Markovian queueing networks models are widely used for performance evaluation of
computer systems, production lines, communication networks and so on. Routing strategies
allocate clients to queues after the end of service. In many situations such as deterministic,
probabilistic, or state dependent like Join the shortest queue routing, the routing function
could be written in terms of index scheduling functions introduced in [?,?]. When the
capacities of queues are finite [?], product form property of the steady-state usually vanishes …