PPI Repository

Expected Waiting Time from activity A to activity B

\mathit{e}\mathit{WTAB}\mathit{^{\times}}(g, \mathit{a}, \mathit{b})

\mathit{\times} \in \{\mathit{s}, \mathit{c}, \mathit{sc}, \mathit{w}\}

Dimension:

Time

Granularity:

Group of cases

Description

Expected sum of elapsed time between the start event of an activity instance and the complete event of the activity instance that precedes it, for all activity instances in a case that occurs between the earliest instantiation of a specific activity and the earliest instantiation of another specific activity that occurs after the former one, including both, for all cases belonging to a group of cases.

Potential use

Event log in which the time spent between every activity instance of a specific subprocess is of relevance.

Assumptions

Minimizing waiting time is desirable.

Desired Value:

MINIMIZE

Required attributes

act
actinst
case
time
type

Equations

Formula for practitioners

$\displaystyle \frac{\sum _{c \in g} \begin{cases} \mathit{WTAB}(c, \mathit{a}, \mathit{b}) & \mathit{if \ } \mathit{fi}(c, \mathit{a}, \mathit{b}) \neq \emptyset \\ 0 & \mathit{if \ } \mathit{fi}(c, \mathit{a}, \mathit{b}) = \emptyset \end{cases}}{ \sum_{c \in g} \begin{cases} 1 & \text{if } \mathit{fi}(c, \mathit{a}, \mathit{b}) \neq \emptyset \\ 0 & \text{otherwise} \end{cases} }$

$\mathit{WTAB}$ returns the waiting time from activity $\mathit{a}$ to activity $\mathit{b}$ in case $c$ .
$\mathit{fi}$ returns the set of first instances of activity $\mathit{b}$ that occur after the first instance of $\mathit{a}$ in case $c$ .

Formal Definition

$\frac{\sum _{c \in g} \begin{cases} \mathit{WTAB}\mathit{^{\times}}(c, \mathit{a}, \mathit{b}) & \mathit{if \ } \mathit{fi}\mathit{^{sc}}(c, \mathit{a}, \mathit{b}) \neq \emptyset \\ 0 & \mathit{if \ } \mathit{fi}\mathit{^{sc}}(c, \mathit{a}, \mathit{b}) = \emptyset \end{cases}}{|\{c \in g \mid \mathit{fi}\mathit{^{sc}}(c, \mathit{a}, \mathit{b}) \neq \emptyset\}|}$
The $\mathit{\times}$ indicates that the PPI can take multiple forms, in this case, $\mathit{\times} \in \{\mathit{s}, \mathit{c}, \mathit{sc}, \mathit{w}\}$ . If $\mathit{\times} = \mathit{s}$ , then the function considers activity instances that were started within the start and end activity instances, i.e., it uses $\mathit{WTAB}\mathit{^{s}}$ ; if $\mathit{\times} = \mathit{c}$ , then the function considers activity instances that were completed within the start and end activity instances, i.e., it uses $\mathit{WTAB}\mathit{^{c}}$ ; if $\mathit{\times} = \mathit{sc}$ , then the function considers activity instances that were either started or completed within the start and end activity instances, i.e., it uses $\mathit{WTAB}\mathit{^{sc}}$ ; if $\mathit{\times} = \mathit{^{w}}$ , then the function considers all activity instances that were active within the start and end activity instances, i.e., it uses $\mathit{WTAB}\mathit{^{w}}$ .

Python implementation

This indicator is implemented in the process_performance_indicators Python library.

process_performance_indicators.indicators.time.groups.expected_waiting_time_from_activity_a_to_b

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