Distributed Partial Information Management (DPIM) Schemes for Survivable Networks - Part II

Transcription

1 IEEE INFOCO Ditributed Partial Information anagement (DPI) Scheme for Survivable Network - Part II Dahai Xu Chunming Qiao Department of Computer Science and Engineering State Univerity of New York at Buffalo fdahaixu, qiaog@ce.buffalo.edu Abtract A major challenge in hared path protection i to elect a jointly optimized pair of active and backup path. While Integer Linear Programming (ILP) baed approache are notoriouly time conuming, exiting heuritic uch a active path firt (APF) can only achieve ub-optimal reult. In thi paper, we propoe a novel heuritic called APF with potential backup cot or APF-PBC under the ditributed partial information management framework decribed in [1] for ultra-fat proceing of on-line requet. We how that APF-PBC can outperform exiting cheme baed on Integer Linear Programming (ILP) with exactly the ame input (either partial or complete information) and contraint. We alo how how an intuitive function to compute the potential backup cot i derived mathematically baed on the tatitical analyi of experimental data. Keyword bandwidth haring, ditributed routing, online heuritic, potential cot, tatitical analyi. I. INTRODUCTION any emerging application including nation-wide collaborative cience and engineering project require that reliable, high-bandwidth connection be dynamically et up and releaed between large computing reource (e.g., torage with terabyte to petabyte of data, clutered upercomputer and viualization diplay). To meet the requirement of thee emerging application in an economical way, a network mut quickly and dynamically proviion bandwidth-guaranteed connection, each with ufficient protection againt poible failure of network component (e.g., link or node). In thi work, we will introduce an ultra-fat and efficient algorithm to proce on-line requet for either etablihing or releaing a connection. Driven by the demand for more flexible ervice from application mentioned above (the puh effect), and ervice provider deire to generate more revenue (the pull effect), much frequent requet for connection etablihment and releae are expected in the near future. Thi, in conjunction with the large (and increaing) ize of the Internet, make ditributed control and ignaling extremely attractive for the purpoe of achieving a good calability. While approache other than path protection may be adopted to achieve a good degree of urvivability, thi work conider hared path protection for it quick retoration peed (even without fat fault localization) and high bandwidth efficiency. In the following, we will limit our dicuion to protection againt a ingle link failure a the problem of protection againt a ingle node failure can be tranformed to that of protection againt a ingle link failure by plitting each node into two halve interconnected with a virtual directed link. With path protection (but no bandwidth haring), a linkdijoint pair of path, called active path (AP) and backup path (BP), repectively, from an ingre node to an egre node i ued to atify each connection. ore pecifically, aume that the connection require w unit of bandwidth. The ame amount of bandwidth (called active bandwidth or ABW) will be ued on each link along the AP to etablih the connection. eanwhile, w unit of bandwidth (called backup bandwidth or BBW) i alo reerved on each link along the BP. Thi BP will be ued to (re- )etablih the connection to re-route the information to be carried after the AP break due to a link failure along the AP (and before the AP can be retored). The concept of hared protection i illutrated uing the following example. Aume that two connection, requiring w 1 and w 2 unit of bandwidth, repectively, are etablihed uing two link dijoint AP. Note that under uch an aumption, the two AP cannot break at the ame time provided that at mot one link in the network can fail at any given time, or more preciely, no additional failure may occur before an exiting failure i repaired (which i a fairly practical and reaonable aumption). Accordingly, their correponding BP need not be ued to (re)-etablih the two connection, repectively, at the ame time either. Hence, if the two correponding BP ue the ame link e, they can hare the backup bandwidth (BBW) without affecting the urvivability of either connection. ore pecifically, with hared protection, the total BBW that need be allocated on link e (for the two BP) i maxfw 1 ;w 2 g (intead of w 1 + w 2 ).

2 IEEE INFOCO The problem of minimizing the total bandwidth or TBW (which i the um of ABW and BBW) needed to atify a given et of requet for etablihing connection i NP-hard. any off-line approache with application to SONET, AT and WD network cae have been propoed (ee for example the work in [2], [3], [4] and the reference contained therein). In the on-line cae to be tudied, not all requet arrive at the ame time, and a deciion a to how to atify a requet for connection etablihment (if poible at all) ha to be made without knowing which requet will arrive in the future, and, for the ake of guaranteed QoS, without being able to rearrange the way exiting connection are etablihed. In uch an cae, a enible approach i to try to allocate minimal TBW when atifying each requet for connection etablihment, and de-allocate a maximal BBW when atifying each requet for connection releae (a the ABW to be de-allocated i alway a fixed amount). The above approach may lead to either the minimization of TBW needed to upport a given number of requet, or the maximization of revenue for a given network capacity, or both (thee two performance metric will be decribed in more detail in Sec. IV). In fact, everal ILP formulation have been propoed to optimize for each requet in the online cae. However, even if one can allocate minimal TBW (and deallocate maximal TBW) for each requet, one cannot guarantee the bet overall performance in term of minimization of TBW needed to upport a given number of requet that have arrived over time, or maximization of revenue for a given network capacity. The above reaon i why an ILP formulation for the online cae may not yield the bet overall performance (unlike in the off-line cae). In fact, one of the pleaantly urpriing reult of thi paper i that our propoed heuritic algorithm can perform better than an ILP formulation (with the ame input). Clearly, reducing the additional BBW needed for a requet for connection etablihment through BBW haring will help reduce TBW needed to atify the requet (but not necearily minimize TBW unle the AP, and thu the total ABW allocated, i already fixed a in the cae where the APF heuritic i ued). However, in order to determine whether or not a new BP choen to atify the requet for connection etablihment (or releae) can hare (or have hared) BBW with an exiting BP on a given link, and conequently, how much BBW on the link need be allocated (or de-allocated) to atify the requet, one need to firt determine whether or not their correponding AP are link dijoint. Such deciion can be made if a controller (either in centralized or decentralized control implementation) maintain complete per-flow information a in the SCI cheme (which ue an ILP formulation) [5] or complete aggregated information a in the SR cheme (which ue the APF heuritic) [6]. In both cae, the amount of information to be maintained by a controller i at leat O(E 2 ), where E i the number of link in a network. Several approache have been propoed which require only O(E) partial (and aggregated) information. They include the SPI cheme (which ue ILP) in [5] and the two DPI cheme in [1]. While all thee approache achieve a lower degree of BBW haring than SCI and SR, a a price paid to maintain at leat an order of magnitude le information, the DPI cheme perform remarkably better than SPI with the ame order of magnitude information. Part I of our work decribe the DPI framework and the two cheme in detail [1]. It alo pecifie, among other, how the partial information i exchanged, how to etablih or releae a connection, and in particular, how to allocate or deallocate bandwidth through ditributed ignaling. In thi paper, which i Part II of a document, we decribe a novel DPI cheme, which ue a heuritic baed on APF but with an intuitive potential backup cot function derived mathematically baed on the tatitical analyi of experimental data. Such a heuritic, named APF-PBC, combine the joint optimization capability of ILP and the ultra-fat peed of APF (which may be conidered a a pecial cae of APF-PBC where the potential cot i et to 0). In fact, it can achieve a better performance than an ILP baed DPI cheme decribed in [1] with the exactly ame input (partial information). It can alo be applied to the cae with complete (per flow or aggregated) information, and achieve a better performance than SCI (alo baed on ILP). The ret of the paper i organized a follow. Section II contain the notation to be ued throughout the paper. Section III decribe the main idea and motivation behind the propoed APF-PBC, and it application to exiting cheme with complete or partial information. Section IV preent the performance evaluation model ued, followed by numerical reult of the comparion between the APF-PBC heuritic and ILP formulation a well a other APF heuritic. Section V through VII decribe in detail the APF-PBC heuritic, including it mathematical derivation, approximation and implification, repectively, baed on the analyi of experimental data. Section VIII conclude the paper. II. NOTATIONS In thi ection, we preent the notation to be ued throughout the paper. ot of thee notation, except the lat two, have alo been defined and ued in Part I of the

3 IEEE INFOCO document [1] but have to be included here in order to decribe previou work a well a the propoed cheme. We conider a network G with E directed link (repreented by et E and V node, which can be claified into two categorie: edge node (ingre or egre), to which uer or terminal device are connected, and core node (which are node other than an edge node). To facilitate our preentation, we will ue a tuple (! d; w) to repreent a new requet for connection etablihment (or releae), where and d are the ource (or ingre) and detination (or egre) of the connection, repectively, and w i the bandwidth (in unit) requeted by the connection. The following additional notation will be ued, where a calligraphic font tyle (e.g., A i ued to denote a et or a vector while a non-calligraphic tyle (e.g., A) i ued to denote a calar value: ffl OUT (n); IN(n) ρ E: Set of link going from and coming into node n 2V, repectively. ffl AP and BP: Set of link along an AP and BP, repectively. ffl A e : Set of connection whoe AP travere link e. ffl A P e = w k : Total (i.e., aggregated) ABW on link e k2a e dedicated to the connection in A e. ffl B e : Set of connection whoe BP travere link e. ffl B e : Total BBW allocated on link e for B e. Due to BBW haring, B e < P w k. k2b e ffl R e : Reidue bandwidth of link e. It initial value i equal to the capacity of link e, C e. R e = C e A e B e (with only protected connection). ffl Sa b = A T a Bb : Set of connection whoe AP travere link a and whoe BP travere link b, where a; b 2E. ffl Sa b = P w k : Total amount of bandwidth required by k2s b a the connection in Sa. b It i a fraction of A a a well a B b that i ued by the AP and BP, repectively, of the connection in Sa. b ffl BCa: b Additional BBW needed on link b in order to ue it a a part of a BP for a new connection whoe AP travere link a. It value depend on which BBW etimation method i ued. While mot of the above notation are imilar to thoe ued in [5], the following notation are pecific to the propoed DPI cheme: ffl BC e : Etimated BBW needed on link e along a new BP. Auming that it correponding AP i known, BC e = max 8a2AP Be a. Whether thi value i the minimum BBW needed on link b or not depend on which BBW etimation method i ued to derive Ba. e In addition, thi i equal to the actual BBW allocated on link b in SCI, SR and SPI but not in the DPI cheme (which may reult in an overetimation but alway allocate the minimal BBW). ffl P B (e) = fsa e ja 2 Eg: Profile of BBW on a given link e. Thi i a vector coniting of a lit of Sa e value, one for each link a. Baically, it pecifie the amount of BBW on link e that i ued to protect againt the failure of every other link (e.g., a 1, a 2 a E 2E) in the network. ffl P B e =max 8a Se a: Thi i the maximum value over all the component in P B (e). It i alo the minimum (or neceary) amount of BBW needed on link e to backup all active path. If a BBW allocation cheme (uch a the DPI cheme to be decribed) alway allocate minimum BBW on link e, then B e = P B e. ffl P A (e) = fse b jb 2 Eg: Profile of ABW on a given link e. Thi i a vector coniting of a lit (or et) of Se b value, one for each link b. It complement P B (e), and pecifie the amount of ABW on link e that i protected by every link (e.g., b 1, b 2 b E 2E) in the network. ffl P Ae = max 8b S b e: Thi i the maximum value over all the component in P A (e). It i alo the ufficient amount of bandwidth that need be reerved on any link in the network in order to protect againt the failure of link e. ffl P Ae : Thi i the average value over all the component of a given ABW profile on link e. It i only ueful to decribe the APF-PBC heuritic. ffl = max 8a;b Sb a: max P B e. 8e Thi i alo equal to max 8e P Ae or III. APF-PBC AND ITS EXTENSIONS TO PRIOR SOLUTIONS In thi ection, we firt decribe the main idea behind APF-PBC, then briefly ummarize recently propoed cheme for on-line hared path protection under ditributed control, and finally decribe their APF-PBC extenion, whenever applicable. In the following dicuion, we conider a requet for connection etablihment, (! d; w). A. ain otivation and Idea of APF-PBC A dicued in the Sec. I, a major challenge in achieving efficient hared path protection in an on-line cae i that, while on each link ued by an AP, a fixed amount of ABW (i.e., w unit) i to be allocated, the amount of BBW to be allocated on each link along a BP depend on many factor including which link are ued by the correponding AP. Thi i why APF i not ideal a it doe not conider (nor care about) the potential cot along the BP yet to be choen when electing the AP. The hortet pair of path (or SPP) algorithm uch a the one in [7] doe not take poible BBW haring into conideration either in that it

4 IEEE INFOCO eentially aume that the cot of each link on a BP i alo equal to w. Therefore, uch an SPP algorithm may be ueful only when each edge node cannot obtain a better (i.e., more accurate) etimation of the additional BBW needed (than w). On the other hand, on-line approache baed on ILP formulation guarantee minimal allocation of TBW for each requet by optimize the joint election of both AP and BP. However, they do not guarantee an optimal reult for all requet that arrive over time. In addition, their computational complexity i uually too high to be calable. The main motivation for our work on APF-PBC i to overcome the diadvantage of the APF and ILP baed approache, while trying to combine the bet of the two. ore pecifically, the path determination algorithm baed on APF-PBC can run a fat a thoe baed on APF. Yet, they can alo take into conideration the BBW to be allocated when determining the AP a in ILP-baed approache. Thi i accomplihed by aigning a potential cot to each link, baed on which the AP i elected uing a hortet-path algorithm. In thi paper, we will ue fi e (w) =c w P Ae (1) a a function to etimate the potential cot of link e, where c i a contant between 0 and 1. Thi function fi e i derived mathematically baed on the tatitical analyi of experimental data, a to be decribed in later ection. Note that many other form of the potential function derived from intuition may be ued. We have teted ome of them and have found that they do not perform a well a that given above. Alo, APF may be conidered a a pecial cae of APF-PBC where the value of fi e (w) i alway et to zero. The main idea of an approach baed on APF-PBC i a follow. Given a network G with ome exiting connection, any link e whoe R e < w will be (logically) removed initially a uch a link cannot be ued by the AP. Each remaining link a will be aigned a cot which i equal to w + fi a (w). A cheapet path i then found for ue a the AP. Afterward, the link along the AP are then removed, but the other link removed initially hould be put back a they may have enough reidual bandwidth for the correponding BP yet to be choen. The BP i choen next, alo uing a cheapet path algorithm but the outcome depend on how much information on exiting AP and BP the algorithm ha. ore information lead to more accurate etimation of the additional BBW (or backup cot) for each link, and accordingly a better election of BP. It i worth noting that once a BP i choen, the actual amount of additional BBW allocated on a link e along the BP may not be equal to the etimated amount at the time of determining the BP. In fact, the unique feature of the DPI cheme i that, while an edge node will obtain a (lightly) over-etimated backup cot baed only on the partial information it ha, the actual amount to be allocated i alway minimal [1]. We now turn to previou olution and their APF-PBC extenion. We claify all the cheme into two main categorie, one with complete information (per flow or aggregated) and the other with partial (and aggregated) information. In each category, either ILP or ome heuritic uch a APF and APF-PBC may be ued. B. With Complete Information We firt review two cheme requiring at leat O(E 2 ) complete information, with an emphai on the main idea behind path determination, before decribing their APF- PBC extenion. The firt, called SCI in [5], ue complete per flow information and an ILP formulation to determine a pair of path. The econd, called SR [6], ue complete and aggregated information and APF intead. Hereafter, thee two will be named a SCI-I, where I i hort for ILP, and SCI-A, where A i hort for APF, repectively. In SCI-I, a controller maintain, for every link e 2 E, both A e and B e, and baed on which, all other parameter decribed in Sec. II and in particular Sa b for every link a and link b (i.e., all combination) can be derived. In SCI- A, each edge node (a ditributed controller) maintain, for every pair of link a; b 2E, the complete and aggregated information on Sa b (which turn out to be what SCI-I really need). In both cheme, if link b i a candidate link for the yetto-be choen BP, whoe correponding AP i already determined (i.e., AP i known), the additional BBW (or backup cot) to be allocated on link b can be calculated a BC b =maxf max 8a2AP (Sb a + w B b); 0g (2) In SCI-I, a pair of path (whoe link have ufficient reidue bandwidth) with a minimal TBW i determined uing ILP. Note that ince BC b i the minimum amount of additional BBW needed on link b, it i alo the actual amount of additional BBW to be allocated to link b in SCI-I. In SCI-A, an AP (with a minimal number of link whoe reidual bandwidth i larger than w) i found firt uing a hortet path algorithm. Thi i equivalent to APF-PBC with fi a (w) = 0. Then, the link ued by the AP i removed, and each remaining link b i aigned a cot equal to BC b given above. Thereafter, each link b with

5 IEEE INFOCO R b < BC b i removed and a cheapet path found next i ued a the BP. A in SCI-I, the actual amount of additional BBW to be allocated to link b i alo equal to BC b (the minimum) in SCI-A. We now decribe a cheme, to be called SCI-P where P i hort for PBC, which i baically an APF-PBC extenion of SCI-I (and SCI-A). The propoed SCI-P cheme require exactly the ame information to be maintained a SCI-A. The only difference between SCI-P and SCI-A i that the former ue APF-PBC (with fi a (w) given in Eq. 1). A urpriing reult i that SCI-P can perform better than SCI-I. A cheme imilar to SR but i mainly for link-baed retoration wa decribed in [8], which could alo ue the propoed APF-PBC heuritic to find a better AP. A mentioned earlier, the main deficiency of thee cheme i the huge amount of information they require to maintain. C. With Partial Information There are two ub-categorie, one containing cheme baed on SPI [5], and the other containing cheme baed on DPI [1]. All thee cheme ue O(E) partial information but thoe baed on DPI perform much better than thoe baed on SPI. C.1 SPI Baed Scheme In SPI, alo to be named SPI-I for conitency, only the value of A e and B e (in addition to R e ) for every link e are maintained by a controller (at each edge node). It ILP formulation i the ame a that ued by SCI-I except that Eq. 2 i replaced by BC b =maxf max (A a + w B b ); 0g (3) 8a2AP Note that, thi can clearly be an over-etimation. Unfortunately, without any further information, the additional amount of BBW o overly etimated will alo be the actual amount of BBW to be allocated to link b. One can apply APF and APF-PBC to SPI and obtain cheme to be named a SPI-A and SPI-P, repectively. SPI-A and SPI-P will be the ame a their SCI counterpart except in term of how additional BBW needed i etimated and allocated. In addition, ince the information on P Ae and are not available in SPI-P, one ha to replace them with A e and max A e repectively, in Eq. 1 in 8e2E order to calculate the potential cot. Our reult, though not hown in thi paper, indicate that SPI-P can alo outperform SPI-I. In addition, all SPI baed cheme perform ignificantly wore than their DPI counterpart, which agree with the reult hown in [1]. Later, we will how the performance of SPI-I only for comparion. C.2 DPI-baed Scheme In one DPI cheme decribed in [1], to be named DPI-SA-I, every node (i.e., edge or core) maintain the following two vector and four calar for each local outgoing link e 2 OUT (n): P A (e), P B (e), and B e, R e, P Ae, and P B e. The lat two calar are for convenience only a they can be derived from the two vector. The total amount of information maintained at each node i practically limited to O(E) a the number of local link at each node i typically bounded by a mall contant. An edge (ingre) node will alo maintain the three calar, B e, R e, and P Ae for each remote (i.e., non-local) link e. The lat one i ueful for obtaining a good etimate of the additional BBW needed on the remote link e. Iti alo ued to determine. Hence, even if every node i an edge node, the amount of information to be maintained at each node i till limited to O(E). How the local information i updated and remote/non-local information i exchanged ha been dicued in [1]. DPI-SA-I ue an ILP formulation imilar to that decribed for SPI in [5] with everal improvement. For example, a more accurate backup cot etimation than that given by Eq. 3 can be obtained a follow: BC b =maxf max 8a2AP minf(p Aa + w B b );wg; 0g (4) Another major improvement made in DPI-SA-I i to allocate only minimal BBW on each link along a choen BP. ore pecifically, a ignaling packet to reerve BBW along the BP will contain AP, and baed on which, each node along the BP update their locally maintained information on the profile of BBW (i.e., P Be ). Thereafter, each node calculate bw = maxfp B e B e ; 0g, which i the minimal BBW to be allocated on link e (ee [1] for detailed explanation). Baed on DPI-SA-I, one can alo apply APF and APF-PBC to obtained two extended cheme, to be named DPI-SA-A and DPI-SA-P, repectively. Thee two cheme will work in the ame way a their SCI counterpart, except that each DPI-SA baed cheme may elect a different BP than that elected by it SCI counterpart (due to different etimation of the back up cot). Another DPI cheme decribed in [1], called DPI- -A, maintain only R e for any remote link e. A a reult, it cannot take advantage of APF-PBC (but can ue, and indeed ue APF a a pecial cae of APF-PBC). Thi i alo true in everal other ditributed cheme with partial information decribed in [9], [2], [10]. where only R e

6 IEEE INFOCO for any remote link i available at the edge. According to our reult obtained from thi tudy (to be hown next), and thoe from [1], both DPI-SA-P and it pecial intance, DPI-SA-A, outperform DPI--A. Since they only require the edge node to maintain two more calar (namely, B e and P Ae ) for each remote link e than DPI--A doe, the cot-effectivene of the propoed APF-PBC (and it pecial cae APF) under the DPI framework i quite obviou. IV. PERFORANCE EVALUATION We are primarily intereted in comparing the performance of the cheme baed on APF-PBC, namely SCI-P, and DPI-SA-P, with their counterpart baed on ILP, namely SCI-I and DPI-SA-I, a well a thoe baed on APF, namely, SCI-A (=SR) and DPI-SA-A. Proceing and ignaling overhead are ignored in thi quantitative comparion tudy. When imulating APF-PBC baed cheme, c in Eq. 1 i et to 0.5. In the ret of the ection, we decribe the network topology aumed, traffic type conidered, and performance metric ued before preenting the reult. A. Network Topology Fig. 1. A 15-node network To facilitate a fair comparion between variou approache, we conider the topology hown in Fig. 1, which i the ame a that ued in [5] and ha 15 node and 28 bidirected edge (for a total of 56 link). The capacity of each link i aumed to be either infinite or limited a to be dicued in the next ubection. Another network with 70 node and 264 link i alo conidered, and relatively conitent performance reult are obtained. B. Traffic Type We conider two type of traffic, one in which an etablihed connection lat forever (i.e., incremental traffic) a in [5], [6], and the other in which it may terminate after a certain duration (i.e., dynamic traffic). In both cae, the ingre and egre of a connection etablihment requet i evenly ditributed among all node, and requet arrive in an on-line fahion. For the cae with incremental traffic, the bandwidth required by the connection i uniformly ditributed between 1 and 10 unit a in [5]. Note that, any requet arrival proce may be aumed. For the cae with dynamic traffic, the bandwidth required varie from 1, 2, 3, 4, 6 and 12 unit with probability being 20%, 10%, 30%, 10%, 10%, 20%, repectively. In addition, requet are aumed to arrive according to a Poion proce, and the connection duration ha a Pareto ditribution. Thi i jut an attempt to model realitic traffic (which may be elf-imilar and whoe bandwidth requirement range from OC-1 to OC-12). Other poibilitie, including uniformly ditributed bandwidth requirement and exponentially ditributed connection duration, have alo been examined, and we have found that they have no ignificant impact on the performance of variou cheme tudied in thi paper. C. Performance etric The following two performance metric are ued, one for each traffic type conidered. C.1 Bandwidth Saving (Ratio) To obtain thi metric, it i aumed that the capacity of each link i infinite (and hence all requet will be atified), and the traffic i incremental. After an appreciable number of requet have been atified, TBW conumed (i.e., um of ABW and BBW on AP and BP, repectively) for each of the cheme i evaluated and conequently, bandwidth aving, in term of TBW conumption ratio over the NS cheme, i determined a imilarly done in [5]. Note that, for a given requet, the BBW needed will be no le than the ABW needed in NS. Hence, even if an ideal cheme that achieve maximum BBW haring i ued, the bandwidth aving ratio will be upper-bounded by 50% (achievable only if no BBW i needed at all). C.2 Total Earning (Ratio) The bandwidth aving meaure may not mean much ince in a practical cae, all link have a finite capacity and thu not all requet can be atified. Accordingly, in thi et of experiment (imulation), we aume that each link ha a finite capacity and dynamic traffic i conidered. For example, in the Fig. 1 above, each dark (bold) link (coniting of two unidirectional link) i aumed to have a capacity of 192 unit in each direction (to model an OC-192 link), and each of the other

