In the E-Commerce network, it is an important issue to reduce the transmission time. The quickest path problem is to find the path in the network to send a given amount of data from the source to the sink with minimum transmission time. This problem traditionally assumed that the capacity of each arc in the network is deterministic. However, the capacity of each arc is stochastic due to failure, maintenance, etc. in many real-life networks. This paper proposes a simple algorithm to evaluate the probability that d units of data can be sent through the E-Commerce network within T units of time. Such a probability is a performance index for E-Commerce networks

Lin, Yi-Kuei

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Association for Information Systems AIS Electronic Library (AISeL) ICEB 2001 Proceedings International Conference on Electronic Business (ICEB) Winter 12-19-2001 Use Quickest Path to Evaluate the Performance for An E-Commerce Network Yi-Kuei Lin Follow this and additional works at: https://aisel.aisnet.org/iceb2001 This material is brought to you by the International Conference on Electronic Business (ICEB) at AIS Electronic Library (AISeL). It has been accepted for inclusion in ICEB 2001 Proceedings by an authorized administrator of AIS Electronic Library (AISeL). For more information, please contact elibrary@aisnet.org. The First International Conference on Electronic Business, Hong Kong, December 19-21,2001 Use Quickest Path to Evaluate the Performance for An E-Commerce Network Yi-Kuei Lin Department of Information Management Van Nung Institute of Technology Chung-Li, Tao-Yuan, Taiwan 320, R.O.C. E-mail: yklin@cc.vit.edu.tw  ABSTRACT  In the E-Commerce network, it is an important issue to reduce the transmission time. The quickest path problem is to find the path in the network to send a given amount of data from the source to the sink with minimum transmission time. This problem traditionally assumed that the capacity of each arc in the network is deterministic. However, the capacity of each arc is stochastic due to failure, maintenance, etc. in many real-life networks. This paper proposes a simple algorithm to evaluate the probability that d units of data can be sent through the E-Commerce network within T units of time. Such a probability is a performance index for E-Commerce networks.  Keywords: Quality Management; E-Commerce Networks; Performance index; Quickest Path.  1. Introduction  It is an important issue to reduce the transmission time in the E-Commerce network. Chen and Chin [5] proposed the quickest path problem to find a path with minimum transmission time to send a given amount of data from the source to the sink, where each arc has the capacity and the lead time [5, 9, 17]. More specifically, the capacity and the lead time of each arc are both assumed to be deterministic. However, due to failure, maintenance, etc., the capacity of each arc is stochastic in many real flow networks such as computer systems, telecommunication systems, etc. This paper is mainly to evaluate the probability that an E-Commerce network can send d units of data from the source to the sink within a given time T in which each arc is stochastic. Such a probability is named the system reliability and is a performance index for E-Commerce networks. A simple algorithm based on minimal paths (MPs) is proposed firstly to find all lower boundary points for (d,T), and then to calculate the system reliability in terms of such points, where a MP is an ordered sequence of arcs from the source to the sink without loops, and a lower boundary point for (d,T) is a vector representing the capacity of each arc.  2. The Algorithm  Let G º (N, A, L, M) denote an E-Commerce network with source s and sink t where N the set of nodes, A º {ai|1 £ i £ n} the set of arcs, L º (l1, l2, …, ln) with li the lead time of ai and M = (M1, M2, …, Mn) with Mi the maximal capacity of ai. The (current) capacity of arc ai, denoted by xi, takes values 0 = bi1 < bi2 <…  <  iirb = Mi. The vector X º (x1, x2, …, xn) denotes the capacity vector of G. We assume that the capacity of each arc is stochastic with a given probability distribution, and that all data are sent through one minimal path. Suppose P1, P2, …, Pm are MPs of G from s to t. With respect to each MP Pj = {aj1, aj2, …, jjna }, j = 1, 2, …, m, the capacity of Pj under the capacity vector X is )(min1 jknkxj££. If d units of data are transmitted through Pj under the capacity vector X, then the transmission time, denoted by y(d, X, Pj), is  lead time of Pj +úúúùêêêéjPd ofcapacity  the = úúúùêêêé+££=åjknknkjk xdljj11 min, where éxù is the smallest integer such that éxù ³ x. Let x(d, X) denote the minimum transmission time for the network to send d units of data from s to t  under the capacity vector X. Then x(d, X) = ),,(min1 jmjPXdy££.  