1. 1 Dr. Ali Amiri TCOM 5143 Lecture 8 Capacity Assignment in Centralized Networks

2. 2 Dr. Ali Amiri Definition of the network problem Given network data: The network topology is given: the locations of the terminals, links and central processor are given. In the simplest case, we assume that link cost is proportional to capacity. A budget to acquire link capacities. This implies that total link capacity is also given. Why?

3. 3 Dr. Ali Amiri Given network data (cont.) The network traffic statistics (requirements) are given; i.e., we know: The average message lengths Average arrival rates of messages: number of messages flowing between any two points in the network.

4. 4 Dr. Ali Amiri Task Assign a capacity (in bps) to each link in the network to provide a specified level of service to users (or any other network performance criteria).

5. 5 Dr. Ali Amiri Network performance criteria There are two main criteria to evaluate the quality of service to users: Minimize the average message time delay in the network. Minimize the largest time delay expected in any link in the network.

6. 6 Dr. Ali Amiri Notes Most of the discussion here is based on the pioneering work of L. Kleinrock. The examples in this lecture are taken from Schwartz.

7. 7 Dr. Ali Amiri Example 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

8. 8 Dr. Ali Amiri Star Network Topology Let’s assume that network has a star topology as shown below. 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.

9. 9 Dr. Ali Amiri … 1 2 6 4 3 7 5 … … … … … …

10. 10 Dr. Ali Amiri Let i = average message rate (messages/sec) for link i. C i = capacity (in bps) for link i. The job is to determine the capacity (in bps) to allocate to each of the seven links in the network.

11. 11 Dr. Ali Amiri We assume that nodal processing delay is negligible compared to the queuing delay incurred by messages waiting for the outgoing link to be made available. Link i C i bps i messages/sec

12. 12 Dr. Ali Amiri The average arrival message rate for link i (that is the average number of messages per second arriving at link i ) is the sum of all incoming messages routed over that link.  In the example, all the terminals have the same message rate (1/30 messages/sec). Therefore, 1 = 10* 1/30 =0.33 messages/sec,  2 = 9* 1/30 =0.3 messages/sec, etc.

13. 13 Dr. Ali Amiri Message arrival rates for the example 6 =0.4 4 =0.2 3 =0.13 7 =0.13 5 =0.2 2 =0.3 1 =0.33 … 1 2 6 4 3 7 5 … … … … … …

14. 14 Dr. Ali Amiri We make the following additional assumptions necessary to model the telecommunication network as a network of independent M/M/1 queues: A Poisson message arrival rate with i messages/sec arriving on average at link i. An exponential distribution of message lengths with 1/  i bits/message on average for link i. Infinitely large buffer capacity. Messages arrive independently.

15. 15 Dr. Ali Amiri Under these assumptions and by applying queuing theory, the average time queuing delay in seconds incurred by messages at link i is given by: Ti= 1/(  i C i - i ), where C i is the capacity of link i that needs to determined. This delay includes the average time taken to transmit a message (1/(  i C i ) sec) plus the message buffering delay.  i = i /(  i C i ) is called the traffic intensity (or utilization factor) for link i. It is value is always less than 1.

16. 16 Dr. Ali Amiri T i can be defined in terms of  i, Therefore, T i increases with:  Traffic intensity (  i ).  The increase in the average time required to transmit a message 1/(  i C i ). In our example, the average message length is 1/  i = 120 bits/message for each link.

17. 17 Dr. Ali Amiri Network design objective The design objective is to minimize the time queuing delay averaged over the entire network with total capacity of the links assumed fixed. (This also means that the budget for acquiring capacity is fixed because we assume that cost is linearly proportional to capacity).

18. 18 Dr. Ali Amiri Let  be the total incoming message rate for the network. In our example, what is the value  of ?  = 0.333+0.3+0.4+0.2+0.133+0.133+0.2=1.7 Using queuing theory, the average message delay in the network is defined as

19. 19 Dr. Ali Amiri Assuming that the total capacity is held fixed, we want to determine the capacity of each link (C i ) in order to minimize T. Min T subject to

20. 20 Dr. Ali Amiri If we solve this minimization problem, we obtain the optimal capacity of link i: where plays the role of traffic intensity for the entire network. The value of  can be easily computed because C is assumed to be known.

21. 21 Dr. Ali Amiri This formulae is called the square root assignment rule. The square root assignment rule first assigns the absolute minimum capacity to each link i (that is, i /  i ) and then allocates the remaining capacity to each link following a square root assignment strategy. The minimum average message delay for the entire network is given by:

22. 22 Dr. Ali Amiri Let's apply this rule to the example. Here  i =  = 1/120 for all links because 1/  i = 120 bits/message. For our example, let's say the budget allows to acquire a total capacity of 4500 bps for all links in the network, that is C=4500 bps.

23. 23 Dr. Ali Amiri Ti= 1/(  i C i - i ),  = total incoming message rate for the network.  = 0.333+0.3+0.4+0.2+0.133+0.133+0.2 =1.7 messages/sec

24. 24 Dr. Ali Amiri Calculation details

25. 25 Dr. Ali Amiri Summary Table

26. 26 Dr. Ali Amiri Case of Discrete capacities Assume now that capacities are available at only two discrete values: 450 bps and 900 bps. Adjust the link capacities shown in the previous table to the closest of these available capacities and recompute the link queuing delays.

27. 27 Dr. Ali Amiri Notes: We can use either formulae in slide 18 or formulae in slide 23 to compute T* for the optimum case. However, we can use only formulae in slide 18 to compute T in the discrete case.

28. 28 Dr. Ali Amiri Very important note if time allows we cover the next section in class and you will be responsible for it in the exam. If time does not allow, we do not cover that section and it won’t be included in the exam.

29. 29 Dr. Ali Amiri Extension: Case of tree topology Example Node Message rate (messages/sec) 10.33 20.3 30.13 40.2 50.2 60.4 70.13

30. 30 Dr. Ali Amiri 0.2 1 2 6 4 3 5 0.33 0.4 0.13 Link 1 Link 2 0.3 Link 6 7 Link 7 0.13 Link 5 Link 4 Link 3

31. 31 Dr. Ali Amiri To analyze this topology and determine the appropriate link capacity allocation, we must take into account in computing queuing delays not only messages from terminal ports at each nodal concentrator, but also messages forwarded from previous links as well. Assuming that messages are independently generated at each node in the network, the average message rate for a particular link is equal to the sum of the previous link message rates and the message rate from external ports.

32. 32 Dr. Ali Amiri For example, the queuing delay for messages required to traverse link 2 is due to messages externally generated, with a rate of 0.3 messages/sec at node 2 (corresponding to messages generated by the 9 terminals located at node 2 Detroit), plus messages at the rate of 0.33 messages/sec coming from node 1 Chicago). Then the message rate through link 2 is 0.3+0.33= 0.63 messages/sec.

33. 33 Dr. Ali Amiri 0.2 6 =0.4 4 =0.2 3 =0.66 5 =0.2 2 =0.63 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

34. 34 Dr. Ali Amiri To minimize the total average queuing delay in the network with tree topology, we apply the same formul developed in the case of star topology. The results of the calculations to minimize the total average queuing delay in the example network with the tree topology using C=4500 bps as the total capacity of all links in the network are shown below. Assume next that capacities are available at only two discrete values: 450 bps and 900 bps. Adjust the link capacities shown in the previous table to the closest of these available capacities and recompute the link queuing delays.

35. 35 Dr. Ali Amiri Ti= 1/(  i C i - i ),  = total incoming message rate for the network. = 0.333+0.3+0.4+0.2+0.133+0.133+0.2=1.7 messages/sec

36. 36 Dr. Ali Amiri T*=0.33 sec

37. 37 Dr. Ali Amiri Notes: We can use formulae in slide either 18, 23 or 35 to compute T* for the optimum case. However, we can use only formulae in slide 18 to compute T in the discrete case. Summary table