7 IEEE INFOCO link ha a capacity of 48 unit in each direction (to model an OC-48 link). A a reult, ome requet will be rejected under a heavy traffic load. The total number of rejected connection etablihment requet (after an initial et of requet are atified) uing each cheme ha been ued a a performance meaure (e.g., in [5]). However, comparion between different cheme baed on uch a meaure (or equivalently blocking probability) may not be fair a it doe not differentiate one requet from another (ee related dicuion in [1]). Thi motivate u to ue the total earning (or revenue) a a metric. To thi end, a cheme-independent Earning Rate matrix for the entire network i ued. An entry at (i; j) repreent earning per bandwidth unit and time unit by a connection from ingre node i to egre node j. The earning from a connection from i to j i thu the product of the earning rate, requeted unit of bandwidth, and the connection duration. In thi tudy, for lack of a better alternative, the earning rate i baed on the cot of uing the cheapet (or hortet) pair of AP and BP in the network from i to j (auming there were infinite capacity in the network), and hence i independent of the current load in the network 1. An important and deirable conequence of uing the aumed earning rate (along with the earning from a connection) i that it tend to dicourage an algorithm that trie to maximize earning from chooing an unnecearily expenive (or long) path to etablih the connection. Becaue chooing an expenive/long path under uch a model may prevent other (future) connection from being etablihed and thu reulting in lot revenue. We compare the total earning of each cheme and in particular, the improvement ratio over the NS cheme. D. Simulation Reult Fig. 2 how the total bandwidth conumed after atifying 200 connection etablihment requet (or demand) in the 15-node network from 10 experiment (imulation run). It can be een from the figure that, given the ame O(E) partial information, DPI-SA-P conitently outperform DPI-SA-I (which in turn conitently outperform SPI-I). In addition, SCI-P alo outperform SCI-I lightly. The difference between SCI-P and DPI-SA-P may not be big enough to warrant the additional overhead involved in maintaining O(E 2 ) complete information a required by SCI-P. Table I how the average bandwidth aving ratio (v. NS) (over the 10 experiment) in the 15-node and 70-node 1 If the earning rate i load-dependent, it will become chemedependent alo Total Network Bandwidth Conumed (After 200 Demand) NS SPI I DPI SA I DPI SA P SCI I SCI P Experiment Number Fig. 2. Bandwidth conumed (from 10 experiment) network. For each network, the firt row i for cheme uing ILP, and the econd and third row are for their correponding cheme uing APF and APF-PBC, repectively. TABLE I AVERAGE BANDWIDTH SAVING RATIO 15-node network 37.2%(SCI-I) 28.0%(DPI-SA-I) 36.4%(SCI-A) 29.3%(DPI-SA-A) 38.7%(SCI-P) 32.1%(DPI-SA-P) 70-node network 35.5%(SCI-I) 26.4%(DPI-SA-I) 34.3%(SCI-A) 27.0%(DPI-SA-A) 36.4%(SCI-P) 29.7%(DPI-SA-P) An intereting obervation i that the propoed DPI- SA-A alo perform lightly better than DPI-SA-I, while SCI-A perform lightly wore than SCI-I. In order to evaluate the performance of variou cheme in network with limited link capacity and dynamic traffic, another 10 experiment have been conducted in which each network i loaded with heavy (dynamic) traffic (under a light load, the difference between variou cheme are inignificant). Fig. 3 how the total earning after proceing 500 demand (not all 500 requet are atified). An intereting obervation i that while the SCI and DPI baed cheme perform well, SPI-I performed poorly and in fact, wore than NS in ome experiment. Thi i becaue BC b given by Eq. 3 may be larger than w (which i needed by NS), that i, SPI-I may overly etimate the BBW needed, and allocate exceive BBW. Thi i epecially problematic in a network with limited capacity a

8 IEEE INFOCO Total Earning (After 500 Demand) 8 x SCI P SCI I DPI SA P DPI SA I NS SPI I chooing the trace collected from SCI-I i that we want to mimic thi almot ideal approach. In APF-PBC, a major challenge i that when trying to determine PBC for link a, the ingre node reponible for path determination doe not even know which link ha been or will be ued by BP to protect againt the failure of link a. If link b were to be ued by a new BP (whoe correponding AP i going to ue link a), and the amount of the BBW already allocated to link b (which i B b ) were known to be x, the additional BBW needed on link b (or backup cot) i BC b a =maxf + w x; 0g (5) Experiment Number Fig. 3. Total earning after 500 demand (from 10 experiment) many connection requet that could have been atified even in NS will now be rejected. Table II how the average total earning ratio (v. NS) for SCI and DPI baed cheme. Thee reult alo how that APF-PBC outperform their ILP counterpart. An intereting obervation i that even the APF cheme perform very well and in fact the difference between APF and APF-PBC are not much. Alo, thee reult ugget that the trade-off between additional overhead and performance gain (for dynamic traffic) may be worthwhile in a large network with limited capacity and heavy load. TABLE II AVERAGE TOTAL EARNING RATIO where = Sa b, which hould be no greater than x. The firt tep toward gueing the potential backup cot i to aume that i known for the time being (thi aumption will be relaxed later), and proceed to calculate an expected backup cot uing a weighted average over all poible value of x (i.e., B b of all b 2E). ore pecifically, we will treat x, for an arbitrary link b, (where = max B b ), a a random variable U between 0 and 1 8b whoe cumulative ditribution function (CDF) F (U ), obtained experimentally, i illutrated in Fig. 4. A can be een, F (U = B b ) can be approximated by a normal ditri- bution function N (μ; ffi), where μ i the mean value, and ffi i it tandard deviation. Let it correponding probability denity function be f (U ) node network 28.7%(SCI-I) 19.3%(DPI-SA-I) 27.6%(SCI-A) 21.1%(DPI-SA-A) 29.6%(SCI-P) 23.1%(DPI-SA-P) 70-node network 30.1%(SCI-I) 12.3%(DPI-SA-I) 31.4%(SCI-A) 17.6%(DPI-SA-A) 31.2%(SCI-P) 17.1%(DPI-SA-P) F(B b /) F(B b /) N(µ=0.335,δ=0.275) V. POTENTIAL BACKUP COST -DERIVATION In thi ection, we will decribe how the PBC function fi a (w) i derived baed on the tatitical analyi of experimental data. All experimental data i baed on the trace collected from the imulation run in which SCI-I i evaluated for the 15-node network hown in Fig. 1 auming infinite link capacity (ee Sec. IV for additional aumption made in thoe experiment). The baic idea behind B b / Fig. 4. Ditribution of B b Baed on Eq. 5 and the definition of a denity function, the expected backup cot of uing any link b by the BP to back up an AP uing link a, provided that the value of S b a i equal to, become

