A store with several registers. People arrive at rate λ, and each register, while someone is there, checks them out at rate μ. Utilization ρ = λ/(cμ) is the fraction of time each register is busy. It is the same whether there is one line or a line at each register. The wait is not. A line at each register is c separate M/M/1 queues: you stay in the line you walked up to, so a register can sit idle while another line still has people. One line feeding every register is an M/M/c queue: the next free register takes the next person. Click a symbol. The store marks the thing that symbol names.
Utilization
The rates are averages
Arrivals and checkouts are messy. λ is how many people arrive per unit time, on average, so the mean gap is 1/λ. μ is how many checkouts one register finishes per unit time, on average, so the mean checkout is 1/μ. The bars are this store. The curve is the exponential those averages imply. The dash is the mean.
Two layouts, same people
Both sides get the same arrivals and the same checkout times. On the left each person picks a register at random and stays. On the right there is one line, and the next free register takes the next person.
People waiting
How many people are standing in line. Orange is every separate line added together. Teal is the one shared line. The dashed line is that layout’s steady mean, Lq, drawn while ρ < 1.
Why the wait gets long
Mean wait Wq against utilization ρ. Both curves climb without a bound as ρ reaches 1. The one shared line stays under the separate lines the whole way. The dots are this store.
Exponential gaps between arrivals, exponential checkout times. The waits are the steady means. The picture is one sample path of both layouts.