Question

2. In the telecommunication industry, telephone companies typically connect each customer directly to a switching center, which is a device that routes calls between the users in the system. Alternatively, to use fewer cables for routing the telephone calls, a company can combine the calls of several customers in a message compression device known as a concentrator and then use a single cable to route all of the calls transmitted by those users to the switching center. Constructing a concentrator at any node in the telephone network incurs a node- specific cost and assigning each customer to any concentrator incurs a “homing cost" that depends on the customer and the concentrator location. Suppose that all of the customers lie on a path and that we wish to identify the optimal location of concentrators to service these customers assume that we must assign each customer to one of the concentrators). Suppose further that the set of customers allocated to any concentrator must be contiguous on the path many telephone companies use this customer grouping policy). Design an efficient algorithm to find an optimal set of concentrators that serve all the customers on the path with a minimum total incurred cost. (15 points)

Answer 1

In a telecommunication network where a large number of widely dispersed users share a limited number of resources, the strategy for access will play a large part in determining the cost and performance of the network. We consider the problem of locating "access facilities," or concentration points, to obtain an economic connection of users to resources. The problem is formulated as the locating of generic access facilities (GAF's) to obtain an economic connection of nodes (users) to a resource connection point (RESCOP). The nodes may be connected through multipoint lines, but with a constraint on the number of nodes which may share a single line. The GAF's are constrained in capacity, expressed as the number of nodes they can support, and have a cost associated with them.

The efficient algorithm to find an optimal set of concentrators that serve all the customers on the path with a minimum total incurred cost characterized by the following four steps:-

1) Simplify the problem to a point-to-point problem by replacing clusters of nodes by single "center-of-mass" (COM) nodes.

2) Partition the reduced set of COM nodes by applying an Add algorithm, resulting in one of the COM nodes selected as a GAF site.

3) Select one of the original nodes as a real GAF site in each partition by examining the original nodes closest to the COM node selected in the Add algorithm, and selecting the best.

4) Apply a line-layout algorithm to each partition, with its selected GAF site serving as the central node.