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Hello may I ask for your help?

First the setting: I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. Customers arrive with intensity $\lambda_{i,1}$ for certain nodes. After they have been served at a node they move to a node with operation intensity $\mu_{k, j + 1}$ for a certain $k$.

This system also has anti-customers, which move through this grid. They move from node to node with intensity $\alpha$ and wait on this node with intensity $v$. If there is a customer on this node (being served -- thus customers in the queue won't be harmed) they delete this customer with chance 1. and directly move with a certain tactic (it is possible to move to the same node again, with the given intensity).

Second foreshadowing: Now I wanted to make an indication of the chance deleting a customer when there is one anti-customer active and one node. Once a customer is deleted the anti-customer moves directly back to the same node. Now I thought the probability would look like the following:

Note: $AC$:= anti-customer on node and $X_{s}$:= exponential distribution with intensity $s$. $$ P(delete) = P(delete|AC)P(AC) + P(delete|\bar{AC})P(\bar{AC}) $$

Now $P(delete|AC) = 1$ from the text.

$P(AC) = P(X_{\lambda_{1}} < X_{v}) = \frac{\lambda_{1}}{\lambda_{1} + v}$, At least I think -- The chance that AC is on the node.

Furthermore $P(\bar{AC}) = P(X_{v} < X_{\lambda_{1}})) = 1 - P(AC)$, The chance that AC is not on the node, thus the anti-customer left earlier than an arrival.

I couldn't get any further, now a friend suggested for $P(delete|\bar{AC}) = P(X_{\alpha} < X_{\mu_{1}}) = \frac{\alpha}{\mu_{1} + \alpha}$.

The Problem: Now I ran a simulation for this case with $\alpha = v = \lambda = \mu = 1$ the probability I found or to put it differently $\frac{delete}{delete + departure}$, with $delete + departure =$ total customers that left the system, was around $0.5715$.

Note on the simulation: I started with an empty grid and initialized the simulation with moving the AC to the node with the intensity $\alpha$ and the customer with intensity $\lambda_{1}$. There is no difference when initialize step of the AC is moving or waiting (already on a node).

When I use those parameters in the probability equation $P(AC) = 0.75$, which is not near the chance given by my simulation. So what is more wrong my reasoning what the chance is or my simulation? Which is basically the same question. Could someone please help me with this (my) headache?

Discussion: I strongly suggest there is some fault in my reasoning, since it is also possible that an AC arrives and waits on the node and is still able to delete a customer. Furthermore, the chance of an AC on the node is different too, I guess, but I can't get my head around how to get that in the equation.

I thought of Limiting Probabilities, but I am not sure how to use that in this problem.

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