does not optimize the ticket probing process so as to find better paths. This paper proposes an Enhanced Ticket-based Routing. (ETBR) algorithm. The ETBR ...
The Enhanced Ticket-based Routing Algorithm Li Xiao, Jun Wang and Klara Nahrstedt Department of Computer Science University of Illinois at Urbana-Champaign Urbana, IL 61801
Abstract— The Delay-constrained least-cost routing problem is to find the least cost path which satisfies a given delay constraint. There are two major difficulties to solve this problem. The first difficulty is the NP-Completeness of this routing problem. The second difficulty is that the networking information used for routing may be imprecise. The ticket-based routing (TBR) algorithm[1], aiming to find a sub-optimal solution, provides a heuristic approach to overcome the above difficulties and solve the routing problem. Although TBR proposes a detailed ticket forwarding method based on imprecise end-to-end information, it does not optimize the ticket probing process so as to find better paths. This paper proposes an Enhanced Ticket-based Routing (ETBR) algorithm. The ETBR improves the effectiveness of ticket probing by two techniques. The first technique uses colorbased ticket distribution for tickets of different colors. The tracing information of green tickets and yellow tickets is kept separately to avoid unnecessary ticket dropping. The second technique uses historical probing results to optimize ticket probing, so that redundant probing paths are eliminated. Through extensive simulations, we demonstrate that the ETBR can find paths which have much lower cost1 than TBR, without decreasing the success ratio or increasing the message overhead.
I. I NTRODUCTION Quality of Service(QoS) routing is to find a path which satisfies given QoS requirements. Two major issues make QoS routing problems hard. First, the routing problem with multiple independent QoS constraints is NP-hard. Second, some network parameters are inaccurate or out-of-date due to periodical updating. Using the source routing approach, [2] and [3] model the imprecise information by probability density functions. The goal is to find a path which has the maximum probability to satisfy the QoS requirements. However, it is difficult to obtain the correct probability functions for such imprecise information. [4] tries to setup probability models by introducing network operating details, such as the triggering policy in link-state protocols. However, too many assumptions make it not very practical. Furthermore, it just optimizes the probability to find a feasible path, and the optimality for QoS metrics, such as cost of the path, is ignored. [5] solves the This work was supported by DARPA under contract number F30602-97-20121 and National Science Foundation under contract number NSF ANI 0073802. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the DARPA or National Science Foundation. 1 The cost, which will be defined in Section II-A, stands for any additive metric for optimization. For example, the price for passing one network link, the buffer size, etc..
inaccurate link state problem with multiple QoS constraints using rate-based control model. The end-to-end delay and delay jitters are guaranteed by controlling rate, but this method still faces the difficulty in finding effective probability function for link state. Moreover, the delay constraint is coupled with the bandwidth. As a result, it is inefficient to guarantee low delay when the bandwidth requirement is low. A novel distributed routing algorithm, the ticket-based routing, is proposed in [1] and [6]. From the source node, probing packets, called probes, are sent toward the destination through different paths to search for a feasible route and optimize given QoS parameters. Imprecise end-to-end network information is considered in ticket distribution polices, and its impact is also offset by multi-path probing. [7] and [8] combine ticket-based routing with resource reservation protocols so as to further minimize the influence of imprecise data. However, they do not consider how to improve the probing ability to tolerate the imprecise information, so that better paths can be found without extra routing message overhead. Using the Delay-Constrained Least-Cost Routing (DCLCR) problem as an example, this paper proposes the Enhanced Ticket-based Routing(ETBR) algorithm to improve the searching ability of tickets. Compared with the original Ticket-Based Routing (TBR) algorithm [1], ETBR can find paths with much lower cost, while it does not sacrifice the success ratio and has nearly the same amount of message overhead. The rest of the paper is organized as follows. Section 2 abstracts the ticket-based routing algorithm [1] and the problem to be solved. Section 3 presents Enhanced Ticketbased Probing algorithm. Section 4 is the simulation results, and Section 5 is the conclusion. II. T HE DCLCR P ROBLEM AND T ICKET- BASED ROUTING A. DCLCR Problem A network is represented by a graph G V; E , where V is the node set, and E is the link set. Each link has two non-negative QoS attributes: delay e and cost e , standing for the delay and the communication cost on link e. The cost (or the delay) of a path is defined as the sum of the cost (or the delay) of all the edges on the path. The Delay-Constrained Least-Cost Routing (DCLCR) problem we are trying to solve is defined as follows. Definition 1—Delay-constrained least-cost routing problem: In graph G V; E , given source node s, destination node t,
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and delay constraint D, find a path P from node s to node t, such that (1) delay P D, and (2)8 path P 2 , where frjr is any path from s to t, and delay r Dg, satisfies cost P cost P . This problem is NP-complete[9].
