Constant lead time

A common approach to lead time quotation is to promise a constant lead time to all customers, regardless of the characteristics of the order and the current status of the system [66] [112]. Despite its popularity, there are serious shortcomings of fixed lead times [63]. When the demand is high, these lead times will be understated leading to missed due dates and disappointed customers, or to higher costs due to expediting. When the demand is low, they will be overstated and some customers may choose to go elsewhere. 

In real physical systems, the populations and h(0p are not truly constant in time, even in the absence of a field, because of relaxation processes. These relaxation processes lead, at sufficiently long times, to thennal... 

This is the steady state compensator. The lead-lag element with lead time constant xFLD and lag time constant XpLG is the dynamic compensator. Any dead time in the transfer functions in (10-7) is omitted in this implementation. 

When a plastic is subjected to an external load the observed stiffness changes with time. In a creep test the load is kept constant leading to an increase in strain. In a stress-relaxation test the deflection (frequently compression) is kept constant so that the stress is observed to relax. The changes will be primarily due to physical effects, and the strains may be reversible if sufficient time is allowed. At long durations the applied load can lead to failure, known as creep-rupture or stress-rupture. 

In case of a homogeneous temperature distribution in the heated area, h corresponds to the temperature coefficient of the heater material, otherwise h includes the effects of temperature gradients on the hotplate. As a consequence of the aheady mentioned self-heating, the applied power is not constant over time, and the hotplate cannot be simply modelled using a thermal resistance and capacitance. Replacing the right-hand term in Eq. (3.28) by Eq. (3.35) leads to a new dynamic equation ... 

To derive the steady-state solution of Eq. (1) we assume that the populations are constant in time and consequently their time derivatives can be set to zero. This leads to a set of 4 equations for the equilibrium populations Pi, i = 1,2, 3 of the three states involved. We are interested in the rate R at which the system emits photons. This rate is evidently given by... 

Equation 4.35 shows that the concentration deviations based on a linearization analysis of the rate laws in Eqs. 1.54a and 1.54c will decay to zero exponentially ( relax ) as governed by the two time constants, r, and r2. These two parameters, in turn, are related to the rate coefficients for the coupled reactions whose kinetics the rate laws describe (Eqs. 4.36c-4.36e and 4.38). If the rate coefficients are known to fall into widely different time scales for each of the coupled reactions, their relation to the time constants can be simplified mathematically (Eq. 4.39 and Table 4.3). Thus an experimental determination of the time constants leads to a calculation of the rate coefficients.20 In the example of the metal complexation reaction in Eq. 1.50, with the assumptions that the outer-sphere complexation step is much faster than the inner-sphere complexation step and that dissociation of the inner-sphere complex is negligible (k b = 0 in Eq. 1.54c), the results for tx and r2 in the first row of Table 4.3 can be applied. The expression for tx indicates that measurements of this parameter as a function of differing equilibrium concentrations of the complexing metal and ligand will produce a straight line whose slope is kf and whose y-intercept is kb. The measured values of l/r2 at these same two equilibrium concentrations then lead to a calculation of kf. 

A steady state is often not achieved in soils, so a steady rate is sometimes used, where the rate of volumetric water depletion is constant, leading to relations considerably more complicated than are Equations 9.8 and 9.9.3 We note that the equations describing water flow in a soil for the onedimensional case (Eq. 9.7), for cylindrical symmetry (Eq. 9.8), and for spherical symmetry (Eq. 9.9) all indicate that the volume flux density of water is proportional to LSGl1 times a difference in hydrostatic pressure (see Table 9-2). 

Although the derivation of Fichthorn and Weinberg only holds for Poisson processes, their method has also been used to simulate TPD spectra. [37] In that work it was assumed that, when At computed with equation (57) is small, the rate constants are well approximated over the interval At by their values at the start of that interval. This seems plausible, but, as the rate constants increase with time in TPD, equation (57) systematically overestimates At, and the peaks in the simulated spectra cire shifted to higher temperatures. In general, if the rate constants are time dependent then it may not even be possible to define the expectation value. We have already mentioned the case of cyclic voltammetry where there is a finite probability that a reaction will not occur at all. The expectation value is then certainly not defined. Even if a reaction will occur sooner or later the distribution Prx(0 has to go faster to zero for t —> oo than 1/t for the expectation value to be defined. Solving equations (48), (52), or (55) does not lead to such problems. 

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