Simulated SA Throughput vs. p

Transcription

1 Problem 1. Simulation of Slotted Aloha network (A) There are n stations in the network that are trying to transmit messages to an access point. (B) All the stations are synchronized and they are using slotted aloha to access the channel. (C) A station transmits a new message in a slot with probability p. (D) The length of a message is one time slot. (E) Messages collide at the access point if two or more stations are transmitting during the same time slot. (F) If a collision takes place, the station will try to retransmit its undelivered message with probability q. (G) Once a backlogged message is delivered successfully, the station will once again start generating new messages in a slot with probability p X:.5 Y:.3664 Simulated SA Throughput vs. p 15 nodes, 1 sim. avg Throughput [packets/time] Node's Prob. of TX (p=q) [] Figure 1. Simulated throughput of Slotted Aloha network In Figure 1, the probability of an not-backlogged node transmitting, p, is equal to the probability of a backlogged node transmitting. For small values of p, the simulated network throughput is equal to the theoretical maximum of e -1 =.368.

2 1 1-1 Simulated SA Throughput vs. p simulated theory Throughput [packets/time] Node's Prob. of TX (p=q) [] Figure 2. Simulated and theoretical throughput of Slotted Aloha network, p=q The theoretical throughput and simulated throughput agree well if p <.2, which makes sense because the theoretical throughput is derived based on the assumption that p and q are small. The theory overestimates the throughput for values of p >.2, and at values of p>.6 the simulation for 1e4 time slots shows the throughput goes to packets/s. If the simulation time was extended, it is feasible that the simulation would yield a non-zero, but very small throughput for p>.6, but the simulation time becomes very long.

3 Table 1. Comparison of theoretical and simulated Slotted Aloha throughput for varied p Simulated SA Throughput over p and q Throughput [packet/s] Not-backlogged node's Prob. of TX (p) [] Backlogged node's Prob. of TX (q) [] 1 Figure 3. Simulated throughput of Slotted Aloha network for varied p and q

4 Simulated SA Throughput over p and q Throughput [packet/s] Backlogged node's Prob. of TX (q) [] Not-backlogged node's Prob. of TX (p) [] Figure 4. Simulated throughput of Slotted Aloha network for varied p and q - another viewing angle In Figure 4, the throughput of the network can be increased from the theoretical maximum of ~.36 when p=q, to ~.5 by making the probability of a not-backlogged node s transmission high (~.9) and making the probability of a backlogged node s transmission low (~.5).

5 Standard Dev. of SA Throughput over p and q Standard Dev. of Throughput [packet/s] Not-backlogged node's Prob. of TX (p) [] Backlogged node's Prob. of TX (q) [] Figure 5. Standard deviation of Slotted Aloha simulated throughput for varied p and q

6 Problem 2. Simulation of a Slotted Aloha network with SNR threshold (A) n = 1 nodes in network (B) nodes located at random locations in square area, side = 4m, AP at center (C) simulation time = t = 1e4 time slots (D) q = p (E) Use SNR model, if SNR of node is > SNR_thr, then transmission is successful. Simulated SA-SNR Model Throughput vs. p X:.9 Y: Throughput [packets/time] 1 SNR_thr =.1 SNR_thr = 1 SNR_thr = 6 X:.9 Y:.6991 X:.9 Y: Node's Prob. of TX (p=q) [] Figure 6. Simulated throughput of slotted aloha network with SNR reception model If the SNR threshold is very low, for example.1 in Figure 6, the throughput of the network gets up to 2.1packets/slot when the p=.9. Imagine that 2 nodes in the network transmit at the same time with approximately the same power, then the SNR at the receiver from each node s transmission will be ~.5, which is greater than the SNR threshold of.1. In this case, the packet from both nodes is received correctly, by the rules of the network model. A receiver that can detect packets with such low SNRs would depend on the packets being coded with orthogonal codes that minimize intersymbol interference, for example in a CDMA scheme. When the SNR threshold is 6, the throughput falls off rapidly for p >.2. This makes sense intuitively because as the p increases, the probability of many packets being transmitted at the same time increases. If the SNR requirement is large, only packets that are transmitted, and can pass the SNR requirement, for example, the packet from the closest node, can be received correctly.

