  Delay-Differential Equations (DDEs) are the most common representation for
systems with delay. However, the DDE representation is limited. In network
models with delay, the delayed channels are low-dimensional and accounting for
this heterogeneity is not possible in the DDE framework. In addition, DDEs
cannot be used to model difference equations. Furthermore, estimation and
control of systems in DDE format has proven challenging, despite decades of
study. In this paper, we examine alternative representations for systems with
delay and provide formulae for conversion between representations. First, we
examine the Differential-Difference (DDF) formulation which allows us to
represent the low-dimensional nature of delayed information. Next, we examine
the coupled ODE-PDE formulation, for which backstepping methods have recently
become available. Finally, we consider the algebraic Partial-Integral Equation
(PIE) representation, which allows the optimal estimation and control problems
to be solved efficiently through the use of recent software packages such as
PIETOOLS. In each case, we consider a very general class of delay systems,
specifically accounting for all four possible sources of delay - state delay,
input delay, output delay, and process delay. We then apply these
representations to 3 archetypical network models.
