Problems

1) (Problem 5.1) Show that the throughput performance vector (TH₁,…,TH_K) belongs to the polytope T, where
and c_k = b_k/m_k.

2) (Problem 5.3) This problem shows that the natural generalization of the product bound for networks with multiservice traffic is not valid. Consider a product-form loss network with J = 2, K = 3, C₁ = C₂ = 2, R₁ = {1,2}, R₂ = {1}, R₃ = {2}, b₁ = 1, b₂ = b₃ = 2 and r₂ = r₃. Thus the narrowband class uses both links whereas the two wideband classes each use one link. Denote b₁ for the probability of blocking of class-1 calls. Also consider a stochastic knapsack with K = 2, C = 2, and b₁, b₂, r₁, r₂ having the same values as for the two-link network. Let be the probability of blocking class-1 calls in the knapsack. Determine values for r₁ and r₂ such that .