On the Positive Harris Recurrence for Multiclass Queueing Networks: a Uniied Approach via Uid Limit Models

A heavy traac limit theorem for networks of queues with multiple customer types. 27 We now compute the principal minors of D k 2 and D k 3. First consider D k 2. All the principal minors of order 2 are equal to 1. The principal minor corresponding to f1; 3; 4g is equal to 1 + 3 1 3 4 == 2 , and the remaining principal minors of order 3 are equal to 1. The determinant of the matrix itself is 1 ? 1 2 3 4. Now consider D k 3. Once again, the principal minors of order 2 are all equal to 1. The principal minor corresponding to f1; 2; 3g is equal to 1 + 1 2 3 1 == 4 , and that corresponding to f1; 3; 4g is equal to 1 ? 1 3 4 , while the remaining two principal minors of order 3 are equal to 1. The determinant of the matrix is 1 ? 1 2 3 4. Thus from the remark preceding the lemma it is clear that under the condition 1 2 3 4 < 1; (5.23) D k 2 and D k 3 are completely-S matrices. Finally consider D k 4. Since every entry in the matrix is nonnegative, and the diagonal elements are all strictly positive (in fact equal to 1), it is easy to see that for every principal submatrix A of D k 4 , A1 > 0, where 1 is the vector of all ones of the corresponding dimension. This directly veriies the completely-S property for D k 4. References 1] R. Atar and P. Dupuis. Large deviations and queueing networks: methods for rate function identiication.