr.-=t)= 1-6)"[1 .i 0ltr,(t-pby-L +o-pby]l otNow eq. 19 can be written:

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p (E ^) = * r rn,,, I-# [, - 6), l, .,t ( ; ) tr r, _" - I r 1 "- I r0 -,) + eWhich has the following expression:

n1z^) = trrn,,r, [' -(;)' l'.i olffi . ffi]]] rtRecall that P(E) = lim P(E^) and lim P(A^(Q)) = \ then from eq. 10:

p (E) = E -, o" l, - (*) " tll u *L*-. ffilll es)Error Interarrival Distribution

In this section we approximate the distribution on the number of good wordsbetween bad words (a bad word has an uncorrectable number of errors). Define thefollowing probability :

P-(r) t Pr{(m+r)h read is the 1"'bad read after the mh read,lthe m'h read is bad}Then define:P(r) = 31r,(r)

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Define the following events for all m : 1,2,..., ild for all r : 1,2,...:E^ t fthe mh word written to the FIFO is in error when read out, i.e. the wordcontains at least 2 errorslN^(r) I [the first error read after the n{ read occurs at read m*r)

Thus P-(r) = P(N.(r)lE ). Recall from the FIFO analysis that P(A^(/)) is theprobability that the n{ word sees /-1 words in the queue upon arrival and thus mustwait / departures before leaving the queue. To determine P^(r), condition on thenumber of reads the mh word spent in the queue. By the law of total probability:

Kp ^(r) = E r(lv,(r)lE^r{,(0o)) p (A^(%)lE ^)b=1We can determine P(A^(%) lE,) from Bayes' theorem:

P(A^(%)|E) = P (A^(oo)) P(E l,{,(oo))P(E^)

Assuming the system is in steady-state when the bad word arrives P(A^(l)) : p,which is given in eq. 10. Using an analysis similar to the derivation of eq. 28 wedetermine:

(31)

P (Enr'{'(,r) =' -(;)" [' .,1 0l- *-tt;fr . ffi]] (32)Returning to P^(r), the probability thr t the rh ord is the next bad ord is theprobabilit that the 1"'through rft-l words are good and the rh word is rad:


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KP^(r) = lfr1fi, n^.,i,n,,.,\E^A^(%))to=l -

KKP ^(r) = E ... E lp


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KKp ^(r) = E ... E lp


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Then A," = Ar*, Pmd 4".2 = A",*r P = (A, P)P = AnP'so in genenl A.r,*t = An PtUse the defined notation to simplify the following expression:

Pt=

experienced by the mh word given that the mh

In general:

word was in error.P(A^(nlE ) ] (3e)

(38)P,,

P*,

E p(E^.|l,.r(0r)) E p(A,r(or) lr{,(h)) p(A^(q)lE^)tr=l !o=1K= E P(E^.l1/,.r(!r)) A, pr,q

(40)

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rElr=l

p^(r)= (;{ p(E^*, A'.,(Q))A", p'-'tr)fr [t ,:

I rf rlp (z^, ;r{,.r( 0r) E I p


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(x \'-rl x \ -p(r) =lE "Un,({,)) Ap'-' p+,J il lt -E rru^lAhe)),ttr-' r,f s6)gtrmmarT

In this analysis we have determined the probability of an uncorrectable number oferrors for the single cell alpha particle. We have made some approximations in orderto study the error interarrival distribution. The next chapter compares the simulationresults with these analysis results. The plausibility of our assumptions for theinterarrival distribution will be apparent from comparison with simulation results.

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mcenroe thesis ch4 analysis

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