Content Based Status Updates
|
|
- Imogene Burns
- 5 years ago
- Views:
Transcription
1 Contnt Basd Status Updats li Najm LTHI, PFL, Lausann, Switzrland mail: Rajai Nassr LTHI, PFL, Lausann, Switzrland mail: mr Tlatar LTHI, PFL, Lausann, Switzrland mail: Abstract Considr a stram of status updats gnratd by a sourc, whr ach updat is of on of two typs: priority or ordinary; ths updats ar to b transmittd through a ntwork to a monitor. W analyz a transmission policy that trats updats dpnding on thir contnt: ordinary updats ar srvd in a firstcom first-srvd fashion, whras th priority updats rciv prfrntial tratmnt. An arriving priority updat discards and rplacs any currntly-in-srvic priority updat, and prmpts with vntual rsum any ordinary updat. W modl th arrival procsss of th two kinds of updats as indpndnt Poisson procsss and th srvic tims as two possibly diffrnt rat xponntials. W find th arrival and srvic rats undr which th systm is stabl and giv closd-form xprssions for avrag pak ag and a lowr bound on th avrag ag of th ordinary stram. W giv numrical rsults on th avrag ag of both strams and obsrv th ffct of ach stram on th ag of th othr. I. INTRODUCTION Whil th classical notion of dlay is a masur of how long a packt spnds in transit, th Ag of Information [] is a rcivr-cntric notion that masurs how frsh th data is at th rcivr. Spcifically, with ut dnoting th gnration tim of th last succssfully rcivd packt bfor tim t, on dfins t = t ut as th instantanous ag of th information at th rcivr at tim t. On can thn considr = lim τ τ τ tdt, as th tim avrag ag. Obsrv that t incrass linarly in th intrvals btwn packt rcptions, and whn a packt is rcivd, t jumps down to th dlay xprincd by this packt. This rsults in a sawtooth sampl path as in Fig.. In [] [6] th proprtis of wr invstigatd undr th assumption that th packts ar gnratd by a Poisson procss, and various transmission policis M/M/, M/M/, gamma srvic tim,.... A rlatd mtric, calld avrag pak ag, was introducd in [3] as th avrag of th valu of th instantanous ag t at tims just bfor its downward jumps. In Fig., K j dnots th instantanous ag just bfor th rcption of th j th succssfully transmittd packt, and hnc, th avrag pak ag is givn by pak = lim N N N K j. 2 j= Th authors in [7] studid th avrag ag whn considring multipl sourcs snding updat through on quu. Thy computd th avrag ag for thr scnarios: all sourcs transmit according to an M/M/ FCFS policy, all sourcs transmit according to an M/M// with prmption policy and all sourcs transmit according to an M/M// with blocking policy. In th M/M// with prmption policy, if a nwly gnratd updat finds th systm busy, th transmittr prmpts th on currntly in srvic and starts snding th nw packt. On th othr hand, in th M/M// with blocking policy, if th gnratd updat finds th systm busy, it is discardd. In [8], th authors also considr multipl sourcs transmitting through a singl quu but in this cas thy assum a gnrally distributd srvic tim. Morovr, thy study two scnarios: all sourcs transmit according to an M/G/ FCFS policy or all sourcs transmit according to an M/G// with blocking policy. For ach on of ths policis, th authors giv th xprssion of th avrag pak ag rlativ to ach sourc. In this papr, w assum updats ar gnratd according to a Poisson procss with rat λ, and that th updats blong to two diffrnt strams whr ach stram i is chosn indpndntly with probability p i, i =, 2. So w hav two indpndnt Poisson strams with rats = λp and = λp 2. Howvr, unlik [7] and [8], w assum a diffrnt transmission policy for ach stram. To th bst of our knowldg, this modl was not studid bfor although it modls a natural scnario. In fact, th two indpndnt strams gnratd by th sourc can b usd to modl diffrnt typs of contnt carrid by th packts of ach stram. For xampl, if th sourc is a snsor, on stram could carry mrgncy mssags fir alarm, high prssur, tc. and thus it nds to b always as frsh as possibl whil th othr stram will carry rgular updats and hnc is not ag snsitiv. Thrfor, it stands to rason to transmit ths two strams in a diffrnt mannr. Th rgular stram will b transmittd according to a FCFS policy whil th high priority stram will b snt by prmption, packts of th high priority stram prmpt all packts including packts of thir own stram. W will furthr assum that th srvic tims rquirmnts of th two strams may b diffrnt; a packt of th rgular stram will b srvd at rat µ, a packt of th priority schm at rat µ 2. W will study th abov modl and answr th qustions: what should th rlation btwn, µ, and µ 2 b for th systm to b stabl? How dos ach stram affcts th avrag ag of th othr on? What ar th ags of ach stram? To answr ths qustions, w will giv a ncssary and sufficint condition for th systm stability, find th stadystat distribution of th undrlying stat-spac, and giv closd
