Th Univrsity of Alabama in Huntsvill Elctrical and Comutr Enginring Homwork # Solution CE 6 Sring 8 Chatr : roblms ( oints, ( oints, ( oints, 8( oints, ( oints. You hav a RAID systm whr failurs occur at idual disks at a constant rat r disk. Th rair tim of disks is xonntially distributd with rat. Suos w ar in an arthquak-ron ara, whr building-dstroying arthquaks occur according to a oisson rocss with rat. If th building is dstroyd, so too is th ntir RAID systm. Driv an xrssion for th robability of data loss for such a systm as a function of tim. Assuming that th man tim btwn such arthquaks is yars, lot th robability of data loss as a function of tim using th aramtrs /, hours and / hour. Modify to only gnrat Markov chain and diffrntial quations. Dfin th stat of th systm by how many disks ar u for i, if data loss has not occurrd, and if it has. i (t is th robability of bing in stat i at tim t, w sk - (t. For convninc, dfin a, b, c, d. Th diffrntial quations for th Markov chain ar: with th initial conditions (, (. d ( t a ( t b ( t d ( t c ( t d ( t. Givn a RAID lvl systm with an orthogonal arrangmnt of d strings and g 8 RAID grous, comar th MTTDL for diffrnt valus of d from to. Assum an xonntial rair rocss for singl disks and for strings of disks with rair rats of /hour and /hour, rsctivly. Also assum failur rats for singl disks and strings of disks of -6 /hour and x -6 /hour, rsctivly. Th rair rat for disks is f d -t and th rair rat for strings is f s -t. d -6 and s * -6. d ( τ disk str ( f ( τ dτ disk
d ( disk st r τ τ ( dτ τ ( * τ dτ t [ ] d(6* dτ (6* t 6 6 (6* (6* [ ( ] d(6 * d(6* d(6 * [ ( ] ( d ( g τ disk st r ss f str ( ( d ( g τ disk st r τ ss ss ot ( τ dτ ( dτ ( d (* d(* d(6 * d i l lss iss lstrss lot iot lstrot ldloss- MTTDL ldloss-o MTTDLo.E- 9.6E- 6.E-.7E-.E-.E-.7E-.E-.E-9 6.66E8.9E-9 7.8E8.E-.E-9 7.8E-.6E- 7.8E- 6.E-.7E- 6.E-.E-9.E8.9E-9.78E8 6.6E-.E-9 9.E-.E-.6E-9 7.8E-.6E- 9.E-.8E-9.E8.9E-9.E8 7.E-.69E-9.E-.7E-.9E-9 9.E-.E-.E-9.7E-9.E8.9E-9.6E8 8.8E-.6E-9.7E-.9E-.7E-9.E-.7E-.6E-9.E-9.9E8.E-9.99E8 9.E-.E-9.E-.E-.7E-9.7E-.9E-.9E-9 6.9E-9.E8 6.7E-9.9E8 6.E-.8E-9.E-.77E-.6E-9.E-.E-.8E-9 7.9E-9.7E8 7.66E-9.E8. Driv xrssions for th rliability and availability of th ntwork shown in Figur. for th cas (r,w (, whr a singl vot is assignd to ach nod in th nonhirarchical organization. In this cas, both rad and writ orations can tak lac if at last thr of th fiv nods ar u. Assum that failurs occur at ach nod according to a oisson rocss with rat, but th links do not fail. Whn a nod fails, it is raird (rair includs loading u-to-dat data and th rair tim is an xonntially distributd random variabl with man /. Driv th rquird xrssions for th systm rliability and availability using th Markov chains (s Chatr shown in Figur.a and b, rsctivly, whr th stat is th numbr of nods that ar down.
Modify to do availability only. Availability is modld by th robability of bing in on of th stats in th following st: {,, } for t. d ( d ( ( d ( ( d ( ( d (
i i A 8. For th xaml shown in Figur. th four nods hav an availability whil th links hav th availabilitis indicatd in th figur. Us Huristic to assign vots to th four nods, writ down th ossibl valus for w and th corrsonding minimal valus of r, and calculat th availability for ach ossibl valu of (r,w. Assum that rad orations ar twic as frqunt as writ orations Rad and writ quorums undr Huristic : v(a. (.7*..7 Assign vots to A v(b. (.7*..9*..*..8 Assign vots to B (now vots v(c. (.9*..7*..6 Assign vots to C v(d. (.*..7*..9 Assign vots to D Total numbr of vots. Sinc is vn, add xtra vot to on of th largst, ick B. w > /, r w > R W Rad Quorums Writ Quorums r w A 6 6 AB, BC, BD, ACD AB, BC, BD, ACD.976.976.976 7 AB, AC, BC, CD, BD BC, ABD, ACD.99.98.9679 8 B, AC, AD, CD ABC, ABD, BCD..8.9 9 AD, B, C ABC, BCD..89.899 A, B, C, D ABCD..886.89 A, B, C, D ABCD..886.89
To calculat th robability of a rad quorum or a writ quorum, look at combinations of th links sinc all nod availabilitis ar.. R6 W6 R W7 R W8 AB BC BD CD RQ WQ RQ WQ RQ WQ.7.68.8..68..6.78.68.9..98..8.78.88.976.976.976.998.98.966.8.9 R W9 R W R w AB BC BD CD RQ WQ RQ WQ RQ WQ.7.68.8..68..6.78.68.9..98..8.78.88.89.899.89.89.886 Th robabilitis ar givn for th rad and writ quorums. Availability occurs whn both a rad and writ quorum occurs and is wightd by th roortion of rads and writs.
. Show how chcksums can b usd to dtct and corrct rrors in a scalar by matrix multilication for th following xaml. Assum a x matrix Show th corrsonding column wightd matrix A C and assum that during th mulitlication of A C by th scalar a singl rror has occurd rsulting in th following outut Th column wightd matrix A C is Aftr multilying by th scalar and with th singl rror, w obtain For columns and, S and S ar both zro. For column w calculat S Σ i a i, - WCS ( 7 -, and S Σ i i- a i, - WCS ( 68-88. Sinc both S and S ar non-zro, w calculat S/S (- imlying that a, is rronous. W corrct th rror using a, a, - S 7-6.