Model Solution for Assignment 4 in CSC8513, S2 2016 Ron Addie October 7, 2016 Question 1 Carry out the experiment described in Practical 6 with 50 traffic streams. Plot the throughput with 5 different averaging windows: 0.01s, 0.05s, 0.1s, 0.5s, and 1s. Observe and record the mean and standard deviation of these five different trace plots and calculate the loss using these parameters. Answer The network from Practical 6, shown in Figure 1, was simulated, as prescribed, with averaging at the levels indicated (in fact some additional windows were included). A plot of the traffic, averaged at windows of 1 sec down to 0.002 seconds is shown in Figure 2. The standard deviation of this traffic was measured by the simulation and is shown in Table 1. A plot of the standard deviations, fitted with to a power-law function (obtained by use of a Libre-office spreadsheet), is shown in Figure 3. The implied loss, calculated by the normal loss formula, is shown in Column 3 of this table. The actual loss, in the simulation, was 986078=2236501 = 0:4409. This loss is very close to all the loss estimates in Table 1. Thus, it turns out that any of the time intervals used in the table would be satisfactory for the purpose of estimating the standard deviation to use in modelling traffic loss. 1Model Solution for Assignment 4 Ron Addie Figure 1: The network of Practical 6 2Model Solution for Assignment 4 Ron Addie Figure 2: Link 1 traffic, averaged at different window widths 3Model Solution for Assignment 4 Ron Addie Window width Standard deviation Implied loss 1 447628 0.4419784 0.5 499867 0.4430493 0.2 533191 0.4440104 0.1 537268 0.4441437 0.05 538561 0.4441866 0.02 538542 0.444186 0.01 545859 0.4444357 0.002 742042 0.4552088 Table 1: Standard deviations and implied loss for different averaging window widths Figure 3: Standard deviation of traffic averaged with different window widths, and the loss this implies for link 3, using the normal loss formula 4Model Solution for Assignment 4 Ron Addie Site A Site B Link Figure 4: Planning a Link Exercise 9.5 Choose a plan for installing transmission equipment on a point-to-point communication link under the following assumptions: (i) traffic is 6 Mbit/s at present, with standard deviation of 4 Mbit/s; (ii) mean traffic and variance of the traffic is doubling every three years; (iii) the link capacity can be selected as 10 Mbit/s, 55 Mbit/s, or 165 Mbit/s, at a cost of $100,000.00, $120,000.00, or $150.000.00 respectively; (iv) when an upgrade is required, the cost of the link which was previously installed will need to be completely written off. (v) a dollar now is worth 12% more than a dollar one year later. The two sites and the alternative connecting links are depicted in Figure 4. Consider also an alternative case (Case (b)) in which the initial traffic offered to the link is 10 Mbit/s instead of 6 Mbit/s. Also assume that the standard deviation of this traffic is 10 Mbit/s at the outset and that the variance of the traffic grows at the same rate as the mean, i.e. doubles every three years. Note: in saying that mean traffic doubles every three years, it should be taken also that the mean traffic increases in the ratio p 3 2 each year. If traffic was increasing in the ratio x over each successive group of m years, we would normally assume that traffic was increasing in the ratio p m x over each year. Of course, all 5Model Solution for Assignment 4 Ron Addie Year Mean s Required capacity (mean + 2 s) Next available module 1 6 4 14 55 2 7.56 4.49 16.5 55 3 9.52 5.04 19.6 55 4 12 5.66 23.3 55 5 15.12 6.35 27.8 55 6 19.04 7.13 33.3 55 7 24 8 40 55 8 30.24 8.98 48.2 55 Table 2: Traffic between the two sites for the next 8 years, in Case (a) such parameters are merely estimates and growth is unpredictable and varies from year to year randomly, so that the assumption that growth is distributed over successive years as uniformly as possible is only an approximation. However, given the imprecision of all the other considerations, this assumption is probably reasonable. Explain your conclusions carefully and provide full workings for both cases in your answer. Answer (a) Let us start by working out the mean and standard deviation of the traffic to be carried, over 8 years. The mean and standard deviation of the traffic, and the implied minimum required capacities, over the next 6 years, are shown in Table 2. From this table it is clear that it will be adequate if a module of 55 Mbits/s is installed in year 1. (b) In this case, the projected traffic and required capacity are shown in Table 3. In this case we have to choose between installing the 165 Mbit/s capacity link in year 1, or installing a 55 Mbit/s link in year 1 and upgrading to 165 Mbit/s in year 5. The discounted cost of the first plan, in year 1 $, is $150,000. The discounted cost of the upgrade plan, on the other hand, is $120,000 + 1:12−5 × 150;000 = $205;114. It is therefore best, clearly, to install the 165 Mbit/s system at the start. 6Model Solution for Assignment 4 Ron Addie Year Mean s Required capacity (mean + 2 s) Next available module 1 10 10 30 55 2 12.6 11.2 35 55 3 15.9 12.6 41 55 4 20 14 48.3 55 5 25.2 15.9 56.9 165 6 31.7 17.8 67.4 165 7 40 20 80 165 8 50.4 22.4 95.3 165 Table 3: Traffic between the two sites for the next 8 years, in Case (b) 7