Consider a grocery store with two queues. At either queue, a customer

has an exponential service time with an expected value of $3$ minutes. Customers arrive at

the two queues as a Poisson process of rate $\backslash$lambda customers per minute. Consider the

following possibilities:

$(1)$ Customers choose a queue at random so each queue has a Poisson arrival process

of rate $\backslash$lambda $/2.$

$(2)$ Customers wait in a combined line. When a customer completes service at either

queue, the customer at the front of the line goes into service.

For each system, calculate the limiting state probabilities. Under which system is the

average number of customers smaller?