A comment on Boucherie product-form results

This is a sketch about Boucherie paper on product form. It is quite informal, for a deeper view we suggest to study the interesting paper [2]. Our aim is just to point out the main main conditions for this product form. 1 The Boucherie result on the product form The starting CTMCs In this section we summarize the results presented in [2]. Let us consider K stable and regular CTMCs. The state spaces S1, . . . SK of the CTMCs are finite or countable. Let qk(nk, nk) be the transition rate from state nk to nk of k-th CTMC. Let πk : Sk → [0, 1] denote the normalized stationary probability function, i.e., ∑ nk∈Sk πk(nk) = 1. Let n = (n1, . . . , nK) with nk ∈ Sk and k = 1, . . . , K be a state of the composed process. Clearly if the CTMCs are independent we have product form, i.e., π(n) = ∏K k=1 πk(nk). The competition Let use introduce an index set I. Intuitively I is a set which labels the resources. In each state a CTMC uses one and only one resource. So we can partition a state space Sk in I mutually exclusive sets Aki such that: ∪i∈IAki = Sk ∧ Aki 6= ∅ If CTMCs k1 and k2 compete over resources i then this means that the combined process does not reach the state n where components nk1 ∈ Ak1i and nk2 ∈ Ak2i. We denote with Cki the set of CTMCs which compete with CTMC k on the resource i ∈ I. The product form condition Let us study the process with state space S = S1× . . .×Sk and let us define the following transition matrix for the CTMC on S: