1. 1 TCOM 5143 Lecture 10 Centralized Networks: Time Delay and Cost Tradeoffs

2. 2 Definition of the problem We want to assign capacities to links in a centralized network in order to minimize the overall link cost, with the total time delay in the network required not to exceed some certain maximum amount.

3. 3 Definition of the problem (cont.) Given The network topology is given: the locations of the user nodes (or terminals), links and central processor are given. We use here a tree topology. Traffic flow on each link. Different capacities available to each link with the corresponding costs. The maximum value of the total time delay allowed in the network.

4. 4 Definition of the problem (cont.) Task Assign a capacity to each link in the network. Goal Minimize total costs.

5. 5 Example (same as that of lecture 8) Suppose that there are seven cities, each with a specified number of terminals (listed below) to be connected to a central computer in Washington, D.C.: Chicago10 terminals Detroit9 Charlotte, NC4 Miami, Fla6 New Orleans6 N.Y.12 Columbus4

6. 6 Example (same as that of lecture 8) Assume that each terminal produces a message, on average, once every 30 seconds and that the average message length is 120 bits. At each city node there is a concentrator that is used to combine incoming messages from terminals and route them over the appropriate outgoing link after some necessary processing and buffering. Let’s assume that network has a tree topology as shown below.

7. 7 0.2 6 =0.4 4 =0.2 3 =0.66 5 =0.2 2 =0.66 1 =0.33 1 2 6 4 3 5 0.33 0.4 0.13 Link 1 Link 2 0.3 Link 6 7 Link 7 7 =0.53 0.13 Link 5 Link 4 Link 3 i =message rate for link i in messages/sec

8. 8 Example (same as that of lecture 8) Four capacity choices are considered in this example. These capacities and their corresponding costs (say in $/month/mile) are listed in the following table. LevelCapacity$/mo/mile 14501.4 29001.7 313501.875 418002.0

9. 9 The average link message delay (in seconds) for messages transmitted over link i is given by Ti= 1/(  i Q i - i ); where Q i is the capacity of link i. (See lecture 8)

10. 10 The table below shows the delay and cost for each available capacity for each link in the network.

11. 11 Problem formulation Parameters n: number of links in the network. m: number of capacity levels. Q ij : Capacity with level j for link i. D: Maximum allowable total link delay in the network. Decision variables

12. 12 Problem formulation The total link cost = Two constraints have to be satisfied: Every link i must be assigned a capacity: The total link delay in the network cannot exceed D:

13. 13 Problem formulation The problem can then be formulated as follows. Minimize Subject to

14. 14 Solution method: Greedy heuristic Step 1: For each link, sort the capacity levels in ascending order of the ratio Step 2: To each link i assign the capacity level with the minimum ratio Let L i be the level of capacity assigned to link i. If the delay constraint is satisfied, stop. The greedy heuristic solution is found. Otherwise go to step 3.

15. 15 Solution method: Greedy heuristic (cont.) Step 3: Select the link i* such that the ratio is the smallest among the same ratios of all other links. If the delay constraint is satisfied, then stop. The greedy heuristic solution is found. Otherwise repeat step 3.

16. 16 Example The maximum allowable total link delay in the network is 1800. Step 1

17. 17 Step 2 Total cost = 329+537+450+534+485+319+529=3183. Total delay = 292+320+325+282+282+298+310=2109 > 1800

18. 18 Step 3 We select i*=6 because its corresponding ratio is the smallest among the same ratios of all other links. Apply the rule:. Thus L 6 = 1+1=2 Total cost = 3252 Total delay =1952 > D=1800. Repeat step 3.

19. 19 Step 3 We select i*=1 because its corresponding ratio is the smallest among the same ratios of all other links. Apply the rule:. Thus L 1 = 1+1=2 Total cost = 3322 Total delay =1799 <= D=1800. Stop; the greedy heuristic solution is found.