There is a room which has fixed spaces for people to sit and eat food (say benches). Each such space has a desirability rating depending upon a) Its location relative to the door and b) the number of occupied spaces around it at a particular time. Also, considering 1 person per space. So, the desirability of a space will be: $$ D(i) = - \biggl(C_i + \sum_{k=0}^n {\operatorname{IsOccupied}(k) C_{i,k}} \biggr) $$ Where $i$ is the selected space, $k$ are numbers denoting the rest of the spaces, $C_{i,k}$ is the steric effect constant on space $i$ due to spaces $k$ being occupied and $\operatorname{IsOccupied}(k)$ is a boolean function which gives $1$ as output when the space k is occupied and $0$ if not. Hence, $0$ being the highest desirability.
Now, consider people start coming in with a time interval $\lambda$ [Poisson distribution] and occupying spaces such that, the every person will choose the most desirable space. (In case of equal desirability, just choose one of those at random)
Now, every person, takes some time to eat food and leave: $\Delta t$~$N[\mu,\sigma^2]$; Let $\mu=15minutes$ and $\sigma = 5 minutes$
If the mess starts at time $t_0$, considering these, can we find the probability of a person coming in at time $t_1 = $ some chosen time, to get a space with a desirability above a certain level (say $-0.5$)?
Layout of 8 benches. Example simulation
Taking from the layout in the picture $n=8$. Following are the values of constants:
$ C_1=0.3;\space C_2=0.2;\space C_3=0.15;\space C_4=0.8;\space C_5=0.6;\space C_6=0.2;\space C_7=0;\space C_8=0.15;\space C_{1,2}=0.7=C_{2,1};\space C_{1,4}=0.5=C_{4,1};\space C_{1,5}=0.2=C_{5,1};\space C_{2,5}=0.5=C_{5,2};\space C_{2,3}=0.7=C_{3,2};\space C_{3,5}=0.5=C_{5,3};\space C_{6,4}=0.45=C_{4,6};\space C_{6,7}=0.7=C_{7,6};\space C_{7,5}=0.45=C_{5,7};\space C_{7,4}=0.44=C_{4,7};\space C_{7,8}=0.5=C_{8,7};\space C_{8,5}=0.4=C_{5,8};\space $ For all unspecified values: $C_{i,k}=0$
I looked for it in some simulation software, but could not see how to get the specific type of answer. Any idea of what kind of path I might take to get to the answer is appreciated.
Edit: Did a quick sanity check and realised there was no way to get a positive $D(i)$.