The variables ε, δ, and tF, are 1-Wire master circuitry related and are not specified in the DS2431 or DS28E07 data sheets.  These numbers will vary from implementation to implementation. Only the electrical characteristics of the 1-Wire slave are specified in the data sheets. 
  • ε is in the DS2431 data sheet’s Figure 11, and it represents the time required for the pullup circuitry to pull the voltage on IO up from VIL to VTH.  The pullup circuitry is external to the slave and not under the slave’s control. 
  • δ in Figure 11 represents the time required for the pullup circuitry to pull the voltage on IO up from VIL to the input-high threshold of the bus master. Again, this is not under the slave's control, but the 1-Wire master.
  • tF is not defined in the 1-Wire slave data sheet but represents the “Fall time” – the time it takes for the 1-Wire master circuitry to bring the voltage down from VPUP to VTL.  Figures 10 and 11 show tF.  You will find tF in 1-Wire master data sheets, such as the DS2485 where it is a “settable” parameter – see Table 38 on page 32 of the DS2485’s data sheet.

Both the DS2431 and DS28E07 data sheets have “maximum duration” notes (notes 17 and 18 in the DS2431's data sheet). 

For example, note 17's “maximum duration" represents the max time it takes for the 1-Wire master to bring the line low to when it releases the line.  See picture below. Looking at the variables, this would be:  tW1LMAX + tF – ε (note in this case tW1L represents tW1LMAX).  The values for tF and ε are dependent upon the 1-Wire master implementation.  Analog usually puts the tF value in the data sheet of the 1-Wire master.  If the customer creates his/her own 1-Wire master from a microcontroller port pin, these values can be measured. 

The data sheet says that ε’s duration “depends on the pullup resistor (RPUP) used and the capacitance of the 1-Wire network attached”.

If the customer wants to know what times are recommended for a generic 1-Wire Master “duration” of a Write-One LOW, please see the following application note titled “1-Wire Communication Through Software.”  Specifically, take a look at Figure 1 and Table 2.