Non-product form equilibrium probabilities in a class of two-station closed reentrant queueing networks

Abstract

While many single station queues possess explicit forms for their equilibrium probabilities, queueing networks are more problematic. Outside of the class of product form networks (e.g., Jackson, Kelly, and BCMP networks), one must resort to bounds, simulation, asymptotic studies or approximations. By focusing on a class of two-station closed reentrant queueing networks under the last buffer first served (LBFS) policy, we show that non-product form equilibrium probabilities can be obtained. When the number of customer classes in the network is five or fewer, explicit solutions can be obtained. Otherwise, we require the roots of a characteristic polynomial and a matrix inversion that depend only on the network topology. The approach relies on two key points. First, under LBFS, the state space can be reduced to four dimensions independent of the number of buffers in the system. Second, there is a sense of spatial causality in the global balance equations that can then be exploited.

To our knowledge, these two-station closed reentrant queueing networks under LBFS represent the first class of queueing networks for which explicit non-product form equilibrium probabilities can be constructed (for five customer classes or less), the generic form of the equilibrium probabilities can be deduced and matrix analytic approaches can be applied. As discussed via example, there may be other networks for which related observations can be exploited.

Appendices

Appendix A: Global balance equations

Here, we obtain the matrix form of the global balance equations and introduce matrix notation omitted in the paper. Similarly to floor 0, we obtain the GBE matrix form for floor 1 as

where

for i=1,…,n−1,

All A _i , B _i , and C ₁’s are m×m matrices. Note that assuming we know the initial conditions Y[−1], Y[1] can be written as Y[1]=S Y[−1] for an appropriate matrix S. Similarly, the general equations for the GBEs can obtained as

2≤k≤N−2, where A _i , B _i and C ₁ are as before.

Letting

We obtain

Appendix B: Matrix definitions for the transition probability matrix

Here, we give notation for the sub-matrices within the transition probability matrix. For convenience, we append virtual states to floors −1 and N−1, so that they too have mn states. Transitions to and from these states occur with probability 0. We can express the transition probability matrix T as

The matrix $P_{00}^{\prime}$ and Θs are m×m matrices.