  In stochastic models for queues and their networks, random events evolve in
time. A process for their backward evolution is referred to as a time reversed
process. It is often greatly helpful to view a stochastic model from two
different time directions. In particular, if some property is unchanged under
time reversal, we may better understand that property. A concept of
reversibility is invented for this invariance. Local balance for a stationary
Markov chain has been used for a weaker version of the reversibility. However,
it is still too strong for queueing applications.
  We are concerned with a continuous time Markov chain, but dose not assume it
has the stationary distribution. We define reversibility in structure as an
invariant property of a family of the set of models under certain operation.
The member of this set is a pair of transition rate function and its supporting
measure, and each set represents dynamics of queueing systems such as arrivals
and departures. We use a permutation {\Gamma} of the family menmbers, that is,
the sets themselves, to describe the change of the dynamics under time
reversal. This reversibility is is called {\Gamma}-reversibility in structure.
  To apply these definitions, we introduce new classes of models, called
reacting systems and self-reacting systems. Using those definitions and models,
we give a unified view for queues and their networks which have reversibility
in structure, and show how their stationary distributions can be obtained. They
include symmetric service, batch movements and state dependent routing.
