A Comparison of Hard-state and Soft-state Signaling Protocols

Transcription

1 A Comprison of Hrd-stte nd Soft-stte Signling Protocols Ping Ji, Zihui Ge, Jim Kurose, nd Don Towsley Computer Science Deprtment, University of Msschusetts t Amherst, ABSTRACT One of the key infrstructure components in ll telecommuniction networks, rnging from the telephone network, to VC-oriented dt networks, to the Internet, is its signling system. Two brod pproches towrds signling cn be identified: so-clled hrd-stte nd soft-stte pproches. Despite the fundmentl importnce of signling, our understnding of these pproches - their pros nd cons nd the circumstnces in which they might best be employed - is mostly necdotl (nd occsionlly religious). In this pper, we compre nd contrst vriety of signling pproches rnging from pure soft stte, to soft-stte pproches ugmented with eplicit stte removl nd/or relible signling, to pure hrd stte pproch. We develop n nlytic model tht llows us to quntify stte inconsistency in single- nd multiple-hop signling scenrios, nd the cost (both in terms of signling overhed, nd ppliction-specific costs resulting from stte inconsistency) ssocited with given signling pproch nd its prmeters (e.g., stte refresh nd removl timers). Among the clss of soft-stte pproches, we find tht soft-stte pproch coupled with eplicit removl substntilly improves the degree of stte consistency while introducing little dditionl signling messge overhed. The ddition of relible eplicit setup/updte/removl llows the soft-stte pproch to chieve comprble (nd sometimes better) consistency thn tht of the hrd-stte pproch. Ctegories nd Subject Descriptors C.2.2 Computer-Communiction Networks] Generl Terms Performnce, Design Keywords signling, soft-stte, hrd-stte This work is supported in prt by DARPA subcontrcts with Northrop Grummn Informtion Technology 2-2 nd 2-9, Ntionl Science Foundtion subcontrct with the University of Florid UF-EIES-253-UMA, nd Ntionl Science Foundtion grnt ANI Permission to mke digitl or hrd copies of ll or prt of this work for personl or clssroom use is grnted without fee provided tht copies re not mde or distributed for profit or commercil dvntge nd tht copies ber this notice nd the full cittion on the first pge. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission nd/or fee. SIGCOMM 3, August 25 29, 23, Krlsruhe, Germny. Copyright 23 ACM /3/8...$5... INTRODUCTION One of the key infrstructure components in ll telecommuniction networks, rnging from the telephone network, to VC-oriented dt networks, to the Internet, is its signling system. Two brod clsses of signling pproches cn be identified: so-clled hrdstte nd soft-stte pproches. Between these two etremes lie signling pproches tht in prctice borrow vrious mechnisms from ech. Despite the fundmentl importnce of signling, our understnding of these two pproches - their pros nd cons nd the circumstnces in which they might best be employed is still not well understood. Brodly speking, we ssocite the term soft-stte with signling pproches in which instlled stte times out (nd is removed) unless periodiclly refreshed by the receipt of signling messge (typiclly from the entity tht initilly instlled the stte) indicting tht the stte should continue to remin instlled. Since unrefreshed stte will eventully time out, soft-stte signling requires neither eplicit stte removl nor procedure to remove orphned stte should the stte-instller crsh. Similrly, since stte instlltion nd refresh messges will be followed by subsequent periodic refresh messges, relible signling is not required. The term soft-stte ws coined by Clrk 3], who described the notion of periodic stte refresh messges being sent by n end system, nd suggested tht with such refresh messges, stte could be lost in crsh nd then utomticlly restored by subsequent refresh messges - ll trnsprently to the end system, nd without invoking ny eplicit crsh-recovery procedures:... the stte informtion would not be criticl in mintining the desired type of service ssocited with the flow. Insted, tht type of service would be enforced by the end points, which would periodiclly send messges to ensure tht the proper type of service ws being ssocited with the flow. In this wy, the stte informtion ssocited with the flow could be lost in crsh without permnent disruption of the service fetures being used. I cll this concept soft stte, nd it my very well permit us to chieve our primry gols of survivbility nd fleibility... Roughly speking, then, the essence of soft-stte pproch is the use of best-effort periodic stte-instlltion/refresh by the stteinstller nd stte-removl-by-timeout t the stte-holder. Softstte pproches hve been tken in numerous protocols, including RSVP 2], SRM 9], PIM 6, 5, 7], SIP] nd IGMP4]. Hrd-stte signling tkes the converse pproch to soft stte - instlled stte remins instlled unless eplicitly removed by the receipt of stte-terdown messge from the stte-instller. Since stte remins instlled unless eplicitly removed, hrd-stte signling requires mechnism to remove orphned stte tht remins

2 fter the stte-instller hs crshed or deprted without removing stte. Similrly, since stte instlltion nd removl re performed only once (nd without stte refresh or stte timeout), it is importnt for the stte-instller to know when stte hs been instlled or removed. Relible (rther thn best-effort) signling protocols re thus typiclly ssocited with hrd-stte protocols. Roughly speking, then, the essence of hrd-stte pproch is the relible nd eplicit instlltion nd removl of stte informtion. Hrd-stte pproches hve been tken in protocols such s ST-II4, 8] nd Q.293b5]. Between the etremes of pure hrd-stte pproch nd pure soft-stte pproch lie mny protocols tht hve dopted elements of ech pproch. Indeed, protocols tht were initilly conceived s pure soft-stte protocols hve dopted number of hrd-stte mechnisms (often s etensions) over time. For emple, in IGMPv4], soft-stte timeout t router ws used to detect the deprture of previously registered hosts; IGMPv2/v3 8, 2] lter dded n eplicit leve messge to llow host to eplicitly inform the stte-holding router of its deprture. In the originl RSVP 2], PATH nd RESV stte instlltion messges were trnsmitted besteffort under the ssumption tht the loss of signling messge would be recovered from lter refresh messge; ACK-bsed relible signling ws introduced s n etension to RSVP in ] nd ws lso suggested in 3]. RSVP hs lso provided for eplicit (lthough optionl) removl of filter specifictions since its conception. Hrd-stte protocols hve dopted elements of the soft-stte pproch s well. In the ST-II hrd-stte signling protocol, periodic HELLO messges serve to inform the HELLO sender tht ll is well with its neighbors, nd tht its own stte tht relies on given neighbor is still vlid - n implicit refreshing of its stte. Given the blurred distinctions between hrd-stte nd soft-stte pproches nd the fct tht protocols flling into one ctegory often dopt mechnisms typiclly ssocited with the other, we believe tht the crucil issue is not whether hrd-stte or soft-stte pproch is better in some bsolute sense. Insted, we believe tht the more fundmentl question is to understnd how the mechnisms tht hve evolved into being included in vrious hrd-stte nd soft-stte signling protocols cn best be used in given situtions, nd why. In this pper, we thus compre nd contrst vriety of signling pproches rnging from pure soft-stte pproch, to soft-stte pproches ugmented with eplicit remote stte removl nd/or relible signling, to pure hrd-stte pproch. We define set of generic protocols tht lie long this spectrum, nd develop unified prmeterized nlytic model tht llows us to quntify key performnce metric ssocited with given signling protocol - the frction of time tht the stte of the stte-instller nd the stte-holder re inconsistent 6]. We lso quntify the cost (both in terms of signling overhed, nd ppliction-specific costs resulting from stte inconsistency) ssocited with given signling pproch nd its prmeter vlues (e.g., stte refresh nd removl timeout intervls). Among the clss of soft-stte pproches, we find tht dding eplicit removl in soft-stte pproch substntilly improves stte consistency, while introducing little dditionl signling messge overhed. The ddition of relible eplicit setup/updte/removl further llows the soft-stte pproch to chieve comprble (nd sometimes better) consistency thn tht of the hrd-stte pproch. Our work here focuses on evluting the performnce of different signling mechnism nd protocols. However, there re other non-performnce-oriented criteri by which to compre vrious signling pproches (e.g., the compleity or robustness of protocol implementtion). These criteri, however, re beyond the scope of this pper. The reminder of this pper is structured s follows. In Section 2, we describe five different signling protocols tht incorporte vrious hrd-stte nd soft-stte mechnisms, nd qulittively discuss the fctors tht will influence performnce. Section 3. presents n nlytic model for emining the performnce of these pproches in the single-hop cse, nd compres their performnce. Section 3.2 considers the multi-hop cse. Section 4 discusses relted work. Finlly, Section 5 summrizes this pper nd discusses future work. 2. SOFT-STATE, HARD-STATE AND PRO- TOCOLS IN BETWEEN In this section, we describe the opertion of five different bstrct signling protocols. These protocols differ in the mnner in which stte is instlled, mintined, nd removed, nd whether selected signling messges re trnsported best-effort or relibly. We will consider single node (henceforth referred to s the signling sender ) tht wishes to instll, mintin, nd eventully remove (or hve removed) stte t remote node (tht we will refer to s the signling receiver ). We consider the simple, but illustrtive, emple of signling sender hving locl stte vlue tht it wishes to instll t one or multiple signling receivers long signling pth. When the signling sender stte vlue equls it of the signling receiver(s), we will sy tht the vlues re consistent 6]; otherwise the sender nd receiver(s) stte vlues re inconsistent. Our gol here is not to model specific signling protocol such s RSVP or Q293b, but rther to cpture the essentil spects of identifibly different pproches towrds signling. After describing the protocols, we then consider the performnce metrics by which these protocols cn be evluted, nd qulittively discuss the fctors tht will impct performnce. We will consider the following five pproches: Pure soft-stte (): In this pproch, the signling sender sends trigger messge ] tht contins stte instlltion or updte informtion to the signling receiver, nd strts stte refresh timer (with vlue ). When the stte-refresh timer epires, the signling sender sends out refresh messge 2] contining the most up-todte signling stte informtion, nd resets the refresh timer. Trigger nd refresh messge re sent in best-effort (unrelible) mnner. When trigger or refresh messge is received t the signling receiver, the corresponding signling stte informtion is recorded nd stte-timeout timer (with vlue ) ssocited with this stte is strted (or restrted if it ws lredy running). Signling stte t the signling receiver is removed only when its stte-timeout timer epires; tht is, stte will be mintined s long s the receiver continues to receive refresh messges before the stte-timeout timer epires. This timeout could occur becuse the signling sender is no longer sending refresh messges (becuse its locl stte hs been removed nd it thus wnts the remote stte to be removed t the signling receiver), or becuse refresh messges hve been lost in trnsmission, nd hve resulted in stte timeout t the signling receiver. We will refer to the ltter cse s flse removl of stte, since the signling sender did not intend for this stte to be removed. Soft-stte with Eplicit Removl (+ER): +ER is similr to the pproch, with the ddition of n eplicit stte-removl messge. When stte is removed t the signling sender, the sender sends best effort (unrelible) signling messge to the signling receiver crrying eplicit stte-removl informtion. Stte refresh nd trigger messges, nd stte-timeout timer re ll employed s in the cse of.