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B. The Ticket-based Routing Algorithm The Ticket-Based Routing (TBR) algorithm is firstly presented in [1] as a heuristic solution to the DCLCR problem. Each path is probed by one probing packet, called a probe. In each probe p, the probe state is recorded, including the path that p has traveled through so far, the accumulated delay delay p and the accumulated cost cost p of the path. When a routing request arrives at the start node s, a probe with G0 green tickets and Y 0 yellow tickets is sent toward the destination t. Green tickets take paths with less end-to-end cost, optimizing the cost metric; yellow tickets take paths with less end-to-end delay, trying to find feasible paths. The probe can be split into several probes as long as there are sufficient amount of tickets in it. A probe, which arrives at the destination t and whose accumulated delay is less than D, represents a feasible solution. The path, which has the least cost in all the feasible solutions is chosen. The resources on this path are then reserved for further data transmission. Three important data are used in the algorithm Ri t : The ticket distribution set, which contains all possible next hops for routing tickets from i to t. Di t : The minimum end-to-end delay from i to t recorded at i. When yellow tickets to t arrive at node i, they are distributed to the elements of R i t in proportion to Dk t 1 , where k 2 Ri t . Ci t : The minimum end-to-end cost from i to t recorded at i. When green tickets arrive at node i, they are distributed to the elements of R i t in proportion to 1 , where k 2 R t . Ck t i Notice that Di t is end-to-end delay from i to t recorded at i, and it is updated periodically by distance vector or link-state protocols. Therefore, its value may be imprecise. In order to get its approximation value, estimation is used. The variation D i t is predicated from previous data by exponential average 2. Therefore, the real value falls into the interval Di t Di t ; Di t Di t . For simplicity, Ci t is used as precise data[1]. The impact of the imprecise information can be offset by using the multi-path probing with tickets.
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different colors may interfere with each other, and redundant searching path may exist. Second, D i t can not be estimated precisely. Third, the selection of final probes at the destination does not take into account the load balancing. Among these limitations, the unoptimized probe distribution is the most important one. Using our ETBR, we can improve the searching ability of the algorithm without increasing the number of tickets, i.e., we can find paths having lower cost with the same amount of message overhead. For example, in TBR, at most one probe is allowed to go through one link. This strategy eliminates infinite cycles in ticket probing, but it unfortunately blocks some useful probes. ETBR optimizes the probe distribution by giving tickets of different colors independent distribution set R i t . Furthermore, probes might be optimized by historical probing information. These improvements of ETBR make it possible to find paths with lower cost without increasing message overhead or losing calladmission ratio.
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B. Color-based Ticket Distribution Set R ig t and Riy t
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Let us assume that probe p with g p green tickets and y p yellow tickets arrives at node i. R i t is the set of possible next hops from i to the destination t. The incoming tickets will be distributed among the elements in R i t according to the value 1 and D t 1 , where of Ck t cost i; k delay i; k k k 2 Ri t . In TBR, both green tickets and yellow tickets have the same Ri t , where Ri t fkjdelay p delay i; k Dk t Dk t D, k is the neighbor of i, and no probe has passed link i; k or link k; i g. This means if one ticket passes link i; k , other tickets can not pass link i; k any more, even though they are in two different colors. This strategy eliminates infinite cycles in probing path. However, it blocks some useful probes. For example, in Fig.1(a), probe p containing two green tickets passes link i; k first. The probe q , which arrives later, will be dropped at node i, because link i; k has been visited by p and no other outgoing links exist. However, if these two yellow tickets in q were sent to k , they might find better paths, because they could take different probing paths from two green tickets in p.
