Distribution generalized Erlang

Figure 9.6 Pseudocompartment configurations generating Erlang (A), generalized Erlang (B), and phase-type (C) distributions for retention times in phenomenological compartments. Retention times are distributed according to A Exp(Ai) and Ai Exp(A2).

In the general field of statistics, the RTD of an n-stage CSTR battery is called an Erlang distribution, or a Gamma distribution when n is not integral. Then (n-1) is replaced by r(n) in the equation given in Section... 

This section proposes the use of a semi-Markov model with Erlang- and phase-type retention-time distributions as a generic model for the kinetics of systems with inhomogeneous, poorly stirred compartments. These distributions are justified heuristically on the basis of their shape characteristics. The overall objective is to find nonexponential retention-time distributions that adequately describe the flow within a compartment (or pool). These distributions are then combined into a more mechanistic (or physiologically based) model that describes the pattern of drug distribution between compartments. The new semi-Markov model provides a generalized compartmental analysis that can be applied to compartments that are not well stirred. 

Dellaert studies two lead time policies, CON and DEL, where DEL considers the probability distribution of the flow time in steady-state while quoting lead times. He models the problem as a continuous-time Markov chain, where the states are denoted by (n, 5)=(number of jobs, state of the machine). Interarrival, service and setup times are assumed to follow the exponential distribution, although the results can also be generalized to other distributions, such as Erlang. For both policies, he derives the pdf of the flow time, and relying on the results in [88] (the optimal lead time is a unique minimum of strictly convex functions), he claims that the optimal solution can be found by binary search. 

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