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Unformatted text preview: EE 555 La ast Homew work(s) th Due: April 30 , 2 2009 Section A A Access Net tworks Question A.1. Consider two differe ent configura ations for a FT TTH distribution plan. The fiber attenuation is 0.3 dB/km. and between n the BDU and d the first hou use) and is 10 00m. The minimal x is the distance between houses (a acceptable receive pow wer at each h house is P (W W). In the first (Plan A) the ere is a home run fiber from the Block D Distribution U Unit to each h house. The BD DU has a 32w way passive o optical splitter. The signal r ratio (input to o each output) for the 32w way splitter is 18dB. In the sec cond (plan B), a single fiber r runs from th he Block Distr ribution Unit past all of the e houses on t the block and there is a passive optical tap that drop ps some of the signal powe er off to each h house. The t taps % of the powe er to the hous se and pass a along 95% of t the power. deliver 4% Compare the two desig gns in terms o of transmit an nd receive po owers, cost, re eliability, con nvenience, and any tors that shou uld be considered. other fact Question A.2. Consider a wireless network consisting of a single base station supporting N mobile terminals. The channel can be modeled as operating in slotted time with the slot duration equal to a packet transmission time. In a slot, node i will have a new packet arrival with probability p (all terminals have the same traffic parameters.) a) Design 1 TDM N slots are grouped in to a frame and each node is assigned one slot in each frame. Let qi (t) represent the number of packets in queue at node i at the end of slot i in frame t. Write down a series of equations governing the evolution of the queues. Solve (analytically) this queueing model for the average delay as a function of the utilization. b) Design 2 Slotted Aloha. A node can be in one of 2 states: S (success) or B (backlogged). A node with packets in its queue transmits in the next slot with probability 1 if it is in state S and with probability if it is in state B. This may result in a collision (two or more simultaneous transmissions) or a success (exactly one transmission). If it is a collision, the node moves to state B. If it is a success the node modes to state S. Initially all nodes are in state S. Write down a series of equations for the random variables governing the evolution of the queues on a slot by slot basis. BONUS: Compare the performance of theses two systems by simulation for N=10. Section B Switches and Routers Question B.1. Consider a shared n input n output shared memory switch. The memory cycle time is t (nanoseconds) (a read or write requires one cycle) and the memory data path width is w bytes. The packets are fixed length b octets (or bytes). (All buffer management pointers are managed in a separate ultrafast memory). What is the fastest (input or output) line speed that this switch can handle (bytes per second) so that there is no overrun? Evaluate for n=16, t=10, b=100, w=50. Question B.2. This problem reconsiders the HOL blocking issue when there is "hot spot" traffic. Consider a 2x2 switch with input queueing. In class we determined that for balanced traffic (both output equally likely) the maximum throughput is 0.75 cells per slot (to each output). Now suppose that the 1 ) and fraction of traffic destined to output 1 is (and the fraction destined to output 2 is that this applies to traffic coming in on either input 1 and input 2. Assume a loaded situation (i.e. the queues never empty). Let the state (i,j) represent the state where the packet at the HOL in input queue 1 is destined to i and the packet at the HOL in input queue 2 is destined to j (state space is (1,1); (1,2); (2,1); (2,2)). i) Write down the transition matrix. ii) Solve for the equilibrium distribution. iii) Find = the throughput to output i (under the heavily loaded condition). Verify that Question B.3. i) Define strictly nonblocking. ii ) Is the following switch strictly nonblocking? If not is it rearrangeably nonblocking? Give reasoning. iv) Plot total throughput as a function of . 2x2 xbar 2x2 xbar 2x2 xbar 2x2 xbar 2x2 xbar 2x2 xbar Question B.4. Show that if two (8x8) banyan switches are connected in tandem, the resulting switch is rearrangeably nonblocking. How many paths are there between any pair of inputoutputs? Question B.5. Consider a multistage NxN switching network comprised of log2N stages of 2x2 crossbars The switch has buffers at the inputs and no buffers at the outputs. In explaining your answers to a) and c) below, specific examples for the case of N=4 should be given. a) One problem that arises is HOL blocking. What causes this and why is it a problem? b) Suggest 2 ways to modify the switch design to reduce the impact of HOL blocking. c) Another problem is internal blocking. What causes this and why is it a problem? d) Suggest two ways to reduce the impact of internal blocking. Question B.6. Draw a diagram of a 4 x 4 optical (electronic control plane, optical data plane) packet switch showing how HOL conflicts are reduced through 4 delay lines. Describe its operation. (Note there are 4 external inputs and outputs). Section C Traffic Management Question C.1. Consider a multiplexer handling fixed length packets using a token bucket burst control mechanism. Packets that arrive when there is no token available are marked. One token is added to the bucket at the end of each slot and the bucket depth is 5. The arrival process is defined by: pk = Pr{k arrivals per slot}. For the case of p0 = 0, p1 = 0.4, p2 = 0.4 a) Draw the complete Markov chain for the number of tokens available at the beginning of a slot. b) Solve the Markov chain and find the rate of marked packets. Question C.2. Consider a Poisson arrival stream of fixed length packets of length x seconds at a rate packets per second. a) The arrival process is passed through a leaky bucket (not Token Bucket) control mechanism. The bucket depth is 3 packets. Packets overflowing the bucket are dropped. Develop a Markov Chain to model this system. (Use the number of packets in the buffer/bucket at departure instants as the state). Find an expression for the packet loss rate and the mean delay for packets that are successfully transmitted. c) The packets that are dropped are eventually retransmitted by the source (some form of selective repeat ARQ is used). Assume that the combination of new and retransmitted packets can still be modeled as a Poisson arrival process, how might you modify your answer to part b) to account for the retransmitted packets? Question C.3. Consider the following stream of packets (all of length 3) to be served from a (very large) shared buffer. Packet id (group.packet#) P1.1 P2.1 P1.2 P1.3 P2.2 P1.4 P2.3 P2.4 Arrival time 1 2 3 4 6 8 9 10 Departure time W1 W2 W Find the departure time, mean waiting times, and mean system times (for each group separately and also overall) for the following queueing disciplines. (If departures and arrivals occur at the SAME instant, assume the departure occurs first). a) FCFS b) Non preemptive priority (group 2 has higher priority) c) Weighted Fair Queueing where group 2 is supposed to get a service rate that is 2 times that of group 1. Question C.4. Consider a buffer processing packets. We assume that the packet generation process is Poisson at a rate and that the packets have exponentially distributed length, mean 1/. The buffer can hold K packets. a) Draw the Markov chain and solve for the blocking probability B and the throughput (successful transmission rate) for the case of / = 1. b) Now consider the following feedback congestion control mechanism. The source is made aware of the current instantaneous buffer occupancy and adjusts its input rate according to the following formula. k = K k K Derive an expression for the throughput. Question C.5. Consider the following simplistic model of a congestion backoff mechanism. A source sends packets at a (Poisson) rate to a multiplexer. The message length can be assumed to be exponential with mean length 1/. The buffer is finite and can hold K messages. When the number of packets in the buffer reaches a threshold U, a rate reduction message is sent back to the source and it reduces its traffic to a rate (0< <1). When the queue drops below a lower threshold L, the source reverts to sending traffic at a rate . Develop a Markov chain model of the buffer occupancy. ...
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 Spring '08
 Silvester

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