Optimal Path Selection for Minimizing the Differential Delay in Ethernet Over SONET

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1 Optimal Path Selection for Minimizing the Differential Delay in Ethernet Over SONET Satyajeet huja and Marwan Krunz Dept. of ECE University of rizona Turgay Korkmaz Dept. of Computer Science University of Texas at San ntonio bstract We consider the problem of minimizing the differential delay in a virtually concatenated Ethernet over SONET (EoS) system by suitable path selection. In such a system, a service provider can dynamically add virtual channels to or drop them from a Virtually Concatenated Group (VCG). new virtual channel can be added to the VCG provided that the differential delays between the new channel and the existing ones are within a certain limit that reflects the available memory buffer of the EoS system. We model the problem of finding such a virtual channel as a constrained path selection problem, where the delay of the required (feasible) path is constrained not only by an upper bound but also by a lower bound. We consider two cases: exactly known link delays and imprecisely known link delays. For the first case, we propose two algorithms for finding a feasible path. The first is based on a link metric that linearly combines the original link weight (the link delay) and its inverse. The theoretical properties of such a metric are studied and used to develop a highly efficient heuristic for path selection. The second algorithm is a backward-forward heuristic in which the nodes in the graph are prelabeled during the backward phase. The labels are then used in the forward phase to identify a feasible path. For the imprecise-link-state case, the problem is modeled as one of finding the most probable feasible path, where link weights are random variables. backward-forward heuristic is proposed which again uses prelabeling of the graph in the backward direction followed by a forward search that attempts to minimize an objective function. Simulations are conducted to evaluate the performance of the proposed algorithms and to demonstrate the advantages of the probabilistic path selection approach over the classic trigger-based approach. Index Terms Ethernet over SONET/SDH, virtual concatenation, differential delay, path selection. I. INTRODUCTION Much of the global network infrastructure in place today is based on the SONET/SDH technology [],[2]. This technology uses a bandwidth hierarchy indicated by STS-n, where n =, 3, 2, 48,.... The basic unit in this hierarchy is the STS- channel, which corresponds to 5.84 Mbps of bandwidth. SONET was originally developed to support voice traffic. s the demand for data services (IP and Ethernet traffic) continues This work was supported by NSF under grants NI , NI , and NI , and by the Center for Low Power Electronics (CLPE) at the University of rizona. CLPE is supported by NSF (grant # EEC ), the State of rizona, and a consortium of industrial partners. ny opinions, findings, conclusions, and recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. n abridged version of this paper was presented at the International Conference on Computer Communications and Networks (ICCCN), Chicago, US, Oct. -3, to rise, there is a need to provide cost-effective solutions for supporting such services over the SONET/SDH infrastructure. Figure depicts the various possible approaches for transporting IP data over SONET. The first approach is to use IPover-TM-over-SONET using L5 (TM daptation Layer 5) [3]. One problem with this approach is its excessive protocol overhead, which can be as high as 25% of the used bandwidth. Packet-over-SONET (POS) addresses this problem by eliminating TM encapsulation and instead using the Pointto-Point Protocol (PPP) [], [4]. Under POS, PPP-encapsulated IP packets are framed using the High-Level Data Link Control (HDLC) protocol and are mapped into SONET. The basic function of HDLC is to provide framing, i.e., delineation of the PPP-encapsulated IP packets across the synchronous transport link. nother method for transporting IP data over SONET is to use the Generic Framing Procedure (GFP), which encapsulates Ethernet frames and then maps them into SONET frames [5]. GFP supports all the basic functions of a framing procedure, including frame delineation (needed to facilitate the transport of bursty traffic over TDM-like systems). It is currently considered the most popular framing procedure for supporting Ethernet-over-SONET (EoS). Fig.. GFP IP PPP SONET / SDH L5 Ethernet HDLC TM Methods for transporting IP traffic over SONET. EoS has been gaining popularity in point-to-point and multipoint LN interconnections [6]. It enables network providers to use their existing SONET infrastructure to provide new data services over regional and national geographic areas. Such services include Ethernet-private-leased-line services that provide dedicated point-to-point bandwidth and Ethernet-

