Bandwidth packing problem with queueing delays: modelling and exact solution approach

Abstract

We present a more generalized model for the bandwidth packing problem with queuing delays under congestion than available in the extant literature. The problem, under Poison call arrivals and general service times, is set up as a network of spatially distributed independent M/G/1 queues. We further present two exact solution approaches to solve the resulting nonlinear integer programming model. The first method, called finite linearization method, is a conventional Big-M based linearization, resulting in a finite number of constraints, and hence can be solved using an off-the-shelve MIP solver. The second method, called constraint generation method, is based on approximating the non-linear delay terms using supporting hyperplanes, which are generated as needed. Based on our computational study, the constraint generation method outperforms the finite linearization method. Further comparisons of results of our proposed constraint generation method with the Lagrangean relaxation based solution method reported in the literature for the special case of exponential service times clearly demonstrate that our approach outperforms the latter, both in terms of the quality of solution and computation times.

Appendix

We briefly present the mathematical model and the Lagrangian relaxation based solution approach reported by [5] for the special case when $cv = 1$ such that the links in the network are modeled as M / M / 1 queues. For this, we introduce an additional set of variables $W_{ij}^m$ as defined below:

$$\begin{aligned} W^m_{ij}=\left\{ \begin{array}{ll} 1 &{} \text{ if } \text{ call } m\hbox { uses link} (i,j) \text{ in } \text{ either } \text{ direction };\\ 0 &{} \text{ otherwise. } \end{array} \right. \end{aligned}$$

The non-linear integer programming model of this problem is :

On dualizing the constraint set (29) using non-negative lagrangean multipliers $\alpha _{ij}^m$ $\forall (i,j) \in E$ and $m \in M$, the problem $[P_{M/M/1}]$ decomposes into two sets of subproblems: (i) [$L1_{LR}^m$] $\forall m \in M$; and (ii) [$L2_{LR}^E$] $\forall (i, j) \in E$, as given below:

The solution algorithms to solve [$L1_{LR}^m$], LP relaxation of [$L2_{LR}^E$] and to generate feasible solutions are presented Algorithms 3, 4, and 5.

The pseudocode to solve the BPP using Lagrangian Relaxation method is outlined in Algorithm 6.