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III. T HE E NHANCED T ICKET- BASED ROUTING A. The Limitations in TBR Algorithm
(a) Ticket Blocking
There are several limitations in the TBR algorithm. First, the probe distribution is not optimized. For example, tickets of
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Fig. 1: Use Rig t and Riy t for green tickets distributing and yellow tickets distributing respectively.
Our solution is to keep track of link visiting information for different colors. We divide R i t into two independent sets Rig t and Riy t for green tickets and yellow tickets respectively, where R ig t fkjdelay p delay i; k Dk t Dk t D, k is the neighbor of i, and no green ticket has passed link i; k or link k; i g: The definition of y y Ri t is similar, and Ri t fkjdelay p delay i; k Dk t Dk t D, k is the neighbor of i, and no yellow ticket has passed link i; k or link k; i g. This solution is demonstrated in Fig.1(b). When probe q arrives at node i, ; because green tickets have passed link i; k before, R ig t and Riy t fkg. Since Riy t fkg, the two yellow tickets can be sent to node k as probe q 20 . However, since R ig t ;, probe q10 , which contains one green ticket, is dropped. The reason is that if q10 is sent to k , q10 will follow the identical probing path as p 0 . Hence, it becomes a redundant probe, which can not find new feasible paths, but increases message overhead. Theorem 1: There are no infinite cycles in probing paths when color-based ticket distribution sets R ig t and Riy t are used. Proof: The proof is omitted because of limited space.
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C. Ticket Optimization Using Historical Probing Results When probe p passes node i, the state of p, including the accumulated delay, cost and the path can be recorded at node i as the historical information. This information may be used to optimize the probes which pass i later. In TBR, cycles might exist in some probing paths, because a ticket can visit a node for multiple times through different edges that connect to the node. Using probe optimization, probing cycles can be removed. This is shown in Fig.2(a). Probe p goes through link i; j , and is split into two probes p1 with a yellow ticket and p 2 with a green ticket. A loop will be formed when p 2 goes back to node i from node k . If it just starts from node i again with the accumulated delay and cost, its state is worse than the state when it leaves i for the first time. To put p2 back to its previous state and remove the cycle in the path, we can update the state of p 2 with the historical probing results, which are stored at node i. This is called ticket optimization, and shown in Fig.2(b). The path of p2 : : : ! i ! j ! k ! i ! l ! : : : is replace by ::: ! i ! l ! ::: . Tickets may also be optimized by using historical probing results left by other tickets. Furthermore, if the starting node is the same, the results from previous routing sessions can also be used. This is an effective way to offset the impact of imprecise routing information. During the time between two adjacent link-state advertising, D i t fluctuates around the precise value. The historical results from previous routing sessions contain some useful routing information which is more up-to-date than D i t which is from the last link- state advertising. However, when using the historical information from previous sessions, we must make sure it is not out-of-date.
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Fig. 2: Use history probing information to remove cycles in the probing path and put ticket in the optimized state.
We define as the time-to-live value for the historical probing records, which should be smaller than the link-state advertising interval and much larger than the maximum time for the ticket to travel through the network (the maximum ticket life time). We define the historical probing record as follows. Definition 2—Historical Probing Record: Historical probing record is defined as HP R fsrc; path; dtpr ; ctpr ; timestampg, where src is the starting node; path is a node sequence from src to the current node, delay path representing the path that ticket travels; d tpr and ctpr cost path ; timestamp is the establishing time of this record. Because probing is optimized for two metrics, cost and delay, two types of records are used for green tickets and yellow tickets respectively. At each node i, two historical probing records are kept for all other nodes, tpr d and tprc . tprd records the path which has the least delay from tpr d :source to i. tprc records the path which has the least cost from tpr c :source to i. When a probe p arrives at a node, the algorithm TPR-T ICKETU PDATING (p) will be executed before p is distributed to the next hops. Given tpr c and tprd have the same source node as p, the pseudo-code of TPR-T ICKET-U PDATING (p) is below.
