# How do I minimize a queue with two servers, or can I?

I have a queue with a constant stream of customers which come in at a rate greater than the system can process them.

The first server is a two register system. The second server is a two man capacity sandwich station for making sandwiches.

There are three employees and I want to know the optimal deployment for 1) keeping the queue as small as possible overall both at the pickup counter and at the register combined and 2) moving people at the fastest rate.

I have it modeled so that the registers process 4 orders per minute with two workers and 2 orders per minute with one worker. Then the customer moves to the pickup counter and waits for the sandwiches. The sandwiches process 3.5 orders per minute with two workers and 1.85 orders per minute with one worker.

I am confused about the proportion of deployment. I have already determined that I should have 2 workers on registers about 41% of the time and 2 workers on sandwiches 59% of the time. What I am trying to understand is, is it better to constantly be moving the workers back and forth to achieve this balance, or does it matter, or is it better to lump this deployment into large chunks of time spent at each so long as the allotments of time average out to 41%? (Assume no transition costs.) And does maintaining these proportions in certain time frames actually decrease wait times for individual customers? Is this kind of a 6 of one vs half dozen the other kind of situation?

## 1 Answer

Since your system is unstable, the queue lengths and hence the waiting time of customers is going to infinity. It would make sense however to ask the question on a finite interval say a day.

Your answer seems to be coming from the solution of a continuous (fluid limit) version of the system where the server can be shared between the two stations to balance the throughput of both stations. In that case the control is the amount of time the extra server spends on each station up to time $t$.

The exact solution of the problem I believe depends on the initial condition, arrival rate and the length of the horizon but assuming a large initial condition and since the system is blowing up I think as long as the time allocations are such that the output rate of stations are balanced you achieve the minimum waiting times. So how you implement it wouldn't make a difference.