9 IEEE INFOCO (w; ; ) R min( +w = ;1) f( 0 ( + w x x) ) P ( y ) d( x )(6) where min( +w ; 1) i ued a the upper bound for x due to the fact that BCa b =0when x + w. In addition, P ( y ), which i equal to F (1) F ( ), denote the probability that a variable y (repreenting the BBW on a link) i between and, and thu need be included ince only thoe link whoe y are valid. Note that (w; ; ) i in fact independent of any particular link. The olid line in Fig. 5 how a typical graph for (w; ; ), where the horizontal and vertical axi are normalized to and w, repectively. The value of (w; ; ) for a given w, and i obtained by uing the adaptive Lobatto quadrature [11] method to evaluate the integral function in Eq. 6, and thoe value form the curve hown in Fig. 5. In Fig. 5, i choen to be 125 unit, which i a typical value after a large number (e.g. 500) of connection have been etablihed in each of the everal experiment conducted. ζ(w,, )/w w=80 w=40 w=20 w= / Fig. 5. Graph of (w; ; ) The econd tep toward gueing the potential backup cot i to relax the aumption that i known by calculating a weighted average value of (w; ; ) over all poible value of on link a. ore pecifically, let u treat PAa a a random variable V between 0 and 1 with a CDF G(V ). Fig. 6 how G(V = ), obtained experimentally, PAa for five randomly elected link in the 15-node network (pecifically, link 0! 1; 2! 6; 8! 3; 9! 5; 9! 12). From Fig. 6, it eem that the CDF G(V ) may be approximated by an exponential function (which matche our intuition/expectation), but uch an approximation i not w=5 w=2 G(/P Aa ) /P Aa Fig. 6. Cumulative ditribution function of P Aa neceary to derive the potential cot function and hence will not be made. Let the correponding denity function be g(v ) (whoe actual ditribution or it approximation i not important). Baed on the above dicuion, the potential backup cot i: Z 1 fi a (w) = (w; ; )g( )d( ) (7) 0 P Aa P Aa VI. POTENTIAL BACKUP COST -APPROXIATION In thi ection, we will implify Eq. 7 with everal reaonable approximation. Firt, from the dahed line in Fig. 5, it eem that each curve (or rather the point on each curve), which correpond to a given value of w can be approximated (or fit) by a line of the form Y = c 1 X +c 2, where Y = (w;;), w X = (which happen to be the ame a x defined earlier), and the value of c 1 and c 2 depend on the given w and (a to be dicued later). Although uch an linefitting approximation i good only up to the point where X reache about 0.75, it will not have much affect on the value of fi a (w) given by Eq. 7 above. Thi i becaue a can be een from Fig. 6, the probability that i PAa larger than 0.75 i already very mall. oreover, ince P Aa», the probability that i larger than 0.75 i even maller, and in fact, almot negligible. Furthermore, a can be een from Fig. 5, we may fit the group of curve with line a follow. Firt, the lope of thee fitting line, to be denoted by Φ(w; ) (intead of c 1 to more clearly indicate their dependency on w and ), may be obtained a dy at a reaonable value of X (or dx ). Fig. 7 how the value of Φ(w; ) (obtained when

10 IEEE INFOCO Φ(w,) =0:75) a a function of w normalized with Φ(w,) η(1 e w/(γ) ), η=1.2 γ = w/ Fig. 7. Graph of Φ(w; ) It i clear from Fig. 7 that Φ(w; ) can be imply approximated by an exponential function (1 e fl w ), where =1:2 and fl =0:2. In addition, a can be een from Fig. 5, c 2 ß 0:2 if w =40, and c 2 ß 0:1 if w =20and o on. In other word, c 2 ß 0:05w. But ince in all thee cae, = 125, we may et c 2 = w, where ß 0:6. In hort, we have the following approximation for (w; ; ): VII. POTENTIAL BACK COST -SIPLIFICATION While one may ue the potential cot function given in Eq. 9, it can be further implified without affecting it uefulne or the performance of a cheme that adopt it. Firt, we note that if Eq. 9 i ued, each edge node need to maintain P Aa which i not needed in the DPI-SA-I nor in DPI-SA-A. Every time an edge node atifie a connection etablihment (or releae) requet whoe AP ue link a, it can imply increae (or decreae) P Aa by wq E, where q i the number of link along the choen BP, and multicat the updated value to all other edge node. To avoid the need to maintain P Aa at the edge node, one may replace P Aa with μ a P Aa, where μ a = P Aa PAa i almot a contant a can be een from it CDF hown in Figure 8, which i obtained experimentally. From the figure, it i clear that the CDF can be fit by a curve with a normal ditribution (whoe mean i around 0.18 and whoe tandard deviation i only 0.043) µ a N(µ=0.186,δ=0.043) (w; ; ) ß Φ(w; ) w + w2 (8) (where the right hand and left hand ide of Eq. 8, normalized with w, are plotted uing the curve and line, repectively, in Fig. 5). We now plug in thi approximation for (w; ; ) into Eq. 7. Since we have: Z 1 g( )d( Z 1 )= g( )d( )=P Aa 0 P Aa P Aa 0 P Aa P Aa PAa» 1 and the expected value of on (thi i becaue link a, i, by definition, P Aa ), and in addition, Z 1 g( )d( )=1 0 P Aa P Aa we have the following approximation for fi a (w): fi a (w) ß Φ(w; ) w P Aa + w2 (9) µ a Fig. 8. Ditribution of μ a To further implify the PBC function, one may omit the econd term in Eq. 9, that i, w2, a thi term i not only mall, but mot importantly, independent of all link a well (a to be hown, thi omiion will not affect the uefulne of the PBC). Finally, we note that for any requet with w > 0, the value of Φ(w; ) i larger than 0 but no greater than 1.2 (ee Fig. 7). In particular, it i independent of the link for which a PBC i being etimated. Although it cannot be eliminated a it i a part of the firt term in Eq. 9 which i link dependent, the reult hown in Fig 9, for example, indicate that any reaonable value of μ a Φ(w; ) between 0 and 1 may be ued to etimate the PBC without any ignificant impact on the performance of the APF-PBC heuritic.

