# Modeling an assembly line as a batch processing $M/D^{(b,b)}/r$ queue

Consider an assembly line where some parts are produced in a station following a Poisson process with a rate of $$\lambda$$.

These parts are put directly in a bin (transfer time to the bin is negligible). We have $$r$$ robot arms that take these parts from the bin and place them in the next station.

Each robot arm is programmed to detect the number of parts in the bin, and it only collects and transfers them when there are precisely $$b$$ units available.

It takes $$t$$ time units for an arm to go from the bin to the next station, and it takes $$t$$ time units for it to return to the bin. The time for grabbing and dropping products is negligible.

If we model the robot arms as $$M/D^{(b,b)}/r$$, is the service time $$t$$ or $$2t$$?

I thought since it takes $$t$$ for a batch to be "served" which is it being taken from the bin and placed in the next station, the service time is $$t$$, and the service rate is $$1/t$$.

Does this make sense?

I assume you want to analyze the service time w.r.t one robot grabs and drops the parts and moves between the stations. By knowing the service time for each part is $$\lambda$$ and we need to box precisely $$\text{b}$$ units, the average waiting time for the robot is $$(\text{AWT}_r = \lambda*b)$$. As long as this time has been greater than the transfer time between the stations, the service time of the robot is $$t$$, and as @Sutanu pointed out The t for return will be part of the average wait time for the queue. In the following, this is depicted as an example: The service time for Assy_line was set around $$20$$ parts per hour, while the transfer time between the stations was assumed to be $$1$$ minute. The number of parts in the bin should be exactly $$3$$ to move by a robot. The robot needs to wait to fill the box and its service time is $$t$$. If robots work simultaneously to transfer the parts between the stations, the service time depends on the number of robots and the average number of parts waiting to box.