3 Figure : Signling sender nd receiver: messges nd mechnisms Soft-Stte with Relible Trigger (+RT): +RT is similr to with two importnt dditions. First, trigger messges re trnsmitted relibly in +RT. Ech time trigger messge is trnsmitted, the sender strts retrnsmission timer (with vlue ). On receiving n eplicit trigger messge, the destintion not only updtes signling stte, but lso sends n cknowledgment to the sender. If no trigger cknowledgment is received before the retrnsmission timer epires, the signling sender resends the trigger messge. Secondly, +RT lso employs notifiction mechnism in which the signling destintion informs the signling sender bout stte removls due to stte-timeout timer epirtion. This llows the signling sender to recover from flse removl by sending new trigger messge. Soft-Stte with Relible Trigger/Removl (+RTR): +RTR is similr to the +RT pproch, ecept tht the +RTR pproch uses relible messges to hndle not only stte setup/updte but lso stte removl. Hrd-Stte () pproch: In the pproch, relible eplicit messges re used to setup, updte nd remove stte t the signling receiver. Neither refresh messges nor soft-stte timeout removl mechnisms re employed. A crucil concern with ny hrd-stte protocol is the removl of orphned stte t the signling receiver. Becuse the hrd-stte protocol does not provide timeoutbsed stte removl, it must rely on n eternl signl to detect tht it is holding orphned stte. This signl cn be generted for emple, by seprte hertbet protocol whose job is to detect when the signling sender crshes nd then inform the signling receiver of this event. Alterntively, the eternl signl might be generted vi notifiction from lower lyer protocol t the signling receiver tht hd n ssocition with lower lyer protocol t the signling sender nd hence detected tht signling sender filed. Once such n eternl notifiction (signl) is received, the hrd-stte signling pproch clens up the orphned signling stte ssocited with the signling sender. One complicting fctor is tht of flse notifiction - the eternl signl my flsely detect signling sender crsh (this would occur, for emple, if series of hertbet messges were lost, but the signling sender ws still opertionl). As in the cse of +RT, flse notifiction cn be repired by hving the signling receiver notify the signling sender (if it eists) tht its orphned stte hs been removed. A signling sender whose stte hs been incorrectly removed cn then send new trigger messge. Figure illustrtes the messges nd mechnisms used by the signling sender nd receiver in the vrious signling protocols. In the following section, we will develop unified prmeterized nlytic model tht llows us to quntify key metric ssocited with given signling protocol - the frction of time tht the stte of the stte-instller nd the stte-holder re consistent (i.e., hve the sme vlue). Clerly, we would like this vlue to be s close to one s possible. In ddition to quntifying consistency, we would lso like to quntify the cost ssocited with given signling pproch nd the level of consistency it is ble to chieve. One spect of this cost will be the signling messge rte itself. A second spect of this cost is the cost ssocited with being in n inconsistent stte. For emple, in IGMP, when n end host leves without signling its deprture to its edge router, multicst dt will continue to flow towrds the receiver (even though the receiving host is no longer in the multicst group) - cost. In the cse of hierrchicl peer-to-peer file-shring system in which client uplods the nmes of the files it shres to server when it joins the P2P network, but then leves the network without signling its deprture, the inconsistent stte t the server will result in other peers ttempting to contct the deprted peer - gin, cost. In Section 3, we model this cost s weighted sum of the signling overhed nd ppliction-specific costs (corresponding to the cost of the unwnted multicst dt flows, or connection ttempts to deprted peer in the emples bove). We conclude this section with qulittive discussion of the fctors tht will influence the performnce of signling protocols: Appliction-specific inconsistency cost. As noted bove, these re the costs ssocited with the signling sender nd receiver being in inconsistent sttes. Clerly, when this cost is high, the signling sender my be willing to incur higher signling overhed in order to keep the signling sender nd receiver sttes s consistent s possible. Refresh timeout vlue. As noted in ], the smller the vlue of the refresh timer, the sooner tht consistent stte will be instlled t the stte-holder, nd consequently the smller the ppliction-specific cost due to stte inconsistency. However, this dvntge comes t the cost of n incresed signling rte. If the ppliction-specific cost of inconsistent stte is high, however, this incresed signling cost my be wrrnted.

4 Soft-stte timeout vlue. Since this timer is ment to remove stte tht is not refreshed, idelly this vlue should be s smll s possible in order to remove orphned stte s soon s the signling sender deprts. However, too smll timeout vlue cn result in flse stte removl. Signling messge loss. As the probbility of messge loss becomes higher, we epect the frction of time tht the signling sender nd receiver sttes re inconsistent lso increse, s it will tke longer for either messge to be delivered relibly, or for best-effort refresh messge to be delivered. In cses of high loss nd high ppliction-specific inconsistency costs, those protocols with eplicit relible trnsfer will be preferble. Number of hops. In certin signling protocols such s RSVP nd AFSP9], signling sender must instll stte t multiple nodes between itself nd the ultimte signling destintion. As the number of hops increses, the frction of time tht ll nodes re in n inconsistent stte will lso increse. In the following sections, we will develop n nlytic model tht will llow us to quntittively eplore these issues. 3. MODELING AND ANALYSIS OF SIGNAL- ING APPROACHES We begin our nlysis by considering the simple emple of single node (the signling sender ) tht cn instll, mintin, chnge, nd eventully remove (or hve removed) single piece of stte informtion t remote node (the signling receiver ). We focus here on single piece (rther thn multiple pieces) of stte, s it is conceptully simpler nd the ltter cn generlly be considered s multiple instntitions of the former. The instlltion, mintennce, chnge, nd removl of stte is ccomplished using one of the five bstrct signling pproches described in the previous section. We ssume tht the signling sender nd receiver communicte over network tht cn dely nd lose, but not reorder, messges. 3. Signling in Single-hop System We first consider single-hop system, in which the signling sender nd receiver re the only two entities involved in the signling protocol. As shown in Figure 2, we cn think of the two entities s being connected through single logicl hop, which my consist S D S () single physicl hop (b) multiple physicl hops with end-to-end signling Figure 2: single-hop signling systems of one or more physicl hops. A number of eisting pplictions nd protocols fit this simple single-hop model. For emple, signling in the IGMP protocol 4] occurs between n end system nd its first-hop router. When the end system joins multicst group, stte indicting this group membership must be instlled in the firsthop router; when the end host leves the multicst group, this stte should be removed from the router. In certin peer-to-peer file shring pplictions such s Kz 2], peer registers its shred files with server ( supernode in the cse of Kz), which then redirects peers seeking given file to peer nodes tht hve tht file. A D peer s registrtion of its files t supernode is single-hop signling process, where the signling sender is the peer, the signling receiver is the supernode, nd the signling stte contins the identities of the shred files nd the fct the peer is in the system nd serving files. 3.. Model Description Before describing our system model, we first briefly discuss the events tht cn occur during the life cycle of signling sender/receiver pir. Signling stte setup. When the signling session first instlls (initilizes) its locl stte, it trnsmits signling messge contining the stte to the receiver. After some dely, the signling messge reches the remote receiver, enbling both sender nd receiver to chieve consistent stte. Signling stte updte. A sender my lso updte its locl stte. As in the cse of stte setup, the sender then instlls the new stte vlue t the receiver. When sender updtes its locl stte, the sender s nd receiver s stte will be inconsistent until the updte successfully propgtes to the receiver. Signling stte removl. At the end of the lifecycle, the sender will remove its stte. At this point, the receiver s stte should lso be removed. Once the sender hs removed its stte, the receiver s stte is stle (inconsistent) until it is removed. A number of protocol-dependent mechnisms (including stte-timeout, nd eplicit removl messges) cn be used to remove receiver stte. Flse signling stte removl. The destintion my incorrectly remove stte, even though the sender is still mintining stte. This cn occur s result of vrious protocol-dependent events. For emple, in soft-stte pproches, the stte-timeout timer could epire t the receiver nd remove stte, even though the sender is still mintining stte. 465 D6E 798 :9;!"$#"% &'$(*),+-./$*,23 This stte does not eist in model for or +RT. >6? <6= B6C Stte setup Stte updte Stte removl Flse removl Figure 3: A continuous time Mrkov model for single-hop signling Given these events in the lifecycle of signling sender nd receiver, we cn develop Mrkov model, shown in Figure 3, to cpture this behvior. The Mrkov model s sttes re defined s follows. Ech stte consists of pir of vlues, FHGJILKMGNPOQK where G I nd G N refer to the sttes of the signling sender nd receiver, respectively: Mrkov stte FSRTKVUWO cptures the initil stge of the lifecycle, when signling stte hs been instlled t the sender but not t the receiver. This is n inconsistent stte, since the sender nd receiver s stte vlues do not mtch.

