This paper presents a new approximation for the response times in a production system with M machines and s operators, where each machine has its own arrival stream of jobs, where no machine can work without the constant attention of an operator and where the operators can serve only one machine at a time. The arrival streams are Poisson with equal rates and all jobs have the same service time distribution. Several strategies are considered for the operators. Extensive simulations show that for the case of exponential service times the differences between these strategies are usually small. The approximation is also quite good for non-exponential service times.