2 form xprssions for th avrag pak ag and a lowr bound on th avrag ag of th rgular stram and compar thm to th avrag ag of th high priority stram. This papr is structurd as follows: in Sction II, w start by dfining th modl and th diffrnt variabls ndd in our study. In Sction III w driv th stability condition of th systm and its stationary distribution. Th closd form xprssions of th avrag pak ag and th lowr bound on th avrag ag of th rgular stram ar computd in Sction IV. Finally, in Sction V w prsnt numrical rsults of our rsults. II. SYSTM MODL W considr a sourc that gnrats packts or updats according to a Poisson procss of rat λ. ach packt, indpndntly of th prvious packts, is of typ with probability p and of typ 2 with probability p 2 = p. W can thus s our sourc as gnrating two indpndnt Poisson strams U and U 2 with rats = λp and = λp 2 rspctivly, λ = + s [9]. As notd in th introduction, th diffrnt strams can b usd to modl packts of diffrnt typs of contnt, for xampl, mrgncy mssags, alrts, rror mssags, warnings, notics, tc. W also assum that th updats ar snt through a singl srvr or transmittr quu to a monitor. Th srvic tim of ach packt is considrd to b xponntially distributd with rat µ for stram U and rat µ 2 for stram U 2. Th diffrnc in srvic rats btwn th two strams is to account for th possibl diffrnc in comprssion, packt lngth, tc., btwn th two strams. Givn this modl, w impos on th transmittr that all packts from stram U should b snt. Hnc th srvr applis a FCFS policy on th packts from stram U with a buffr to sav waiting updats. On th othr hand, w assum that th information carrid by stram U 2 is mor tim snsitiv or has highr priority and thus w aim at minimizing its avrag ag. To this nd, th transmittr is allowd to prform packt managmnt: in this cas w assum th srvr applis a prmption policy whnvr a packt from U 2 is gnratd. This mans that if a nwly gnratd packt from stram U 2 finds th systm busy srving a packt from U or U 2, th srvr prmpts th updat currntly in srvic and start srving th nw packt. Morovr, if th prmptd packt blongs to U, this packt is placd back at th had of th U -buffr so that it can b srvd onc th systm is idl again. If th prmptd packt blongs to U 2 thn it is discardd. Howvr, if a nwly gnratd U -packt finds th systm busy srving a U 2 -packt, it is placd in th buffr and srvd whn th systm bcoms idl. This choic of policy for th ag snsitiv stram is motivatd by th conclusion rachd in [] that for xponntially distributd packt transmission tims, th M/M// with prmption policy is th optimal policy among causal policis. Ths idas ar illustratd in part in Fig. which also shows th variation of th instantanous ag of stram U. In this plot, t i and D i rfr to th gnration and dlivry tims of t Q K t D t 2 t Q 2 K 2 Q 3 K 3 Q 4 Z 3 Z 4 X 4 T 4 X 3 T 3 K 4 t 3 D D2 t 2 D 2 t 3 D 3 D3 t 4 t4 D 4 D4 Systm busy srving stram U 2 Fig.. Variation of th instantanous ag of stram U. th i th packt of stram U whil t i and D i ar th start and nd tims of th i th priod during which th systm is busy srving packts from stram U 2 only. III. SYSTM STABILITY AND STATIONARY DISTRIBUTION Th fact that w wish to rciv all of stram U updats and that stram U 2 has highr priority and prmpts stram U might lad to an unstabl systm. In ordr to driv th ncssary and sufficint condition for th stability of th systm w study th Markov chain of th numbr of packts in th systm in srvic and waiting shown in Fig. 2. In this chain, q is th idl stat whr th systm is compltly mpty. Stats q i, i >, in th uppr row rfr to stats whr th quu is srving a packt from stram U whil stats q i, i >, in th row blow corrspond to th quu srving a packt from stram U 2. In both cass thr ar i stram U updats waiting in th buffr. Th systm lavs stat q at rat to stat q whn a packt from stram U is gnratd first and it lavs q at rat to stat q whn a packt from stram U 2 is gnratd first. Howvr, whn th systm ntrs stat q i, i >, thr xponntial clocks start: a clock with rat µ which corrsponds to th srvic tim of th stram U packt bing srvd, a clock with rat which corrsponds to th gnration tim