Joint Traffic Blocking and Routing under Network Failures and Maintenances

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1 Jont Traffc Blockng and Routng under Network Falures and Mantenances Chao Lang ECE Dept. Polytechnc Unversty Brooklyn, NY, 2 Emal: clang@photon.poly.edu Zhu Ge AT&T Labs - Research 8 Park Avenue, POB 97, Florham Park, NJ 7932 Emal: gezhu@research.att.com Yong Lu ECE Dept. Polytechnc Unversty Brooklyn, NY, 2 Emal: yonglu@poly.edu August, 26 Abstract Under devce falures and mantenance actvtes, network resources reduce and congeston may arse nsde networks. As a result, users experence degraded performance on packet delays and losses. Tradtonal approaches focused on reroutng traffc to allevate network congeston and mprove users performance. However, due to the network capacty reducton, traffc reroutng alone cannot always satsfy users performance requrements. In ths paper, we study a dual approach that combnes traffc blockng (rate-lmtng) at the edge of a network and traffc reroutng nsde the network. Whle ngress traffc blockng reduces the utlty of network users, overall network performance can be sgnfcantly mproved due to allevated congeston and shorter routng paths. Therefore, t s mportant to desgn ngress traffc blockng and routng jontly to acheve a good balance between the two factors. Workng towards ths goal, we formulate a jont ngress blockng and routng optmzaton problem. We develop mechansms to ntroduce blockng and routng dfferentatons among users wth dfferent servce prortes and wth dfferent level of mpact to network congestons. Usng network topologes and traffc demands collected from a Ter- ISP network, we evaluated our approach. Our result shows that by blockng only a small fracton of traffc, one can greatly reduce network congeston under severe falures and mantenance actvtes. Our soluton effcently dentfes the optmal blockng and routng dfferentaton among heterogeneous users and acheves much better performance n comparson wth proportonal traffc blockng. The proposed algorthms can be easly adopted by network servce provders n ther traffc engneerng practces. I. INTRODUCTION In a large scale system such as an ISP or an enterprse network, t s unrealstc to assume or to ensure the well-functonng of all components. Unexpected falures, such as a cut on a fber or a condut, a breakdown of network devce (electronc or optcal amplfer, lne card, port, router, etc.) may occur at any tme and can take a few hours up to several days to repar. In addton, mportant mantenance actvtes, such as a router IOS upgrade, can also take network resources out of servce n the meantme. Therefore, t s mportant to desgn the network topology and select the routng strategy n a way such that the network can be robust to sporadc component falures, even though t means added cost and reduced effcency n many cases. On the other hand, t s equally mportant to have the optmal adaptaton strategy so that network resources can be reallocated to best accommodate traffc demand under falure condtons. Ths s done by adjustng the routng of the traffc n the presence of falures. There are two classes of ntra-doman routng protocols that are commonly used n the Internet. The frst class s lnk-state-routng, represented by Open Shortest Path Frst (OSPF []) and Intermedate System-to-Intermedate System (IS-IS [2]) routng. In OSPF and IS-IS, each lnk s assocated wth a postve weght and traffc s routed along the shortest paths defned by the lnk weghts. In cases of tes where several outgong lnks are on the shortest paths to the destnaton, the flow s splt evenly among them. In the second class, represented by Mult-protocol Label Swtchng (MPLS [3]), routng can be more flexbly defned. In MPLS, routng path and splttng fracton can be arbtrarly chosen based on the source and destnaton of the traffc flows. In both cases, routng confguratons (lnk weghts, splttng fractons) are determned n tune wth the normal traffc and network condtons. Under falures or mantenance actvtes, network bandwdth resource may be greatly reduced. Even though routng protocols can converge to new routes that bypass falures, the routes are not talored for the survvng bandwdth resource, causng

2 congeston nsde the network. As a result, network users would experence performance degradaton on packet delays and losses. To address ths problem, network operators have developed proactve and reactve countermeasures. For planned mantenance actvtes, a set of routng confguratons can be computed based on the expected survvng bandwdth and demand matrx. These new confguratons are then deployed n network before the planned ntrusons take place. For hardware falures, ths procedure s trggered by network alarms and operaton tckets n a reactve manner, where the up-to-date network topology and capacty nformaton s used n dervng new routng confguratons that allevate network congeston. Adaptng to new routng paths under network capacty losses allows full utlzaton of the avalable bandwdth. However, n some cases, survvng network capacty can no longer accommodate the traffc demand due to serous falures that take down a sgnfcant porton of the network bandwdth. As congeston s unavodable n such cases, sgnfcant packet delays and massve packet losses on congested lnks are expected. An alternatve to ths unavodable traffc loss at the nternal network lnks s to drop the traffc at the edge of the network. Ths s acheved by confgurng rate lmtng ACLs (access control lst) at the perphery of