5 Y e m c c Mrkov sttes FMU$KR,O correspond to cses where the sender hs removed the stte, but the receiver hs not. These sttes re lso inconsistent. When the sender nd receiver hve consistent signling stte, the stte of the Mrkov chin is X. When the sender nd the destintion hve different signling stte (i.e., both hve instlled stte, but the stte vlues re different), the Mrkov chin is in sttes X Y. When the signling stte is removed from both the sender nd the receiver, the system enters n bsorbing stte represented by Mrkov stte FMU$KUWO. Note tht ech of the inconsistent sttes, FSRTK6UWO, FMU$KMRZO, nd XY re further divided into two seprte Mrkov sttes distinguished by subscripts nd 2, the purpose of which is to cpture protocoldependent detils tht we will describe shortly. In Figure 3, shded rrow indictes the initil stte of the Mrkov chin, nd the double circled stte FMU$KVUWO is the bsorbing Mrkov stte. The trnsitions mong the Mrkov sttes re illustrted in Figure 3 with different line styles indicting the different events (stte setup, stte updte, stte removl nd flse removl) tht cuse stte trnsitions. The system prmeters considered in the stte trnsitions re: \ `_ N : ]6^ : signling stte updte rte N is the sender s men signling stte lifetime : flse stte removl rte t receiver : signling chnnel dely bdc : signling chnnel loss rte In ddition, we hve the following previously-discussed protocol specific prmeters: : soft-stte refresh timer vlue : soft-stte stte timeout timer vlue. : messge retrnsmission timer vlue for relible trnsmission We model the signling stte lifetime nd the intervl between signling stte updtes s eponentilly \ distributed rndom vribles (with mens ]6^ N nd ]9^ _, respectively), flse removl s Poisson process with rte, nd messge losses s independent Bernoulli trils with prmeter b`c. Furthermore, we pproimte the soft-stte refresh intervl, stte-timeout intervl, messgeretrnsmission intervl nd chnnel dely s eponentilly distributed rndom vribles with mens,, nd K respectively. In Section 2, we discussed five different pproches towrds signling. Ech of these pproches cn be modeled using the model shown in Figure 3, with different trnsition rtes (nd in some cses disbled trnsitions) for ech of the pproches. We net describe the model trnsitions for ech of these different signling pproches. These trnsitions re shown either in the model digrm or in Tble. Soft-Stte () model. The initil stte of the model, FSRTKVUWOe, corresponds to the cretion of new signling stte t the sender. As discussed erlier, this results in trigger messge being sent to instll stte t the receiver. After chnnel dely, one of two events cn occur. First, the trigger messge cn successfully rech the destintion. This event occurs with probbility FM]$U b`c O, nd is modeled by the trnsition from stte FSRdK6UWO e to stte X with rte FM]WU bc OM^. The second possibility is tht the trigger messge is lost. This event occurs with probbility b`c, nd is represented by the trnsition from FSRdK6UWOQe to FSRdK6UWOMf with rte b c ^. Eventully refresh messge will rech the destintion. Since refreshes re sent periodiclly with intervl, nd ech messge reches the destintion with probbility (]U bc ), there is trnsition from FSRTK6UWO f to stte X with rte FM] U bc OM^L. The updte process is similr to the setup process. When the stte is consistent, i.e., the Mrkov chin is in stte X, stte updte \ cuses the Mrkov chin to trnsit from = to stte XY e t rte. The trigger messge successfully rrives t the receiver with probbility (]gu bc ) nd verge dely, which corresponds to trnsition bck to X t rte FM]U bc OM^. While in XY e, the loss of the trigger messge cuses the Mrkov chin to trnsit to stte Xhf Y t rte bc ^ c. With rte FM]U bc OM^L, the Mrkov chin trnsits from stte X$f Y bck to stte X. Note tht n updte my lso occur when the system is in stte FSRTKVUWO f or stte XY f, which cuses the Mrkov chin \ to trnsit to stte FSRdKVUWO e or stte XY e respectively with rte. Our model serilizes events in the signling process. For emple, it does not llow stte updte while trigger messge is on its wy to the receiver. We ssume tht n updte hppens either before previous trigger messge is sent out or fter the trigger messge hs lredy reched the receiver (or hs been lost). Sender signling stte is removed t rte N, i.e., sender hs session of men length ]6^ NZi If the signling stte is removed t the sender before the receiver hs obtined the stte, the Mrkov chin simply trnsits from FSRdKUWO f to the bsorption stte FMU$KjUWO. However, if the receiver hs lredy instlled stte informtion either consistently or inconsistently, i.e., the system in stte XY f or stte X, the Mrkov chin trnsits to stte FMU$KMRZOje. Therefter, the receiver must wit for the stte-timeout timer to epire in order to remove the orphned stte. We model this by letting the Mrkov chin trnsit from stte FMU$KR,O e to stte FMU$KUWO with rte ]6^L. Note tht the Mrkov model for does not include the FMU$KMR,O f stte in Fig. 3. Finlly, stte cn be removed t the destintion due to the lck of refresh messges before the stte-timeout timer epires. This is modeled by Mrkov chin trnsition from sttes X, XY f, to stte FSRdKUWOMf _ with rte i Since such flse removl only hppens when ll refresh messges within stte timeout timer durtion hve been lost, we `_ pproimte the probbility J_ of this event s blknmporqds. Therefore, cn be epressed s X b ktmpouqds. Note tht, the model does not llow stte trnsition from Xve Y to FSRdK6UWOQeVK due to the seriliztion considertions noted bove. Soft-Stte with Eplicit Removl (+ER) model: Recll tht in +ER, signling messge crries eplicit stte removl informtion (in ddition to the stte-timeout mechnism) to remove signling stte. We model this eplicit removl by modifying the stte removl process in the model s follows. When the Mrkov chin enters stte FMU$KMR,OQe s result of sender stte removl, n eplicit stte removl messge is sent out. With probbility (]gu b`c ) nd fter chnnel dely, this messge rrives t the destintion nd triggers the removl of the corresponding stte. We model this by letting the Mrkov chin trnsit from FMU$KMR,O e to the bsorbing stte FMU$KVUWO with rte FM]wU bc OM^. The loss of the eplicit removl messge cuses the Mrkov chin to trnsit from FMU$KMR,O e to FMU$KMR,O f. From there, the system trnsits to the bsorbing stte FMU$KUWO t rte ]9^9, cpturing the stte removl cused by the stte-timeout timer epirtion. Soft-Stte with Relible Trigger messges (+RT) model: The Mrkov model for +RT differs from the model for in tht, when trigger messge crrying stte setup/updte informtion is lost, either successful refresh messge or successful retrnsmission of the trigger messge cn bring the Mrkov chin from stte X f or stte FSRTKVUWO f to stte X with rte FM]9^9 yz]9^{ $O} ~FM]U bc O. Soft-Stte with Relible Trigger/Removl messge (+RTR) model: The Mrkov model for +RTR differs from the model for +RT in tht, when n eplicit removl messge is lost, the system