11 IEEE INFOCO In other word, we may replace μ a Φ(w; ) with a contant c to arrive at Eq. 1, which i rewritten here a fi a (w) =c w P Aa (10) A can be een from Fig 9 which how the bandwidth aving ratio of DPI-SA-P over NS in the 15-node network, the value of c doe not matter much a long a c» 1. The figure alo how that with or without the econd term in Eq. 9, (i.e., with =0:6 or =0), the performance of DPI-SA-P i pretty much the ame. Average Bandwidth Saving Ratio V. NS θ = 0 θ = c Fig. 9. The effect of contant c and on the performance of APF-PBC Note that uch a PBC function, though derived mathematically, i quite intuitive a the larger the w, the higher the PBC. In addition, for a given w, the larger the P Aa, the more likely that a larger amount of additional BBW need be allocated on BP (and hence a larger PBC). VIII. CONCLUSION In thi paper, we have propoed a novel heuritic called APF-PBC to determine a pair of active and backup path for each on-line requet to etablih a connection uing hared path protection. It baic idea i to attach a potential backup cot (PBC) to each link o that one may elect an active path firt (APF), while till taking into conideration the impact of bandwidth haring along a yet-to-be-choen backup path. We have mathematically derived an intuitive potential backup cot (PBC) function baed on the tatitical analyi of experimental data, which can be ued to minimize the total bandwidth or maximize the total earning. Although the APF-PBC heuritic i developed primarily for the ditributed partial information management (DPI) framework, it can alo be applied to other cheme with ditributed or centralized routing. In fact, it baic idea may alo be extended to other joint optimization problem. One of the intereting reult obtained from thi tudy i that the propoed APF-PBC, of which APF i a pecial cae, can not only run much fater than Integer Linear Programming (ILP) baed approache, but alo outperform them. Our performance evaluation reult have revealed that thi i true in all the on-line cae we have conidered in thi paper, a long a APF-PBC and ILP have the ame information (either complete or partial), traffic load (either incremental or dynamic) and contraint (uch a no rearrangement of exiting connection). We have explained the reaon for thi pleaantly urpriing reult, which hould encourage people to look for more practical and efficient heuritic when tackling imilar on-line optimization problem intead of imply reorting to ILP. REFERENCES [1] Chunming Qiao and Dahai Xu, Ditributed partial information management (dpim) cheme for urvivable network - part i, in ubmitted to INFOCO 2002, [2] B. Dohi et. al, optical network deign and retoration, Bell Lab Technical Journal, pp , Jan-ar [3] Yijun Xiong and Lorne G. aon, Retoration trategie and pare capacity requirement in elf-healing atm network, in IEEE/AC Tran. on Networking, vol. 7, no. 1, 1999, pp [4] Ramu Ramamurthy et. al, Capacity performance of dynamic proviioning in optical network, Journal of Lightwave Technology, vol. 19, no. 1, pp , [5] urali Kodialam and T V. Lakhman, Dynamic routing of bandwidth guaranteed tunnel with retoration, in Proceeding - IN- FOCO, 2000, pp [6] Yu Liu, D. Tipper, and P. Siripongwutikorn, Approximating optimal pare capacity allocation by ucceive urvivable routing, in Proceeding - INFOCO, 2001, pp [7] J.W. Suurballe and R.E. Tarjan, A quick method for finding hortet pair of dijoint path, Network, vol. 14, pp , [8] Ching-Fong Su and Xun Su, An online ditributed protection algorithm in wdm network, in Proceeding - ICC, [9] C. Dovroli and P. Ramanathan, Reource aggregation for fault tolerance in integrated ervice network, in AC Computer Communication Review, vol. 28, no. 2, 1998, pp [10] Ramu Ramamurthy, Sudipta Sengupta, and Sid Chaudhuri, Comparion of centralized and ditributed proviioning of lightpath in optical network, in Optical Fiber Communication Conference - OFC 2001, 2001, pp. H4 1. [11] W. Gander and W. Gautchi, Adaptive quadrature - reviited, in BIT, Vol. 40, 2000, Thi document i alo available at pp