Content Based Status Updates

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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: