Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 2 Introducton One of the defnng prncples of the netw

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1 Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes Λ Khaled Harfoush Azer Bestavros John Byers Computer Scence Department Boston Unversty Boston, MA 225 Techncal Report BUCS-2-3 Abstract Current Internet transport protocols make end-to-end measurements and mantan per-connecton state to regulate the use of shared network resources. When two or more such connectons share a common endpont, there s an opportunty to correlate the end-to-end measurements made by these protocols to better dagnose and control the use of shared resources. We develop packet probng technques to determne whether a par of connectons experence shared congeston. Correct, effcent dagnoses could enable new technques for aggregate congeston control, QoS admsson control, connecton schedulng and mrror ste selecton. Our extensve smulaton results demonstrate that the condtonal (Bayesan) probng approach we employ provdes superor accuracy, converges faster, and tolerates a wder range of network condtons than recently proposed memoryless (Markovan) probng approaches for addressng ths opportunty. Keywords: End-to-end measurement; Packet-par probng; Shared losses; TCP/IP; Performance evaluaton. Λ Ths work was partally supported by NSF research grant CCR

2 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 2 Introducton One of the defnng prncples of the network protocols used n the Internet les n ther ablty to manage and share network resources farly across competng connectons. Ths s a notable engneerng achevement, especally n lght of the fact that ndvdual connectons exert dstrbuted control over ther transmsson rates. But ths fne-graned autonomy that connectons exert coupled wth our lmted understandng of the nteractons that multple (TCP) connectons mpose lmts the degree to whch network resources can be tghtly controlled. In our ongong work as part of the Mass project [26], we nvestgate crcumstances n whch better dagnoss of network resources can be obtaned, whch we hope wll lead to mproved control mechansms. In ths paper, we explore the effects of concurrency on dagnosng network condtons. As an example, a popular Internet server (e.g. Web server, proxy server, content dstrbuton outlet, streamng meda server, etc.) may potentally command a large number of concurrent connectons. Whle most of these connectons are lkely to be to dfferent clents, many may n fact be traversng the same set of congested resources. If connectons sharng common congested resources can be dentfed, then mproved network resource usage can be acheved through judcous allocaton of bandwdth. In partcular, rather than controllng connectons traversng congested network resources ndependently, an Internet server could apply an aggregate control mechansm to such connectons. Examples of such mechansms nclude the aggregate congeston management technque proposed under the Congeston Manager framework [2] and the ATCP protocol [3]. Applcatons of ths technque could extend well beyond the doman of congeston control to QoS admsson control, selectng multple mrror stes n parallel [5] and mproved connecton schedulng at webservers. But n order for any such control strateges to be practcal, an endpont must be able to quckly and accurately dentfy whether or not a set of ts connectons to remote locatons traverse the same set of congested resources. Helpfully, the end-to-end measurements made n the course of normal operatons by most transport protocols provde a wealth of nformaton about the end-to-end characterstcs of a path n the network. For example, although the nodes comprsng the path may not be known, end-to-end bottleneck bandwdth rates, round-trp tmes and packet loss statstcs can all be nferred from the dynamcs of a TCP connecton []. In ths paper, we show that n addton to the above connecton-specfc parameters, end-to-end measurements from dfferent connectons can be correlated n order to dentfy connectons that share smlar network condtons. What consttutes smlar condtons" depends on the purpose of the dentfcaton process. For the purpose of ths paper, we specfy two possble problem statements, defned below. In each of the problem domans, we consder a scenaro n whch there s a sngle server, whch has actve connectons (e.g. TCP flows) to two dstnct clents, both experencng steady-state packet loss rates of at least ffl, for some constant ffl >. We assume that the paths from server to the clents form a tree, whch from the server's perspectve conssts of a sequence of shared lnks followed by a sequence of dsjont lnks, n whch the shared porton of the sequence may be empty. Loss Sharng: For these two connectons, determne f the ncdence of packet loss on the shared porton of the tree s at least ffl, for a fxed constant k. k Bottleneck Equvalence: For these two connectons, determne f the ncdence of shared loss s greater than the ncdence of dsjont loss for both of the connectons.

3 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 3 We have formulated these problem statements as yes-no questons, but note that the technques we develop extend to the related queston of estmatng the ncdence of shared loss. Also, t should be clear that whle Bottleneck Equvalence mples Loss Equvalence, the converse does not hold. Paper Contrbutons and Overvew: Ths paper proposes an analytcal technque for the robust determnaton of both loss and bottleneck equvalence for pars of uncast connectons emanatng from the same server. Our technque reles solely on end-to-end loss nformaton avalable at the server as a result of passve montorng or of actve probng. We present extensve smulaton results that demonstrate the effectveness of our approach as compared wth the recently proposed approach of Rubensten, Kurose, and Towsley [25] and the robustness of our technque to a wder varety of network and cross-traffc characterstcs than prevous work consdered. The remander of ths paper s organzed as follows. In Secton 2, we revew exstng lterature and related work. In Secton 3, we present our basc model and assumptons, and we derve an analytcal soluton to the loss sharng and bottleneck equvalence problems stated above. In Secton 4, we descrbe detals of our mplementatons, we propose novel metrcs for the evaluaton of dagnostc accuracy and convergence characterstcs, and we present results of extensve smulaton experments that we have conducted to compare our technque to that proposed n [25]. We conclude ths paper n Secton 5 wth a summary and a descrpton of our ongong research. 2 Related Work 2. A Taxonomy of Efforts to Characterze Network Condtons Inference and predcton of network condtons s of fundamental mportance to a range of networkaware applcatons, so t s no surprse that numerous research efforts are underway n ths space. We classfy and survey these research efforts n the context of our current work. One wdely adopted strategy s to mne the data collected by network-nternal resources, such as BGP routng tables, to generate performance reports [2, 4, 7, 9, 5]. Ths approach s best appled over long-tme scales to produce aggregated analyses such as Internet weather reports, but does not lend tself well to provdng answers to the fne-graned questons we propose here. Another approach s statstcal nference of network nternal characterstcs based on end-to-end measurements of pont-to-pont traffc [4, 7, 27, 6, 23, 22, 9]. We adopt ths general approach because nformaton s gathered at the approprate granularty (on a per-connecton bass) and at the approprate tme scale to address the questons we study. These approaches can be further classfed as actve approaches, whch ntroduce addtonal probe traffc nto the network, and passve approaches, whch make nferences only from exstng network traffc. The beneft of the former approach s flexblty: one can make measurements at those locatons and tmes whch are most valuable; whle the beneft of the latter approach s that no addtonal bandwdth and network resources are consumed solely for the purpose of data collecton. Cuttng across other dmensons, one can also classfy approaches as ether recever-orented or sender-orented, dependng on where nferences are made; and multcast-drven or uncast-drven, dependng on the model used to transmt probe traffc. Use of multcast traffc s appealng, as losses and delay wthn the multcast tree nduces correlated behavor at recevers, whch can streamlne nferencemakng and produce results wth hgher confdence. Unfortunately, passve probng n an envronment

4 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 4 Network Measurement End-to-End Measurement Actve Passve Actve Passve Multcast [7] [2, 4] [7, 6, 23, 27] Uncast [5, 9] [2, 4] [6, 4, 25], [X] [22, 27, 25], [X] [25], [X] [25] Sender Recever Sender Recever Table : A Taxonomy of Efforts to Characterze Network Condtons. where multcast traffc s not present makes such a strategy nfeasble. Table llustrates the above taxonomy wth references to studes and projects that fall wthn each of ts dfferent categores. The work we present n ths paper s dentfed as [X]; t s sender-based and s targeted for uncast envronments. It works under both passve and actve probng assumptons, albet wth dfferent accuracy and convergence propertes. 2.2 Packet-Par Probng One of the essental technques n our constructons s the use of packet-par" technques, orgnally used by Keshav [6], and subsequently refned by Carter and Crovella [9] and Paxson [2, 22, 2], to determne bottleneck bandwdth on a network path. In our work, we use a packet par probe to a par of dfferent recevers to ntroduce loss and delay correlaton, much the same way a multcast packet to these two recevers ntroduces correlaton. A challenge assocated wth ths approach, especally n passve probng, s nter-packet spacng and the tme scales over whch we can expect correlatons to be present. The strateges we employ follow early work by Bolot [4] and recent work by ajnk, Moon, Kurose and Towsley [27] whch study the temporal dependence n uncast and multcast packet losses, respectvely. 2.3 Estmaton of Network Parameters Usng End-to-End Measurements The specfc problem of dentfyng and characterzng bottleneck equvalence classes s motvated n part by recent work on topologcal nference over multcast sessons [8, 6,, 23]. By makng purely end-to-end observatons of packet loss at endponts of multcast sessons, Ratnasamy and McCanne [23] and Cáceres et al. [6] have demonstrated how to make unbased, maxmum lkelhood estmaton nferences of (a) the multcast tree topology and (b) the packet loss rates on the edges of the tree, respectvely. They demonstrate that an observer wth access to a complete record of arrvals and lost packets for each destnaton can make unbased nferences about the underlyng tree from that record. Ther work s made possble by the fact that only one copy of a packet traverses any edge of the multcast tree. Thus, f two recevers share a common edge n the multcast tree, and the packet s dropped n the queue pror to traversng that shared edge, both downstream recevers wll lose that partcular packet. Wth suffcently many measurements, ths correlated behavor makes the nferences above possble. The work most closely related to ours s that of Rubensten, Kurose and Towsley[25]. Ther work uses end-to-end probng to detect shared ponts of congeston (POCs). By ther defnton, a pont of congeston s shared when a set of routers are droppng and/or delayng packets from both flows. Ther technque for dentfyng POCs uses Posson probe traffc to both remote endponts and cross-

5 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 5 correlaton measures computed between pars of packets from these flows. Our technques dffer from thers by usng packet pars to explot temporal dependence, our strateges for estmatng parameters of the bottleneck queue and our ablty to make accurate assessments when multple congested gateways may exst along a path. In the expermental work secton, we also demonstrate the mproved accuracy and faster convergence of our approach. 3 Dagnoss of Shared Losses usng Bayesan Probng In ths secton, we descrbe the technque we propose for detectng shared losses. We start by descrbng the basc defntons, revewng the overall objectve and provdng the motvaton for the technques that we propose. Then we provde the algorthmc and analytcal detals of the underlyng technque, whch we llustrate on a one-server, two-clents scenaro. 3. Basc Defntons and Motvaton Consder the set of lnks used to route uncast traffc between a server and two dfferent clents. Together these lnks form a tree T rooted at the server, wth the clents at the leaves and routers at the nternal nodes. The flows of packets sent from the server to each of the two clents share some of T 's lnks and then contnue on separate lnks en route to the dfferent clents. A lnk L s the lnk whose downstream node s node as llustrated n Fgure. We refer to the set of lnks en route to clent A as L A, the set of lnks en route to clent B as L B and the set of lnks that they share as L S. Our objectve s to defne a bnary dagnostc test that would dentfy whether or not sgnfcant packet loss s occurrng on the set of lnks shared by clent flows. To calbrate the level of loss whch warrants a shared losses dagnoss, we defne the followng parameter of our BP approach. Defnton For a dagnostc procedure, the senstvty constant c s the maxmum loss probablty allowed on the shared porton of the paths to multple recevers whle producng a no shared loss" dagnoss. The value of the senstvty constant c determnes the tolerable level of shared losses that the BP technque wll allow under a no shared losses" dagnoss. Thus, n effect, the value of the senstvty constant c can be used to tune the eagerness of our BP technque to reach a shared losses" dagnoss. To acheve our objectve we ntroduce two types of probe sequences: Defnton 2 A -packet probe sequence S ( ) s a sequence of packets destned to clent such that any two packets n S ( ) are separated by at least tme unts. Defnton 3 A 2-packet probe sequence S ;j ( ;ffl) s a sequence of packet-pars where one packet n each packet-par s destned to and the other s destned to j, and where the ntra-par packet spacng s at most ffl tme unts and the nter-par spacng s at least tme unts. The ntuton behnd the -packet probe sequence s to provde a baselne loss rate over each of the two end-to-end paths whle the 2-packet probe sequence s used to provde a dstngushng mechansm

6 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 6 to measure correlated loss over the shared lnks. The key nsght s that because of ther temporal proxmty, we expect packets wthn a packet par to have a hgh probablty of experencng a shared fate on the shared lnks. If the ncdence of shared loss on the shared lnks s hgh, ths leads to an ncreased probablty of wtnessng coupled losses wthn a packet par. The values of and ffl n the above defntons of probe sequences are chosen emprcally to make t lkely that the probes experence ndependent and dependent packet loss events, respectvely. Whle we wll descrbe approprate settngs of and ffl n our expermental secton, we wll generally requre ffl to be on the order of a mllsecond and to be on the order of a second, to acheve hgh dependence and ensure ndependence, respectvely. Server (Sender) L Shared Segment of Paths from Server to Clents L 2 L 3 2 L 5 Dsjont Segment of Paths from Server to Clents Clent A (Recever) L L 6 6 Clent B (Recever) Fgure : Notaton used to descrbe the topology between a server and two clents 3.2 The Bayesan Probng Technque We now return to the tree depcted n fgure to llustrate the basc premse of our proposed uncast probng technque and ts assocated analyss. Wth our packet probe sequences, there are four expermental outcomes whch we use n our analyss: successful probes n the -probe sequences, successful packet-par probes n the 2-probe sequence, and unsuccessful probes n the 2-probe sequence n whch both packets n a par are lost. The followng notaton wll be useful throughout our analyss. Let g A and g B denote the fracton of the -packet probes n S A ( ) and S B ( ) respectvely whch were successfully receved. Smlarly, let g A;B denote the fracton of the 2-packet probes n S A;B ( ;ffl) that were successfully receved by both clents A and B and let b A;B denote the fracton of the 2-packet probes that were lost en route to both clents A and B. Note that g A;B + b A;B may be less than due to pars of probes n whch one probe s lost en route to one clent whle the other probe arrves successfully at the other clent. To establsh a relatonshp between outcomes of probes and network queues, we use the followng termnology and notaton. Any ndvdual queue can accomodate zero, one, or more than one fxed-sze probe packets at any tme nstant. In general, we defne p k be the steady-state probablty that the queue at L can store exactly k probe packets, and p k+ be the probablty that the queue at L can store

7 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 7 k or more probe packets. From ths defnton, p + s the probablty that a sngle probe packet sent over L at an nstant chosen at random wll successfully traverse L and p s the probablty that such a probe wll be lost over L. Wth ths notaton, we can establsh the followng relatonshps between probe sequences and queue szes. Fact The quanttes g A and g B are unbased estmators for and, respectvely. p + p + 2L A 2L B Fact 2 The quantty g A;B s an unbased estmator for p 2+ 2L S : 2(L A S LB )nl S p + Fact holds because a sngle probe successfully arrves at the destnaton f and only f each queue en route has avalablty for at least one probe packet. Lkewse, Fact 2 follows snce a packet par successfully arrves at the destnaton f and only f each shared queue has avalablty for both packets n the par and dsjont queues have avalablty for at least one probe packet. Establshng a smlar relatonshp for b A;B s consderably more complex by vrtue of the number of ways n whch both packets n a packet par may be lost. Ether both packets are lost on the shared lnks; or exactly one packet s lost on the shared lnks, whle the other s lost on the dsjont part of the tree; or both are lost ndependently on the dsjont lnks. Lettng q A be a shorthand for the probe loss probablty over only the dsjont lnks to clent A,.e. defnng q A = p + ; 2L A nl S and defnng q B smlarly, we can enumerate these possbltes to establsh the fact that: Fact 3 The quantty b A;B s an unbased estmator A A (qa + q B ) + p + 2L S p + p 2+ 2L S 2L S 2L S p 2+ q A q B : From these three facts, we can obtan an unbased estmate for a quantty whch occupes a central locaton n Fact 3 and whch we defne as follows: X = p + p 2+ : () 2L S 2L S X can be nterpreted as the probablty of a packet par encounterng a stuaton on the shared lnks n whch all shared queues have space for one probe packet, but not all queues have space for two packets. We next prove that we can obtan the followng (surprsngly smple) estmate for X:

8 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 8 Lemma The quantty g A + g B + b A;B g A;B s an unbased estmator for X. Proof: We start by usng Fact 2 to relate g A;B to X as follows: E[g A;B ] = p 2+ p S + 2L S 2(L A LB )nl S = = p + X 2L S A p + 2L A nl X p + 2L B nl S p S + p + p + 2L A LB 2L A nl S 2L B nl S p S + X( q A )( q B ) (2) 2L A LB A Combnng ths equaton wth Facts and 3 and by the lnearty of expectaton, we can wrte: E [b A;B + g A + g B g A;B ] = = p + + p + p + + X(q A + q B ) + p 2+ q A q B p S + + X( q A )( q B ) 2L A 2L B 2L S 2L S 2L A LB = X + q A q B X + p + + p + p + p S + + p 2+ q A q B 2L A 2L B 2L S 2L A LB 2L S It now suffces to demonstrate that the quantty q A q B X cancels wth the remanng terms. By applyng the defntons we have: q A q B X = q A q A = = p + p 2+ 2L S 2L S p + 2L A nl S p + 2L A nl S p + 2L B nl S p + + 2L B nl S p + p + p + + 2L S 2L A 2L B p + 2L S A p 2+ 2L S C p S + A 2(L A LB )nl S p S + p 2+ q A q B 2L A LB 2L S q A q B p + 2L S p 2+ 2L S q A q B Therefore the desred cancellaton does take place, yeldng the result.

9 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes The X Factor We now motvate the reason for whch obtanng an unbased estmate of the value of X s valuable. As we mentoned n the precedng secton, an estmate of X s an estmate of the probablty that one of the queues on the shared lnks has room for a sngle probe packet. As the followng example clearly demonstrates (and as one mght magne), the magntude of ths value, whch s an analogue of p, tends to be hghly correlated wth the magntude of packet loss on that lnk. Fgure 2 (left) shows how the values of p, p and p 2+ on a sngle lnk nteract n an M=M==K queueng system wth queue sze K = 2 as a functon of the traffc load ρ. Under lght load (ρ not much larger than ), the values of p and p are almost dentcal. Under heavy load (ρ much larger than ), the value of p becomes larger than the value of p. The experment depcted n Fgure 2 (rght) demonstrates smlar phenomena n a bursty traffc model whch we descrbe n detal n Secton 4. The fgure suggests that the value of p ncreases n tandem wth the value of p as the background traffc rate ncreases. Ths trend s a key to our proposed technque and ponts to the value of an unbased estmate for X..8.8 Probablty.6.4 Probablty p p p Utlzaton.2 p p p Utlzaton Fgure 2: Values of P, P and P 2+ when K = 2 for dfferent values of ρ: M/M//K Analyss (left) and ns smulaton results wth 64 Pareto ON/OFF UDP flows (rght). To summarze, we can effcently compute a runnng estmate of X usng the -packet and 2-packet probe sequences sent from the server to the two clents. If X > c (for some emprcally-determned senstvty constant c) we conclude that there are sgnfcant" losses on the shared part of the path between the server and the clents. Otherwse we conclude that losses are prmarly due to packet losses on the dsjont part of the path between the server and the clents. 3.4 Basc Assumptons A basc premse of our work s that whle we assume the loss rate on all lnks n our topology may have substantal short-term varablty (as s to be expected wth self-smlar background traffc), the mean packet loss rate on each lnk s statonary over longer tme scales. Ths statonarty requrement s needed to allow a dagnostc procedure to converge. Thus, statonarty s requred only over tme scales that are comparable to the tme t takes the dagnostc procedure to converge. In the next secton we show that the BP technque possesses superor convergence, makng t qute effectve even when statonarty can only be assumed for short ntervals (on the order of few seconds).

10 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes In the analyss presented above, we have made the followng addtonal assumptons whch we enumerate and dscuss here:. Losses on the lnks occur only due to queue overflows. 2. 8; j : Losses on lnk L are ndependent from losses on lnk L j. 3. A relable feedback mechansm enables the sender to determne wth certanty whether a gven probe packet was lost. 4. The temporal constrants mposed on probe sequences (whether -packet or 2-packet probe sequences) are preserved throughout the journey of the probes from sender to recevers. Assumpton reflects the current DropTal behavor present n most Internet routers today. We consder the negatve consequences of RED gateways on our technque n the expermental secton. Assumpton 2 allows us to gnore any spatal correlaton between lnk losses, and thus gnore any addtonal correlaton terms. Assumpton 3 enables us to assume that the server s able to accurately dentfy the outcome of the probng process,.e. whch packets of a -packet or 2-packet probe sequences were lost. Assumpton 4 s our most sgnfcant assumpton, snce t ensures that the ndvdual packets wthn each packet-par of a 2-packet probe sequence S A;B ( ;ffl) are separated by at most ffl tme unts on all traversed lnks. Moreover, we must be assured that ffl s suffcently small that two packets of a packet-par are close enough to each other on all traversed lnks to enable an accurate samplng of the state of a queue at the tme the 2-packet probe reaches that queue. In partcular, we need to use p 2+ as the probablty that the two packets of a packet-par have traversed lnk. Ideally, we would desre that the two packets reach the queue wth an nter-arrval tme ffl =. If the packets n a par become substantally separated from one another n flght, our estmates g A;B and b A;B wll be based. We have studed the effects of ffl > on the performance of our BP technque. Our fndngs (presented n the next secton) confrm that the bas ntroduced by small amounts of separaton and/or long paths s not excessve. 4 Expermental Results In ths secton we present results of extensve smulatons that () compare our Bayesan Probng (BP) technque to the Markovan Probng (MP) technque proposed and evaluated n [25], and (2) establsh the robustness of the BP technque to varous parameters and condtons. 4. Technques Evaluated Bayesan Probng Technque: Recall from our presentaton n secton 3, the BP technque requres the specfcaton of the and ffl parameters of the temporal constrants mposed on -packet and 2-packet probe sequences. We have measured the separaton between probe packets for paths consstng of a large number of hops. The results consstently ponted to the valdty of our temporal constrant assumpton above and, consequently, the robustness of our BP technque.

11 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes In the experments we present n ths secton, probes were sent at a mean rate R. However, to allevate synchronzaton effects, we mposed addtve random nose on the nterpacket spacng so that t was unformly dstrbuted over the range [ R 5ms; R +5ms], thus = 5ms. In our experments, R we set the value of ffl to ; that s packets wthn a packet-par were sent back-to-back, wth no tme separaton. Also, to normalze the losses on the shared lnks experenced by both recevers, the 2-packet probes n S A;B ( ;ffl) alternate between the two possble packet orderngs. Another parameter of our BP technque s the value of the senstvty constant (c). Recall that the value of the senstvty constant c determnes the level of shared losses that the BP technque wll tolerate whle ndcatng a no loss sharng" dagnoss. In our experments, the value of c was fxed at.4. Ths value was chosen emprcally based on experments dscussed later n ths secton. Markovan Probng Technque: The MP technque descrbed n [25] reles on the use of two Posson processes for sendng probe sequences f and f 2 from the sender to the two recevers. To detect shared losses the MP technque depends on the calculaton of the Auto-Correlaton and the Cross-Correlaton functons. The Auto-Correlaton functon (M a ) s the condtonal probablty that a packet from f s lost, gven that the prevous packet from f s lost. The Cross-Correlaton functon (M x2 ) s the condtonal probablty that a packet from f s lost, gven that the precedng packet from f 2 was lost. Gven M a and M x2, the MP technque descrbed n [25] suggests the followng test for dentfyng shared losses. MP test: f and f 2 are dagnosed to havng shared losses f M x2 > M a ; they are dagnosed as havng no shared losses otherwse. It s mportant to note that the MP test (above) could be reformulated by reversng the roles of the probes sent on the f and f 2 paths. MP2 test: f and f 2 are dagnosed to havng shared losses f M x2 > M a2 ; they are dagnosed as havng no shared losses otherwse. In our experments, we noted that the MP and MP2 tests yelded smlar dagnoss when losses were symmetrc along the non-shared lnks (or equvalently when losses are ether all shared or all on the ndependent lnks a central assumpton of the MP technque descrbed n [25]). However, the MP and MP2 tests yelded qute dfferent dagnoss when ths condton sezed to hold true. Snce we were nterested n loosenng ths assumpton (by allowng losses on both the shared and dependent portons of the paths), we combned the above tests nto the followng alternatve test. MP* test: f and f 2 are dagnosed to havng shared losses f M x2 > M a2 OR M x2 > M a ; they are dagnosed as havng no shared losses otherwse. 2 Our expermental results, whch we present later n ths secton, show that the MP* test mproved the accuracy of the MP technque sgnfcantly. Thus n the remander of ths paper, and unless otherwse noted, we wll use the MP* test as the default" test for the Markovan Probng (MP) technque. 2 In prvate communcatons wth the frst author of [25], we also consdered a conjunctve test for the dentfcaton of shared losses as opposed to the dsjunctve test we propose here. Our expermental evaluaton ndcated that a dsjunctve test yelded better results, and s thus adopted n ths paper.

12 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes Expermental Setup We used the Network Smulator (ns) [8] to smulate the topology llustrated n Fgure 3. Ths topology represents a server and two clents. The shared porton of the paths between the server and the two clents s modeled by a sngle lnk (L), whereas the dsjont portons of the paths between the server and the two clents are modeled by lnks (L2) and (L3), respectvely. Both technques were smulated from the server sde by mplementng a new ns agent" that sends 2-byte probe packets to the recevers and wats for an acknowledgment for each probe sent. Probes are annotated wth sequence numbers. The agent uses the absence of a probe acknowledgment as an ndcaton of the probe's loss on the way to ts destnaton. Also, the agent keeps some statstcs about the probe losses and based on these statstcs esmates whether there are shared losses or not by usng ether the BP, MP or the MP* technques descrbed above. L 2 L L 3 Cross Traffc Lnk Buffer Node 2 Network 3 Fgure 3: Topology used n our experments Baselne Model: Each one of the three lnks n Fgure 3 s modeled by a sngle DropTal queue. The lnk delays were all set to 4ms and the lnk buffer szes were all set to 2 packets. Each of these lnks was subjected to background traffc resultng from a set of Pareto ON/OFF UDP sources wth a constant bt rate of 36Kbps durng the ON tmes wth a packet sze of 2 bytes. The average ON and OFF tmes were set to 2 seconds and second, respectvely. The Pareto shape parameter (ff) was set to.2. After a warm-up" perod of seconds, the probng processes (and assocated dagnostc processes) are started. To represent the varous levels of congeston that any of these lnks may exhbt, we have chosen three sets of parameters that result n Hgh", Mld", and Low" levels of congeston. The baselne parameter settngs for these congeston levels (and the resultng loss rates) are tabulated n Table 2. Parameter Congeston Level Settng Hgh Mld Low Lnk Bandwdth Mb/sec Mb/sec Mb/sec # of background flows Observed Loss Rate 7-5% < 7% <.% Table 2: Settngs used (and resultng loss rates) for the three levels of congeston consdered

13 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 3 Basc Test Cases: In order to evaluate the dagnostc abltes of the above technques, we defne four possble scenaros, featurng dfferent levels of congeston along the shared and dsjont portons of the paths between the server and ts clents. Table 3 enumerates these four scenaros. 3 Scenaro Lnk Congeston Level Condton Correct # L L2 L3 Shared Losses Shared Bottleneck Dagnoss (I) Hgh Low Low es es es (II) Hgh Mld Low es es es (III) Mld Hgh Low es No es (IV) Low Mld Hgh No No No Table 3: Test case scenaros consdered n ths paper Scenaro (I) represents a stuaton n whch a hghly congested lnk exsts on the shared porton of the path to the two clents, and no congeston exsts on the dsjont porton of the paths. Scenaro (IV) represents a stuaton n whch losses are only possble on the dsjont porton of the path to the two clents. Scenaros (I) and (IV) represent the ltmus test" cases that must be dagnosed correctly by any technque that ams at dentfyng shared losses (or lack thereof). Scenaro (II) represents a stuaton n whch a hghly congested lnk exsts on the shared porton of the path to the two clents, and a lesser congested lnk exsts on one of the dsjont porton of the paths. Scenaro (III) represents a stuaton n whch a hghly congested lnk exsts on one of the dsjont portons of the paths to the clents, and a lesser congested lnk exsts on the shared porton of the path to the two clents. It s mportant to note that scenaros (II) and (III) volate one of the assumptons of the Markovan Probng technque of Rubensten, Kurose, and Towsley namely, that losses on a gven path are the result of exactly one congested lnk on that path. We have ncluded these scenaros to hghlght the robustness of our Bayesan probng technque n partcular ts ablty to converge to a correct dagnoss when losses on a gven path are the result of multple congestons along that path. Notce that the exstence of multple congested gateways on a sngle path over an extended perod of tme s qute possble (due n part to the documented scalng phenomena of network traffc) [, 24]. 4.3 Performance Metrcs We consder three man metrcs: () Accuracy, (2) Settlng tme, and (3) Convergence Rato. We defne each of these metrcs next. In each of the defntons below, we assume that the dagnoss process starts at tme t = and that»» N refers to the dagnoss experment under consderaton. Defnton 4 The accuracy of a dagnostc strategy at tme t s defned as the probablty that the dagnostc strategy wll yeld a correct dagnoss at tme t. To measure the accuracy of a dagnostc strategy at tme t, we measure the percentage of smulaton experments n whch a correct dagnoss was reached at tme t. 3 Results from addtonal scenaros we have tested were consstent wth the results we present for the four scenaros n Table 3.

14 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 4 Defnton 5 For an experment, the settlng tme S (t) of a dagnostc strategy s defned as the latest tme t» t at whch a wrong (or nconclusve) dagnoss was made for that experment. The mean settlng tme S(t) of a dagnostc strategy s defned as the expected value of the settlng tme at tme t. The above defnton mples that the (mean) settlng tme s a monotoncally non-decreasng functon of t. To measure the mean settlng tme S(t), we averaged the settlng tme for all smulaton experments at tme t. P N= S (t) S(t) = N In the remander of ths paper, we use settlng tme to mply mean settlng tme. Ths settlng tme as a functon of t can be used to characterze the convergence of a dagnostc strategy (or lack thereof). We do so next. Defnton 6 For an experment, the convergence rato C (t) of a dagnostc strategy s defned as the rato between the tme ellapsed snce settlng and t namely C (t) = t S (t) t = S (t) t The mean convergence rato of a dagnostc strategy C(t) s defned as the expected value of the convergence rato at tme t. One can easly show that a random dagnoss strategy yelds a convergence rato that approaches as t ncreases. Thus, one can vew the convergence rato as a measure of how much better" a dagnostc strategy s compared to a random dagnoss. The closer the convergence rato s to zero, the slower the convergence; and, the closer the convergence rato s to one, the faster the convergence. The value of the convergence rato for large enough values of t can be used to characterze the lkelhood of convergence. In partcular, f the convergence rato approaches a constant r (» r» ) as t approaches nfnty, then t follows that the probablty that the dagnostc strategy wll converge n an nfntely long experment s r. In our presentaton below, and unless otherwse specfed, we use the term convergence rato" to mean the convergence rato at tme t = T max, where T max = 3 seconds s the smulaton tme of our experments. 4.4 Baselne Results for BP versus MP Technques Accuracy: Fgure 4 shows the accuracy acheved over tme for the four basc scenaros we consdered. Clearly, our Bayesan Probng (BP) approach yelds a consstently hgher accuracy than that acheved by the Markovan Probng (MP) approach. For scenaros (I), (II), and (III) n whch shared losses exst, our BP approach converges to % accuracy wthn a very short perod of tme. Ths s n sharp contrast to the MP approach, whch oscllates consderably around the 75-9% accuracy range under scenaro (I), around the 6-8% accuracy range under scenaro (II), and around the 7-75% accuracy range under scenaro (III).

15 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 5 (I) (II) BP MP* MP Tme(sec).2 BP MP* MP Tme(sec) (III) (IV) BP MP* MP Tme(sec).2 BP MP* MP Tme(sec) Fgure 4: Accuracy of BP vs MP for the basc test scenaros under the baselne model For scenaro (IV) n whch there are no shared losses, our BP approach agan converges rather quckly to % accuracy. Intally, the MP approach performs qute poorly (actually droppng to a 2-3% accuracy as late as seconds nto our experments). However, over tme, the MP approach does converge to almost % accuracy as well. Convergence Characterstcs: To apprecate the convergence characterstcs of our BP approach (compared to the MP approach), we plot the settlng tme and convergence rato for both approaches under our four test cases. These metrcs are shown n Fgures 5 and 6. Fgure 5 ndcates that the settlng tme of our BP approach s decdedly lower than that of the MP approach under all test scenaros. Moreover, n three out of the four test scenaros namely (I), (II), and (III) the settlng tme functon of the MP approach does not seem to level off, whereas the settlng tme functon of the BP technque levels off n all four scenaros. The superor convergence propertes of our BP approach are further confrmed n Fgure 6. Table 4 shows the values of the mean settlng tme and the convergence rato at tme t = 3 for both the BP and the MP approaches. It shows that, on average, the settlng tme for the BP approach s around 5:54 seconds, compared to 36:75 seconds for the MP approach. Also, t shows that, on average, the BP approach converges n 98% of the cases compared to 55% for the MP approach.

16 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes BP MP* MP (I) ln(tme(sec)) BP MP* MP (II) ln(tme(sec)) BP MP* MP (III) ln(tme(sec)) BP MP* MP (IV) ln(tme(sec)) Fgure 5: Settlng tme of BP vs MP for basc test scenaros under the baselne model (log-log plot) Test Mean Settlng t = 3 sec Convergence t = 3 sec Scenaro MP Technque BP Technque MP Technque BP Technque (I) (II) (III) (IV) Average Table 4: Settlng Tme and Convergence Rato at tme t = 3 for both the MP and BP approaches 4.5 Robustness of BP Technque In the remander of ths secton we evaluate the robustness of our BP technque to a host of parameters that may mpact ts performance characterstcs (namely accuracy and convergence). Effect of the BP Senstvty Constant: As we noted earler, the value of the senstvty constant (c) used throughout our experments was.4. We used ths value after comparng the effect of c on the accuracy and the convergence rato for the four baselne scenaros. Ths comparson s shown n fgure 7.

17 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 7 (I) (II).8 BP MP* MP.8 BP MP* MP Tme(sec) Tme(sec) (III) (IV).8 BP MP* MP.8 BP MP* MP Tme(sec) Tme(sec) Fgure 6: Convergence rato of BP vs MP for basc test scenaros under the baselne model As these fgures ndcate, settng c to.4 was a compromse between the accuracy and convergence ratos of scenaros (III) and (IV). Reducng the value of c leads to low performance for scenaro (IV) (.e. when losses are ndependent) snce the BP approach tends to dentfy more false postves". On the other hand, ncreasng the value of c leads to lower performance for scenaro (III) (.e. when shared losses exst but are not domnant for one of the recevers) snce the BP approach's senstvty to shared losses s reduced, resultng n a msdagnoss. Effect of Temporal Separaton ffl: As we dscussed n Secton 3, an mportant assumpton of our BP technque s that the separaton (n tme) between packet pars n a 2-packet probe sequence (.e., ffl) s suffcently small so as to keep the two packets of a packet-par close enough to each other on all traversed lnks. Ths enables an accurate samplng of the state of a queue at the tme the 2-packet probe reaches that queue. Fgures 8 and 9 show the accuracy and convergence rato of our BP technque under the four baselne models and for varous values of the BP senstvty constant c. In general, our experments show that the BP technque's accuracy and convergence are qute robust for values of ffl less than msec. Notce that a separaton larger than msec s unlkely 4 even when packet-pars traverse long paths. 4 If the two packets n a packet-par are sent back-to-back, then t would be necessary for 2.5MB of cross traffc to nterveen between these two packets on a Mbps lnk to acheve a separaton of msec.

18 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes Accuracy (I).84 (II) (III).8 (IV) Threshold c Convergence Rato (I) (II) (III) (IV) Threshold c Fgure 7: Effect of c on accuracy (left) and convergence (rght) for the four basc scenaros. The results n Fgures 8 and 9 ndcate that as the separaton between packet-pars n 2-packet probe sequences ncreases (.e., as ffl grows larger), the BP technque's ablty to dagnose shared losses,.e. under scenaros (I)-(III), decreases. Under scenaro (IV) The accuracy and convergence of the BP technque are unaffected by ffl. Ths s expected snce 2-packet probe sequences are nstrumental only for the detecton of shared losses (whch are not present under scenaro (IV). An nterestng observaton from the results shown n Fgures 8 and 9 s the trade-off between the senstvty constant (c) and the temporal separaton between packet-pars (ffl). When ffl s small (e.g. ffl < msec), a larger value of c yelds better accuracy and convergence for all scenaros. However, as ffl grows larger (.e. as the effectveness of 2-packet probe sequences decreases), a decrease n c lead to better accuracy and convergence for scenaros (I), (II), and (III).e. when a dagnoss of shared losses" s warranted but lead to a deteroraton n both accuracy and convergence for scenaro (IV).e. when a dagnoss of ndependent losses" s warranted. Thus, f the value of ffl cannot be guaranteed to reman wthn the [; msec] range, then the value of c should be chosen based on whch msdagnoss s safer namely, msdagnosng shared losses as ndependent, or msdagnosng ndependent losses as shared. Effect of Traffc Burstness: In our baselne experments, traffc burstness was moderate wth the ON/OFF tmes of the constant-bt-rate UDP background flows set to a Pareto dstrbuton wth ff = :2. Traffc burstness may negatvely mpact the accuracy and convergence of a dagnostc strategy, snce t may reduce (or ncrease) loss correlatons. Table 5 shows the accuracy and the convergence rato of the BP technque for the four baselne scenaros under varous values of ff (.e. under dfferent levels of background traffc burstness). These fgures show that the BP technque's accuracy and convergence are unaffected by traffc burstness, except for very small values of ff (namely ff = :) under scenaro (IV). 5 Effect of Buffer Szes: Another parameter that may affect the correlatons between losses s the szng of lnk buffers. In our baselne experments, the lnk buffer sze was fxed at 2 packets. Table 6 shows the accuracy and the convergence rato of the BP technque for the four baselne scenaros under varous buffer szes. These fgures show that the BP technque's accuracy and convergence are unaffected by buffer szng, except for large buffer szes under scenaro (III) and under small buffer szes under scenaro (IV). 5 Note that n ths experment, our packet-par probes were sent back-to-back" (.e. ffl = ). The mpact of traffc burstness s lkely to be more pronounced when ffl >.