2 virtual-private-line services that use statistical multiplexing to share bandwidth among various streams. EoS with virtual concatenation [] utilizes the existing SONET infrastructure, with only edge nodes (source and destination) being required to support such concatenation. By using the LCS (Link Capacity djustment Scheme) [7], virtual concatenation facilitates dynamic link upgrade and on-demand bandwidth adjustment without requiring additional hardware. In virtual concatenation, several STS-n channels, belonging to possibly different Optical Carriers (OCs), can be concatenated between the source and destination to support Ethernet connectivity. These channels, which we will simply refer to as virtual channels (VCs), form a Virtually Concatenated Group (VCG). The absolute difference between the end-to-end delay of the new VC and the delay of each existing VC in the VCG must not exceed some upper bound that is determined by the amount of high-speed memory available at the destination node. In this case, the new VC is said to be feasible. In this paper, we consider a point-to-point EoS LN application and focus on the problem of selecting a feasible VC to be added to an existing VCG. First, we consider the path selection problem under the assumption of exact linkstate (delay) information. The problem in this case reduces to that of finding a path whose delay lies between two strictly positive bounds, C and C 2. We show that this problem is NPcomplete, and accordingly propose two approximate solutions to it. We then consider the case when the link delays are not exactly known to the source node. In practice, several factors contribute to such imprecision [8], [9], [0], including information staleness due to the periodic nature of link-state protocols (e.g., OSPF) and sampling errors. lso, there may be cases where some of the link delays are not available because the participating nodes are managed by different Network Management Systems (NMS) that do not share link information between each other for policy reasons (in these cases, the physical distance between any two nodes can be used as a course estimate of the link delay). s demonstrated in [0], relying on outdated information and considering it as exact can significantly degrade the performance of a path selection algorithm. We cope with link imprecision by modeling link delays probabilistically, and develop a highly efficient heuristic for finding the most probable feasible VC subject to the constraints C and C 2. The rest of the paper is organized as follows. In Section II, we describe virtual concatenation in detail. We algorithmically formulate the problem of adding a VC to a VCG in Section III, and present our path selection algorithms for the case of exact link delays. In Section IV, we formulate the problem of finding the most probable feasible path under inaccurate linkstate information and provide a heuristic for finding such a path. Section V presents simulation results. Finally, the paper is concluded in Section VI. II. VIRTUL CONCTENTION IN EOS SYSTEMS To motivate the significance of virtual concatenation in EoS systems, let us consider a typical scenario of interconnecting two Gigabit Ethernet LNs over a SONET network. The smallest SONET payload slot that can carry such traffic is STS-48 (2.5 Gbps); this results in bandwidth wastage of about 60%. solution to avoid this wastage is to use concatenated payloads. Two methods for concatenation are available []: Contiguous and Virtual. Both methods provide aggregate bandwidth in multiples of VCs at the termination. The difference between the two is in the way in which the total bandwidth of the concatenated VCs is accumulated. Contiguous concatenation maintains bandwidth along the same transport path. Constituent VCs of the concatenated payload cannot be individually and independently routed. ny bandwidth upgrade, i.e., the addition of an VC, has to be done along the same path [2]. If no VC is available over any of the links along this path, then the upgrade cannot be done. In contrast, virtual concatenation splits the aggregate bandwidth into several VCs that are independently established between the two endpoints, as shown in Figure 2. The routes of 0/00BT Ethernet Corporate LN xsts n TIME SEQ No PYLOD STMP =T = 0 DT SONET End Node TIME STMP =T SEQ No PYLOD = DT new STS n Fig. 2. PTH (D ) SONET Network PTH2 (D 2 ) xsts n TIME SEQ No PYLOD STMP =T2 = 0 DT EoS setup with virtual concatenation. SONET End Node 0/00BT Ethernet TIME SEQ No PYLOD STMP = DT =T3 Corporate LN new STS n these VCs may or may not overlap. Whereas contiguous concatenation requires concatenation functionality at each network element, virtual concatenation requires such functionality only at the path termination. Virtual concatenation in EoS systems is performed by the virtual concatenation protocol. This protocol allows the bandwidth demand to be split into k portions. The source node sends traffic down k members of the VCG and the sink node reconstructs the data stream back. For example, a - Gbps demand can be mapped into seven STS-3 channels (or 2 STS- channels), incurring only 8% wastage. VCG members are allowed to be on noncontiguous time slots and they can be independently routed. Data are byte-interleaved over the multiple VCs of the VCG, i.e., if the first byte is transmitted on the rth VC, then the next byte is transmitted on the (r+)th VC, and so on. Control packets that contain necessary information for reassembling the original data stream are inserted in some of the currently unused SONET header fields. This information includes the sequence order of channels and a frame number, which is used as a time-stamp []. The receiving endpoint is responsible for re-assembling the original byte stream and compensating for the differential delay between different paths taken by the VCs. The compensation for such a delay is implemented using a very highspeed memory, which stores the payload bytes of VCs from successive frames. When the frames of all the VCs with the same frame number (i.e., time-stamp) arrive at the receiving node, the Ethernet packets can be extracted from them. It is obvious that the higher the differential delay, the larger the high-speed memory requirement.