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TPR-T ICKET-U PDATING(p) 1 if p is green ticket 2 then if current time tprc :timestamp > 3 then tprc p 4 else if tprc :cost < p:cost 5 then p tprc 6 else tprc p 7 if tprd :delay > p:delay 8 then tprd p 9 else if current time tprd :timestamp > 10 then tprd p 11 else if tprd :delay < p:delay 12 then p tprd 13 else tprd p 14 if tprc :cost > p:cost 15 then tprc p 16 tprd :timestamp current time 17 tprc :timestamp current time
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Basically, there are two tasks in the algorithm above. First, use the information in the incoming probe to update historical probing records. Second, optimize the probe using historical probing records. Because green tickets are aimed to find a path with lowest cost, they are optimized according to tpr c which stands for the lowest cost found so far. Similarly, yellow tickets use tprd . When a ticket is to be updated, its cost, delay , and sequence of visited node are copied from the corresponding historical probing record. If a record is too old, i.e, current time tpr:timestamp > (recall is the time-to-live value for historical probing records), it should not be used to optimize tickets. Once a ticket passes through a node, the timestamp of related historical probing records are updated. As mentioned before, is much greater than the maximum time for one ticket traveling through the network and less than the link-state advertising interval. Theorem 2: There is no cycle in ticket probing path if tickets are optimized by the historical probing records. Proof: By contradiction. Assume the ticket p is green. Suppose there is a cycle in the probing path of p. Then there must be at least one node i that has been visited twice by p. When p visits i for the first time, the historical information should be updated by p. If i appears twice in the path of p, only two cases may make this happen. When p visits i for the second time, (1) the historical probing record at i has expired, or (2) p has a lower cost than tpr c :cost. Because is longer than the whole life cycle of tickets and all the cost metrics on the edges have positive value, both cases are not possible. Therefore, there is no cycle in the probing path. IV. S IMULATION R ESULTS In order to evaluate the performance of the ETBR, extensive simulations have been conducted to compare the performance of ETBR with TBR, flooding and the shortest path algorithms[1]. The flooding algorithm is actually a TBR algorithm with infinite yellow tickets and zero green ticket. The shortest path algorithm uses one probe searching along the least-delay path based on the last advertised delay values. Our simulations use the same curves of the ticket numbers and the same network topology as in TBR [1] so as to show the improvement obtained from the color-based ticket distribution and ticket optimization. Most of the simulation parameters, such as link delay, link cost, etc., also have the same values as in the TBR algorithm. is 500 times of the maximum time for a ticket to travel through the network. Imprecision rate [1], written , is the relative change of link delay, and it ranges from 5% to 50% in simulations. In each simulation case, given an imprecision rate and an average delay requirement, 5000 routing requests are tested. We define three metrics for the routing algorithms as follows. Average Path Cost: The ratio of the total cost of all established connection paths to the number of established
connections, which is the same as in [1]. Success Ratio: In [1], the success ratio is defined as the number of accepted routing requests divided by the total number of requests. However, this definition does not reflect the relative routing ability compared with the optimal algorithm, because the optimal success ratio may be changed in different network configurations. In this paper, success ratio is re-defined as success ratio =
number of accepted routing requests optimal number of accepted routing requests
The optimal value can be obtained by using flooding algorithm, which can definitely find a feasible path if there exists one. Therefore, according to our new definition, the success ratio of the flooding algorithm is 100%. Message Overhead: A probe passing from one node to another is one message. Message overhead is defined as the number of messages consumed by one routing request. Because ETBR and TBR are both approximation solutions to DCLCR problem, they can only provide sub-optimal results. In order to prove ETBR has better searching ability than TBR, three arguments must be established. First, ETBR should achieve lower average path cost. Notice that only the routing requests which can be accepted by both algorithms should be considered in the simulation. The reason is that, in some cases, the requests which can not be accepted by both may influence the average path cost substantially. This influence should be avoided, because it reflects the difference in success ratio, not the path cost optimization, which makes the results incomparable. Second, based on a large number of randomly generated routing requests, ETBR has comparable or better success ratio than TBR. Third, ETBR has comparable or lower message overhead than TBR. The simulation results confirm that these three arguments are well-established. The following subsections show the details case-by-case. A. Average Path Cost Because most of the requests that are accepted by the shortest path algorithm can also be accepted by the other three algorithms, we only consider the requests that can be accepted by the shortest path algorithm. The results are shown in Fig.3 and Fig.4. The x-axis is the average delay requirement D. In each simulation case, the delay constraint D is generated ;D . The y-axis is the randomly from the interval D average path cost. The lower the curve, the better the algorithm. These figures show that ETBR can find the least cost paths among these four algorithms. Fig. 5 compares ETBR and TBR directly by only using the routing requests that can be accepted by both ETBR and TBR. It is demonstrated more clearly that ETBR obtains substantial improvements in average path cost when compared with TBR. In TBR, the steady value 3 of average path cost is 300. In ETBR,
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3 The steady value means the convergent value as the value of average delay requirement increases.
