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Definition 10. Let P = (σ, Qs , Qf , ρ) and let L be a lumping with a collector matrix V , and a corresponding τ -distributor W . The τ -lumped Markov Reward Chain with Fast Transitions P = (σ, Qs , Qf , ρ) is defined as σ = σV, Qs = W Qs V, Qf = W Qf V, ρ = W ρ. We say that P τ -lumps to P and write P L P. , they depend on the choice of the distributor. The restriction to τ -distributors does not change this. Subsequently, the τ -lumped process depends on the choice of the τ -distributor. The motivation for restricting to τ -distributors is that all τ -lumped processes are then equivalent in the limit.

Sm ≤ n the indexes for which vector h strictly increases. Then we define z0 = 1, and for i such that 1 ≤ i ≤ m we define zi as the last index just before vector h strictly increases: zi = si − 1, 1 ≤ i ≤ m. Finally, if zm < n then we define zm+1 = n. 6) we have: s1 = 2, s2 = 4, s3 = 5 and z0 = 1, z1 = 1, z2 = 3, z3 = 4, z5 = 5. i = 0, r1 = h1 while (zi < n) do a = minj>i hzj −hzi zj −zi , k = arg minj>i hzj −hzi zj −zi for (u = zi + 1 to zk ) do ru = a u + rzi i=k end Remark 1 (Complexity of Algorithms 1 and 2).

Rn = hn , ri = min{hi , ri+1 }, i < n. 44 A. Buˇsi´c and N. Pekergin Increasing Convex Order. Consider first the upper bounding case. We will suppose that vector h is increasing. Note that if this is not the case, then an increasing vector h such that h ≤ h ≤ 1 should be first computed as described in the strong stochastic upper bound case described above. Then we need to find a vector r such that h ≤ r ≤ 1. For an increasing vector h, an increasing and convex vector r such that h ≤ r ≤ 1 can then be easily obtained as follows: rn = hn , rn−1 = hn−1 , ri = max{hi , 2ri+1 − ri+2 }, i ≤ n − 2.

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