Consider an $\frac{M^{[x]}}{M}\frac{?}{1}$ model where the batch sizes are $1,2,$ or $3$ with equal probability, such that $5$ batches arrive per hour on average. The mean service time is $4$ min $.$ Draw the state transition diagram of the queueing model and hence write down the system of balance equations. Also evaluate the traffic intensity, expected system size and expected time spent by a customer. Compare the expected system size of this model with that of an M$/$M$/1$ model with average arrival rate same as the effective arrival rate of the given $\frac{M^{[x]}}{M}\frac{?}{1}$ model.

$14$