Consider our checkout counter example. Assume that there are two types of customers who arrive according to independent Bernoulli processes with rates p1 belongs to (0,1) and p2 belongs to (0,1), respectively. The overall arrival process of all customers follows a merged Bernoulli process of the two separate Bernoulli processes. All customers who arrive join a single queue, which has a capacity of 10 customers. We are interested in making predictions about the length of the queue at any point in time. For each of the following parts, choose the correct statement.
1. Assume that service times are not type-dependent and are modelled as independent geometric random variables with parameter q belongs to (0,1) for all customers in the queue. One can model this queue using the same transition probability graph as in the previous video with \\( p=\\left(p_{1}+p_{2}\\right) / 2 \\) and \\( q \\). One can model this queue using the same transition probability graph as in the previous video with \\( p=1-\\left(1-p_{1}\\right)\\left(1-p_{2}\\right) \\) and \\( q \\). One can model this queue using the same transition probability graph as in the previous video with some other appropriate choice of \\( p \\) and \\( q \\). There are no values of \\( p \\) and \\( q \\) for which one can model the queue using the same transition probability graph as in the previous video. 2. Assume now that service times are type-dependent and are modelled as independent geometric random variables with parameters \\( q_{1} \\in(0,1) \\) and \\( q_{2} \\in(0,1) \\), respectively, for the two types of customers. One can model the queue using the same transition probability graph as in the previous video with \\( p=\\left(p_{1}+p_{2}\\right) / 2 \\) and \\( q=\\left(q_{1}+q_{2}\\right) / 2 \\). One can model the queue using the same transition probability graph as in the previous video with \\( p=1-\\left(1-p_{1}\\right)\\left(1-p_{2}\\right) \\) and \\( q=\\left(p_{1} q_{1}+p_{2} q_{2}\\right) /\\left(p_{1}+p_{2}\\right) \\). One can model this queue using the same transition probability graph as in the previous video with some other appropriate choice of \\( p \\) and \\( q \\). There are no values of \\( p \\) and \\( q \\) for which one can model the queue using the same transition probability graph as in the previous video.
1. One can model this queue using the same transition probability graph as in the previous video with \\( p=1-\\left(1-p_{1}\\right)\\left(1-p_{2}\\right) \\) and \\( q \\).
2. One can model the queue using the same transition probability graph as in the previous video with \\( p=\\left(p_{1}+p_{2}\\right) / 2 \\) and \\( q=\\left(p_{1} q_{1}+p_{2} q_{2}\\right) /\\left(p_{1}+p_{2}\\right) \\).