Two Parallel Queues Created by Arrivals with Two Demands II

We consider the double queue that arises when arriving customers simultaneously place two demands handled independently by two servers. It is assumed that the customer arrivals form a Poisson process with mean 1, the servers have exponential service times with rate α ,β, and 1 < α β, which insures stability of the queue. Let X₁ ,X₂ be the respective lengths of the α- and β-queues, and Pₘₙ = P [ X₁ = m,X₂ = n ] at equilibrium. In a previous paper with the same title we obtained a formula for the generating function P( z,w ) = ∑ pₘₙ zᵐ wⁿ. We use this to derive the asymptotic behavior of pₘₙ as m,n → ∞. The asymptotic results are employed to study the interdependence of X₁ ,X₂. We derive limit laws for the expectation and distribution for either of these random variables conditioned on the other.