Wireless Network Design: Optimization Models and Solution ...

8 The Design of Partially Survivable Networks 179
versions are developed in Sections 8.3 and 8.4, respectively. Conclusions are in
Section 8.5.
8.2 The Single Period Problem
All cells in a cellular network must be able to communicate with each other and
with the MTSO. This communication is achieved by electronic connection of the
base stations to the MTSO through a fixed network, which may use copper cable,
fiber optics, microwave radio, or (less frequently) satellite links. Transmission rates
for this fixed part of the network tend to be high — for example, as early as 2000,
AirTouch Cellular had built its own OC-12 and OC-48 optical rings to backhaul
cellular traffic among hubs and MTSOs in the Los Angeles metropolitan area.
In the single period problem we are given a backbone network comprised of a
self-healing SONET ring of known capacity. The locations of the nodes (i.e., the
hubs and the MTSO) on the ring are assumed to be known. The traffic on the ring is
assumed to be fully protected as the ring is self-healing. The demand at a cell is the
number of DS-0 channels needed during “busy hour”. In order to increase reliability,
cells are often connected to multiple nodes on the ring; the number of nodes to which
a cell is to be connected is its diversity requirement. It is assumed that a cell’s traffic
is split equally among the nodes it is connected to — for instance, a cell with a
diversity requirement value of 2 would have half its traffic routed through one node
and the remaining half through another node. This ensures partial survivability in
case of call-node link failure, as only 50% of the traffic from this cell will be lost.
The objective is to find a least-cost cell-to-hub interconnection network that satisfies
the diversity requirements.
Figure 8.1 shows an example network. Nodes a–e are hubs which are connected
by a high capacity ring while the cells u–z need to be assigned. In this example, the
diversity requirements of nodes v, x and y are 1, 2 and 3, respectively. The arrows
denote one possible assignment of cells to nodes.
8.2.1 Problem Formulation
Dutta and Kubat [4] formulated the problem as (P1) below, after defining the following.
V is the set of cells to be connected to the ring, while H is the set of hubs (excluding
MTSO). R is the set of nodes including the MTSO (i.e., R = H ∪ {MTSO}.
The capacity of the ring is K, while the demand from cell i is Di; both are in numbers
of DS-0 circuits. Si is the diversity requirement of cell i, and ci j is the cost of
connecting cell i to hub j, in dollars per year. The decision variable xi j is 1 if there
is a link from cell i to hub j, and 0 otherwise.