Having read the Queuing Theory Cookbook and having seen the calculations that are used for different characteristics of the models, specifically the M$/$M$/1$ model; we can use the formula for the number of people we can expect to see in a waiting line given a certain arrival rate and service rate, $($lambda and mu respectively.$)$ Given that $$p$$ is equal to lambda$/$mu in the queuing theory cookbook we can use the formula below substituting lambda and mu for $$p$.$

Lq $=$ Lambda $2$

$-----------------------$

Mu $($Mu $$ Lambda$)$

One of the rules of queuing theory is that the service rate MUST exceed the arrival rate. In this exercise we will illustrate with a graph why that is necessary. In the accompanying Excel spreadsheet the A column holds arrival rates and the B column holds service rates. The C column indicates that there is one service channel. In the column indicated, place a calculation reflecting the formula above substituting the number in the A column for Lambda and the B column for Mu$.$ Use the same formula for each row.

The first group shows the service rate approaching the arrival rate, and the bottom set essentially shows the opposite.

As you fill in formulas the graphs will take shape.

$1.$ What do the graphs indicate to you? Describe in a couple sentences below:

$2.$ Now, at the bottom of each section place matching numbers at the bottom of the series, eg$.30$ and $30$ at the bottom of the first group. What happens to the graph?

$3.$ Remember now the graph indicates the size of the waiting line. $($Think of the early voting lines you saw on television during the $2020$ US election season.$)$ When arrivals match or exceed the service rate, what will happen to the waiting line?