of stram U packts and a clock with rat which corrsponds to th gnration tim of stram U 2 packts. If th µ -clock ticks first, th systm gos to stat q i : this mans that th currnt stram U packt was dlivrd and th quu bgins th srvic of th nxt on in th buffr if thr is any. Howvr, if th -clock ticks first, a nw stram U updat is gnratd and addd to th buffr and hnc th systm gos to stat q i+. On th othr hand, if th -clock ticks first, th systm prmpts th packt currntly in srvic and placs it back at th had of th buffr and starts th srvic of th nwly gnratd stram U 2 updat. Thus th systm gos to stat q i+. Whn th systm ntrs a stat q i, i >, two xponntial clocks start: th clock with rat and a clock with rat µ 2 which corrsponds to th t
3 q q q 2 q 3 µ 2 µ µ 2 µ q q 2 q 3 q 4 µ µ 2 µ 2 Fig. 2. Markov chain govrning th numbr of packts in th systm. srvic tim of a stram U 2 packt. If th -clock ticks first, th nwly gnratd stram U packt is placd in th buffr and th stram U 2 updat is continud to b srvd. Hnc th systm gos to stat q i+. Howvr, if th µ 2-clock ticks first, th stram U 2 packt has finishd srvic and th systm starts srving th first stram U packt in th buffr if thr is any. Thus th systm gos to stat q i. This nxt thorm givs th ncssary and sufficint condition for th abov systm to b stabl as wll as its stationary distribution. Thorm. Th systm dscribd in Sction II is stabl, i.. th avrag numbr of packts in th quu is finit, if and only if µ > + λ 2. 3 µ 2 In this cas th Markov chain shown in Fig. 2 has a stationary distribution Π = [π, π,..., π i,..., π,..., π i,... ], whr π i dnots th stationary probability of stat q i, i, and π i dnots th stationary probability of stat q i, i >. This stationary distribution is dscribd by th following systm of quations, C = π = µ 2, 4 µ 2 + µ A i = [ I ] H i whr λ = +, [ ] [ ] πi C D A i = π i, H =, I [ + λ µ µ2λ2 λ µ µ2λ2 µ +µ 2 +µ 2 µ µ 2+ µ2λ µ µ 2+ µ 2+ µ 2+ π, i 5 ] [, D = µ I is th 2 2 idntity matrix and is th 2 2 zro matrix. ], Proof. Th distribution givn by 4 and 5 satisfy th dtaild balanc quations of th Markov chain shown in Fig. 2. Morovr, 3 is th condition ndd to hav π >. Th condition in 3 can b intrprtd as follows: dfin th map f from th stat-spac of th chain as fs = if s is in {q, q,... } and fs = if s {q, q 2,... }. For ach s and s for which fs = and fs = th transition rat from s to s is th sam and similarly for s and s with fs =, fs =, µ 2. Consquntly F t = fst, with st bing th stat at tim t, is Markov which would not b th cas for an arbitrary F, and it is asily sn that F t = a fraction φ = µ 2 / + µ 2 amount of tim, F t = a fraction φ = / + µ 2 amount of tim. Thus, whil th Markov chain in Fig. 2 movs right at rat, it movs lft at a rat µ φ. Th systm is stabl only if th rat of moving lft is largr than th rat of moving right; which givs th condition 3. A. Prliminaris IV. AGS OF STRAMS U AND U 2 In this sction, unlss statd othrwis, all random variabls corrspond to stram U. W also follow th convntion whr a random variabl U with no subscript corrsponds to th stady-stat vrsion of U j which rfrs to th random variabl rlativ to th j th rcivd packt from stram U. To diffrntiat btwn strams w will us suprscripts, so U i corrsponds to th stady-stat variabl U rlativ to stram U i, i =, 2. In addition to that, w dfin: i X i to b th intrarrival tim btwn two conscutiv gnratd updats from stram U i, so f X ix = λ i λix, i =, 2 ii S i to b th srvic tim random variabl of stram U i updats, so f S it = µ i µit, i =, 2 iii T j to b th systm tim, or th tim spnt by th j th stram U updat in th quu. In our modl, w assum th srvic tim of th updats from th diffrnt strams to b indpndnt of th intrarrival tim btwn conscutiv packts blonging to th sam stram or not. Givn th dscription of th modl in Sction II, w s that from th point of viw of stram U th systm bhavs as an M/G/ quu, with ach packt j having an indpndnt virtual srvic tim Z j physical srvic tim S j tim Z j as follows: which could b diffrnt from its. W dfin th virtual srvic Z j = D j maxd j, t j, 6 whr D j is th dlivry tim of th j th packt and t j is its gnration tim. Fig. shows th virtual srvic tim for packts 3 and 4. B. Avrag pak ag of stram U For stram U, givn that th avrag ag calculations sms to b intractabl, w will comput its avrag pak ag and giv a lowr bound on its avrag ag. W start by dfining th vnt Ψ j = {packt j finds th systm in stat q } and its complmnt Ψ j. Thn w nd th following lmmas.