6 X N ] š š wits for the stte-timeout timer to epire or successful retrnsmission of the removl messge to go into stte FMU$KUWO. Thus the trnsition rte from stte FMU$KMRZO f to stte FMU$KUWO is ]6^L FM]$U b c OM^{. Hrd-Stte () model: The model is similr to the +RTR model, ecept tht the trnsition rtes ssocited with refresh messges nd stte-timeout timers re ecluded. In ddition, s discussed in Section 2, the pproch must rely on n eternl signl to recover from sender filure. Accounting for the relted cost of such n eternl signl is difficult, since it depends on the underlying scheme tht performs the filure-detection for the hrd-stte pproch. For instnce, link-lyer sensing mechnism provides filure detection to signling without introducing etr signls; wheres filure-detection relying on n underlying hert-bet echnging mechnism my hve n overll overhed comprble to tht of +RTR. Nonetheless, we consider the filure-detection s seprte component in the system rchitecture with the signling mechnism. Therefore, in our pper, we eclude this prt from the nlysis. However, we ssume tht the eternl signl cn be flsely generted with rte `, which cuses fulty removl of signling stte in the pproch. We summrize the protocol-specific stte trnsition of the Mrkov chin for different signling pproches in Tble, where J denotes the stte trnsition rte from Mrkov stte to Model Solution nd Performnce Clcultions Using this model, we cn now study the performnce of the signling pproches discussed in Section 2. We re interested in the following metrics: the inconsistency rtio,, defined s the frction of time tht the signling sender nd receiver do not hve consistent stte vlues; nd the normlized verge signling messge rte, ˆ, defined s ˆ N{, where is the totl number of signling messges required during the lifetime of signling session (i.e., time from when the signling stte is initited until it is removed from the system), nd ]9^ N is the epected lifetime of the sender s signling session. Since the lifetime of the signling session t the receiver vries with the signling pproch while ]9^ is invrint, the normliztion provides fir comprison between different signling pproches. To obtin the inconsistency rtio,, we need to know the frction of time tht the system spends outside stte FXŠOQK before it eventully trnsits to the bsorbing stte FMU$KVUWO. This is equivlent to evluting the sum of the sttionry probbilities of the inconsistent sttes in the recurrent Mrkov model where the bsorption stte be the st- FMU$KUWO nd the strting stte FSRdKUWOje re merged. Let tionry probbility of the recurrent Mrkov model in stte. We hve the following epression for : ŒX Žt V J H wž V t ` š œ š œ wžs H ŽS j X ]guž œ () To obtin the totl signling messge overhed,, we need to compute the verge lifetime of signling stte, Ÿ, nd the men signling messge rte ˆ : X Ÿ jˆ (2) Here, Ÿ is derived from clculting the men time to bsorption for stte FSRTK6UWO e in the trnsient Mrkov model, nd ˆ is obtined by considering in which stte nd with wht rte ech of signling messges - eplicit trigger nd removl messges, soft-stte refresh messges, retrnsmission nd cknowledgment messges - re generted during the signling process. We proceed s follows. With successfully trnsmitted trigger messge, the Mrkov chin trnsits from stte FSRdKUWOje or X e Y to stte X, nd if trigger messge is lost, the Mrkov chin trnsits from stte FSRTK6UWO e to FSRdKVUWO f or from Xve Y to X$f Y. Thus the men messge rte for eplicit triggers, ˆ, M, is, ˆ, X wž V t ` H Žt V J H u œ š œ œ Q œ Ž V t ` H Žt V J H u tžt t ` y œ š œ u œ š (3) Similrly, the men messge rte for eplicit removl, ˆ M, is ˆ M X ŽS j H ŽS J S j tž t ` ŽS j H ŽS H Q tžs J j (4) Soft-stte refresh messges re generted t men rte ]9^9 when the Mrkov chin is in sttes FSRdK6UWO f, X, or XY f. Therefore the men messge rte for refresh messges, ˆ, cn be epressed s, ˆ X PF* Žt t J œ y œ š O (5) If trigger messges re trnsmitted relibly, retrnsmissions will be generted t rte ]9^L when the chin is in sttes FSRdK6UWOrf nd X$f Y, nd cknowledgment messges will be generted for every trnsition to stte X. Therefore, the men messge rte for relible triggers, ˆ, u, cn be computed by, `_ ˆ u X ] F* Žt V J š œ O œ F* œ š œ O (6) The third term of ˆ, u is cused by flse removl, since relible trigger scheme requires the signling destintion to send messge to the signling sender notifying it of the removl. Similrly, for relible removl, the men messge rte ˆ u is: ˆ, u X ] wžs J j wžs J S Ž j H Q tžs J ` wžs ŽS J j V ŽS t ` (7) In summry, the overll men messge rte for different signling protocols re s follows: +ER +RT +RTR ªˆ«X ˆ, M J ˆ, ªˆ«X ˆ M ˆ ˆ M ªˆ«X ˆ, M J ˆ, l ˆ, u ªˆ«X ˆ, M J ˆ, l ˆ, M lyˆ, u ` ˆ, u ªˆ«X ˆ M ˆ Q yˆ u yˆ 3..3 Model Evlution We now compre nd contrst the performnce of the five different signling pproches using our modeling frmework. In order to use representtive prmeter vlues, let us consider s n emple, the signling process between Kz regulr peer (herefter, simply referred to s peer) nd its supernode (s described in the beginning of Section 3.). Unless otherwise noted, we use the following defult prmeters: bc X i P, X± P L² ³, ]9^ \ X± { P³, ]9^ Ń Xµ]6 P { ³, X ¹P³, ºX± {, X¼», nd X± i P {,]. These prmeter vlues re chosen to cpture the behvior of Kz session: signling stte is dded when the peer strts the Kz ppliction, nd is updted when the peer chnges its collection of shred files (e.g., new file is downloded into its shred directory). When the peer eits the Kz ppliction, the peerstte mintined by the supernode should be deleted. If this stte is not removed t the supernode, n inconsistent stte will occur. As result, the supernode my respond to other peers incorrectly (e.g., directing them to n lredy deprted peer; these other peers my then fruitlessly contct the deprted peer, decresing the usbility of the ppliction). Impct of session length (]9^ N ). We first study the performnce of different signling pproches s function of the e-

7 U N Trnsition rtes +ER +RT +RTR Ž V t ` H Q tžt t J nd œ š j œ š bdc ^ bc ^ bc ^ bc ^ bc ^ Žt t J H u œ nd œ š j œ FM]gU bc OM^ FM] U bc OM^ FM]U bdc OM^ FM]U bdc OM^ FM] U bc OM^ Žt t J j œ nd œ š V œ FM]gU bc OM^L FM] U bc OM^9 FM]9^9 yz]9^l ŠO} PFM] U bc O FM]6^L ]9^L ho} PFM] U bdc O FM]U bc OM^L ŽS J j H j ŽS j U bc ^ bc ^ bc ^ ŽS j H Q tžs J t J ]6^L FM]U bc OM^ ]6^L FM]U bdc OM^ FM] U bc OM^ ŽS j tžs J t J U ]9^9 U ]6^L ½zFM] U b c OM^L FM]U b c OM^L `_ b knmporqds c ^L b knmporqds c ^L b ktmpouqts c ^9 b ktmpouqts c ^9 Tble : Model trnsitions.7 Inconsistency rtio. +ER +RTR +ER +RT +RTR +RT Averge signling messge rte RTR +RT +ER +ER +RT +RTR. Men lifetime of signling stte t () ¾ ÁÀ* signling sender (seconds): Figure 4: Comprison ginst session length, ]6^ Men lifetime of signling stte t (b) signling sender (seconds): à ÄnÅHÆ pected mount of time tht signling stte is instlled t the signling sender, FM]9^ N~O. In our Kz emple, this corresponds to peer s verge session length. In Figure 4 (), we plot the inconsistency rtio, nd in Figure 4 (b), we plot the normlized verge signling messge rte ˆ for the different signling pproches. Figure 4 provides number of insights into the single-hop signling system: When the epected session length increses, both the inconsistency rtio nd the verge signling messge rte decrese for ll signling pproches. In the contet of our Kz emple, this implies tht if Kz is used mostly by peers who tend to turn themselves off shortly fter strting, s opposed to remining on for long periods, (e.g., peers use Kz for 5 minutes every hour versus 2 hours every dy), the system is likely to incur more signling overhed, with supernodes responding to queries bsed on stle informtion. Compring +ER to, we note tht the improvement of +ER over (using the inconsistency rtio s the performnce metric) becomes more significnt s the verge session length decreses. Even when the verge session length is on the order of thousnds of seconds, the benefit of dding eplicit removl is still non-negligible. This is due to the fct tht removing orphned stte requires reltively long wit for the timeout timer to epire, in the bsence of eplicit removl. More importntly, considering the verge messge rte in Figure 4, we find tht when the verge session length is on the order of thousnds of seconds, the ddition of eplicit removl introduces negligible signling messge overhed compred to the pproch. While the cost of including this cpbility is so low, our model indictes tht it is very useful to include eplicit removl in soft-stte signling in such circumstnces. This is becuse tht the penlty of not using eplicit removl is so high. Figure 4() indictes tht the performnce gin (in terms of reduced inconsistency rtio) chieved by introducing relible triggers becomes significnt when the peers verge session length is long. This is evidenced by the fct tht when the verge session length is long, the five pproches re differentited ccording to whether or not they provide relible triggers. Conversely, when the verge session length is shorter (towrds the left of Figure 4()) the five pproches re grouped on the bsis of how stte removl is performed: those without eplicit removl (, +RT), those with eplicit removl (+ER) nd those with relible removl (+RTR, ). We note tht for long sessions, when the differences in trigger messge relibility is most pronounced, the inconsistency rtio is reltively low for ll pproches. Also, s shown in Fig. 4(b), the limited benefit of +RT over comes with non-trivil dditionl signling overhed. Thus, for these ppliction prmeters, providing relible trigger messges does not pper to be very crucil. +RTR provides essentilly the sme inconsistency rtio s. This suggests tht beyond eplicit removl nd relible trnsmission, ny enhncements to soft-stte refresh mechnism cn provide only modest gins (if ny) in the inconsistency rtio. Indeed, in some cses +RTR lredy performs slightly better thn. Impct of messge loss nd dely. Figure 5 plots the inconsistency rtio for different signling pproches for vrious loss

8 Ê Ë Inconsistency rtio ER +RT +RTR +ER +RT Signling chnnel loss rte: ÇSÈ.25.3 () +RTR Inconsistency rtio ER +RTR É +ER +RT +RTR +RTR Signling chnnel dely (in seconds): (b) Figure 5: Comprison ginst link loss rte b c nd link dely Inconsistency rtio. +ER +RT +RTR +RT +RTR +ER.. Soft-stte refresh timer (in seconds): () Averge signling messge rte. +ER +ER +RT +RTR +RT +RTR. Soft-stte refresh timer (in seconds): (b) Figure 6: Comprison ginst soft-stte refresh timer ( ) rtes () nd delys (b). Figure 5() indictes tht even for modest loss rtes (e.g., 5%), relible trnsmission significntly improves the performnce of soft-stte protocols. Figure 5(b) plots the inconsistency rtio versus the one-wy sender-to-receiver dely. We observe n pproimtely liner increse in the inconsistency rtio under ll signling pproches. However, signling pproches with relible trnsmission ehibit slightly lrger slope. This is becuse the vlue of the retrnsmission timer is generlly proportionl to the chnnel dely. Thus, to recover from loss, pproches with relible trnsmission suffers longer ltencies in n environment with longer trnsmission delys, while soft-stte pproches tht only rely on refresh mechnism do not. Impct of timer configurtion. There re three different timers used in the five signling pproches we consider: the soft-stte refresh timer, the soft-stte stte-timeout timer nd the retrnsmission timer. Figure 6 eplores the performnce of different soft-stte signling pproches under different soft-stte refresh timer settings. Since does not employ refresh mechnism, it is shown s Ì on the y-is. When the refresh timer vlue chnges, we set the stte-timeout timer to be 3 times the vlue of the refresh timer. Figure 6 revels n interesting trdeoff between hving short refresh timer (to reduce the inconsistency rtio) nd long refresh timer (to keep signling messge overhed low). Overll Cost. As discussed erlier, there re two components of overll cost: signling messge cost, nd ppliction-specific costs rising from inconsistent stte in the sender nd receiver. For emple, we sw erlier tht for IGMP, this ltter cost ws the trnsmission of unwnted multicst dt; in the cse of Kz, this ltter cost ws the dditionl overhed cused by the supernode providing peers with pointers to lredy deprted peers. To evlute the cost of both signling overhed nd ppliction-specific costs re- Integrted cost (C) +ER +RT +RTR +RT Í +RTR.. Soft-stte refresh timer vlue (in seconds): +ER Figure 7: Soft-stte refresh timer ( ) sulting from stte inconsistency, we define n integrted cost (Î ) s ÎÏX Ð V žˆ (8) where Ð indictes the reltive weight of ppliction-specific cost due to inconsistent signling stte. In Kz, for emple, Ð might be interpreted s the number of signling messges ssocited with fruitless queries tht re cused by inconsistent file-shring stte t the supernode. In the following, we set Ð to be (msg/sec). In Figure 7, we plot the integrted cost ssocited with different signling pproches versus the soft-stte refresh timer vlue, ( ). From this figure, we observe tht there eists reltively sensitive optiml operting points for nd +RT, bove which the inconsistency cost increses substntilly nd below which the messge signling cost increses significntly. Such n optiml operting

9 Ñ Ò Inconsistency rtio. +RT +ER +RT +RTR +ER Inconsistency rtio ER +RT +RTR +RT +ER +RTR +RTR.. Stte timeout timer (in seconds): ().4.2. Retrnsmission timer (in seconds): (b) Figure 8: Stte timeout timer ( ) for soft-stte bsed pproches nd retrnsmission timer for relible trnsmissions point lso eists for +ER, lthough the integrted cost is not very sensitive to rther longer refresh timer vlues. Lst, for +RTR, longer timer vlue is preferred, nd when the timer is lrge enough (on the order of s of seconds), it provides comprble performnce to the hrd-stte pproch. Figure 8 () eplores the impct of different stte timeout timer vlues on the inconsistency rtio of soft-stte pproches. Here we fi the stte refresh timer to be 5 seconds nd vry the stte-timeout timer. The results indicte tht, when the stte-timeout timer is shorter thn the refresh timer, ll soft-stte bsed pproches perform poorly, since refresh messges rrive too lte to keep live the signling stte t the signling receiver. Once the stte-timeout timer vlue is greter thn the refresh timer vlue, the different pproches behve very differently: +RTR does well with long timeout vlues, since the longer the timeout timer, the less likely it is tht stte is flsely removed due to loss of refresh messges. nd +ER do best when the stte-timeout timer is pproimtely twice the length of the refresh timer, so tht the probbility of flse removl is reduced. However, since longer timeout timers dd lrger delys to remove orphned stte, nd +ER lso require the stte timeout timer to be short enough to void such problems. Recll tht +RT employs notifiction mechnism in which the signling receiver informs the signling sender bout stte removls nd the signling sender recovers from flse removl by sending nother trigger messge. Since +RT is the most sensitive to the process of removing orphned stte nd its notifiction mechnism reduces the penlty of flse removl, it works best with timeout timer vlue tht is just slightly lrger thn tht of the stte-refresh timer. Figure 8 (b) eplores the impct of different retrnsmission timer vlues on the inconsistency rtio of the five signling pproches. Since depends only on eplicit relible trnsmissions for stte setup/updte/removl, it is the most sensitive to chnges in the retrnsmission timer i Trdeoff between inconsistency rtio nd verge signling messge rte. By vrying the soft-stte refresh timer, one cn crete trdeoff between the inconsistency rtio nd the verge signling messge rte of different signling pproches. Since pure hrd-stte signling does not use the refresh timer, neither the inconsistency rtio nor the verge signling messge rte vry with ; in Figure 9, hrd-stte is shown s single point Ì. Figure 9 lso indictes tht the inconsistency rtio of +RTR is insensitive to soft-stte refresh rte (which is determined by the refresh timer), wheres the inconsistency rtio of the other soft-stte pproches chnge with the signling overhed. We lso emined the trdeoffs between inconsistency rtio nd signling overhed bsed on other system or design prmeters (e.g, signling chnnel dely, etc). Due to spce limits, we omit these results. Interested reders cn find the corresponding nlyses nd results in ]. While our model ssumes eponentilly distributed timer vlues, in prctice, signling protocols usully use deterministic timers. To investigte the impct of our eponentil ssumption on the timer vlues, we built simultions tht use deterministic timers under the sme system settings. Our simultion results indicte tht using deterministic timers does not ffect our observtions nd conclusions. For emple, in comprison to the evlution results shown in Figure 4, the inconsistency rtio differs slightly (Ó ]6Ô ) between the nlyticl nd the simultion results with deterministic timers. For the verge signling messge rte, the difference between the nlyticl nd the simultion results is between ¹{Ô to ]9¹{Ô, while the qulittive reltive behvior mong different signling protocols remins unchnged. See ] for more detil. Averge signling messge rte. +RTR +RT +ER +ER +RT +RTR.. Inconsistency probbility Figure 9: Trdeoff between inconsistency rtio nd verge signling messge rte, derived by vrying 3.2 Signling in Multi-hop System In this subsection, we consider the cse in which not only the end systems (the signling sender nd receiver) but lso intermedite routers or rely nodes must mintin signling stte. This would occur, for emple, in the cse the intermedite routers need to mintin bndwidth reservtion for communiction between the end systems. In our bstrct model of multi-hop signling system, there is single signling sender nd chin of signling destintions. We require tht signling stte generted t the signling sender be mintined t ll nodes long the pth between the signling sender nd the finl destintion.