2.1 Lower boundary points for (d,T)  If X is a minimal capacity vector such that the network can send d units of data from the source to the sink within T units of time, then X is called a lower boundary point for (d,T). That is, X is a lower boundary point for (d,T) if and only if (i) x(d, X) £ T and (ii) x(d, Y) > T for any capacity vector Y with Y < X. Hence, the system reliability Rd,T to meet such a requirement is Pr{X|x(d, X) £ T}= Pr{X|X ³ Xj for a lower boundary point Xj for (d,T)}. Several methods such as inclusion-exclusion rule [8,13-16,19], disjoint -event method [8,20] and state-space decomposition [1,2,10,12] can be applied to calculate Pr{X|X  ³ Xj for a lower boundary point for Xj (d,T)}.  2.2 The Algorithm to evaluate the system reliability  As those algorithms in [12-14,16,19,21], the proposed algorithm supposes that all MPs have been precomputed.  Step1. For each Pj = {aj1, aj2, …, jjna }, find the minimal capacity vector Xj = (x1, x2, …, xn) such that the network sending d units of data within T units of time. 1.1 Find the minimal capacity v of Pj such that d units of data can be sent through Pj within T units of time. That is, find the smallest integer v such that  úúùêêé+å= vdljnkjk1 £ T.      (1)  The First International Conference on Electronic Business, Hong Kong, December 19-21,2001 1.2 If v £ )(min1jknkMj££, then Xj can be obtained according to ïîïíìÏ=Ï³=jiijiijkPaxPavuaumx     0   with  of capacity  inimal . (2) Otherwise, Xj does not exist.  Step2. If Xj exists, then Bj = {X|X ³ Xj}. Otherwise, Bj = f. Then the system reliability Rd,T is Pr{UmjjB1=}.  3. An example  We use the network in figure 1 to illustrate the proposed algorithm. The capacity and the lead time of each arc are both shown in table 1. There are six MPs: P1 = {a1, a4}, P2 = {a1, a5, a8}, P3 = {a1, a2, a6}, P4 = {a1, a2, a7, a8}, P5 = {a3, a6} and P6 = {a3, a7, a8}. If 8 units of data are required to be sent from s to t within 9 units of time. Then all lower boundary points for (8,9) and the system reliability R8,9 to meet such a requirement can be derived as follows.             Figure 1. A benchmark network  Step1. 1.1 The lead time of P1 = {a1, a4} is l1 + l4 = 5. Then v = 2 is the smallest integer such that (5úúùêêé+v8 ) £ 9.  1.2 The maximal capacity of P1 is only 1. Hence, X1 does not exist.  1.1 The lead time of P2 = {a1, a5, a8} is l1 + l5 + l8 = 4. Then v = 2 is the smallest integer such that (4úúùêêé+v8 ) £ 9.  1.2 The maximal capacity of P2 is only 1. Hence, X2 does not exist.  1.1 The lead time of P3 = {a1, a2, a6} is l1 + l2 + l6 = 5. Then v = 2 is the smallest integer such that (5úúùêêé+v8 ) £ 9.  1.2 The maximal capacity of P3 is 3. Hence, x1 = x2 = x6 = 2 and xi = 0 for others. So we obtain X3 = (2,2,0,0,0,2,0,0).  1.1 The lead time of P4 = {a1, a2, a7, a8} is l1 + l2 + l7 + l8 = 6. Then v = 3 is the smallest integer such that (6úúùêêé+v8 ) £ 9.  1.2 The maximal capacity of P4 is 3. Hence, x1 = x2 = x7 = x8 = 3 and xi = 0 for others. So we obtain X4 = (3,3,0,0,0,0,3,3). ¼ Table 1. The arc data of figure 1 Arc Capacity Probability Lead time  3* 0.80  a1 2 0.10 2  1 0.05   0 0.05   3 0.80  a2 2 0.10 1  1 0.05   0 0.05   2 0.85  a3 1 0.10 3  0 0.05  a4 1 0.90 3  0 0.10  a5 1 0.90 1  0 0.10   4 0.60   3 0.20  a6 2 0.10 2  1 0.05   0 0.05   5 0.55   4 0.10  a7 3 0.10 2  2 0.10   1 0.10   0 0.05   3 0.80  a8 2 0.10 1  1 0.05   0 0.05  *Pr{the capacity of a1 is 3} = 0.80.  Step 2. Three lower boundary points for (8,9) are generated by step 1. Let B3 = {X|X ³ X3}, B4 = {X|X  ³ X4} and B5 = {X |X ³ X5}. The system reliability R8,9 = Pr{B3 È  B4 È  B5} = 0.91275 by applying inclusion-exclusion rule.  5. Conclusions  This paper uses the capacity vector X and minimal paths to describe the flows in E-Commerce network. An algorithm is proposed to evaluate the probability that d units of data are sent from the source to the sink through the E-Commerce network within T units of time. The idea of lower boundary s t a6 a3 a2 a8 a5 a4 a1 a7 The First International Conference on Electronic Business, Hong Kong, December 19-21,2001 points for (d,T) is proposed, which are the minimal capacity vectors satisfying the requirement. At most m lower boundary points for (d,T) are generated if there are m minimal paths. The system reliability can be computed in terms of all lower boundary points for (d,T). From the point of view of quality management, we can treat the system reliability as a performance index, and conduct the sensitive analysis to improve the most important component (e.g., critical transmission line) which will increase the system reliability significantly.  References [1] Alexopoulos C. A Note on state-space decomposition methods for analyzing stochastic flow networks. IEEE Trans. Reliability 1995;44:354-357. [2] Aven T. Reliability evaluation of multistate systems with multistate components. IEEE Trans. Reliability 1985;34:473-479. [3] Chen GH, Hung YC. 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Use Quickest Path to Evaluate the Performance for An E-Commerce Network

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Use Quickest Path to Evaluate the Performance for An E-Commerce Network

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