the network. In ths way, traffc that would route through the congested lnks and hence are doomed lost can be blocked as early as possble. On the other hand, even f n some cases a set of the routes can be found to accommodate the overall traffc demand, the qualty of the resultant routng paths (e.g., path delay) may deterorate. For many delay senstve traffc (e.g., voce of IP, gamng), such degradaton may be ntolerable. Enablng ngress blockng for some low prorty traffc, although reducng network utlty due to the blocked traffc, can lead to mproved delay performance of the feasble routng paths, therefore mproves the overall qualty of the network servce. In ths paper, we study the problem of dentfyng the optmal congeston mtgaton strategy under crtcal network condtons by utlzng both ngress traffc blockng and traffc routng. Our goal s to mnmze the mpact of network capacty reducton due to falures and mantenances on the overall network servce qualty. In partcular, we formulate a jont ngress blockng and routng optmzaton problem that dentfes the best balance between the network performance experenced by the admtted traffc and the utlty loss due to the blocked (rate-lmted) demand. To the best of our knowledge, ths s the frst work that jontly consder both rate control for network demand pars and adaptve route optmzaton under network falures and mantenances. Our formulaton ncludes both blockng and routng dfferentaton among user demands wth dfferent servce prortes. We propose soluton technques for both the source-destnaton (MPLS) and the lnk-state (OSPF/IS-IS) ntra-doman routng protocols. By usng realstc network and traffc demand data collected from a Ter- ISP network, we evaluate our approach and study the tradeoff between traffc blockng and the path delay qualty. We fnd that our proposed soluton can effectvely dentfy the desred tradeoff. Compared wth the state-of-the-art approach where only traffc reroutng s consdered under severe network capacty loss, our soluton can sgnfcantly reduce (by 6% n our experment) the average path delay of the survvng network by blockng a small amount (4%) of low prorty traffc. Ths s encouragng snce t suggests that by consderng ths new dmenson of control (ngress blockng), the overall network performance under falures and mantenances can be greatly enhanced. The remander of the paper s organzed as follows. In Secton II, we revew related work. We formulate the jont ngress blockng and routng problem n Secton III. The methodology of solvng the jont optmzaton problem s descrbed n Secton IV. We present n Secton V expermental results to demonstrate the effectveness of the proposed approach. We conclude the paper and lst drectons for future work n Secton VI. II. RELATED WORK Most ntra-doman routng n the Internet s carred out by dynamc routng protocols, such as OSPF [], IS-IS [2], and MPLS [3]. Ideally, an optmal routng should smultaneously provde good QoS perceved by end users (e.g., small delay, low loss, and hgh throughput) and good network-wde performance (e.g., balanced load, robustness aganst network falures, and reslence to malcous attacks). Even though the routng optmzaton problem has been well-studed n lterature [4], [5], t s stll very dffcult to operate a large network n ts optmal state. When the network topology s large, t s challengng to calculate and mplement optmal routes, especally n a dstrbuted fashon. Approxmaton algorthms have been proposed to fnd routes that acheve close to optmal performances [6], [7]. Furthermore, t has been shown that Internet traffc presents hgh varablty n multple tme-scales [8], [9]. Dynamc update of optmal routes n response to traffc changes s prohbtvely expensve []. Meanwhle, tmely

3 estmaton of traffc demand matrx s a challengng task [], [2], [3], where the derved traffc matrx usually has lmted accuracy. To address these ssues, extensve research has been made on dentfyng routes that are robust to traffc varatons [], [4], [5] and traffc demand estmaton errors [6], [7]. Random router and lnk falures are encountered on a daly bass n large IP networks [8]. Network operators also take down routers and lnks for routne network mantenances. The state-of-the-art soluton for network under falures and mantenances s to reroute traffc, where network routes are re-confgured locally or globally to route the orgnal traffc demand n the reduced network topology. Backup paths can be pre-confgured for fast routng restoraton [9], [2], [2], [22]. Research has also been conducted to make routng confguratons, such as lnk weghts, robust to network resource reducton [23], [24] so that routng algorthms can quckly converge to new routes under falures and mantenances. However, due to the reducton n network resources, users experence degraded performance after routng reconfguratons. Large scale falures can cause the network ncapable of supportng the suppled traffc demand, leadng to long network delays and hgh packet losses. To avod excessve network congeston, our soluton blocks traffc demand at the network ngress ponts, so as to mantan good performance for admtted users. To the best of our knowledge, ths s the frst work on combnng traffc blockng and reroutng for crtcal network condtons. Ingress traffc blockng s essentally a rate control problem for network demand pars. Classc network rate control studes the optmal rate allocaton for users wth fxed routes. A utlty-prce based framework was proposed to study the farness and stablty of congeston control schemes [25], [26], [27], [28]. Solutons to the dstrbuted optmzaton problem wth dfferent utlty functons lead to dfferent levels of farness among users, such as max-mn farness and proportonal farness. Recently, the nteracton between TCP congeston control and dynamc IP routng has been studed as a jont utlty maxmzaton problem [29], [3]. Jont rate control, routng and resource allocaton algorthms have been proposed to acheve the optmal resource utlzaton n wreless and sensors networks [3], [32], [33], [34], [35]. We study the jont ngress blockng and routng problem n the context of network falures and mantenances. The goal s to mantan good network-wde delay performance at the prce of mnmal network traffc loss. We also consder the blockng and routng dfferentaton among users wth dfferent servce prortes. III. FORMULATION: JOINT BLOCKING AND ROUTING In ths secton, we formulate the jont optmzaton problem wth both ngress blockng and routng. We start wth a baselne model desgned for MPLS type of flow routng. We then augment the baselne model wth addtonal constrants and varatons of ts objectve functon to ntroduce ngress blockng and routng dfferentaton among users wth dfferent servce prortes. The baselne model s also smplfed to utlze source-destnaton based routng. At the end of the secton, we show how to use the baselne model n lnk-state shortest routng. A. MPLS Baselne Model Consder a network G = (V, E ) shared by a set of users D. Let I and E denote the ngress and egress ponts for user D respectvely. The expected traffc volume of user s d. Under falures or mantenances, the network topology s reduced to G = (V, E), wth survvng capacty c j for each lnk j E. To mantan a good network-wde delay performance, ngress routers conduct admsson control to block excessve traffc. Ingress router I admts d d amount of traffc from user. We quantfy users satsfacton usng utlty functons of ther admtted traffc rates. Let U (d ) denote user s utlty at rate d. We assume U ( ) s an ncreasng and concave functon. It follows that user s utlty loss due to traffc blockng s U (d ) U (d ). Let B {d, D} be the vector of admtted traffc for all users. The aggregate network utlty loss s J U (B) = D ( U (d ) U (d ) ). () Snce U (d ) s ncreasng and concave n d, J U (B) s decreasng and convex n B. Admtted traffc wll be routed to mnmze network-wde delay. We start wth a baselne model for MPLS type of flow routng. Lnk weghts based shortest path routng, such as OSPF/IS-IS, wll be dscussed later. For a flow routng, we denote by τ l the traffc rate of user allocated to lnk l, l E. A legtmate set of routes for user

4 {τ l, l E} should satsfy flow conservaton on all nodes n the network: τ l d f v = I τ l = d f v = E, (2) l O(v) l P(v) otherwse where O(v) s the set of outbound lnks and P(v) s the set of nbound lnks at node v. Gven all users routes R {τ l, D, l E}, the total traffc rate on lnk l s f l = D τ l. Smlar to [5], [4], we approxmate the congeston delay on lnk l usng M/M/ formula c l f l. If the propagaton delay on lnk l s p l, the average end-to-end delay for user s φ (B, R) = ( ) τ l + p l. (3) d c l f l l E The network-wde aggregate delay s thus the weghted sum (by traffc volume) of all users, as shown below. J D (B, R) = d φ = ( ) fl + f l p l (4) c l f l D l E Obvously J D (B, R) ncreases wth the admtted traffc B gven any fxed routes R. J D (B, R) s also a convex functon of B and R. Our objectve s to fnd a jont soluton of blockng B and routng R that acheves a good balance between the network delay J D and utlty loss J U. Workng towards ths, we formulate a weghted sum optmzaton problem: subject to (2) and mn ( α)j D(B, R) + αj U (B, R), (5) {B,R} d d, D, (6) f l c l, l E, (7) τ l, D, l E, (8) d, D, (9) where the weght < α < controls the balance between the network delay performance and utlty loss n the objectve functon. Constrants (6) state that admtted traffc of a user cannot exceed offer demand. Constrants (7) state that the aggregate traffc over a lnk cannot exceed the survvng capacty of the lnk. From () and (4), the objectve functon s convex n B and R. The constrants defned n (2) and (6) (9) are lnear. Overall, system () (9) defnes a convex optmzaton problem. For a gven α, by solvng ths convex optmzaton problem, we can obtan the optmal jont soluton of blockng B (α) and routng R (α). The network delay s JD (α) and the utlty loss s J U (α). We can tune α to acheve the desred balance between the network delay and the utlty loss. We wll demonstrate ths through experments n Secton V. B. Dfferentaton between Users A smple blockng strategy s to block all users proportonally,.e., B(η) {d = ηd, < η <, D}. Let R (η) be the mnmal delay routng under proportonal blockng B(η): R (η) argmnj D (B(η), R). {R} Under B(η) and R (η), the network-wde delay s JD (η) and the aggregate user utlty loss s J U (η). Snce J D (η) ncreases wth η and JU (η) decreases wth η, one approach to achevng balance between J D (η) and J U (η) s to try to satsfy the network delay performance requrement by blockng as lttle traffc as possble. In ths case, the jont blockng and routng problem can be formulated as: max η, subject to J D(η) B D, <η

5 where B D s the upper bound requrement on the network-wde delay. In proportonal blockng, the optmal blockng rato η s ndependent of the utlty functons of ndvdual users and can be dentfed effcently through bnary search. In real network envronments, network users are heterogeneous. Network servce provders gve them dfferent servce prortes accordng to the servce level agreement (SLA). Users wth hgher prortes should be blocked less and assgned to routes wth smaller delays. In addton, users wth the same prorty have dfferent ngress and egress ponts. They contrbute dfferently to network congeston even f they have the same traffc rates. Consequently they should be blocked (.e., rate-lmted) and routed dfferently. Therefore, user dfferentaton n blockng and routng s necessary to acheve the optmal network-wde performance. Blockng Dfferentaton. To ntroduce blockng dfferentaton between users, we ntroduce dfferent lower bounds for the admtted traffc rates of users wth dfferent prortes. In other words, we modfy the constrant n (6) to d m d d for user D. In the extreme case, f we set d m = d, no traffc of user can be blocked. The varable d wll be replaced by the constant d n (5). We can also ntroduce blockng dfferentaton n our framework by manpulatng user utlty functons. Assgn a hgher-value utlty functon to a user wll make hm less lkely to be blocked at hs ngress pont. Routng Dfferentaton. To mplement routng dfferentaton, we allow dfferent users between the same par of ngress-egress ponts to take dfferent routes. In addton, we use a weghted network delay performance: J D D w d φ = l E ( zl c l f l + z l p l where w s the weght for user, and z l = D w τ l. A user wth a hgher weght s lkely to be assgned to a shorter route. The optmzaton problem changes to mn {d,τ l } ( α) l E ( zl c l f l + z l p l ), ) + α D ( U (d ) U (d ) ). () The lnear fracton n the frst term makes t no longer convex. To obtan an approxmaton soluton, one may stll approxmate the frst term by a combnaton of lnear functons of (z l, f l ) and the second term by a set of lnear functons of d. Then () can be solved approxmately as a lnear programmng problem. Even wthout delay performance weghts, t s possble that the jont optmzaton returns multple paths for a group of users sharng the same ngress-egress par. Then routng dfferentaton wthn ths group can stll be acheved by assgnng users wth hgher prortes to shorter paths. We wll elaborate on ths n the followng secton. C. Smplfcaton: Source-Destnaton Routng In an MPLS routng formulaton, every user has a set of routng varable {τ l, l E}. Thus, dfferent user demands wth the same ngress-egress ponts can have dfferent routng paths dependng on ther SLA. However, n a source-destnaton based routng, all users sharng the same ngress-egress pont par are enforced to use the same set of routes. Ideally, one would lke to adopt source-destnaton based routng n the optmzaton problem (5), snce ths largely reduces the number of routng varables. The concern s whether we sacrfce the qualty of the routng soluton by dong so. In fact, as we proved n Appendx A, f there s no dfferentaton n users delay performance, the optmal source-destnaton routng can acheve the same performance as the optmal MPLS routng. For source-destnaton routng, we can adopt routng varables {x l (s,d), (s, d) V V, l E}, where xl (s,d) denotes the fracton of traffc from ngress node s to egress node d routed to lnk l. Then flow routng varables {τ l, D, l E} n (5) can be replaced by τ l = d x l (I,E ). However, optmzng varables d and x appear n product forms and makes t dffcult to solve. We can put all users sharng the same ngress-egress pont par (s, d) nto one group D(s, d) = { D I = s and E = d}. For each group D(s, d), we defne a group utlty functon as the soluton of the followng optmzaton problem U (s,d) (z) max U (d ), () {d, D(s,d)} D(s,d)

6 subject to D(s,d) d z, (2) d m d d. (3) U (s,d) (z) represents the maxmal aggregate utlty f the total traffc rate [ allocated to user group D(s, d) s z. It can be shown U (s,d) (z) s an ncreasng and concave functon of z D(s,d) dm, ] D(s,d) d, f U (d ) s concave n d [d m, d ], D(s, [ d) (See proof n Appendx B). We then treat all users n D(s, d) as one super user wth a varable rate z D(s,d) dm, ] D(s,d) d and utlty functon U (s,d) (z). We can frst solve the blockng and routng problem as n (5) to obtan the optmal group rate z(s,d) and routes {xl (s,d)} for users n that group. The admtted traffc rates for all users n group D(s, d) are: {d, D(s, d)} = argmax and the flow routng varables for user can be calculated as: τ l = x l (s,d) d. D(s,d) d=z (s,d) U (d ) If there are multple paths between an ngress-egress par, one can assgn the shorter ones to users wth hgher prortes to acheve routng dfferentaton. Table I shows the algorthm that constructs a set of routes for a gven ngress-egress par from the routng varables n the ncreasng order of the path delay. The set of lnks carryng non-zero traffc of the specfed user group G can be determned from the optmzaton results (τ l ). In each teraton, the shortest path s found from ths lnk set. The traffc rate usng ths path s set as the mnmum of all routng varables on the path. We then update the routng varables on the path by deductng ths traffc rate and remove the lnks whose assocated routng varable becomes zero from the canddate lnk set. Ths repeats for the next shortest path untl the lnk set becomes empty. Fnally, we assgn paths to users n the reverse order of ther prorty (shorter path for hgher prorty). D. Lnk State Routng TABLE I MULTI-PATH ALLOCATION ALGORITHM Defnton: G specfed user group E set of lnks carryng traffc of the user group G φ l lnk delay on lnk l, l E τ l group traffc rate on lnk l, l E Routne: whle E { } M = subset of lnks on OSPF path based on delay {φ l, l E} τ mn=mn {τ l, l M} Output the traffc rate and path(τ mn, M) for each l M τ l =τ l -τ mn f τ l equal to Update E = E l Lnk weght based shortest path routng algorthms, such OSPF []/IS-IS [2], are wdely used n network traffc engneerng practce. It has been shown n [6] that lnk weght optmzaton s an NP-hard problem. However, heurstc algorthms can often effcently fnd lnk weghts that acheve performance that s close to the optmal of ts counterpart (.e., MPLS routng). Based on ths observaton, we develop approxmaton algorthm for the jont blockng and routng problem under OSPF/IS-IS. Whle the couplng between traffc blockng and routng largely ncreases the complexty of the heurstc search, we can decouple ths complexty by adoptng our MPLS baselne model to fnd the optmal blockng confguraton

7 frst. More specfcally, let B (α) and R (α) be the optmal MPLS soluton for some α that acheves the desred level of balance between the network delay JD (α) and utlty loss J U (α). A heurstc algorthm, such as that n [6], s consequently appled to search for the best lnk weghts to route the suppled traffc demand B (α) = {d, D} (.e., the admtted traffc demand calculated by the MPLS model). Network delay under the routng returned by the heurstc algorthm mght be hgher than that of the optmal MPLS routng JU (α). Ths can be addressed by ntroducng a delay margn n the MPLS baselne model to compensate for the performance gap between the optmal MPLS routng and the heurstc lnk-state routng. IV. SOLUTION METHODOLOGY To solve the jont blockng and routng problem formulated n the prevous secton, we frst need to choose approprate user utlty functons. We assume the number of end-to-end connectons n a demand par s proportonal to ts expected traffc volume d. Under ths assumpton, we choose utlty functons as ( U (d ) Md exp( k d ) ), (4) d wth M and k are constants. One nterpretaton s that the utlty of a demand par s the summaton of the utltes of all connectons n that demand par, and the utlty of a connecton s an ncreasng and concave functon of ts traffc admsson rato. Consequently, larger demand pars wll have hgher weghts n the utlty optmzaton. In order to transform the convex optmzaton problem nto a lnear programmng problem, we use pece-wse lnear functons to approxmate lnk congeston delays and user utlty functons. Congeston Delay: The total congeston delay on lnk l s θ l = fl c l f l, where c l s ts lnk capacty and f l s ts total traffc load. Ths lnk delay functon s a convex and ncreasng functon of lnk load. Smlar to [6], we approxmate t by a collecton of pece-wse lnear functons: (k f l 6 θ l = max c l + b ), where k = and b = , {,..., 6}. Utlty Loss: For the utlty functon defned n (4), we choose M = 25 D d, the utlty loss functon and k = 5. For traffc demand L (d ) = U (d ) U (d ) = Md (exp( k d d ) exp( k)) s a convex and decreasng of the traffc admsson rato. Smlar to the congeston delay approxmaton, we approxmate the utlty loss functon L (d ) by a set of lnear functons: L (d ) = Md max m 6 (k m d() d () + b m), where parameters {(k m, b m ), m {,, 6}} determne the turnng ponts of the pece-wse lnear approxmaton. Fgure shows the approxmaton curve for the normalzed utlty loss as a functon of the traffc blockng rato. Wth the above lnear approxmatons, the jont blockng and routng optmzaton can be solved effcently as a Lnear Programmng problem by the AMPL/CPLEX [36] toolkt. A. Evaluaton Network Settngs V. EXPERIMENTS The network traffc data n our experments are collected from AT&T s North Amercan commercal backbone network, whch conssts of tens of Pont of Presence(PoPs), hundreds of routers, thousands of lnks. The PoP-level network topology on June 23, 26 s used n our evaluaton. The router-level topology s frst obtaned usng the method [37]. We then reduce the router-level topology to a PoP-level topology by congregatng the router-level lnks between the same par of PoPs nto a sngle PoP-level lnk. The capacty of a PoP-level lnk s set as the sum of the capactes of all underlyng router-level lnks. The traffc matrxes are estmated usng the tomo-gravty method [3], whch can provde accurate estmates, especally for large traffc matrx elements. The fnal PoP-level

8 .9.7 Utlty Loss Traffc Blockng Rato Fg.. Utlty Loss Pece-wse Lnear Approxmaton traffc matrx contans around 4 Orgn-Destnaton flows at rates rangng from tens of Mbps to tens of Gbps. We emulate network falures or mantenance actvtes by reducng lnk capactes ether randomly or accordng to the shared rsk groups [38] presented n the lower layer topology graph. In the next subsecton, we present our result based on one synthetc falure settng where we dentfy the node wth the hghest lnk degree and cut the capactes of all ts adjacent lnks by half. The result shown s representatve for other severe network falure scenaros. B. Numercal Results ) Tradeoff between Traffc Blockng and Network Delay: The parameter α n the optmzaton objectve functon controls the balance between the network delay and utlty loss from blockng. We vary the tradeoff parameter α from. to.99 and solve the jont optmzaton problem numercally. We plot n Fgure 2 the average traffc Network Delay Average Traffc Blockng Rato Network Delay Average Traffc Blockng Rato α Fg. 2. Average Traffc Blockng Ratos and Network Delays at Dfferent Tradeoff Parameters blockng ratos and network delays as a functon of α, where the delay cost has been normalzed by the network delay wthout ngress blockng. Recall that n equaton (5), when α s small, the user utlty loss has a small weght n the overall objectve functon. As a result, a large porton of traffc has to be blocked n order to quckly brng down the network delay. As α ncreases, the user utlty loss takes a larger weght and less traffc s blocked. Consequently, the network delay ncreases. When α approaches, the utlty loss domnates the network delay n the jont optmzaton, the traffc blockng rato quckly drops close to. Meanwhle, network delay shoots up dramatcally. Ths corresponds to the state-of-the-art approach where only traffc reroutng s used n responses to network falures. To llustrate the tradeoff between network delay and traffc blockng, we plot n Fgure 3 network delay as a functon of traffc blockng rato. The horzontal dashed lne represents the network delay before the falure. Fgure 3 shows that the network delay drops quckly wth a lttle bt of traffc blockng. Wth a blockng rato of 4%,

9 the network delay can be reduced to the level before the falure. By blockng around 2% traffc, the network delay can be reduced about 5 tmes. Ths demonstrates that ngress traffc blockng s very effectve n allevatng network congeston under falures and mantenances. One can sgnfcantly mprove network delay performance by sacrfcng a lttle on user utlty loss. Furthermore, the small percentage of blocked traffc s from low prorty users. The delay decreasng trend slows down when the blockng rato gets larger. By examnng the tradeoff curve, a network operator can choose an approprate α parameter to acheve a desred level of balance between the network delay and traffc blockng. The soluton of the correspondng jont optmzaton wll tell them how much traffc to block for each demand par and how to route admtted traffc n the networks..9 No Blockng.7 Network Delay.5 α =.99 α =.9 Delay Cost before Falure.3 α = α = α = α = Average Traffc Blockng Rato Fg. 3. Network Delay vs. Average Traffc Blockng Rato 2) Blockng Dfferentaton between Demand Pars: The objectve of jont blockng and routng s to reduce the network delay wth mnmal traffc blockng. Dependng on ther traffc rate and ngress-egress ponts, dfferent demand pars contrbute dfferently to network congeston. In the soluton of the jont blockng and routng optmzaton, the demand pars are blocked dfferently. The general trend s that demand pars wth larger endto-end delays are lkely to be blocked more than those wth smaller delays. Fgure 4 demonstrates the correlaton between traffc blockng ratos and average delays of demand pars at four dfferent α values. For each α, after solvng the jont optmzaton problem, we ndex all demand pars n the ncreasng order of ther end-to-end delays. We then plot ther traffc blockng ratos. In these fgures, traffc blockng ratos only take several dscrete values. Ths s because we use a pece-wse lnear functon of blockng rato to approxmate the network utlty loss. The soluton of the resulted lnear programmng problem corresponds to the extreme ponts n the blockng and routng desgn space. From those plots, we observe a postve correlaton between traffc blockng ratos and end-to-end delays of demand pars. Demand pars wth larger end-to-end delays are lkely to be blocked more than demand pars wth smaller delays. In Fgure 4(a), the demand wth the largest end-to-end delay s blocked more than 9%, the demand wth the smallest delay s only blocked around 25%. Wth the ncrease of α, the average traffc blockng rato decreases. Fgure 4 also shows that the maxmal traffc blockng rato decreases as α ncreases. These results suggest a rule for some heurstc algorthms to select demand pars for blockng wthout explctly solvng the jont optmzaton problem: blockng the long delay demand pars frst. To dentfy those long demand pars, one can calculate the optmal routng wthout any blockng to get the optmal routes and queung delays on all the lnks. Based on the algorthm descrbed n Table I, one can dentfy all paths of a demand par and use the end-to-end delay on ts longest path as ts lkelhood of beng blocked. Fgure 5 llustrates the correlaton between the above defned longest path delays and the lkelhoods of beng blocked of demand pars n our experments. We frst sort all demand pars n the ncreasng order of ther longest path delays. We solve the jont blockng and routng optmzaton at dfferent α values. For each α, we calculate the average traffc blockng rato y(α). If a demand par x s blocked, we plot one dot at coordnates (x, y(α)) n Fgure 5. The fgure verfes that the optmal blockng strategy prefers to block demand pars wth larger delays frst. However, the gaps between ponts on horzontal lnes, especally for small average blockng ratos, also ndcate that the optmal blockng strategy cannot be replaced by the smple heurstc algorthm that blocks demand n the decreasng order of longest path