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Fig. 5: Relative Average Cost Imprecision rate: 50% 80 Number of messages per routing request
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the steady value of average cost is 250. ETBR obtains 17% improvement over TBR. This confirms that ETBR has more effective searching ability, which is achieved by two means: (1) The color-based probe distribution avoids unnecessary ticket dropping; (2) Ticket optimization removes redundant paths and cycles by using historical probing records. B. Success Ratio and Message Overhead As shown in Fig.6, when the imprecision rate is 10%, success ratio curves of ETBR and TBR are very close; when imprecision rate is 50%, ETBR is even better than TBR. This is because ticket optimization increases the possibility to find a feasible path if the value of average delay requirement is small. From this result, the second argument is well supported. In Fig.7 and Fig. 8, ETBR, TBR, flooding, and the shortest path algorithms compare their message overhead. The shortest path algorithm has the least message overhead, and flooding has the largest overhead. The curves of ETBR and TBR are very close to each other, and they make a trade off between the message overhead and the searching of optimum paths. V. C ONCLUSION In this paper, the Enhanced Ticket-based Routing (ETBR) algorithm is proposed to solve delay-constrained least-cost routing problem. Based on its antecedent algorithm TBR, ETBR improves the effectiveness of ticket probing and the tolerance to imprecise end-to-end data by using color-based ticket distribution and ticket optimization techniques. The
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simulation results show that ETBR obtains much lower average path cost than TBR without losing success ratio or increasing message overhead. R EFERENCES [1] Shigang Chen and Klara Nahrstedt, “Distributed qos routing with imprecise state information,” in Proceedings of 7th IEEE International Conference on Computer, Communications and Networks, Lafayette, LA, Oct. 1998, pp. 614–621. [2] Dean H. Lorenz and Ariel Orda, “Qos routing in networks with uncertain parameters,” IEEE/ACM Transactions on Networking, vol. 6, no. 6, pp. 768–778, Dec. 1998. [3] Roch A. Guerin and Ariel Orda, “Qos routing in networks with inaccurate information: Theory and algorithms,” IEEE/ACM Transactions on Networking, vol. 7, no. 3, pp. 350–364, June 1999. [4] George Apostolopoulos, Roch Guerin, Sanjay Kamat, and Satish K. Tripathi, “Improving qos routing performance under inaccurate link state information,” in Proc. 16th International Teletraffic Congress, Edinburgh, UK, June 1999, pp. 7–11. [5] Zhong Fan and E. S. Lee, “Multiple qos contrained routing with inaccurate state information,” Electronics Letters, vol. 35, no. 21, pp. 1807–1808, Oct. 1999. [6] Shigang Chen and Klara Nahrstedt, “Distributed quality-of-service routing in ad-hoc networks,” IEEE Journal on Selected Areas in Communications, vol. 17, no. 8, pp. 1488–1505, Aug. 1999. [7] Xin Yuan, Hui Ding, Yuan Zhong, and Jie Zhang, “Resource reservation mechanisms for distributed multi–path quality of service routing,” in The 9th IEEE International Conference on Computer Communication and Networks, Las Vegas, Nevada, Oct. 2000, pp. 9–13. [8] Yuan Zhong and Xin Yuan, “Impact of resource reservation on the distributed multi-path quality of service routing scheme,” in The Eighth International Workshop on Quality of Service (IwQoS2000), Pittsburgh, PA, June 2000, pp. 95–104. [9] Michael Garey and David Johnson, Computers and intractability: a guide to the theory of NP-completeness, New York: W.H. Freeman, 1979.
