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CS-350 - Fundamentals of Computing Systems
Problem 1
The nancial institutions at the Bank Block in Downtown Sector B3 are all connected to a central processing hub. Due to the recent rise in cyber-crime (especially after Dec. 10, 2020), the banks have adopted measures to harden the security of the hub that handles all the transactions. From the outside, and average of 7 requests arrive every millisecond. These have always to go through a congestion control (S0) system rst.
S0 is a single-processor system with an exponentially distributed service time. It operates with average service rate of 7 requests per millisecond. Congestion control is performed by setting a hard cap N on the number of requests at the S0 system. Requests that arrive when N requests are already in S0 are simply rejected. Any request that is not rejected moves on to be handed by the hub.
An accepted request rst reaches the authentication (S1) system: a single-processor system capable of servicing, on average, 8.33 requests every millisecond. Typically, 20% of requests fail authentication and their processing is halted. Valid requests advance to the processing server (S2). This is a single-processor system capable of serving a single request in around 0.1 ms.
After processing, a request is completed and leaves the system with 40% probability. Al- ternatively, the request is routed to a post-processing single-core server (S3) to determine what type of follow-up request needs to be generated to complete handling of the current request. This post-processing step requires on average 0.15 ms. Follow-up requests can be of two types: inter-bank and intra-bank. An inter-bank follow-up request needs to go through authentication, which happens only in 30% of the cases. Conversely, an intra-bank request can go directly to the processing phase skipping authentication altogether. Follow-up re- quests then follow the steps of normal requests i.e. they can themselves require (inter- or intra-bank) follow-up requests and so on. But under no circumstance a follow-up request is sent back to S0.
(a) [3 points] Provide a diagram of the system where you specify the models to be em- ployed to solve the system. Explicitly list any assumption you are making to solve the system that is not already stated in the description above.
(b) [7 points] While keeping the rate of requests from the S0 server as a generic param- eter λa , compute an expression for the average throughput (in terms of requests per millisecond) at the servers S1, S2, and S3. Show your work for full marks.
(c) [3 points] Which server (S1, S3, or S3) represents the bottleneck of the hub? Motivate your answer.
(d) [7 points] The goal is to maximize the overall throughput of the hub while preventing it from buckling-up due to congestion. For this purpose, we can adjust the value of N at S0. Exploring the range of values between 2 and 10, what is the ideal value of N?
Problem 2
You have been assigned to analyze a black-box image compression co-processor. You are trying to understand a few important characteristics of its behavior by providing in input a few images of di erent sizes (in KB). For each experiment, you have recorded the time it took to perform image compression and the total energy (in Joules) consumed by the co-processor to perform compression over the given image. You have also measured that the processor consumes exactly 0 Joules/sec when idle. The measurements you have collected are reported in Table 1.
|
# |
KB |
Time (ms) |
Energy (J) |
|
# |
KB |
Time (ms) |
Energy (J) |
|
1 |
454 |
11.5 |
1.3 |
|
11 |
721 |
9.6 |
0.9 |
|
2 |
952 |
4.6 |
0.2 |
|
12 |
306 |
25.4 |
6.5 |
|
3 |
805 |
21.7 |
4.7 |
|
13 |
572 |
17.9 |
3.2 |
|
4 |
141 |
70.3 |
49.4 |
|
14 |
156 |
75.1 |
56.4 |
|
5 |
194 |
15.9 |
2.5 |
|
15 |
651 |
42.6 |
18.1 |
|
6 |
205 |
11.2 |
1.3 |
|
16 |
171 |
43.3 |
18.7 |
|
7 |
152 |
76.6 |
58.7 |
|
17 |
871 |
13.4 |
1.8 |
|
8 |
47 |
51.2 |
26.2 |
|
18 |
48 |
31 |
9.6 |
|
9 |
534 |
16.3 |
2.7 |
|
19 |
490 |
59.3 |
35.2 |
|
10 |
689 |
15.5 |
2.4 |
|
20 |
818 |
17.9 |
3.2 |
(a) [3 points] Based on the acquired data, can we conclude that the compression of larger images takes longer?
(b) [3 points] Based on the collected data, can we conclude that the compression of smaller images yields lower power consumption? Recall that power is measured in Watts as the total consumed energy over the considered time window (i.e. Joules/second).
(c) [5 points] With what con dence we can build a 士13 ms con dence interval around the mean of the image compression service time?
(d) [5 points] Consider a sequence of image compression requests that arrive (and are potentially queued) at the co-processor under analysis with an average rate of 29.4 requests per second. What will be the average response time of a generic request?
Problem 3
The chemical plant Wharmful Corp. receives three types of raw materials M1, M2 and M3 to be stored in special pressurized cabinets C1, C2, and C3. Materials arrive in packages Pi , with each package containing a given type of material indicated with Pi (Mx). The material Mx can only be stored in Cx. As they move on the conveyor belt, they rst reach the sorting area which can hold up to 4 packages. Here, a robotic sorter can re-arrange the order of the packages. The sorted packages are then sent to a robotic arm that stores them into the corresponding pressurized cabinets. Refer to the Piazza post for a visual overview of the organization of the plant.