10 Û á Ü Ž f ú e e Ž Ž ] ] ] Ž 3.2. Model for Multi-hop Systems In Section 3., we identified vrious fctors tht influence the performnce of signling protocols in the single-hop scenrio. Mny of the results re directly pplicble here s well. In this section, we focus on the unique issues tht rise in the cse of multiple hops. We focus on the sttionry process in multi-hop signling system, in which stte is updted t the signling sender, nd chnges must be propgted to ll receivers. To simplify our model, we consider the lifetime of stte to be infinity ( NÖÕØ ). We model stte updtes s Poisson process with rte `\. Ý ÙwÚ Þ äwå ß æwç à èwé Figure : A five-hop system with three consistent hops Our modeling frmework for multi-hop system is n etension of the single-hop model. Let ê be the totl number of hops (links) in multi-hop system nd ë the number of consistent hops (links). We ssume tht these hops re homogeneous, i.e., they hve identicl chnnel loss rte ( b c ) nd men chnnel dely ( ), nd ssume tht chnnel losses re independent. Here, we define consistent hop to be hop (link) with two ends hving consistent stte informtion. Figure illustrtes five-hop (êìxï¹ ) system with three consistent hops (ëíx ). We define the stte spce s î X FHëpKj³LOr, where ï ë ï ê is the number of consistent hops, nd ³ is specil vrible tking on vlues of or to indicte whether the Mrkov chin is in fst pth Mrkov stte (³žXð ) or slow pth (³ÖXº] ) Mrkov stte. The distinction between fst pth Mrkov stte nd slow pth Mrkov stte will be mde cler lter. In ddition to the FHë KQ³LO sttes, there is specil stte for signling, ñ, tht models the period when the system recovers from flse removl. We net describe the model trnsitions for the pure soft-stte protocol (), the soft-stte protocol enhnced with hop-by-hop relible trnsmission (+RT), nd the hrd-stte protocol (). Modeling protocol trnsitions. Under, stte updtes re crried by trigger messges sent to signling receiver(s). A trigger messge my be lost t ny of the hops. If trigger messge is lost, refresh messge crrying identicl informtion will eventully rech the signling receiver(s) nd mke the receiver(s) stte consistent. A stte is removed if the timeout timer epires due to losses of ll refresh messges sent during the timeout intervl. Note tht if timeout occurs in one node, ll receivers beyond this node in the liner topology will lso time out, becuse they too will not receive the refresh messges. We use the Mrkov model in Figure to model the protocol in the multi-hop environment. Consider the stte updte process. When signling stte is updted t the signling sender, the Mrkov chin trnsits to stte F* ZKQ ~O from other Mrkov sttes. Stte F* ZKQ ~O represents the cse in which no hop is consistent (ëžx¼ ) nd trigger messge is on its wy to the net hop (i.e., the model is in the so-clled fst pth stte, ³X ). After one-hop chnnel dely, two things my hppen to the trigger messge. First, it my successfully rech the net receiver, mking this hop consistent. Thus, the system trnsits from F*,KQ O to FM]{KQ ~O, or more generlly from stte FH rku ~O to stte FH wï]pkq O, with rte FM]WU bc OM^. The second possibility is tht the trigger is lost, in which cse the model trnsitions from F* ZKu ~O to F* ZKV]LO, or more generlly from stte FH ukr ~O to stte FH rkv]lo, with rte b c ^ nd wits for subsequent refresh messge (i.e., the model is in slow pth stte, ³X ] ). ãâ Since we ssume tht refresh intervls re eponentilly distributed with men, nd the probbility tht refresh messge, the generted t the sender reches cross the -th hop is FM]òU b`c O trnsition rte from stte FH pu ]PKV]9O to stte FH ukr ~O is FM]$U b c O ^L. Eventully, the system cn trnsit to stte F*êíKV]LO, either vi fst pth stte F*ê½U ]PKQ O or vi slow pth stte F*êµU ]PKV]9O. In ddition, the stte-timeout timers t signling destintions my epire due to lost refresh messges. Assume hop is the first hop in the chin where stte timeout occurs, (in this cse, the timer of the corresponding sttes t the v¼] -th to the ê -th hops will lso epire). When this hppens, we let the Mrkov chin trnsit to stte Fn PKV]9O Ž with rte e H tž e Ž. Trnsition rte e S tž e cn be clculted by Eqution (9). Ž e H tž e px ôôó õ öôô c*þ ÿ m¼ 9ø ù úüû*ùlúhý c*þ m Lø ù úüû*ùlúhý e k s k s ù otherwise ÿ The epression FM]gU FM]U b`c O e O k s UzFM] U FM]gU bc O (9) O k s pproimtes the probbility tht the timeout hppens t the Fn d ]LO -th signling receiver, but not t ny preceding hop. Model trnsitions for +RT protocol. Under +RT protocol, when the system is trpped in slow pth stte, both successful retrnsmission of the trigger messge nd successful refresh messge cn mke the corresponding hop consistent. This is becuse tht relible trnsmission is used. Therefore, in +RT, the trnsition rte from stte FH wu ]{K6]LO to stte FH rkq ~O becomes e e S tž u X FM]U b c O FM]U b c O i () Model trnsitions for Hrd-stte () protocol. In, relible trigger messges (propgted relibly hop-by-hop) re used to updte stte long the signling pth. Neither refresh messges nor soft-stte timeout removl is employed. Thus, stte trnsition from stte FH U ]{KV]LO to stte (i,) is chieved vi retrnsmission only, nd the trnsition rte is e e S tž u px FM] U bc O i () As in the cse of the single hop system, we model flse removls, J t ech receiver, s n independent Poisson process with rte. Thus, the system trnsits from slow pth stte to the recovery stte (ñ ) with rte ê `. As discussed in Section 2, receiver is notified by n eternl signl when ny filure (e.g., link filure) hppens. This receiver then sends messges to inform other receivers nd the sender of the filure. On receipt of such messge, other receivers remove their ssocited stte(s). If the sender receives such messge, it sends trigger signling messge to re-instll stte. We model this by letting the system trnsit from the ñ stte to the F*,Ku O stte with rte, n pproimtion tht cptures the epected ltency for the sender to initite the recovery process. Further detils of the multi-hop model cn be found in ] Multi-hop Model Solution nd Results The solution of the multi-hop model is similr to tht of the single-hop model. Due to spce constrints, we omit the detils of the solution; interested reders cn consult ]. We focus insted on the results themselves, emining the inconsistency rtio nd signling messge overhed of the three multi-hop signling pproches. In choosing model prmeters, we consider the process of reserving bndwidth long multi-hop pth s n emple. Unless otherwise specified, we use the following defult prmeters: ê Xª {, b c Xª i nd Xª { {² ³ t ech hop, ]9^ \ XP P³,