Sequencing two servers on a sphere

Abstract

We analyze a service system in which two servers move independently on the surface of the n-dimensional sphere. Requests for service arrive independently and uniformly over the surface. The ith request is to be served completely before service of the (i+1)st request begins. In an earlier paper the authors showed that the nearer server (NS) policy is optimal among all server selection policies in the sense that it minimizes the equilibrium expected angular distance E(D) which a server moves to process a request. In the present paper we obtain for all n the integral equation satisfied by the equilibrium measure for the angular distance Φ between servers under the NS policy. The equation is solved numerically. We also show that E(D)+E(Φ) = π, and unse this to compute E(D).