003 Coferece o Iformatio Scieces ad Systems, The Johs Hopkis Uiversity, March 4, 003 Icreasig timig capacity usig packet colorig Xi Liu ad R Srikat[] Coordiated Sciece Laboratory Uiversity of Illiois e-mail: {xiliu,rsrikat}@uiucedu Abstract We provide a lower-boud o the timig capacity of a flow i the presece of ucotrollable iterferece traffic for a sigleserver queue with expoetially distributed service times Usig this result, we show that the timig capacity of a sigle flow ca reach the server capacity asymptotically by splittig the sigle flow ito multiple idepedet sub-flows I INTRODUCTION I [], the authors aalyze the timig capacity of cotiuous-time /G/ queues They showed that, whe the service time distributio is expoetial, the timig capacity is e ats per average trasmissio time, lowest amog all service time distributios I other words, if the service rate is B ats/sec ad each packet is B ats, the the timig capacity of the expoetial server is e ats/sec To achieve this timig capacity, packets are set to the server accordig to a Poisso process of rate e Now suppose that the seder has the freedom to decide the cotet of the packets set through the server A atural questio to ask is whether we ca icrease the timig capacity of the user We show that the timig capacity of a sigle flow ca reach the server capacity ie, B ats/sec asymptotically by splittig the sigle flow ito multiple idepedet sub-flows of differet colors The basic idea is to split the sigle flow to multiple sub-flows ach sub-flow is assumed to be idetified by a uique id, which we refer to as the color of the flow Thus, packets with the same color belog to the same sub-flow, ad iformatio is coded ito the iter-arrival times betwee packets of the same color Because each sub-flow has a smaller arrival rate tha the overall packet flow, the iter-arrival times of packets from a sub-flow is larger, ad thus the distortio caused by the queueig ad the radomess of the service distributio becomes relatively small Thus, the overall timig capacity icreases The tradeoff here is that the cotets of the packets are used for colorig i order to distiguish differet sub-flows The scheme is preseted i detail i the rest of the paper II MAIN RSULTS We start by studyig the case where a flow shares the server with some iterferece traffic The existece of iterferece traffic results i additioal radomess i the departure times of packets The, we study a sigle flow that has bee divided ito sub-flows usig packet colorig ad study the amout of timig iformatio carried i each sub-flow by cosiderig the other sub-flows as iterferece traffic A Iterferece Traffic Suppose that a seder cotrols the iput process of flow 0 to the server The server is shared with iterferece traffic, deoted as flow I, which is ot cotrolled by the seder The service disciplie is Research supported by DARPA through grat F3060-00--054 ad AFOSR through URI grat F4960-0--0365 FIFO The seder caot cotrol or detect the arrivals of the iterferece packets, ad vice-versa The arrival rate of flows 0 ad I are ad λ I, respectively It is iterestig to ote that if the service time of flow 0 s packet is determiistic, the the seder ca achieve ifiite timig capacity, regardless the existece of the ucotrollable iterferece traffic The reaso is that despite the presece of iterferece traffic, the probability that packets i flow 0 icur empty queues upo arrival is o-zero, ad thus, with a positive probability, the decoder ca observe the timig of the arrivals without ay radomess Oe simple ot ecessary the best strategy to achieve this capacity is to trasmit M dummy packets with the same iter-arrival time The iter-arrival time should be large eough so that the probability of a arrival fidig o packets i the queue is positive For simplicity of argumet, we assume the service time of iterferece packets are cotiuously distributed, ad thus, the probability of ay two packets have the same o-zero waitig time is zero Give ay ɛ > 0, let M be large eough so that the probability that at least three packets from flow 0 icur zero waitig i the queue is greater tha ɛ Because there are at least three packets that have departed without icurrig ay radomess i other words, there are at least two iter-departure times that are exactly equal to the correspodig iter-arrival times, the decoder ca measure the iter-arrival time exactly with probability, ad thus, receive ifiite amout of iformatio From ow o, we assume the service times of packets from both flows 0 ad I are iid radom variables with expoetial distributio The arrival process of flow I is a Poisso process with rate λ I The variables A i ad D i deote the iter-arrival ad iter-departure times of the ith packet of flow 0, as show i Figure I the figure, dashed lies with arrows idicate the arrivals ad departures of iterferece packets Let i be the umber of effective iterferece packets for the ith packet of flow 0 To elaborate, if the ith packet arrives before the departure of the i th packet, i is the umber of packets from the iterferece traffic arrived durig A i If the ith packet arrives after the departure of the i th packet, i is the umber of iterferece packets i the system whe the ith packet arrives I other words, the ith packet from flow 0 has to wait for i iterferece packets to be served after the departure of the i th packet I Figure,, 0, ad 3 We defie as effective service time of the ith packet, which is the time elapsed betwee the latter of the arrival of the ith packet ad the departure of the i th packet, ad the departure of the ith packet I other words, is the sum of the service time of the ith packet ad that of i iterferece packets, as show i Figure The iterferece traffic affects the timig iformatio coveyed by flow 0 packets by icreasig the radomess of the effective service time The /M/ queue studied i [] ca be cosidered as a special case where i 0 From the viewpoit of flow 0, packets from iterferece traffic that depart whe there is o flow 0 packet i the queue do ot cause ay distortio o the timig, ad thus, eed ot be co-
D D D 3 S S S 3 coverges i probability to /, ad A A Figure : Demostratio of iter-arrivals, iter-departures, ad effective service times sidered I Figure, the first, third, ad fourth iterferece packets do ot have ay effect o the departure times of flow 0 packets while the secod iterferece packet does Propositio The timig capacity of flow 0 with arrival rate, C, satisfies C log λi Proof: Let the iput process of flow 0 be a Poisso process with rate ad idepedet of the arrival process of iterferece traffic Thus, the iput process of the server is Poisso with rate + λ I Assume the queue is iitially i equilibrium Burke s theorem for a M/M/ queue [] states that the output process is Poisso with rate + λ I i steady state Because the service disciplie is FIFO, each output packet belogs to flow 0 with probability / + λ I idepedetly Thus, the output process of flow 0 is a Poisso process with rate The ituitio for the lower boud is clear: IA ; D hd + ha hd, A hd + ha hs, A hd hs A hd hs hd h log + log λi, A3 log + where holds because D, A cotais the same iformatio as S, A ; holds because s are expoetially distributed with mea / as show i Lemma below, ad D is are idepedet ad expoetially distributed with mea / because the output process of flow 0 is Poisso with rate Thus, IA ; D log λi Oe way to prove that the lower boud o the mutual iformatio is a lower boud o the capacity is to take the approach i [5]; ie, to show that for some iput process, the if-iformatio rate is greater tha the lower boud We ca do this alog the lies of the proof of Theorem 7 i [] To do this, we eed to show that D i coverges i probability to / Because the output process of flow 0 is a Poisso process with rate, by the law of large umbers, we have coverges i probability to / Further, eve though the s are ot idepedet, Lemma shows coverges i probability to / The other steps of the proof follow as i [] Lemma The effective service time,, is expoetially distributed with mea, ad coverges i probability to Proof: Let q 0 be the probability a packet belogs to flow 0, q 0 / +λ I Let ρ +λ I/ Because flow 0 ad flow I have idepedet Poisso arrivals with rate ad λ I, a packet belogs to flow 0 with probability q 0, ad it is idepedet of previous packets Let πj be the steady state probability that there are j packets i the queue whe the ith packet arrives ad pk j be the probability that i k give that there are j packets i the queue whe the ith packet arrives We have P i k πjpk j jk πkpk k + D i jk+ ρρ k q 0 k + πjpk j ρρ j q 0 k q 0 jk+ ρ k q 0 k ρ q 0 k λi λi, k 0,,, I other words, i + is a geometrically distributed radom variable with mea /, ad thus i + Next, we use Chebychev s iequality to prove the covergece Let gi {j : the ith ad jth packets of flow 0 belog to the same busy period}
Note that ad are idepedet if i ad j do ot belog to the same busy period Let B i be the busy period that cotais packet i [ ] [ Si ] j j gi [B i] + Bi + + Si [] B + j gi j gi Thus, [ ] [ ] σ S i B B + Because B, the secod momet of a busy period of a M/M/ queue, is fiite see eg, p 4, q 54 i [3] whe +λ I <, we have lim σ 0 By Chebychev s iequality, P ɛ σ ɛ Thus, Si coverges i probability to / λi So far we have provided a lower boud o the timig capacity of a flow at the presece of idepedet Poisso iterferece traffic whe the service time distributio is expoetial To achieve the lower boud, the iput process is a idepedet Poisso process For flow 0, the server behaves as if it is processor-sharig where the iterferece traffic gets a service rate of λ I ad flow 0 gets a service rate of We ext use the results of this sectio to defie a packet colorig scheme that icreases the timig iformatio carried by a sigle flow B A Colorig Scheme Let us cosider a system where packets have differet colors Iformatio is coded ito the time-itervals of packets with the same color From the viewpoit of a flow with a certai color, all packets with other colors behave as iterferece traffic Suppose each source i, i,,,, geerates packets with a specific color accordig to a idepedet Poisso process with rate λ i Let R Lλ,, λ be the lower boud o the timig capacity By Prop, we have R Lλ,, λ λ i log j i λj λ i Lemma The timig capacity of the -color system satisfies Cλ,, λ λ i log j i λj λ i λ i log λi Further, the lower boud R Lλ,, λ is maximized whe λ i λ j for all i ad j Proof: Whe, the lower boud o the capacity is R Lλ, λ λ log λ λ + λ log λ λ The Hessia matrix of R L is, [ D λ R L λ λ λ λ λ λ λ λ λ It is easy to see that the Hessia matrix is egative defiite, ad thus, R L is a cocave fuctio of λ, λ Further, R L is a symmetric fuctio of λ, λ Thus, we have R Lλ, λ αr Lλ + λ, 0 + αr L0, λ + λ R Lλ + λ, 0 λ + λ log, λ + λ where α λ /λ + λ I words, to split a sigle flow ito two flows icreases the timig capacity Further, because R L is symmetric, λ + λ λ + λ R Lλ, λ R L, Thus, the lower boud is maximized whe λ λ By iductio, it is easy to see that Last, we show that R Lλ,, λ λ i log λi λ R Lλ,, λ λ, λ where λ λi/ Without loss of geerality, we assume that λ λ, λ Note that R Lλ,, λ R Lλ,, λ, where λ,, λ is a permutatio of λ,, λ We have R Lλ, λ,, λ, λ λ + λ R L, λ,, λ, R L λ, λ,, λ, λ λ + λ λ + λ R L, λ,, λ, λ + λ R L λ,, λ λ log λ, λ ]
where λ, λ,, λ, λ is a permutatio of λ + λ /, λ,, λ, λ + λ / i decreasig order I other words, the lower boud is maximized whe all flows have the same arrival rates Note that λi log/ λi is the timig capacity of a sigle-color flow with rate λi [] The lemma states that a -color system achieves higher timig-capacity tha a sigle-color flow with the same total arrival rate Thus, by splittig a sigle flow ito multiple idepedet Poisso flows with differet colors, we icrease the timig capacity Of course, the tradeoff is that we eed to distiguish differet flows by colorig I the ext propositio, we evaluate the icrease i the timig capacity due to such splittig Lemma 3 Suppose that there are flows The the overall timig capacity of the -color flow, C, satisfies C x, where x satisfies x e x e To achieve the lower boud, the iput processes of flows are idepedet Poisso flows with the same rate λ x + x Proof: Let the iput process of each flow be a idepedet Poisso process with rate λ, such that λ < ach flow cosiders the other flows as iterferece traffic By Propositio, we have C λ log λ, λ < λ To fid the value of λ that maximizes the lower boud, we ca take derivative of the right-had side with respect to λ ad set it equal to zero Deote λ x λ The lower boud is maximized whe x e x e, ad we have C x The optimal arrival rate of each flow is λ x + x Because xe x is a icreasig fuctio of x, we have x Thus, we verify that λ < / Propositio Let C be the timig capacity of a sigle flow with packet size B ats It satisfies B log B C B Proof: We ca distiguish N e B flows by usig all B ats i each packet for colorig Followig Lemma 3, we have CN N N x, where x satisfies x e x N e Note that xe x is a icreasig fuctio of x Let x 0 B log B We have ie, Thus, x 0e x 0 B log B e B B N N eb N e x e x, x x 0 B log B C CN B log B Next, we show that the