10 .9.9 Blockng Rato of Traffc Blockng Rato of Traffc Demand Index Sorted by Average Delay Demand Index Sorted by Average Delay (a) α =. (b) α = Blockng Rato of Traffc Blockng Rato of Traffc Demand Index Sorted by Average Delay Demand Index Sorted by Average Delay (c) α =.5 (d) α =.75 Fg. 4. Traffc Blockng Rato v.s. Average End-to-End Delay delays. 3) Comparson wth Proportonal Blockng: The smplest traffc blockng strategy s to block all demand pars proportonally. To demonstrate the effectveness of our jont blockng and routng method, we compare our results wth the smple proportonal blockng. After proportonally cuttng traffc of all demands, we calculate the optmal routng for the blocked traffc matrx to get the network delay cost under proportonal blockng. Fgure 6 compares the network delay cost of the two methods wth the same average traffc blockng rato. We vary the traffc blockng rato from to around.55. From ths fgure, we can see our optmzaton method outperforms the smple proportonal blockng. Wth the same average traffc blockng rato, jont blockng and routng acheves lower network delay than proportonal blockng. The dfference s qute large when the blockng rato s small. The gap closes down when more and more traffc get blocked. Fgure 7 compares the Cumulatve Dstrbuton Functon (CDF) of the delays of all demand pars n three cases: no blockng, proportonal blockng at rato 4% and optmal blockng at average rato 4%. From the curve of no blockng we observe that under network falure, the performance of part of the network degrades severely. There s a large delay varance. Whle the proportonal blockng decreases the average delay, we can see some demand pars stll have large delays. It ndcates that the network congeston can not be allevated effectvely by proportonal blockng. The jont optmzaton method not only decreases the average delay of demand pars, but also effectvely reduces the delay varance. Ths verfes that our jont blockng and routng method can mantan a good network-wde delay performance wth mnmal ngress traffc blockng. VI. CONCLUSION AND FUTURE WORK In ths paper, we study the jont ngress blockng and routng problem under network falures and mantenances. We have shown that ngress traffc blockng s very effectve n allevatng network congeston caused by falures and mantenances. The proposed jont blockng and routng approach can effcently fnd the optmal dfferentaton

11 5 Average Traffc Blockng Rato Demand Index Sorted by Longest Path Delay.9 Fg. 5. Evoluton of Demand Par Blockng Proportonal Optmal.7 Network Delay Average Traffc Blockng Rato Fg. 6. Comparson wth Proportonal Blockng among users wth dfferent servce prortes and ngress-egress combnatons. Our approach can be used to acheve a desred level of balance between network delay and user utlty. Our study opens up a new desgn space for network traffc engneerng. Future research can be pursued n the followng drectons. Lnk-State Routng. Our evaluaton n ths paper s based on MPLS routng. We have shown n Secton III-D that MPLS baselne model can be used to solve the blockng problem for a lnk-state routng. We wll evaluate the performance of our proposed approach usng popular lnk-state routng protocols, such OSPF and IS-IS. Router-Level Soluton. Lmted by the speed of the lnear programmng solver, we were only able to obtan the optmzaton soluton for the PoP-level topology. How to map a PoP-level soluton to a router-level confguraton deserves further study. Another drecton to obtan a router-level soluton s to reduce the sze of the routerlevel jont optmzaton problem by sacrfcng the optmalty of the soluton. We wll study the tradeoffs n developng such smplfcatons. Weghted Delay Performance. We formulated n Secton III-B the jont optmzaton problem when users have dfferent delay performance weghts. It s dffcult to solve exactly the resultant fractonal programmng problem. We wll approxmate t by a lnear programmng problem to obtan approxmaton soluton. Optmzaton for Packet Losses. In ths paper, we focused on the network delay performance. Another mportant metrc under network congeston s lnk packet loss probabltes. We plan to modfy the current framework to account for packet losses. The decson to be made by a network provder s to ether drop packets reactvely on routers nsde hs network or drop packets proactvely on ngress routers. One challengng ssue s that packet losses wll make traffc flows non-conservatve nsde the network.

12 .9.7 CDF.5.3. No Blockng Optmal 4% Proportonal 4% Average End to End Delay x 4 Fg. 7. CDF of Demand Average End to End Delay APPENDIX A Clam: If there s no delay dfferentaton among users, jont blockng and source-destnaton based routng has the same optmum as the baselne model n (5). Proof: Assume B = {d, D} and R = {τ l, D, l E} s the optmal soluton of (5). The traffc rate on lnk l s fl. For any ngress-egress pont par (s, d) (V V ), let D(s, d) = { D I() = s and E() = d} be the set of users wth ngress pont s and egress pont d. Calculate the aggregate traffc rate of users n group D(s, d) on lnk l: y(s,d) l = τ l. D(s,d) Snce {τ l, l E} satsfy flow conservaton (2) and mplement demand d. It s easy to verfy that {yl (s,d), l E} satsfy flow conservaton and mplement the aggregate demand between s and d, d (s,d) D(s,d) d, y(s,d) l d (s,d) f v = s y(s,d) l = d l O(v) l P(v) (s,d) f v = d. (5) otherwse Therefore Y = {y(s,d) l, (s, d) (V V ), l E} s a legtmate source-destnaton routng for the admtted demand B = {d, D}. The traffc rate on any lnk l under Y equals f l. Therefore the aggregate network delay under Y equals that of R. The jont blockng and routng soluton B and Y acheve the optmum of the baselne MPLS model n (5). Snce any source-destnaton based routng Y can be represented by a flow routng R, the optmal jont blockng and source-destnaton routng wll have exactly the same performance as the optmum of (5). APPENDIX B Clam: The group utlty functon defned n equaton () s an ncreasng and concave functon of the aggregate group traffc rate z. Proof: Let {d (j), D(s, d)} denote the optmal rate allocaton among users n group D(s, d) f the aggregate group rate s z j : {d (j), D(s, d)} argmax U (d ), (6) D(s,d) d=zj Then U (s,d) (z j ) = D(s,d) D(s,d) U (d (j) ).

13 If {d (), D(s, d)} s the rate allocaton for z, z 2 > z, construct another rate allocaton as d + = d () + (z 2 z ) d d() j D(s,d) (d j d() j ). Obvously, {d (+), D(s, d)} > {d (), D(s, d)} (element-wse vector comparson). Snce U ( ) s ncreasng functon, we have D(s,d) U (d + ) > U (s,d)(z ). Now that D(s,d) d(+) = z 2, through the defnton of U (s,d) ( ), U (s,d) (z 2 ) U (d + ) > U (s,d)(z ). D(s,d) In other words, the group utlty functon ncreases wth the aggregate group rate. Gven the optmal group rate allocatons {d () } and {d (2) } for aggregate group rate z and z 2, construct a rate allocaton {d (α) } for the aggregate group rate αz + ( α)z 2 as follows: Snce U ( ) s concave functon, Consequently, D(s,d) U (d α ) d (α) = αd () + ( α)d (2). U (s,d) (αz + ( α)z 2 ) D(s,d) ( ) αu (d () ) + ( α)u (d (2) ) = αu (s,d) (z ) + ( α)u (s,d) (z 2 ). D(s,d) U (d α ) αu (s,d) (z ) + ( α)u (s,d) (z 2 ). Therefore, the group utlty functon U (s,d) (z) s an ncreasng and concave functon of the aggregate group rate z. REFERENCES [] J. Moy/IETF, OSPF Verson 2(RFC 2328), 998. [2] R.Callon/IETF, Use of OSI IS-IS for routng n TCP/IP and dual envronments(rfc 95), 99. [3] D. Awduche, J. Malcolm, J. Agogbua, M. O Dell, and J. McManus/IETF, Requrements for Traffc Engneerng Over MPLS (RFC 2328), 999. [4] D. G. Cantor and M. Gerla, Optmal Routng n a Packet-Swtched Computer Network, IEEE Transactons on Computer, vol. C-23, no., pp , 974. [5] R. G. Gallager, A Mnmum Delay Routng Algorthm Usng Dstrbuted Computaton, n IEEE Transacton on Communcatons, january 977, pp [6] B. Fortz and M. Thorup, Internet Traffc Engneerng by Optmzng OSPF Weghts, n Proceedngs of IEEE INFOCOM, 2, pp [7] T. Ye and S. Kalyanaraman, A recursve random search algorthm for large-scale network parameter confguraton, n Proceedngs of ACM SIGMETRICS, 23. [8] W. Leland, M. Taqqu, W. Wllnger, and D. V. Wlson, On the self-smlar nature of Ethernet traffc, IEEE/ACM Transactons on Networkng, vol. 2, no., February 994. [9] V. Paxson and S. Floyd, Wde Area traffc: The falure of posson modellng, n Proceedngs of ACM SIGCOMM, 994. [] B. Fortz and M. Thorup, Optmzng OSPF/IS-IS Weghts n a Changng World, n IEEE Journal on Selected Areas n Communcatons, 22, pp [] A. Medna, N. Taft, K. Salamatan, S. Bhattacharyya, and C.Dot, Traffc matrx estmaton: Exstng technques and new drectons, n Proceedngs of ACM SIGCOMM, 22. [2] M. Roughan, A. Greenberg, C. Kalmanek, M. Rumsewcz, J. Yates, and Y. Zhang, Experence n measurng backbone traffc varablty: Models, metrcs, measurements and meanng, n Proceedngs of Internet Measurement Workshop, 22. [3] Y. Zhang, M. Roughan, N. Duffeld, and A. Greenberg, Fast Accurate computaton of large-scale IP traffc matrces from lnk loads, n Proceedngs of ACM SIGMETRICS, 23. [4] C. Zhang, Z. Ge, J. Kurose, Y. Lu, and D. Towsley, Optmal Routng wth Multple Traffc Matrces: Tradeoff between Average Case and Worst Case Performance, n Proceedngs of IEEE Internatonal Conference on Network Protocols (ICNP), 25. [5] C. Zhang, Y. Lu, W. Gong, J. Kurose, R. Moll, and D. Towsley, On Routng Optmzaton wth Multple Traffc Matrces, n Proceedngs of IEEE INFOCOM, 25.

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