19 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 9 (I) (II).8.8 Accuracy.6.4 Accuracy c=. c=.2 c=.3 c= Epslon(msec).2 c=. c=.2 c=.3 c= Epslon(msec) (III) (IV).8.8 Accuracy.6.4 Accuracy c=. c=.2 c=.3 c= Epslon(msec).2 c=. c=.2 c=.3 c= Epslon(msec) Fgure 8: Effect of ffl on the accuracy of the BP Technque (I) (II).8.8 Convergence Rato c=. c=.2 c=.3 c= Epslon(msec) Convergence Rato c=. c=.2 c=.3 c= Epslon(msec) (III) (IV).8.8 Convergence Rato c=. c=.2 c=.3 c= Epslon(msec) Convergence Rato c=. c=.2 c=.3 c= Epslon(msec) Fgure 9: Effect of ffl on the convergence of the BP Technque

20 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 2 Test Mean Accuracy for 5» t» 3 Convergence t = 3 sec Scenaro ff ο : ff = : ff = :2 ff = :8 ff ο : ff = : ff = :2 ff = :8 (I) (II) (III) (IV) Table 5: BP accuracy and convergence for varous values of ff Test Mean Accuracy for 5» t» 3 Convergence t = 3 sec Scenaro B = B = 4 B = 7 B = B = B = 4 B = 7 B = (I) (II) (III) (IV) Table 6: BP accuracy and convergence for varous buffer szes The frst of the above two cases s an anomaly. It could be explaned by notng that for large buffer szes, the probablty of both packets of a packet-par beng lost on the shared porton of the path decreases sgnfcantly under scenaro (III). 6 As a result, scenaro (III) sezes to qualfy for a loss sharng" dagnoss accordng to the value used for the senstvty constant c. The second of the above two cases could be explaned by notng that for very small buffer szes, the probablty of both packets of a packet-par beng lost on the dsjont portons of the paths (.e. on lnks L2 and L3) s hgher. 7 Such losses tend to bas the BP approach nto msdagnosng these ndependent losses as dependent hence a hgher lkelhood of a false postve dagnoss. Effect of Probng Rate: Another mportant parameter of the BP technque s the probng rate R. A hgher probng rate s desrable because t mples a faster" dagnoss (.e. a shorter settlng tme). However, a hgher probng rate results n smaller tme separaton between probes, and thus threatens to volate the assumpton of probe ndependence, whch s central n our dervaton of the BP dagnostc test. Fnally, n the context of actve probes, a hgher probng rate mples more probe traffc, whch s not desrable. Table 7 shows the accuracy and the mean settlng tme of the BP technque for the four baselne scenaros under varous probng rates (recall that the probng rate used n our baselne experments was 5 probes per second). These fgures show that BP's accuracy s qute robust (even for the hghest rates we attempted). The advantage of hgher probng s evdent n the overall trend of lower settlng tmes when probng rates are ncreased, especally for postve loss sharng dagnoses. For example, by quntuplng the probng rate from 5 to 25, the mean settlng tme under scenaro (I) s reduced by a factor of 5 from 2.34 to 2.29 seconds. 6 Snce all other parameters remaned unchanged, a larger buffer sze would result n a lower loss probablty. 7 Snce all other parameters remaned unchanged, a smaller buffer would result n a hgher loss probablty.

21 Harfoush, Bestavros, and Byers, Robust Identfcaton of Shared Losses Usng End-to-End Uncast Probes 2 Test Mean Accuracy for 5» t» 3 Mean Settlng t = 3 sec Scenaro R = 5 R = R = 2 R = 25 R = 5 R = R = 2 R = 25 (I) (II) (III) (IV) Table 7: BP accuracy and settlng tme for varous probng rates Effect of Queung Dscplne: The BP technque reles on an mportant property of the queung dscplne used on lnk buffers. Namely, t reles on the hgh probablty of back-to-back losses of packetpars n a 2-packet probe when the lnk buffer s full (.e. congested). Ths property s lkely to hold for a DropTal queueng dscplne, whch s the dscplne we have assumed for lnk buffer management n our experments so far. Fgure shows the accuracy and convergence of our BP technque when a Random Early Detecton (RED) [3] queung dscplne s used. In these experments, the parameters of RED that we used were: mnthresh=5, maxthresh=5, and maxp=.. The fgure shows a defnte deteroraton n performance under loss sharng scenaros,.e. scenaros (I), (II), and (III). Ths s expected snce RED tends to reduce loss correlaton and thus s lkely to adversely affect the effectveness of 2-packet probes (snce losses of the two packets n a packet par wll tend to be less well correlated). Ths results n a tendency of the BP technque to be based towards makng a no loss sharng" dagnoss. Fgure shows that despte RED's negatve mpact, the BP technque was stll robust enough to yeld acceptable accuracy and convergence for all scenaros, except scenaro (III), for whch BP's performance was almost undnstngushable from a random dagnoss. It s mportant to note that the MP technque we evaluated earler n ths secton suffers from the same dsadvantage when a RED queung dscplne s deployed [25], whch leaves the problem of robust shared loss dentfcaton n the presence of RED gateways an mportant open problem to the best of our knowledge. Accuracy (I).2 (II) (III) (IV) Tme(sec) Convergence Rato (I).2 (II) (III) (IV) Tme(sec) Fgure : Effect of RED Queueng dscplne on the accuracy (left) and convergence rato (rght).