Mean Project Completion Time in Dynamic Markov PERT Networks

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Amir Azaron, Hideki Katagiri, Kosuke Kato, Masatoshi Sakawa Proof. The expected length of arc (l,j) is 774 1 1 2 N lj m , s ,..., 1 m s 2 mN λ ( s , because the ) duration of activity (l,j) has exponential distribution with parameter 1 2 N λ ( s , s ,..., s ) . The probability that the state vector of system lj m1 m2 mN 1 2 N 1 2 N changes from ( s , s ,..., s ) in node l to ( s , s ,..., s ) , after doing m1 m2 mN the activity (l,j), would be 1k1lj m m mN 2k2lj m1m2 ... mN k1 k2 kN Nk Nlj m1m mN P P P ... 1 2... ... 2 , because the continuous-time Markov processes corresponding to the transitions of the social problems are independent. If the arc (l,j) belongs to the path with maximum expected length, which approximately considered as the longest path, then, the state vector of system in node j actually changes to 1 2 N ( s , s ,..., s ) with the indicated probability. Finally, by conditioning on k1 k2 kN the state vector of system in node j, Theorem 1 is proved. Without losing the generality, we assume that the nodes of the network are numbered from 1 to n, in which there should be no directed path from node j to node l for j>l. The following algorithm can be used to approximate the mean project completion time in dynamic Markov PERT networks. Algorithm 1 2 N Step 1. Begin from node l=n. It is clear that V ( s , s ,..., s ) = 0 m = 1, 2,..., N , i = 1, 2,..., N . for i i n m1 m2 mN
Amir Azaron, Hideki Katagiri, Kosuke Kato, Masatoshi Sakawa i 1 2 N Step 2. Set l=n-1. Then, compute V − ( s , s ,..., s ) for i 775 n 1 m1 m2 mN m = 1, 2,..., N , i = 1, 2,..., N , from the recursive function (5). In this case, the second term of (5) would be equal to 0, because A ( n − 1) = n . i Step 3. Compute (t) φ mik for m i i = 1, 2,..., N i , k i 1, 2,..., N i = , ie i = 1, 2,..., N , by getting the inverse Laplace of φ (s) in equation (3). i Step 4. Set l=l-1. Then, compute i miki ikilj m1 m2 mN for m i 1, 2,..., N i P ... = , k = 1, 2,..., N , i = 1, 2,..., N and j ∈ A(l) , except for j=n, from equation (4). 1 2 N Step 5. Compute ( s , s ,..., s ) V for i i l m1 m2 mN i = 1, 2,..., N , from the recursive function (5). Step 6. If l>1, then, go to 4. Otherwise, go to 7. m = 1, 2,..., N , Step 7. Stop. The approximation of mean project completion time would be 1 2 N equal to V ( s , s ,..., s ) . 1 m1 m2 mN In the worst case, assume that we have a complete stochastic network, in which for each l, A(l)={l+1,l+2,…,n}. In this case, the time complexity of the N N ⎞ ⎛ 2 ⎞ algorithm in steps 2 and 3 would be clearly O⎜ N i ⎟ and O⎜∑= N i ⎟ , ⎝ i 1 ⎠ ⎝ i 1 ⎠ ⎛ ∏=
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Mean Project Completion Time in Dynamic Markov PERT Networks

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