3 III. PTH COMPUTTION UNDER EXCT LINK DELY. Problem Formulation INFORMTION Consider a SONET network with each multiplexer having a fully nonblocking cross-connect. Suppose that a new VC is to be added to an existing VCG that already contains m VCs with respective path delays D, D 2,..., D m. The delay associated with the newly added VC (D new ) should satisfy: max D i D new. () i m Figure 3 shows two examples for m = and m = 2. In (a) m = 0 (b) m = 2 C C2 D 0 C C 2 Fig. 3. Differential delay problem for a VCG that already contains: (a) one path, and (b) two paths. general, D new should satisfy C D new C 2 for some positive constants C and C 2 that depend on D, D 2,...,D m, and. To determine an appropriate path for the new VC, we start by removing from the network graph all the links whose available bandwidth is less than the bandwidth of the prospective VC. In the pruned graph, each link (i, j) is associated with a delay parameter w(i, j). Now we need to find a feasible path; one that satisfies the delay bound in (). In this section, we assume that the w(i, j) s are exactly known. The underlying problem is modeled as follows. Problem : Two-Sided Constrained Path (TSCP) Problem: Consider a network G(V, E), where V is the set of nodes and E is the set of links. Each link (u, v) E is associated with an additive parameter w(u, v) 0. Let C and C 2 be any two positive numbers, with C < C 2. The problem is to find a path P from s to t such that C W (P (s, t)) = D D2 (u,v) P (s,t) w(u, v) C 2. (2) This problem is actually a mixture of both the shortest path problem and the longest path problem. If only the upper bound is considered, then the problem is solved by using Dijkstra s shortest path algorithm. If only the lower bound is considered, then the problem is equivalent to finding the longest path in the network, which is known to be NP-complete. We now prove that the TSCP problem is also NP-complete. Theorem : TSCP problem is NP-complete. Proof: Consider the corresponding decision problem for TSCP, where the goal is to find if there exists a path P from s to t which satisfies (2). First, we show that TSCP belongs to the class of NP problems. The certificate for the verification algorithm is chosen as the path P (s, t) itself. The verification algorithm affirms that C W (P (s, t)) C 2 and that P (s, t) is a path in the network. This verification can be performed in polynomial time. We prove that TSCP is NP-hard by showing that the NPcomplete hamiltonian path problem (HM-PTH) [] can be reduced in polynomial time to the TSCP problem. The reduction approach takes as input an instance {G(V, E), s, t} of the HM-PTH problem. For a graph G(V, E), let C = n, C 2 = n, and w(i, j) = (i, j) E, where n = V. The output of the reduction algorithm {G(V, E), s, t, C = n, C 2 = n, w(i, j) = } is an instance of the TSCP problem. We now show that the output of this instance is yes (positive), if and only if the output of HMPTH on {G(V, E), s, t} is also yes (positive). The output of the TSCP algorithm will be a path P with n W (P ) n. Since we have fixed w(i, j) = (i, j) E, then P will traverse every node exactly once and hence is a hamiltonian path. On the other hand, suppose that G has a HM-PTH Q from s to t. Then, Q will have exactly n hops. ccordingly, W (Q) = (i,j) Q w(i, j) = n, which also satisfies n W (Q) n and is thus a solution to the TSCP problem. The reduction only requires fixing the values of w(i, j) (i, j) E, C, and C 2, and thus can be performed in polynomial time. B. lgorithms B. K-Shortest-Path (KSP) lgorithm: Researchers have extensively investigated the KSP problem and proposed various algorithmic solutions of varying computational complexities (e.g., [2], [3]). KSP algorithm tries to find the K lowest-cost paths in the network. It starts from the source node and explores the underlying graph by maintaining (at least) the best K (sub)paths at each node. For the TSCP problem, we can iteratively use a KSP algorithm with a sufficiently large K until a path whose cost is between C and C 2 is found. The algorithm terminates if it finds a feasible path or if all K returned paths are infeasible. large K renders the algorithm impractical, given the O(Kmlog(Kn)+ K 2 m) complexity of the KSP algorithm [4], where n is the number of nodes and m is the number of links. To avoid this highly complex solution, we propose two algorithms, one based on the KSP algorithm but with a modified link weight and the other based on the backward-forward heuristic. B.2 Modified-Link-Weight K-Shortest-Path (MLW-KSP) lgorithm: Our first algorithmic solution considers a Lagrangian (linear) composite of the link delay and its inverse as a basis for the optimization. This is similar to the Lagrangian solutions often used for multi-constrained path selection problems [5]. Formally, consider the following cost function for any path p from s to t: r(p) = αw (p) + W (p) where α > 0. Suppose there is an algorithm X that minimizes the cost function (3) for a given α > 0. The following theorem shows that if the graph contains a feasible path, then algorithm X will find it using α = C C 2. (3)

