Analytical Solution Sample Clauses
Analytical Solution. Molecular diffusion in a two-component variable density mixture can be described using ▇▇▇▇’▇ law of diffusion (Bird et al., 1960): J = −cD∇xhe where J is the molar flux, c is the molar concentration, D is the molecular diffusivity coefficient and xhe is the helium molar fraction. The molar concentration and molecular diffusivity coefficient can be assumed to be constant such that for a one-dimensional problem Equation (2.12) leads to the unsteady, one-dimensional diffusion equation: ∂xhe ∂t = ∂2xhe ∂y2 where t is time and y is the vertical coordinate. Equation (2.13) can be solved using the method of separation of variables to obtain (Crank, 1975): h 2 ∞ nπh n2π 2 nπy xhe = xhe,0 l + π ∑ sin l exp−
Analytical Solution. An important aspect is the analytical tractability of the underlying stochastic process, used for performance evaluation in SPAs. The underlying CTMCs in MTIPP and PEPA, as well as SMCs in EMPA, are treated analytically, but these continuous time SPAs have interleaving semantics. GSPA, Sπ and GSMPA are the continuous time models, for which a non-interleaving semantics is constructed, but for the underlying GSMPs in GSPA and GSMPA, only simulation and numerical methods are applied, whereas no performance model for Sπ is defined. sACP and TCP dst are the discrete time models with the associated analytical methods for the throughput calculation in sACP or for the performance evaluation, based on the underlying DTMRCs in TCP dst, but both models have interleaving semantics. dtsiPBC is a discrete time model with a non-interleaving semantics, where analytical methods are applied to the underlying SMCs. Hence, if an interleaving model is appropriate as a framework for the analytical solution towards performance evaluation then one has a choice between the continuous time SPAs MTIPP, PEPA, EMPA and the discrete time ones sACP, TCP dst. Otherwise, if one needs a non-interleaving model with the associated analytical methods for performance evaluation and the discrete time approach is feasible then dtsiPBC is the right choice. The existence of an analytical solution also permits to interpret quantitative values (rates, probabilities, weights etc.) from the system specifications as parame- ters, which can be adjusted to optimize the system performance, like in dtsPBC, dtsiPBC and parametric probabilistic transition systems (i.e. DTMCs whose transi- tion probabilities may be real-value parameters) [56]. Note that DTMCs whose transition probabilities are parameters were introduced in [32]. Parametric CTMCs with the transition rates treated as parameters were investigated in [41]. On the other hand, no parameters in formulas of SPAs were considered in the literature so far. In dtsiPBC we can easily construct examples with more parameters than we did in our case study. The performance indices will be then interpreted as functions of several variables. The advantage of our approach is that, unlike of the method from [56], we should not impose to the parameters any special conditions needed to guarantee that the real values, interpreted as the transition probabilities, always lie in the interval [0; 1]. To be convinced of this fact, just remember that, as we have demonstrat...
Analytical Solution. Central Limit Approximating Ap- proach Taking the Ising Model, we’ll try to draw some conclusions about the relations be- tween variables P∞, P0, J and Nr.