It's not just the arrival rate that is a poisson distribution, but it's also the processing time being a poisson distribution that really sets the table for the scenario the article is addressing. For me, it's easier to understand in a grocery store situation, where customers arrive in a poisson distribution at the checkout line, and that each cart has a poisson distribution on how long it takes to process the items in the cart. That second level of variance is key to the queue buildup, especially as the grocer checker reaches 100% utilization.