4 Theorem 2: Consider the TSCP problem. ssume that there is at least one feasible path f in the network, i.e., C W (f) C 2. Let p be a path that minimizes the cost function r for α = C C 2. Then, C W (p ) C 2. Proof: ssume that p is not feasible. We have two cases to consider: (i) W (p ) < C, and (ii) W (p ) > C 2. Consider the first case. Let W (p ) = C ɛ, where 0 < ɛ < C. Since algorithm X returns the path p, it must be true that r(p ) r(f). That is, W (P) 2 Feasibility region W (p ) + C C 2 W (p ) W (f) + C C 2 W (f). (4) Because C W (f) C 2, the right-hand side of (4) is bounded above by C + C 2. Then, the following must be true (C ɛ) + C C 2 (C ɛ) +. C C 2 Further manipulation of the above equation leads to the following: (C ɛ) 2 + C C 2 C C 2 (C ɛ) C2 2ɛC + ɛ 2 + C C 2 C C 2 (C ɛ) C2 2ɛC + ɛ 2 + C C 2 (C ɛ) C + C 2 C + C 2 C C 2 C + C 2 which can be reduced to C ɛ C 2. But this cannot be true because we know that C C 2 and 0 < ɛ < C. For the second case, let W (p ) = C 2 + ɛ, where ɛ > 0. Using a similar argument to the one above, we end up with C 2 + ɛ C. This cannot be true either, because we know that C C 2. So W (p ) should be less than or equal to C 2. Unfortunately, there is no exact polynomial-time algorithm that can minimize the cost function (3). Hence, one has to rely on approximations. We now introduce one such approximate solution to the minimization of the cost function r. Let w (i, j) = w(i, j) and w 2 (i, j) =. For a path P from s to t, let W k (P ) = (i,j) P w(i,j) w k (i, j), k =, 2. graphical representation of the paths from s to t with respect to the cost metrics W (.) and W 2 (.) is shown in Figure 4. The shaded region in the figure contains all feasible paths that satisfy (2). The plot of W (P ) corresponds to the nonlinear component in the cost function (3). Notice that all paths from s to t lie in the region above this curve because W (P ) W 2(P ), which follows from the inequality w(i,j) for w(i, j) > 0. Hence, by w(i,j) (i,j) P (i,j) P using an appropriate linear combination of the link weights w and w 2, a highly efficient heuristic can be developed for minimizing r. Define the following link cost function: l(i, j) = αw (i, j) + w 2 (i, j) (5) Fig. 4. /C /C 2 C C 2 /W (P) W (P) Geometric representation of paths in the (W,W 2 ) parameter space. where α > 0. Minimization of this function is illustrated in Figure 4, where each dashed line represents a contour of equalcost paths. The search for a path is done by sliding the line from the origin outwards towards the top-right corner of the figure. The key issue here is how to set the value of α, which determines the slope of the search line. Clearly, the best value should allow the sliding line to touch the curve that lowerbounds the feasibility region. Because the curve meets this region at the points (C, C ) and (C 2, C 2 ), with corresponding slopes of C 2 the range [ C 2 2 and C 2 2, C 2, respectively, the best value of α falls in ]. From Theorem 2, a good choice of α is C C 2. Using Dijkstra s algorithm with respect to (w.r.t.) the cost function (5), we can compute a path P with path metric: L(P ) = α w (u, v) + w 2 (u, v) (u,v) P (u,v) P = αw (P ) + W 2 (P ). If P is feasible, i.e., C W (P ) C 2, the algorithm terminates. Otherwise, we use the KSP algorithm with the link metric in (5) to continue the search. The algorithm terminates when a feasible path is found or K paths have been returned. B.3 Backward-Forward (BF) lgorithm: lthough the MLW-KSP algorithm provides a significant improvement over the standard KSP (as shown in the simulations), it sometimes requires a large number of iterations before returning a feasible path. We now present another algorithm, called the backwardforward algorithm, for the TSCP problem. The basic idea of the BF algorithm is derived from the H MCOP algorithm [6] for the multiconstraint path selection problem. feasible s t path is heuristically determined at any node u based on the already traversed segment s u and the estimated remaining segment u t. The estimated remaining segment u t is taken as the shortest path from u to t w.r.t. w(.,.). Since the algorithm considers complete paths, it can foresee several paths before reaching the destination. If one of the foreseen paths is feasible, the algorithm terminates. The BF algorithm operates in two phases: backward phase (from

5 t to all other nodes), which is used to estimate the cost of the remaining segment, and a forward phase, which is used to find the most promising path in terms of the nonlinear cost function (3). Figure 5 shows the basic operation of the algorithm. In the backward phase, every node is pre-labelled with the shortestpath distance from itself to node t. The forward step starts from the source s and tries to determine a path that is most likely to be feasible. s S[u] u lready explored nodes Fig. 5. w(u,v) D[v] v Unexplored nodes Exploring the graph in the BF algorithm. pseudocode for the algorithm is shown in Figure 6. For each node u, the algorithm maintains the following labels: the cost of the shortest path from u to t (D[u]), the predecessor of u on the shortest path from u to t (π r [u]), cost of the already travelled segment from s to u (S[u]), and the predecessor of u on this already travelled segment (π s [u]). BF algorithm(g(v, E), s, t, w(.), C, C 2 ). D=Reverse Dijkstra(G, t, w(e)) 2. if D[s] > C 2 3. return: No feasible path 4. else 5. T = Forward Heuristic(G, s, t, w(e), C, C 2, D) 6. if T > C and T < C 2 7. return: T is feasible path 8. else 9. return: T is infeasible path Fig. 6. Pseudocode for the BF algorithm. In the backward direction, the algorithm uses the Reverse Dijkstra s algorithm (RD) [7] to find the shortest path to t w.r.t. w(.,.) from every node u. Initially, we set D[u] = and π r [u] = NIL u V. The algorithm then starts from t by setting D[t] = 0. It explores the graph and eventually finds the shortest path from u to t u V. The algorithm returns an array D that contains the weights of the shortest paths from u to t u V. Before proceeding further, the algorithm checks whether D[s] > C 2. If so, then there is no feasible path and the algorithm terminates. Otherwise, the algorithm proceeds to the forward phase. pseudocode for the forward phase is shown in Figure 7. The forward phase is essentially a modified version of Dijkstra s algorithm. This search uses the information provided by RD to find the next node for the already travelled path segment. Consider a search that has already traversed from t s to u, as shown in Figure 5. The next node v is determined from the unexplored graph by finding the node that minimizes the cost function (3) for the path s u v t. The weight of the path from v to t is taken as the cost of the shortest path w.r.t. w(.,.) provided by RD. Forward Heuristic(G(V, E), s, t, w(.), C, C 2, D). for all i V, i s, do S[i] = π s [i] = NIL end for 2. S[s] = 0, α = C C 2 3. Insert Heap(s, α(s[s] + D[s]) + /(S[s] + D[s]), Q) 4. while Q is not empty, 5. u = ExtractMin(Q) 6. if (u == t) return S[t] 7. for each edge (u, v) outgoing from u, do 8. if (C < S[u] + w(u, v) + D[v] < C 2 ) return S[u] + w(u, v) + D[v] 9. if {α(s[v] + D[v]) + /(S[v] + D[v])} > {α(s[u] + w(u, v) + D[v]) + /(S[u] + w(u, v) + D[v])} 0. S[v] = S[u] + w(u, v). π s [v] = u 2. Insert Heap(v, α(s[v] + D[v]) + /(S[v] + D[v]), Q) end for Function u = ExtractMin(Q) removes and returns the vertex u in the heap Q with the least key value. Function Insert Heap(v, x, Q) inserts the node v in the heap Q with a key value x. If the node is already present in the heap then this function decreases its key to x. Fig. 7. Pseudocode for the forward phase of the BF algorithm. To improve the performance, the forward phase can be used with a KSP implementation of Dijkstra s algorithm presented. IV. PTH COMPUTTION UNDER INCCURTE DELY. Problem Formulation INFORMTION To model the uncertainties in the delay parameter, we follow a probabilistic approach, where we assume that for each link (u, v) the delay parameter w(u, v) is a nonnegative random variable with mean µ(u, v) and variance σ 2 (u, v). We assume that link delays are mutually independent. Each node u can compute the parameters µ(u, v) and σ 2 (u, v) associated with its outgoing links and disseminate them throughout the network. For more discussion on the feasibility and effectiveness of capturing and disseminating such delay information, we refer the reader to [0]. We use the same probabilistic model that was used in [0] to deal with the most-probable delay-constrained path (MP-DCP). In our work, however, the

6 problem at hand involves both upper and lower bounds. The problem can be formally stated as follows. Problem 2: Most-Probable Two-Sided Constrained Path (MP-TSCP) Problem: Consider a network G(V, E). Each link (u, v) E is associated with an additive parameter w(u, v). ssume that w(u, v) is a nonnegative random variable with mean µ(u, v) and variance σ 2 (u, v). Let be the set of all paths from s to t. For a path p, let W (p) = (u,v) p w(u, v). Given two constraints C and C 2, the problem is to find a path that is most likely to satisfy both constraints. Specifically, the problem is to find a path r such that for any other path p from s to t, π(r ) π(p) (6) where π(p) = Pr[C W (p) C 2 ] for any path p. Before presenting our solution to the MP-TSCP problem, we first briefly discuss how to compute the probability measure π. Without loss of generality, we assume that the probability density function (pdf) of w(u, v) is continuous and differentiable over some domain (a, b). The smoothness assumption, which is satisfied by many distributions, enables us to apply the central limit theorem (CLT) approximation (see [8, ]), which results in a path delay that is approximately normally distributed. Note that functions with cutoffs, such as the uniform distribution, are also smooth functions. The CLT approximation also holds for heavy-tailed distributions (e.g., lognormal) as long as they have a finite variance. For a given path p from s to t, let µ(p) = (u,v) p µ(u, v) and σ 2 (p) = (u,v) p σ2 (u, v). Then π(p) is approximately given by π(p) F (C 2, µ(p), σ 2 (p)) F (C, µ(p), σ 2 (p)) = Φ(C 2, C, µ(p), σ 2 (p)) where F (x, µ(p), σ 2 (p)) is the cumulative distribution function (CDF) of a Gaussian random variable with mean µ(p) and variance σ 2 (p) evaluated at x. For a path p, µ(p) and σ 2 (p) are exactly known at the source node. The Gaussian CDF can be evaluated with sufficient accuracy using tables [9]. Hence, π(p) can be evaluated by O() lookups from the table of a Gaussian CDF. mong all paths from s to t, we need to find a path that maximizes π(p) or, equivalently, minimizes π(p). B. Backward Forward lgorithm Under Inaccurate Link-State Information (BF Inaccurate) We now present a heuristic solution for MP-TSCP, which uses the same philosophy of the BF algorithm discussed in Section III. Each node u maintains the following labels: D[u] = {D [u], D 2 [u]}, where D [u] is the mean and D 2 [u] is the variance of the shortest path from u to t w.r.t. µ; E[u] = {E [u], E 2 [u]}, where E [u] is the mean and E 2 [u] is the variance of the shortest path from u to t w.r.t. σ 2 ; π s [u], the predecessor node of u on the already travelled segment s u; µ(s u) and σ 2 (s u); mean and variance of the already travelled segment s u. The feasible path s t is heuristically determined at any node u based on the already travelled segment s u and the estimated remaining segment u t. This remaining segment is estimated either from the shortest path w.r.t. variance or the shortest path w.r.t. mean from u to t, based on which of these has the smaller π(p) value. Since the algorithm considers complete paths, it can foresee several paths before reaching the destination. Let r be the shortest u t path w.r.t. µ, i.e., D [u] = (u,v) r µ(u, v) and D 2[u] = (u,v) r σ2 (u, v). Let q be the shortest u t path w.r.t. σ 2, i.e., E [u] = (u,v) q µ(u, v) and E 2 [u] = (u,v) q σ2 (u, v). Similar to the BF algorithm, BF Inaccurate has two phases: backward phase from t to all the nodes to prelabel all the nodes with D [u], D 2 [u], E [u], and E 2 [u] using RD, and a forward phase to find the most likely path that minimizes π. pseudocode for the BF Inaccurate algorithm is presented in Figure 8. RD is executed first over the graph w.r.t. µ to calculate D [u] and D 2 [u], and then w.r.t. σ 2 to calculate E [u] and E 2 [u]. The forward phase (Figure 9) is a modified version of Dijkstra s algorithm. It uses the information provided by the RD to find the next node from the already traveled segment. Consider a search that has already traversed from s to u. The next node v is determined from the unexplored graph by finding the node that has the smallest cost function π for the path s u v t. Note that in this case, for each such node v, there are two possible paths from v to t, as calculated in the backward step. BF Inaccurate(G(V, E), s, t, µ(e), σ 2 (E), C, C 2, w (.)) a. D=Reverse Dijkstra(G, t, µ(e)) b. E=Reverse Dijkstra(G, t, σ 2 (E)) 2. p = Fwd Inaccurate(G, s, t, µ(e), σ 2 (E), C, C 2, D, E) 3. if C < (i,j) p w (i, j) < C 2 4. return: p is feasible 5. else 6. return: p is infeasible Fig. 8. Pseudocode for the main loop of BF Inaccurate. To improve its performance, the forward heuristic can be used with a KSP implementation of Dijkstra s algorithm. The series of K paths can be tried for establishing the connection and if the differential delay after establishing the path satisfies the constraint (), then the path returned by the algorithm is feasible. V. SIMULTION RESULTS In this section, we describe the simulation model and study the performance of the algorithms proposed in Sections III and IV. In our simulations, a network of N nodes is generated randomly using Waxman s model [20]. The link weights w(i, j), the source and destination nodes, and the constraints C and C 2 are randomly generated.. TSCP with Exact Link Delay Information We compare the performance of the KSP, MLW-KSP, and BF algorithms. First, we compare the KSP and MLW-KSP

7 Fwd Inaccurate(G, s, t, µ(e), σ 2 (E), C, C 2, D, E). for all i V, i s, S[i] = π s [i] = NIL 2. S[s] = 0, µ(s, s) = 0, σ 2 (s, s) = 0 3. Insert Heap(s, S[s], Q) 4. while Q is not empty, 5. u = ExtractMin(Q) 6. if (u == t) return: The path traversed s t 7. for each edge (u, v) outgoing from u, 8a. µ D (v) = µ(s u) + µ(u, v) + D [v] 8b. µ E (v) = µ(s u) + µ(u, v) + E [v] 9a. σ 2 D (v) = σ2 (s u) + σ 2 (u, v) + D 2 [v] 9b. σ 2 E (v) = σ2 (s u) + σ 2 (u, v) + E 2 [v] 0. if {S[v] > Φ(C 2, C, µ D (p), σ 2 D (p))} S[v] = Φ(C 2, C, µ D (p), σ 2 D (p)) π s [v] = u, µ(s v) = µ(s u) + µ(u, v) σ 2 (s v) = σ 2 (s u) + σ 2 (u, v) Insert Heap(v, S[v], Q). if {S[v] > Φ(C 2, C, µ E (p), σ 2 E (p))} S[v] = Φ(C 2, C, µ E (p), σ 2 E (p)) π s [v] = u, µ(s v) = µ(s u) + µ(u, v) σ 2 (s v) = σ 2 (s u) + σ 2 (u, v) Insert Heap(v, S[v], Q) Function u = ExtractMin(Q) removes and returns the vertex u in the heap Q with the least key value. Function Insert Heap(v, x, Q) inserts the node v in the heap Q with a key value x. If the node is already present in the heap then it decreases its key to x. Fig. 9. Pseudocode for the forward phase of the BF Inaccurate algorithm. algorithms in terms of the miss probability, ined as the probability that an algorithm cannot find a feasible path for the connection request under a given value of K. We then study the performance of the BF algorithm. We also compare the average number of iterations required to find a feasible path under the three algorithms. For each link (i, j), w(i, j) is sampled from a uniform distribution in the range [0, 50]. The values of C and C 2 fall into the following three cases: ) The shortest path between s and t w.r.t w(.,.) is greater than C 2. In this case, there is no feasible path. 2) The shortest path between s and t w.r.t w(.,.) is greater than C but less than C 2. This is a trivial case, as the shortest path w.r.t. w is a feasible solution. 