Your job is to devise a strategy to program the robotic sorter. We know the following about the system. (1) the sorter always lls up the sorting area unless there are not enough packages on the belt. Thus, initially it will take in the rst 4 packages; then as one is pushed out, a new one is taken in and so on. It has no knowledge of other packages outside of the sorting area; (2) the three pressurized cabinets C1, C2, and C3 are initially closed and whenever closed they maintain a pressure of 32 psi, 64 psi, and 96 psi respectively; (3) any cabinet needs to be depressurized down to 16 psi before opening and depressurizes at exactly -16 psi per time unit; (4) any cabinet will re-pressurize instantly after it has been closed; (5) the arm can pick up the next packet and bring it to any cabinet in a single time unit; (6) only once the arm has brought a package in front of the corresponding cabinet, the cabinet can start de-pressurization; (7) after a cabinet is at 16 psi it can be opened and the arm can store the current package instantly; and (8) only one cabinet at a time can be in depressurized state, so when a cabinet initiates depressurization, the system automatically closes and re-pressurizes any other opened cabinet.
(a) [6 points] De ne a notion of readiness to adopt FR-FCFS as the sorting strategy with goal of minimizing overall storage time. After that, describe and motivate the order- ing produced by FR-FCFS if the packages arrive on the belt in the following sequence:
P1 (M2), P2 (M3), P3 (M1), P4 (M2), P5 (M1), P6 (M3), P7 (M3), P8 (M1), P9 (M2), P10 (M1), P11 (M3). You do not need to draw the time plot of the system in operation.
(b) [6 points] By using the same notion of readiness de ned above, de ne a new strategy that (1) still prioritizes ready packages but that (2) tries to minimize waiting time when no ready packages are available in the sorting area. Once again describe the ordering produced by the new strategy with the same arrival sequence provided above.
(c) [6 points] Compute the total amount of time for the system to fully store all the 11 packages under the two policies. No need to provide timed graphs, but show your reasoning. You can also assume that sorting does not introduce any delay.
Problem 4
A sensor node needs to acquire and pre-process the data from 4 di erentonboard sensors. It is crucial that pre-processing completes always before a new data sample is available at a given sensors to prevent loss of samples. The sensors are the Gyroscope (G), Magnetometer (M),
Accelerometer (A), and Baroscope (B). The gyroscope produces data at a rate of 200 Hz, with each data sample requiring 2 ms for pre-processing in the worst case. Pre-processing for the magnetometer takes no longer than 1 ms with samples being produced at 125 Hz. A new accelerometer sample is produced every 19 ms and pre-processing takes at most 4 ms. Finally, pre-processing a barometer sample takes at most 6 ms, but the sampling period is con gurable.
(a) [6 points] Without drawing the schedule, perform a back-of-the-envelope calculation to determine a suitable con guration for the sampling period of the barometer such that the system always operates correctly under RM. For this part, consider only values of periods that are integers when expressed in milliseconds.
(b) [6 points] Because the calculation above returned a value of sampling period which is too large, you decide to set the barometer sampling period to 29. Will the system operate correctly under RM?
(c) [8 points] You decide to set the barometer sampling period to 30 ms and stick to RM. You notice however that once in a while (very rarely) the pre-processing of the accelerometer takes 5 ms instead of its nominal 4 ms worst-case execution time. When this happens, the system can still operate correctly if the faulty accelerator pre- processing job is killed when it misses the deadline. Killing any other job is not acceptable. Also, you can apply a utilization-preserving transformation to all or some of your tasks. Speci cally, you can release any pre-processing task twice as fast but only perform half of the required processing in each release. Explain how you can make the system operate correctly under RM.
Problem 5
Two processes, namely the Printer and the Shifter, co-operate to respectively print and update the value of an array of integers that has a xed size of 5 elements. The code executed by the Printer process is provided in Listing 1; the code of the Shifter process is given in Listing 2. If a variable has the same name in both processes, it is to be considered a shared variable. A [FRAGMENT x] statement in the code refers to additional lines of code (a fragment) to be inserted in place of the statement. At the beginning of time, both processes start executing at their start line and will not run again once they reach their end line. The initial state of the shared data structures is as follows: array = [4, 9, 5, 3, 6] ; queue = [8, 1, 7, 2] (NOTE: 8 is at the head of the queue.)
(a) [4 points] Assume that P2 and S2 are empty and that (a) fragment P1 expands to wait(go); and (b) S1 expands to if(queue .empty()){signal(go);}. Here, go is a semaphore initialized to 0. What is the maximum value that could ever be printed at line 13 by the Printer process?
(b) [8 points] Assume that (a) fragment P1 and S1 expand to wait(m); and that (b) P2 and S2 expand to signal(m);. Here, m is a semaphore initialized to 1. What is the maximum value that could ever be printed at line 13 by the Printer process?
(c) [8 points] Assume that fragments P1, P2, S1, and S2 are all empty. What is the maximum value that could ever be printed at line 13 by the Printer process?