Allowing an empty (epsilon) transition in a PDA I'm trying to allow an empty transition in a PDA for the following language:


Alphabet: $Σ = \{a, b, c\}$
Language: $L = \{ a^ib^j \mid i \neq j \} \cdot \{ c \}^\ast$


Examples of words in $L$:


$\varepsilon$
$aabccc$
$abbccc$


Not in $L$:


$abcc$
$aabbc$


Here is what I came up with:



The diagram above uses JFLAP - where the symbol $Z$ reflects the empty stack. The symbol $λ$ is the empty symbol $ϵ$.

It accepts everything as it should, but I don't know how to let epsilon get through. q7 to q8 is when there is more b than a. So there should be a way to allow q7 to q9 where a is more than b but also epsilon can get through. Thoughts? I would like to simply set epsilon through but than aabbc can get through easily enough.
Some key points: 


The only case where you want to accept $\epsilon$ is when you haven't had any other input and your stack is empty. Therefore you need an accepting state right at the beginning.
You need to take into consideration inputs such as: $bac, aac,...$ that will never result in an accepting state, or $abbb$ that results in an accepting state without containing any $c$'s
If you get a $b$ and your stack is empty you have more $b$'s than $a$'s so you need an accepting state
If you get a $c$ and and your stack is not empty you have more $a$'s than $b$'s so you need an accepting state
You reject everything else.