Use of Peer-to-Peer (P2P) service networks introduces a new communication paradigm because peers are both clients and servers and so each peer may provide/request services to/from other peers. Empirical studies of P2P networks have been undertaken and reveal useful characteristics. However there is to date little analytical work to describe P2P networks with respect to their communication paradigm and their interconnections. This paper provides an analytical formulation and optimisation of peer connection efficiency, in terms of minimising the fraction of wasted connection time. Peer connection efficiency is analysed for both a uni- and multi-connected peer. Given this fundamental optimisation, the paper optimises the number of connections that peers should make use of as a function of network load, in terms of minimising the total queue size that requests in the P2P network experience. The results of this paper provide a basis for engineering high performance P2P interconnection networks. The optimisations are useful for reducing bandwidth and power consumption, e.g. in the case of peers being mobile devices with a limited power supply. Also these results could be used to determine when a (virtual) circuit should be switched to support a connection.