From a differential geometric perspective, Information Geometry (IG) aims to characterise the structure of statistical geodesic models. The research done for this paper offers a novel method for modelling the IG of a queuing system. From the perspective of IG, the manifold of the temporary M/M/∞ queue is described in this context. The Fisher Information Matrix (FIM) as well as the Inverse of FIM, (IFIM) of transient M/M/∞ Queue Manifold (QM) are devised. In addition to that, new results that uncovered the significant impact of stability of M/M/∞ QM on the existence of IFIM are obtained. Moreover, the Geodesic Equations (GEs) of motion of the coordinates of the underlying transient M/M/∞ are obtained. Also, it is revealed that stable M/M/∞ QM is developable (i.e., has a zero Gaussian curvature) and has a non-zero Ricci Curvature Tensor (RCT). The Information Matrix Exponential (IME) is devised. Also, it is shown that stability of the devised IME enforces the instability of M/M/∞ QM. More interestingly, novel relativistic info-geometric queueing theoretic links are revealed. A summary combined with future research work are given.