In this paper the steady-state behaviour of many symmetric queues, under the head of the line processor-sharing discipline, is investigated. The arrival process to each of n queues is Poisson, with rate λ, and each queue has r waiting spaces. A job arriving at a full queue is lost. The queues are served by a single exponential server, which has a mean rate μn, and splits its capacity equally amongst the jobs at the head of each nonempty queue. The normal traffic case ρ=λ/μ<1 is considered, and it is assumed that n>>1 and r=O(1). A 2-term asymptotic approximation to the loss probability L is derived, and it is found that L=O(n’-r), for fixed ρ. If δ=(1-ρ)/ρ•1, then the approximation is valid if nδ2>>1 and (r+1)2•nδ, and in this case L∼r!/(nδ)r. Numerical values of L are obtained for r=1,2,3,4 and 5,n=1000, 500 and 200, and various values of ρ<1. Very small loss probabilities may be obtained with appropriate values of these parameters.