3) The shortest path between s and t w.r.t w(.,.) is less than C. The third case is the nontrivial case and is the one considered in our simulations. ccordingly, C and C 2 are generated such that they are always greater than the length of the shortest path between s and t w.r.t. w(.,.). Specifically, we let C = W (p ) + + U(0..50) and C 2 = C + L, where p is the shortest path between s and t w.r.t. w(.,.), and and L are positive constants. Notice that L reflects the allowable differential delay. We first compare the performance of the KSP and MLW- KSP algorithms. Figures 0 and depict the miss probability of the two algorithms versus K for a network of 50 and 00 nodes, respectively. In both figures, we see that the MLW- Miss Probability KSP MLW KSP Iterations permitted Fig. 0. Miss probability vs. number of permitted iterations for KSP and MLW-KSP algorithms (N = 50, = 20, L = 00). Miss Probability KSP MLW KSP Iterations permitted Fig.. Miss probability vs. number of permitted iterations for KSP and MLW-KSP algorithms (N = 00, = 20, L = 00). KSP algorithm has a significantly lower miss rate than the KSP algorithm for the same computational cost. Figure 2 depicts the hit probability versus for the BF algorithm under various values of L and N and with only one permitted iteration during the forward phase (K = ). The plot shows that with only one iteration, the algorithm is able of finding a feasible path in more than 99% of the instances. We report the average number of iterations (average value of K) required to find a feasible path in the KSP, MLW-KSP, and BF algorithms. To obtain this average, we execute each algorithm with a very large K and terminate the algorithm

8 once it finds a feasible path. The results are shown in Table I. s increases, MLW-KSP algorithm performs much better (a) N = 50 KSP algorithm MLW-KSP algorithm BF algorithm TBLE I VERGE NUMBER OF ITERTIONS FOR DIFFERENT VLUES OF (L = 00, N = 00). than the KSP algorithm in terms of computational cost. This is actually expected because when is increased (i.e., constraints get far away from the cost of the shortest path), the KSP algorithm needs to consider more paths to reach the feasibility region, while the MLW-KSP algorithm immediately reaches that region via the modified link weights. The BF algorithm performs much better than the other algorithms and requires significantly fewer iterations to find the feasible path under all scenarios. This is attributed to the fact that BF algorithm works on complete paths and directly optimizes the nonlinear cost function in (3) (b) N = Fig. 2. (c) N = 200 Hit probability vs. for the BF algorithm. B. TSCP with Inaccurate Link Delay Information Next, we study the performance of the proposed BF Inaccurate algorithm. To demonstrate the effectiveness of this algorithm, we compare its performance with the classic trigger-based approach [0], in which the current delay value of a link is advertised once its absolute difference from the last advertised value exceeds a given threshold T. Clearly, the smaller the value of T, the more accurate is the estimate of the link delay and the better is the performance of the path selection algorithm. This performance gain, however, comes at the expense of higher advertisement overhead. For the triggerbased approach, we use the BF algorithm in Section III-B-3, executed using the last advertised link delay values (treated as exact from the standpoint of the BF algorithm). For a given topology, each link (i, j) is assigned a random delay w(i, j). We assume that w(i, j) is normally distributed with mean µ(i, j) and variance σ 2 (i, j). To produce heterogeneous link delays, we also randomize the selection of µ(i, j) and σ 2 (i, j) by taking µ(i, j) uniform[0, 50] and σ 2 (i, j) uniform[0, 50]. We fix the exact value of the link delay (w (i, j)) of a link (i, j) by sampling from a Gaussian distribution with mean µ(i, j) and variance σ 2 (i, j). Note that w is not provided to the BF Inaccurare algorithm and is merely used to check if the path returned by the algorithm is feasible. We set C and C 2 as in Section V-, where p is now the shortest path between s and t w.r.t. w (.,.). BF Inaccurate heuristically attempts to return the most probable feasible path r. Let W (r) be the weight of this path w.r.t. w (.,.). If C W (r) C 2, then we count this a hit. Using only one permitted iteration of the BF Inaccurate algorithm (i.e., K = ), Figure 3 shows the hit probability for various values of and L and a network of 50 and 00 nodes. Figure 4 shows the average number of iterations needed to find a feasible path when no limit is imposed on K. Notice that as

9 the differential delay increases, the average number of required iterations is reduced. Because the difference between the last advertised link delay w last (i, j) and the present (exact) link delay cannot be more than T, we have w last (i, j) uniform[w (i, j) T, w (i, j) + T ]. (7) We vary the value of and L to show the effectiveness of the BF Inaccurate algorithm compared with a trigger-based approach. Figure 5 shows the hit probability versus (T = 5) for a network of 00 nodes. We conducted simulations for other values of T and observed similar trends. BF Inaccurate, which uses a one-time advertised statistical information, performs significantly better than the trigger-based approach as is increased..02 verage number of Iterations required (a) N = verage number of Iterations required (a) N = 50 Fig (b) N = 00 verage number of iterations vs. for the BF Inaccurate algorithm T = Fig. 3. (b) N = 00 Hit probability vs. for the BF Inaccurate algorithm (Threshold based) (Threshold based) (BF_Inaccurate) (BF_Inaccurate) VI. CONCLUSIONS ND SUMMRY In this paper, we considered the problem of finding a feasible VC to be added to a VCG in an EoS system that uses virtual concatenation. The memory requirement of the termination node imposes an upper bound on the differential Fig. 5. Hit probability vs. for the BF Inaccurate algorithm and BF algorithm using trigger-based approach (T = 5, N = 00).