maximum timig capacity is upper bouded by B Let X {X,, X } be the iformatio set through the packets, which is received error-free IX, D ; X, A ID ; X, A + IX ; X, A D ID ; A + IX ; X where holds because X cotais o additioal iformatio regardig D other tha that i A From [], we kow that if B bit, the maximum iformatio capacity of the chael is B Thus, C sup IX, D ; X, A B, X,A which cocludes the proof We ote that the above theorem shows that the ratio of the upper boud to the lower bouds goes to oe as B becomes large Thus, if we have a lot of available colors, the above bouds are tight We have show that, by splittig a sigle flow ito multiple subflows, we ca achieve a timig capacity equal to the capacity of the server This colorig scheme ca be applied to ay combiatios of sigle seder/receiver, multiple seders/receivers The upperboud/lower-boud provided i Prop bouds the overall timig capacity To achieve this capacity lower-boud, a radom ecodig strategy for each sub-flow is to assig codewords that are idepedet realizatios of a Poisso process with rate λ, ad each sub-flow ecodes iformatio idepedetly Further, if there exists Poisso cross-traffic with rate λ I ad the service time distributio is expoetial with rate, the we obtai the capacity boud simply by replacig by i Note that the colorig scheme studied i the paper is differet from a multi-user case i the traditioal sese, eg, the case studied i Sectio 33 i [4] I the multi-user case, the receiver caot distiguish betwee packets from differet trasmitters I the colorig scheme, differet flows have differet colors ad are thus distiguishable Thus, the result here does ot cotradict with the result i [4] Last, we examie the amout of covert iformatio coveyed i the system Let us cosider a two-flow case Let A deote the iterarrival times of flow, B m the iter-arrival times of flow, D +m the iter-departure times of the aggregated flow, ad N +m the ID sequece of the departures of the aggregated flow; ie, if N k i, the the kth departure packet belogs to flow i Let λ ad λ be the arrival rates of flow ad flow, respectively Suppose there is a eavesdropper that observes the packets passig through the server The eavesdropper records the sequece ad cotets of the packets, but ot their timigs I the colorig scheme, because packets are used to idetify differet flows, the eavesdropper extracts certai amout of iformatio by observig the ID sequece I what follows, we quatify the amout of iformatio that is covert, ie, caot be observed by the eavesdropper
Let R I λ + λ + m IA, B m ; N +m be the iformatio rate observed at the eavesdropper Note that IA, B m ; N +m HN +m HN +m A, B m HN +m, where the secod equality holds if the ID sequece observed by the eavesdropper is determied by the arrival times of packets this is the case whe the service disciplie is FIFO Let R t λ + λ + m IA, B m ; N +m, D +m be the total iformatio rate observed at the receiver The, is the covert iformatio rate We have R c R t R I IA, B m ; N +m, D +m ha, B m + hn +m, D +m ha, B m, N +m, D +m ha, B m + hn +m, D +m ha, B m, D +m ha, B m + hn +m, D +m ha, B m, S +m hn +m, D +m hs +m hd +m hs +m + HN +m Note that hd +m hs +m is the amout of mutual iformatio betwee the iter-arrival times ad iter-departure times of a sigle flow with rate λ + λ Thus, we coclude that the covert iformatio rate of multiple flows caot exceed that of a sigle flow whose arrival rate equals the aggregated arrival rate Note that this boud o the covert iformatio rate holds for all service-time distributios III CONCLUSIONS I this paper, we first studied the case whe a flow shares a expoetial server with some ucotrolled, idepedet iterferece traffic Whe the iterferece traffic is Poisso, we obtaied a lower boud o the timig capacity Further, we showed that by splittig a sigle flow ito multiple sub-flows, we ca achieve the capacity of the server asymptotically by usig timig iformatio RFRNCS [] V Aatharam ad S Verdu, Bits through queues, I Trasactios o Iformatio Theory, 4:4 8, 996 [] D Bertsekas ad R Gallager, Data Networks, Pretice-Hall Ic, 987 [3] L Kleirock, Queueig Systems, Volume I: Theory, volume I, Wiley Itersciece, New York, 975 [4] R Sudaresa, Coded Commuicatio over Timig Chaels, PhD thesis, Priceto Uiversity, 999 [5] S Verdu ad T Ha, A geeral formula for chael capacity, I Trasactios o Iformatio Theory, 404:47 57, July 994