4 start s a Fig. 3. Smi-Markov chain rprsnting th virtual srvic tim Y j. Lmma. Lt Y j b th virtual srvic tim of packt j givn that this packt dos not finds th systm in stat q, i.. P Y j > t = P Z j > t Ψ j. Thn, in stady stat, φ Y s = sy = s u v s 2 µ µ 2 s s 2 sµ 2 + µ + + µ µ 2. 7 Similarly, lt Y j b th virtual srvic tim of packt j givn that this packt finds th systm in stat q, i.. P Y j > t = P Z j > t Ψ j. Thn, in stady stat, φ Y s = sy = µ µ 2 s 2 sµ 2 + µ + + µ µ 2. 8 Proof. W start by proving 7. For that w will us th dtour flow graph mthod. Fig. 3 shows th smi-markov chain rlativ to th virtual srvic tim Y j of th j th packt of first stram U. Whn th j th packt gts at th had of th buffr, th systm is in th idl stat s. Hnc with probability it gos immdiatly to stat s whr it starts srving th j th packt. Du to th mmorylss proprty of th intrarrival tim of th scond stram X 2, two clocks start: a srvic clock S and a clock X 2. Th srvic clock ticks first with probability a = P S < X 2 and its valu A has distribution P A > t = P S > t S < X 2. At this point th stram U packt currntly bing srvd finishs srvic bfor any packt from th othr stram is gnratd and th systm gos back to stat s. This nds th virtual srvic tim Y j. On th othr hand, clock X 2 ticks first with probability v = a = P X 2 < S and its valu V has distribution P V > t = P X 2 > t X 2 < S. At this point, a nw stram U 2 updat is gnratd and prmpts th stram U packt currntly in srvic. In this cas th systm gos to stat s 2, whr th prmptd stram U updat is placd back at th had of th buffr and th systm starts srvic of th stram U 2 updat. Whn th systm arrivs in stat s 2, this mans a nw stram U 2 packt was just gnratd and is starting srvic. Thus, two clocks start: a srvic clock S 2 and a clock X 2. Th srvic clock ticks first with probability u = P S 2 < X 2 and its valu U has distribution P U > t = P S 2 > t S 2 < X 2. At this point, th packt currntly bing srvd finishs srvic bfor any nw stram U 2 packt is gnratd and th systm gos back to stat s whr th j th packt of stram U starts srvic again. Howvr, clock X 2 ticks first with probability b = u and its valu B = has distribution P B > t = P X 2 > t X 2 < S 2. At b this point, a nw stram U 2 updat is gnratd and prmpts th on currntly in srvic. In this cas th systm stays in stat s 2. From th abov analysis w s that th virtual srvic tim is givn by th sum of th valus of th diffrnt clocks on th path starting and finishing at s. For xampl, for th path s s s 2 s s 2 s 2 s s in Fig. 3, th virtual srvic tim Y = V + U + V 2 + B + U 2 + A, whr all th random variabls in th sum ar mutually indpndnt. This valu of Y is also valid for th path s s s 2 s 2 s s 2 s s. Hnc Y dpnds on th variabls A j, B j, U j, V j and thir numbr of occurrncs and not on th path itslf. Thrfor, th probability that xactly i, i 2, i 3, i 4 occurrncs of A, B, U, V happn, which is quivalnt to th probability that Y = i k= A k + i 2 k= B k + i 3 k= U k + i 4 V k k= is givn by a i b i2 u i3 v i4 Qi, i 2, i 3, i 4, whr Qi, i 2, i 3, i 4 is th numbr of paths with this combination of occurrncs. Taking into account th fact that th {A k, B k, U k, V k } ar mutually indpndnt and dnoting by {I, I 2, I 3, I 4 } th random variabls associatd with th numbr of occurrncs of {A, B, U, V } rspctivly, th momnt gnrating function of Y is, φ Y s = sy I, I 2, I 3, I 4 = i, i 2, i 3, i 4 = [ a i b i2 u i3 v i4 Qi, i 2, i 3, i 4 i,i 2,i 3,i 4 s i k= A k + i 2 k= B k + i 3 k= U k + i 4 k= V k ] = i,i 2,i 3,i 4 [ a i b i2 u i3 v i4 Qi, i 2, i 3, i 4 sa i sb i 2 su i 3 sv i 4 ]. 9 Howvr 9 is nothing but th gnrating function H D, D 2, D 3, D 4 of th dtour flow graph shown in Fig. 4a, whr D, D 2, D 3, D 4 ar dummy variabls s [, pp ]. Simpl calculations giv H D, D 2, D 3, D 4 = i,i 2,i 3,i 4 [ Qi, i 2, i 3, i 4 a i b i2 u i3 v i4 D i D i2 2 Di3 3 Di4 4 = ad bd 2 bd 2 ud 3 vd 4. Thus φ Y s = H sa, sb, su, sv. From [7, Appndix A, Lmma 2], w know that A, B, U and V ar xponntially distributd with sb = su = +µ 2 λ and 2+µ 2 s sa = sv = λ2+µ +µ s. Simpl computations show that a = µ µ +, b = λ2 µ 2+, u = µ2 µ 2+, v = λ2 µ +. Finally, rplacing th abov xprssions into, w gt our rsult. To prov 8, w us th sam mthod as bfor but in this cas w notic that th j th packt from stram U finds th ]
