Queueing with alternating priorities, treated as random walk on the lattice in the plane

The queueing system with alternating priorities has been discussed recently in numerous articles (Maxwell [1], Avi Itzhak, Maxwell, Miller [2], Neuts, Yadin [3], and others). Consider a system which consists of several queueing units. Each unit is independent in the sense that it has its own waiting line which is generated by an independent Poisson stream of customers. However, all of the units are served by a single server who allocates his time between the units.