10 delay between the new VC and each existing VC of the VCG. We modeled the problem algorithmically as that of finding a path whose delay is both upper and lower bounded. The resulting TSCP problem is shown to be NP-complete. We proposed two efficient heuristic algorithms (MLW-KSP and BF) for the TSCP problem when link delays are exact. Both algorithms achieve a significant improvement in the required computational cost compared to a standard KSP algorithm. The BF algorithm was found particularly efficient, often requiring only one iteration to return a feasible path. For the case of inaccurate link-state information, we modeled the problem as that of finding the most probable path that satisfies the two bounds. We proposed the BF Inaccurate algorithm for this case and verified its performance by comparing it with the standard trigger-based approach. Simulations indicate that BF Inaccurate achieves significant performance improvements over the trigger-based approach. REFERENCES [] ITU-T Standard G.707, Network node interface for the synchronous digital hierarchy, [2] NSI T , Synchronous optical network (SONET): Basic description including multiplexing structure, rates and formats, 200. [3] IETF RFC 577, Classical IP and RP over TM, June 994. [4] IETF RFC 265, PPP over SONET/SDH, June 999. [5] ITU-T Standard G.704, Generic framing procedure, Feb [6] V. Ramamurti, J. Siwko, G. Young, and M. Pepe, Initial implementations of point-to-point Ethernet over SONET/SDH transport, IEEE Communications Magazine, vol. 42, pp , [7] ITU-T Standard G.7042, Link capacity adjustment scheme for virtually concatenated signals, 200. [8]. Shaikh, J. Rexford, and K. G. Shin, Evaluating the impact of stale link state on quality-of-service routing, IEEE/CM Transactions on Networking, vol. 9, no. 2, pp , pril 200. [9] D. H. Lorenz and. Orda, QoS routing in networks with uncertain parameters, IEEE/CM Transactions on Networking, vol. 6, no. 6, pp , Dec 998. [0] T. Korkmaz and M. Krunz, Bandwidth-delay constrained path selection under inaccurate state information, IEEE/CM Transactions on Networking, vol., no. 3, pp , June [] Michael R. Garey and David S. Johnson, Computers and Intractability: Guide to the Theory of NP-Completeness, W. H. Freeman, [2] Gang Liu and K. G. Ramakrishnan, *prune: n algorithm for finding k shortest paths subject to multiple constraints, Proceedings of the IEEE INFOCOM Conference, pp , 200. [3] G. Eppstein, Finding the k shortest paths, Proceedings of the 35th nnual Symposium on Foundations of Computer Science, vol., pp , Nov 994. [4] E. I. Chong, S. R. Sanjeev Rao Maddila, and S. T. Morley, On finding single-source single-destination shortest paths, The Seventh International Conference on Computing and Information (ICCI 95), pp , July 995. [5] Fernando Kuipers, Turgay Korkmaz, Marwan Krunz, and Piet Van Mieghem, n overview of constraint-based path selection algorithms for QoS routing, IEEE Communications Feature Topic on IP-Oriented Quality of Service (QOS), pp , Dec [6] T. Korkmaz and M. Krunz, Routing multimedia traffic with QOS guarantees, vol. 5, pp , Sep [7] R. K. huja, T. L. Magnanti, and J. B. Orlin, Network flows: Theory, lgorithm, and pplications, Prentice Hall Inc., 993. [8] D. E. McDysan, QoS & traffic management in IP & TM networks, McGraw-Hill, [9] thanasios Papoulis and S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill Higher Education, [20] B. M. Waxman, Routing of multipoint connections, IEEE Journal on Selected reas in Communications, vol. 69, pp , Dec 988.