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8 Generating Droplet Sequences
Used Notation
For the droplet sequence generation method, a notation for paths, non-default
successors, and the time a droplet needs to flow through the path is introduced.
Definition 8.1 The path of the payload p is predefined by the experiment to
be executed and can be described as a sequence over channels and modules,
i.e. path p ∈ (C ∪ M) n with 1 ≤ n ≤ |C| + |M|. A header h k (with k ∈ N
as its identifier) can take multiple paths through the network until it reaches the
channel which should be blocked. This set of path options for h k is described with
P h k = {path
h k
0 , path
h k
1 , . . .} and each path
h k
i is described as a sequence over
channels, i.e. path
h k
i ∈ C n with 1 ≤ n ≤ |C|.
Whenever a payload or header path contains a non-default successor of a
bifurcation, the corresponding default successor has to be blocked by a header.
The non-default successors along a payload-path path p and a header-path path
h k
i
are collected in the sets nonDef ault path p ⊆ C and nonDef ault path
h k
i
⊆ C,
respectively.
The required time which the payload p needs in order to flow from the injection
point into a channel/module along its path path p is recursively defined by
T path p (path
p
[1]) = 0
T path p (path
p
[j ]) = T path p (path
p
[j − 1]) + pSteps(path
p
[j − 1]),
where the index notation path p [j ] is used to access a channel/module at position j
in the corresponding path. In a similar fashion, the required time for a header h k to
flow along path path
h k
i is defined by using the function hSteps instead of pSteps.
Generating Candidates
Using the discrete model and the notation introduced above allows to generate
possible sets of headers and their paths, so-called candidates, which can be used
to realize a desired experiment. Therefore, the path of the payload path p serves
as starting point. For this path, all non-default successor channels are collected
in nonDef ault path p . For each channel in nonDef ault path p , a header h k is required
which blocks the corresponding default successor. To this end, all possible paths to
each of the default successor are determined. These paths form possible options for
the header h k and are collected in P h k . Furthermore, these paths possibly contain
non-default successor channels again. Therefore, in order to route these headers
along the non-default successors, further headers are required. This procedure is
recursively repeated for all determined paths until all nonDef ault sets are equal
to ∅.
8 Generating Droplet Sequences
Used Notation
For the droplet sequence generation method, a notation for paths, non-default
successors, and the time a droplet needs to flow through the path is introduced.
Definition 8.1 The path of the payload p is predefined by the experiment to
be executed and can be described as a sequence over channels and modules,
i.e. path p ∈ (C ∪ M) n with 1 ≤ n ≤ |C| + |M|. A header h k (with k ∈ N
as its identifier) can take multiple paths through the network until it reaches the
channel which should be blocked. This set of path options for h k is described with
P h k = {path
h k
0 , path
h k
1 , . . .} and each path
h k
i is described as a sequence over
channels, i.e. path
h k
i ∈ C n with 1 ≤ n ≤ |C|.
Whenever a payload or header path contains a non-default successor of a
bifurcation, the corresponding default successor has to be blocked by a header.
The non-default successors along a payload-path path p and a header-path path
h k
i
are collected in the sets nonDef ault path p ⊆ C and nonDef ault path
h k
i
⊆ C,
respectively.
The required time which the payload p needs in order to flow from the injection
point into a channel/module along its path path p is recursively defined by
T path p (path
p
[1]) = 0
T path p (path
p
[j ]) = T path p (path
p
[j − 1]) + pSteps(path
p
[j − 1]),
where the index notation path p [j ] is used to access a channel/module at position j
in the corresponding path. In a similar fashion, the required time for a header h k to
flow along path path
h k
i is defined by using the function hSteps instead of pSteps.
Generating Candidates
Using the discrete model and the notation introduced above allows to generate
possible sets of headers and their paths, so-called candidates, which can be used
to realize a desired experiment. Therefore, the path of the payload path p serves
as starting point. For this path, all non-default successor channels are collected
in nonDef ault path p . For each channel in nonDef ault path p , a header h k is required
which blocks the corresponding default successor. To this end, all possible paths to
each of the default successor are determined. These paths form possible options for
the header h k and are collected in P h k . Furthermore, these paths possibly contain
non-default successor channels again. Therefore, in order to route these headers
along the non-default successors, further headers are required. This procedure is
recursively repeated for all determined paths until all nonDef ault sets are equal
to ∅.