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By Pearn W. L., Lin G. H.

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Note by the definition of σ that t P (σ ≥ t) ≤ P 0 t =P 0 [k(s) − z] ds ≤ ∆ ¯ ds + (k¯ − z)t ≤ ∆ . [k(s) − k] Since k¯ > z by Assumption (A4), we have, for t ≥ ∆/(k¯ − z), t P (σ ≥ t) ≤ P 0 ¯ ds ≥ t(k¯ − z) − ∆ [k(s) − k] t ¯ − z) − ∆ 1 ¯ ds ≥ t(k√ [k(s) − k] t+1 0 t+1 t 1 t(k¯ − z) − ∆ ¯ ds . 11), we have that, for t ≥ 2∆/(k¯ − z), =P √ t(k¯ − z) ∆ P (σ ≥ t) ≤ C1 exp − √ +√ t+1 t+1 t(k¯ − z) t(k¯ − z) ∆ = C1 exp − √ + − √ +√ 2 t+1 2 t+1 t+1 t(k¯ − z) . ≤ C1 exp − √ 2 t+1 Therefore, ∞ Eσ r = r 0 ≤r tr−1 P (σ ≥ t) dt ¯ 2∆/(k−z) 0 tr−1 dt + rC1 ∞ t(k¯ − z) tr−1 exp − √ ¯ 2 t+1 2∆/(k−z) dt ≤ Cr (∆r + 1), for some constant Cr > 0, which is independent of ∆.

We set σ0 = 0 and τ0 = 0. The control u(·) can be characterized as follows: Use the maximum available production rate u(t) = k(t) to move the surplus process from 0 or x to x+K, and then use the zero production rate until the surplus process drops to the level x. 1. It is obvious by our construction that x(σn ) = x, n = 1, 2, . .. Furthermore, by the strong Markov property of k(·), we have P (τ > σn ) ≤ P (k(σn ) = k, k(σn−1 ) = k, . . , k(σ1 ) = k) = P (k(σ1 ) = k)P (k(σ2 ) = k|k(σ1 ) = k) × · · · × P (k(σn ) = k|k(σn−1 ) = k).

Bai [8] develops a hierarchical approach to solve the problem, and Srivatsan, Bai, and Gershwin [132] apply the approach to the scheduling of a semiconductor fabrication facility. A rigorous analysis of the model was given in Presman, Sethi, Zhang, and Bisi [102], and Presman, Sethi, Zhang, and Zhang [105, 106]. These works will be taken up in Chapter 4. An asymptotic analysis of the model carried out by Sethi, Zhang, and Zhang [122] will be reported in Chapter 8. 4 and describe a model of a dynamic jobshop.

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