Random walks in a queueing network environment

Models of Markov processes with a random transition mechanism

“…It has to be stressed again that our results depend on a special choice of the transition mechanism (inherited from Ref. [4]) where the SE and the basic MPs have a particular influence on each other.…”

“…The problem can be stated in the following form. Suppose we are given a family of 'basic' -----------1 St Petersburg State University of Architecture and Civil Engineering, 190005 RF 2 IITP, Moscow-127994, RF; 5 DPMMS, University of Cambridge, CB3 0WB UK; 3 Math Department, PSU, State College, PA16802 USA; 4 IHES, Bures-sur-Yvette, 91440 France; E-mail: yms@statslab.cam.ac.uk; ims14@psu.edu continuous-time Markov processes (MPs) X (z) (t) on a state space X where z is a parameter describing a 'state of environment' (SE) which can be varied within a space Z. Assume that ∀ z ∈ Z, process X (z) has an invariant measure (IM) ν (z) .…”

“…The IM is a product of geometric distributions identified in terms of λ, µ and P. Making triple (λ, µ, P) dependent on parameter z varying randomly within a finite or countable set Z leads to a number of interesting applications, viz., networks with 'distinguished customers'; cf. [4] and [17]. Our construction for this example is carried in Section 2; it leads to a 'modified' product-form IM for the combined MP and -after the normalization -to a combined equilibrium probability distribution (EPD).…”

“…The combined generator. We now give the description of the model modifying the basic example from [4]. Let Z be a finite or a countable set.…”