$3.$ Consider a semi$-$automated packaging machine. A worker loads the products into the machine, the machine packs the product, and the worker unloads the packed product.

$-$Loading takes $2$ minutes.

$-$Packaging takes $5$ minutes.

$-$Unloading takes $1$ minute.

$-$The machine can only pack one product at a time.

$-$The worker must finish loading or unloading when they start it$.$ They cannot leave these tasks unfinished.

$-$The packing starts automatically as soon as the product is loaded.

$-$The unloading can be done anytime after the packing is finished. The packed product can wait in the machine until the worker unloads it$.$ The worker must unload the packed product before loading a new product into the machine.

a$.$ Model the process with a Petri net. Draw the Petri net. Label transitions and places.

b$.$ Extract the transition matrix for your Petri net for one complete cycle.