  Flow of molecular gas into a complex vacuum system is investigated by a
lumped parameter model to estimate the time evolution of gas pressure $p_g$,
which for the first time takes into account the realistic effect of time-delay
arising due to multiple reasons such as valve response, pumping and transport
of gas, conductance of the pipe network, etc. The net effect of all such delays
taken together into a single (constant) delay term gives rise to a scalar delay
differential equation (DDE). Analytical solutions of a linear DDE in presence
of external forcing due to the (pulsed) injection of gas and outgassing from a
stainless steel (SS) wall are then derived using the Laplace transform method,
where the transcendental characteristic equation is approximated by means of
rational transfer functions such as diagonal Pad{\'e} approximation. A good
agreement is obtained with the numerical results and it is shown from these
solutions that a reasonably good match with experimental results is obtained
only in the presence of a nonzero time-delay. An attempt is also made to
simplify the DDE into an ordinary differential equation (ODE) by using a
low-order Taylor series expansion of the time-delay term. A singular
perturbation method is used to solve the resultant initial value problem (IVP)
type second-order ODE. It is found that unlike the Laplace transform method,
the ODE closely approximates the DDE only if the delay is small. Emergence of
stable periodic oscillations in $p_g$, as a generic supercritical Hopf
bifurcation, if time-delay exceeds a critical value is then established by
numerical and Poincar{\'e}-Lindstedt method.