7 Standard Dev. of Simulated SA-SNR Model Throughput vs. p 1.4 Standard Dev. of Throughput [packets/time] SNR_thr =.1 SNR_thr = 1 SNR_thr = Node's Prob. of TX (p=q) [] Figure 7. Standard deviation of throughput of slotted aloha network with SNR model simulation

8 Problem 3. Simulation of Pure (unslotted) Aloha network (A) For this problem consider a network that is not synchronized. (B) Each station periodically transmits messages that are one time unit long. (C) New messages arrive at a station according to a poisson arrival process with rate λ a. (D) If a message is not delivered to the access point, the station tries to retransmit the same message after a random period of time. This random waiting period is also distributed exponentially with arrival rate λ r. (E) While a station is trying to retransmit its backlogged message, no new messages arrive at the station. Once a backlogged message is delivered, the station once again generates a new message with rate λ a. (F) n = 1 nodes in the network (G) simulation time = 1e4 time slots Throughput [packets/time] Simulated Pure Aloha Throughput vs. λ r X:.645 Y:.1876 X:.3478 Y:.5983 λ a =.1 λ a =.1 λ a = Node's Re-transmission arrival rate, λ r [] Figure 8. Simulated throughput of Pure Aloha network over lambda_r for various lambda_a's When λ a =.1 and λ r ~.5 the simulated throughput is.188, which is close to the theoretical value of e -1 /2 ~.184. The arrival rate of packets for the whole network, G, changes as the backlog of the network changes. For λ a =.1, with n=1 nodes in the network, the network has, on average λ a *n =.1*1 = 1 packet per time slot arriving to the network, and the vulnerability period for unslotted Aloha is 2 time slots (given that 1 packet = 1 time slot) so the throughput for sufficiently small λ r is.184. As the λ r is

9 increased, the number of re-transmitted packets increases which causes more collisions and more re-transmissions, so the throughput drops off. When λ a =.1, the amount of traffic presented to the network is so low that collisions occur infrequently, and the re-tranmission rate, λ r, has little impact on the throughput. With such a low λ a, the network throughput remains low at.1, which is ~n*λ a = 1*.1 =.1, indicating that nodes do not become backlogged and all the traffic presented to the network transmits successfully. σ(throughput [packets/time]) Standard Dev. of Simulated Pure Aloha Throughput vs. λ r λ a =.1 λ a =.1 λ a = Node's Re-transmission arrival rate, λ r [] Figure 9. Standard deviation of throughput for simulated Pure Aloha network (n=1)

10 Appendix : Software and hardware configuration All problems are answered with Matlab scripts and functions; the version of Matlab used to develop the scripts is as follows: The code was developed on a PC with the following hardware and OS:

11 Appendix 1: Run Slotted Aloha simulation in debug mode and check algorithm works correctly. Check my algorithm in function calcsathroughput(n, p, q, t), which calculates the throutput of a slotted Aloha network with n nodes, not-backlogged nodes transmit with probability p, backlogged nodes transmit with probability q, simulated for t timeslots. Use this simplified simulation setup, just to check if algorithm is correct: This is the order of outputs: A little explanation on what the outputs mean: column 1: i = time, number of time slots elapsed column 2: trial result for each node with probability success = p column 3: trial result for each node with probability success = q column 4: state of each node, 1 = backlogged, = not backlogged column 5: 1 if node is backlogged and attempting to send, if node is backlogged and not attempting to send column 6: 1 if node is not backlogged and attempting to send, if not is not backlogged and not attempting to send column 7: how many backlogged nodes are attempting to send column 8: how many not backlogged nodes are attempting to send column 9: how many packets are successfully transmitted column 1: how many packets have failed to transmit column 11: the throughput up to that point in the simulation

12 At time=, all 4 nodes try to transmit, the backlogged mask, blogged_mask is updated to 1111 to indicate that all 4 nodes are backlogged and will thereafter attempt to transmit with probability q, until they successfully transmit, after which they will attempt again with probability p. At time=1, column 2 is meaningless because all the nodes are backlogged, so result of trial with probability p doesn t do anything. Column 3 shows that trial with probability q is so no backlogged nodes try to transmit. The state of nodes doesn t change. At time=2, column 3 shows 1 so exactly 1 backlogged node is trying to transmit, and it is successful, so the number of successful TXs is incremented to 1, and that node s entry in the blogged_mask is changed from 1 for backlogged to for not-backlogged, which is shown in column 4 = 111. At time=3, colunn 3 again has 111 so the trials of the first 3 nodes indicate an attempt to TX, but the blogged_mask in column 4 shows 111, so only the first and third nodes are relevant backlogged nodes trying to transmit. Since there are 2 backlogged nodes trying to transmit, the transmissions are failures (notice the number of failed transmissions increases from 4 to 6). Algorithm looks functionally Ok.