This paper presents an accurate model for evaluating the mean system size of parallel queues with the Join the Shortest Queue (JSQ) policy. The system considered consists of N identical queues with infinite buffers, and each of the queues has one server. The job arrival process is assumed to be Poisson. The service times are assumed to be exponentially distributed. When a job arrives at the system, it is sent to the queue with the smallest number of jobs. Ties are broken by randomly selecting one of the queues with the minimal number of jobs. Exact analysis of the system is known to be very difficult, and our model gives very accurate results. A birth-death Markov process is used to model the evolution of the number of jobs in the system. An iterative procedure is devised to estimate the state transition rates. The mean job response time can then be calculated. Extensive simulations are performed and compared with the analytical results. Our results show that this method provides very accurate estimates (within 3.5%) of the mean job response times for N up to 64.