# Effect on wait time of combining queues

This is a real-world question. I'm trying to convince people to consider changing this process to improve service.

There are 20 servers, each fed by it's own queue. Each queue has 30 customers when the process starts. No new customers are added, and none leave or switch queues, so that at the end, 600 customers will have been served. Service time is exponential. Currently, when a server finishes its 30 customers, it goes out of service.

If I combine the queues so that we have one queue feeding all 20 servers in FIFO order, what is the impact on average wait time?

When I simulate this situation, I get that the isolated queue approach takes around 41 time units, while the communal queue takes around 32.50 They remain in this ratio regardless of what the exponential parameter $\lambda$ is. (The time is inversely proportional to $\lambda$.)