5 bd 2 s 2 vd 4 ud 3 s s ad s a ud 3 s s 2 s s ad b Fig. 4. Dtour flow graphs for a Y and b Y. vd 4 bd 2 systm busy srving a packt from stram U 2. This translats in th dtour flow graph shown in Fig. 4b. Th gnrating function of this graph is H 2 D, D 2, D 3, D 4 = ad ud 3 bd 2 vd 4 ud 3. For D, D 2, D 3, D 4 = sa, sb, su, and rplacing a, b, u and v by thir valus in, w gt 8. sv Lmma 2. Th first and scond momnts of th virtual srvic tim Z ar givn by λ + λ2 + µ2 Z = + + µ 2 µ 2 + µ + µ, 2 Z 2 = 2 + µ µ µ 2 + µ + 2µ 2 µ 2 µ2 λ + µ2 λ2 + µ2 2 Proof. For any packt j of stram U, conditioning on th vnt Ψ j, w gt Z j = P Ψ j Z j Ψ j + P Ψ j Zj Ψ j = P Ψ j Y j + P Ψj Yj, 3 whr Y j and Y j ar dfind as in Lmma. From Thorm w dduc that P Ψ j = π = λ2 +µ 2 π. So in stady-stat, Z = π Y + π Y. Morovr, using 7 and 8 w gt Y = µ 2 +, Y = µ + µ 2 +. µ µ 2 µ µ 2 Similarly, Z 2 = π Y 2 + π Y 2. Using 7 and 8 w gt Y 2 = 2 µ µ µ µ 2 2 and Y 2 = 2 µ + µ µ µ 2 µ µ 2 2. It is worth noting that th condition givn in 3 lads to th inquality Z <, which implis th stability of th M/G/ quu. Thorm 2. Th avrag pak ag of stram U is givn by pak, = + Z + Z 2 2 Z, 4 whr Z is th stady stat quivalnt of Z j dfind in 6, and is th gnration rat of updats blonging to U. With th xprssions for Z and Z 2 givn in Lmma 2, w thus obtain th avrag pak ag of stram U in closd form. Proof. As w hav sn bfor, th systm from stram U point of viw acts lik an M/G/ quu with srvic tim Z. Applying [8, Proposition 2] for a singl stram M/G/ systm with updat rat and srvic tim Z givs 4 as th avrag pak ag for stram U. C. Lowr bound on th avrag ag of stram U Th avrag pak ag is an obvious uppr bound, hnc in this sction w will comput a lowr bound of th avrag ag. From 8, w can dduc that Y = X 2 + Y with X 2 and Y bing indpndnt. In fact, sy = sx2 sy, whr sx2 = µ2 µ and 2 s sy givn by 7. This xplains why Y = Y + µ 2 > Y. Sinc Z = π Y + π Y, thn Y < Z < Y. Hnc th avrag ag of an M/G/ quu with srvic tim Y will giv us a lowr bound to th avrag ag of our M/G/ systm with srvic tim Z. Lmma 3. Assum an M/G/ quu with intrarrival tim X xponntially distributd with rat and srvic tim Y whos momnt gnrating function is givn by 7. Th srvic tim and th intrarrival tim ar assumd to b indpndnt. Thn th distribution of th systm tim T is f T t = C α t C 2 α 2t, t, 5 whr α > α 2 > ar th roots of th quadratic xprssion s 2 sµ + µ µ µ 2 µ 2, C = ρµ µ 2 α α α 2, C 2 = ρµ µ 2 α 2 α 2 α and ρ = Y = λµ2+λ2 µ µ 2. Proof. From [2, p. 66], w know that φ T s = ρφ Y s s + φ Y s.
6 Rplacing φ Y s by its xprssion in 7 w gt ρµ µ 2 s φ T s = s 2 sµ + µ µ µ 2 µ 2 = C s α + C 2 s α 2, 6 by partial fraction xpansion. Morovr, du to condition 3, and α + α 2 = µ + µ 2 + > α α 2 = µ µ 2 µ 2 >. This provs that both roots α and α 2 ar positiv. Without loss of gnrality, w tak α > α 2. Taking th invrs Laplac transform of φ T s w gt 5. From [], w know that th avrag ag of th M/G/ quu with intrarrival tim X and srvic tim Y is LB = 2 X 2 j + T j X j, 7 whr for th j th packt w hav T j = T j X j + + Y j, fx = x + = x {x } and {.} is th indicator function. So T j X j bcoms T j X j = X j T j X j + + Y j X j, 8 whr th scond trm is du to th indpndnc btwn Y j and X j. Proposition. X j T j X j + = ρµ α + α 2 2 α α 2 µ2 + 2 µ 2 α α 2 µ 2 + µ 2 2 α α ρµ α + α 2 µ 2 2 α α 2 λ 2 α α 2 µ 2 + µ 2 2 α α 2 2, 9 whr α + α 2 = µ + µ 2 +, α α 2 = µ µ 2 µ 2 and ρ = Y = λµ2+λ2 µ µ 2. Proof. Givn that T j and X j ar indpndnt thn X j T j X j + = x xt xf T t λx dtdx Rplacing f T t by its valu in 5, w gt 9 aftr som computations. Finally, using 9, Y j = Y = µ2+λ2 µ µ 2 and X j = X =, w can find a closd form xprssion for T j X j. Rplacing this xprssion in 7 and using th fact that X 2 j = 2, w obtain a closd form xprssion of th avrag ag LB of an M/G/ quu Ag Avrag Ag lowr bound for stram Avrag Pak Ag for stram Avrag Ag for stram 2 Avrag Ag for M/M/ quu with srvic rat μ Fig. 5. Plot of th avrag ag for stram U 2 and avrag pak ag and lowr bound on th avrag ag for stram U, with µ =, µ 2 = 5, = 2 and < µ µ 2 µ 2 = 2. with intrarrival tim X and srvic tim Y. This is also a lowr bound on th avrag ag of th M/G/ quu with intrarrival tim X and srvic tim Z. D. Avrag ag of stram U 2 By dsign, stram U 2 is not intrfrd at all by stram U and hnc bhavs lik a traditional M/M// with prmption quu with gnration rat and srvic rat µ 2. Th avrag ag of this stram was computd in [2] to b λ2 U2 = µ V. NUMRICAL RSULTS Fig. 5 shows th avrag pak ag pak, and th lowr bound on th avrag ag LB as computd in th prvious sction for stram U as wll as th avrag ag U2 of stram U 2. In this plot, w fix µ =, µ 2 = 5, = 2 and w vary. As w can s, for stram U both th lowr bound and th avrag pak ag blow up whn gts clos to µµ2 µ 2. This obsrvation is in lin with our rsult in Thorm and th stability condition 3. It is asy to s via a coupling argumnt that if w incras, th ag procss U t of th U stram will stochastically incras. W s from th plots that th lowr bound on U and its avrag pak xhibit th sam bhavior. On th othr hand, th avrag ag of stram U 2 is dcrasing in from 2. Consquntly, minimizing U2 and minimizing U ar conflicting goals. W hav sn that th avrag ag of stram U 2 is not affctd by th prsnc of th othr stram. Howvr, Fig. 5 shows th ffct of stram U 2 on th avrag ag of stram U. For that, w plot th avrag ag rf of an M/M/ quu with gnration rat and srvic rat µ givn in []. W obsrv that th lowr bound on th avrag ag of stram U is vry clos to th valu of th avrag ag of this stram if stram U 2 is not prsnt. Only at high valus of around 5 dos LB clarly start to divrg from rf. Howvr, pak, starts divrging much soonr as xpctd from an uppr bound. Nonthlss, for a non ngligibl intrval of spcially at low valus, th uppr bound pak, and th lowr bound LB ar clos and thy
7 giv a tight approximation of th avrag ag of stram U. This mans that at ths valus of, th avrag ag achivd by stram U is vry clos to that achivd by th sam stram with th absnc of stram U 2. Thus stram U 2 has almost no ffct on stram U for ths valus of from an ag point of viw. Th ffct of stram U 2 on th first on is spcially flt at high valus of, which is an xpctd bhavior. VI. CONCLUSION In this papr w studid th ffct of implmnting contntdpndnt policis on th avrag ag of th packts. W considrd a sourc gnrating two indpndnt Poisson strams with on stram bing ag snsitiv and having highr priority than th othr stram. Th high priority stram is snt using a prmption policy whil th rgular stram is transmittd using a First Com First Srvd FCFS policy. W drivd th stability condition for th systm as wll as closd form xprssions for th avrag pak ag and a lowr bound on th avrag ag of th rgular stram. W also dducd that on can t hop to minimiz both strams if w can only control th gnration rat of th high priority stram. ACKNOWLDGMNTS This rsarch was supportd in part by grant No / of th Swiss National Scinc Foundation. RFRNCS [] S. Kaul, R. D. Yats, and M. Grutsr, Ral-tim status: How oftn should on updat? in Proc. INFOCOM, 22. [2], Status updats through quus, in Conf. on Information Scincs and Systms CISS, Mar. 22. [3] M. Costa, M. Codranu, and A. phrmids, Ag of information with packt managmnt, in Proc. I Int l. Symp. Info. Thory, Jun 24, pp [4] C. Kam, S. Komplla, and A. phrmids, Ag of information undr random updats, in Proc. I Int l. Symp. Info. Thory, 23, pp [5]. Najm and R. Nassr, Th ag of information: Th gamma awakning, in Proc. I Int l. Symp. Info. Thory, 26, pp [6] R. D. Yats and S. Kaul, Ral-tim status updating: Multipl sourcs, in Proc. I Int l. Symp. Info. Thory, Jul. 22. [7] R. D. Yats and S. K. Kaul, Th ag of information: Ral-tim status updating by multipl sourcs, CoRR, vol. abs/ , 26. [Onlin]. Availabl: [8] L. Huang and. Modiano, Optimizing ag-of-information in a multiclass quuing systm, in Proc. I Int l. Symp. Info. Thory, Jun. 25. [9] S. M. Ross, Stochastic Procsss Wily Sris in Probability and Statistics, 2nd d. Wily, Fb [] A. M. Bdwy, Y. Sun, and N. B. Shroff, Ag-optimal information updats in multihop ntworks, CoRR, vol. abs/7.57, 27. [Onlin]. Availabl: [] B. Rimoldi, Principls of Digital Communication: A Top-Down Approach. Cambridg Univrsity Prss, 26. [2] J. Daigl, Quuing Thory with Applications to Packt Tlcommunication, sr. Quuing Thory with Applications to Packt Tlcommunication. Springr, 25. [Onlin]. Availabl:
The Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More information4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.
PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also
More informationRandom Access Techniques: ALOHA (cont.)
Random Accss Tchniqus: ALOHA (cont.) 1 Exampl [ Aloha avoiding collision ] A pur ALOHA ntwork transmits a 200-bit fram on a shard channl Of 200 kbps at tim. What is th rquirmnt to mak this fram collision
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationComputing and Communications -- Network Coding
89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationEstimation of apparent fraction defective: A mathematical approach
Availabl onlin at www.plagiarsarchlibrary.com Plagia Rsarch Library Advancs in Applid Scinc Rsarch, 011, (): 84-89 ISSN: 0976-8610 CODEN (USA): AASRFC Estimation of apparnt fraction dfctiv: A mathmatical
More informationPrinciples of Humidity Dalton s law
Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationThe Equitable Dominating Graph
Intrnational Journal of Enginring Rsarch and Tchnology. ISSN 0974-3154 Volum 8, Numbr 1 (015), pp. 35-4 Intrnational Rsarch Publication Hous http://www.irphous.com Th Equitabl Dominating Graph P.N. Vinay
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationCS 361 Meeting 12 10/3/18
CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationEquidistribution and Weyl s criterion
Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss
More informationBrief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
More informationAnswer Homework 5 PHA5127 Fall 1999 Jeff Stark
Answr omwork 5 PA527 Fall 999 Jff Stark A patint is bing tratd with Drug X in a clinical stting. Upon admiion, an IV bolus dos of 000mg was givn which yildd an initial concntration of 5.56 µg/ml. A fw
More informationRandom Process Part 1
Random Procss Part A random procss t (, ζ is a signal or wavform in tim. t : tim ζ : outcom in th sampl spac Each tim w rapat th xprimnt, a nw wavform is gnratd. ( W will adopt t for short. Tim sampls
More information2.3 Matrix Formulation
23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationPHA 5127 Answers Homework 2 Fall 2001
PH 5127 nswrs Homwork 2 Fall 2001 OK, bfor you rad th answrs, many of you spnt a lot of tim on this homwork. Plas, nxt tim if you hav qustions plas com talk/ask us. Thr is no nd to suffr (wll a littl suffring
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationSquare of Hamilton cycle in a random graph
Squar of Hamilton cycl in a random graph Andrzj Dudk Alan Friz Jun 28, 2016 Abstract W show that p = n is a sharp thrshold for th random graph G n,p to contain th squar of a Hamilton cycl. This improvs
More information4.2 Design of Sections for Flexure
4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More informationTwo Products Manufacturer s Production Decisions with Carbon Constraint
Managmnt Scinc and Enginring Vol 7 No 3 pp 3-34 DOI:3968/jms9335X374 ISSN 93-34 [Print] ISSN 93-35X [Onlin] wwwcscanadant wwwcscanadaorg Two Products Manufacturr s Production Dcisions with Carbon Constraint
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationCOUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM
COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationWhat are those βs anyway? Understanding Design Matrix & Odds ratios
Ral paramtr stimat WILD 750 - Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow. 000. Applid logistic rgrssion. John Wily & ons Inc.
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More information1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:
Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationElements of Statistical Thermodynamics
24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationThe pn junction: 2 Current vs Voltage (IV) characteristics
Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n
More informationContinuous probability distributions
Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More informationSequential Decentralized Detection under Noisy Channels
Squntial Dcntralizd Dtction undr Noisy Channls Yasin Yilmaz Elctrical Enginring Dpartmnt Columbia Univrsity Nw Yor, NY 1007 Email: yasin@.columbia.du Gorg Moustaids Dpt. of Elctrical & Computr Enginring
More informationA Simple Formula for the Hilbert Metric with Respect to a Sub-Gaussian Cone
mathmatics Articl A Simpl Formula for th Hilbrt Mtric with Rspct to a Sub-Gaussian Con Stéphan Chrétin 1, * and Juan-Pablo Ortga 2 1 National Physical Laboratory, Hampton Road, Tddinton TW11 0LW, UK 2
More informationThus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.
Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn
More informationData Assimilation 1. Alan O Neill National Centre for Earth Observation UK
Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal
More informationOn spanning trees and cycles of multicolored point sets with few intersections
On spanning trs and cycls of multicolord point sts with fw intrsctions M. Kano, C. Mrino, and J. Urrutia April, 00 Abstract Lt P 1,..., P k b a collction of disjoint point sts in R in gnral position. W
More informationMATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationCalculus concepts derivatives
All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving
More information2. Laser physics - basics
. Lasr physics - basics Spontanous and stimulatd procsss Einstin A and B cofficints Rat quation analysis Gain saturation What is a lasr? LASER: Light Amplification by Stimulatd Emission of Radiation "light"
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More informationTypes of Transfer Functions. Types of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr x[n] X( LTI h[n] H( y[n] Y( y [ n] h[ k] x[ n k] k Y ( H ( X ( Th tim-domain classification of an LTI digital transfr function is basd on th lngth of its impuls rspons h[n]:
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationA Prey-Predator Model with an Alternative Food for the Predator, Harvesting of Both the Species and with A Gestation Period for Interaction
Int. J. Opn Problms Compt. Math., Vol., o., Jun 008 A Pry-Prdator Modl with an Altrnativ Food for th Prdator, Harvsting of Both th Spcis and with A Gstation Priod for Intraction K. L. arayan and. CH. P.
More informationHomogeneous Constant Matrix Systems, Part I
39 Homognous Constant Matrix Systms, Part I Finally, w can start discussing mthods for solving a vry important class of diffrntial quation systms of diffrntial quations: homognous constant matrix systms
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts
More informationSuperposition. Thinning
Suprposition STAT253/317 Wintr 213 Lctur 11 Yibi Huang Fbruary 1, 213 5.3 Th Poisson Procsss 5.4 Gnralizations of th Poisson Procsss Th sum of two indpndnt Poisson procsss with rspctiv rats λ 1 and λ 2,
More informationHomotopy perturbation technique
Comput. Mthods Appl. Mch. Engrg. 178 (1999) 257±262 www.lsvir.com/locat/cma Homotopy prturbation tchniqu Ji-Huan H 1 Shanghai Univrsity, Shanghai Institut of Applid Mathmatics and Mchanics, Shanghai 272,
More informationDeift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
More informationIntroduction to Condensed Matter Physics
Introduction to Condnsd Mattr Physics pcific hat M.P. Vaughan Ovrviw Ovrviw of spcific hat Hat capacity Dulong-Ptit Law Einstin modl Dby modl h Hat Capacity Hat capacity h hat capacity of a systm hld at
More information4037 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Lvl MARK SCHEME for th Octobr/Novmbr 0 sris 40 ADDITIONAL MATHEMATICS 40/ Papr, maimum raw mark 80 This mark schm is publishd as an aid to tachrs and candidats,
More informationMutually Independent Hamiltonian Cycles of Pancake Networks
Mutually Indpndnt Hamiltonian Cycls of Pancak Ntworks Chng-Kuan Lin Dpartmnt of Mathmatics National Cntral Univrsity, Chung-Li, Taiwan 00, R O C discipl@ms0urlcomtw Hua-Min Huang Dpartmnt of Mathmatics
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationInference Methods for Stochastic Volatility Models
Intrnational Mathmatical Forum, Vol 8, 03, no 8, 369-375 Infrnc Mthods for Stochastic Volatility Modls Maddalna Cavicchioli Cá Foscari Univrsity of Vnic Advancd School of Economics Cannargio 3, Vnic, Italy
More informationCO-ORDINATION OF FAST NUMERICAL RELAYS AND CURRENT TRANSFORMERS OVERDIMENSIONING FACTORS AND INFLUENCING PARAMETERS
CO-ORDINATION OF FAST NUMERICAL RELAYS AND CURRENT TRANSFORMERS OVERDIMENSIONING FACTORS AND INFLUENCING PARAMETERS Stig Holst ABB Automation Products Swdn Bapuji S Palki ABB Utilitis India This papr rports
More informationON A CONJECTURE OF RYSElt AND MINC
MA THEMATICS ON A CONJECTURE OF RYSElt AND MINC BY ALBERT NIJE~HUIS*) AND HERBERT S. WILF *) (Communicatd at th mting of January 31, 1970) 1. Introduction Lt A b an n x n matrix of zros and ons, and suppos
More informationAlpha and beta decay equation practice
Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationAnother view for a posteriori error estimates for variational inequalities of the second kind
Accptd by Applid Numrical Mathmatics in July 2013 Anothr viw for a postriori rror stimats for variational inqualitis of th scond kind Fi Wang 1 and Wimin Han 2 Abstract. In this papr, w giv anothr viw
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationContent Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles.
Tachr: CORE TRIGONOMETRY Yar: 2012-13 Cours: TRIGONOMETRY Month: All Months S p t m b r Angls Essntial Qustions Can I idntify draw ngativ positiv angls in stard position? Do I hav a working knowldg of
More informationInheritance Gains in Notional Defined Contributions Accounts (NDCs)
Company LOGO Actuarial Tachrs and Rsarchrs Confrnc Oxford 14-15 th July 211 Inhritanc Gains in Notional Dfind Contributions Accounts (NDCs) by Motivation of this papr In Financial Dfind Contribution (FDC)
More information