# Solutions to Chapter 4

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1 Solutions to Chapter 4 1. A television transmission channel occupies a bandwidth of 6 MHz. a. How many two-way 30 khz analog voice channels can be frequency-division multiplexed in a single television channel? / = 00 channels b. How many two-way 00 khz GSM channels can be frequency-division multiplexed in a single television channel? / = 30 channels c. Discuss the tradeoffs involved in converting existing television channels to cellular telephony channels? The biggest advantage of using existing television channels to provide cellular telephony channels is the very large bandwidth that they occupy. In theory, one could divide each television channel into many cellular telephone channels, as shown in parts (a) and (b). However, a significant trade-off would soon become apparent.. Cellular telephone networks use low-power base transmitters to create relatively small cells so that frequency re-use is possible. This dramatically increases the number of users that a network can accommodate. Television stations transmit at a very high power over very long distances (as one would expect for a broadcast medium). Thus, in the region that currently receives a given television network broadcast (say, the larger part of a city), only 00 analog channels or 30 GSM channels could be accommodated.. A cable sheath has an inner diameter of.5 cm. a. Estimate the number of wires that can be contained in the cable if the wire has a diameter of 5 mm. Ignoring empty space between the wires: 5 π ( ) 5 π ( ) = 5 wires b. Estimate the diameter of a cable that holds 700 wire pairs. Ignoring empty space between the wires:

2 π ( d ) = 700 wires d 60 5 π ( ) 3. Suppose that a frequency band W Hz wide is divided into M channels of equal bandwidth. a. What bit rate is achievable in each channel? Assume all channels have the same SNR. Each user uses W/M bandwidth. Using Shannon s Channel Capacity formula: Bit rate = ) log 1+ SNR) bps M b. What bit rate is available to each of M users if the entire frequency band is used as a single channel and TDM is applied? In this case, the total bit rate afforded by the W Hz is divided equally among all users: W log (1 SNR) Bit rate = + bps M c. How does the comparison of (a) and (b) change if we suppose that FDM requires a guard band between adjacent channels? Assume the guard band is 10% of the channel bandwidth. Because of the guard band we expect that the scheme in (b) will be better since the bit rate in (a) will be reduced. In (a), the bandwidth usable by each channel is 0.9W/M. Thus, we have: W Bit rate = ( 0.9 )log (1 + SNR) bps M 4. In a cable television system (see Section 3.7.), the frequency band from 5 MHz to 4 MHz is allocated to upstream signals from the user to the network, and the band from 550 MHz to 750 MHz is allocated for downstream signals from the network to the users. a. How many MHz upstream channels can the system provide? What bit rate can each channel support if a 16-point QAM constellation modem is used? The system can provide: (4 5) MHz Mhz Channel By Nyquist, we have: = 18 upstream channels. R = 6 pulses bits ( 10 ) 4 = 16 Mbps sec ond pulse

3 b. How many 6 MHz downstream channels can the system provide? What bit rates can each channel support if there is an option of 64-point or 56-point QAM modems? Similarly, the system can provide: ( ) MHz 6 Mhz Channel = 33 upstream channels. R 46-point = 6 pulses bits (6 10 ) 6 = 7 Mbps sec ond pulse R 56-point = 6 pulses bits (6 10 ) 8 = 96 Mbps sec ond pulse 5. Suppose a system has a large band of available bandwidth, say 1 GHz, which is to be used by a central office to transmit and receive from a large number of users. Compare the following two approaches to organizing the system: a. A single TDM system. The channel will provide a maximum bit rate, say b bps. Each user will use b/n bps where n is the number of users. The number of users will be limited by the noise on the channel and the switching speed of the hardware. b. A hybrid TDM/FDM system in which the frequency band is divided into multiple channels and time division multiplexing is used within each channel. Although in theory, the total bit rate provided by the system is the same, the hybrid solution may be practically better because, if we assume again n users, each user timeslot is at a lower bit rate and for a longer duration. Therefore, TDM synchronization and switching speed problems are less significant than in the pure TDM version. By leveraging both the time and frequency domain, the technological requirements posed by either one are lessened. 6. Suppose an organization leases a T-1 line between two sites. Suppose that 3 kbps speech coding is used instead of PCM. Explain how the T-1 line can be used to carry twice the number of calls. If a 3 kbps speech coding (such as Adaptive Differential Pulse Code Modulation (ADPCM)) is used, the bit rate required for each user is reduced to half of that required for PCM (64 kbps). The format in which a T-1 system carries information needs to be carried. For example a frame could now carry 48 4-bit samples. Aside: Chapter 1 explains how advanced speech coding schemes reduce the bit rate while maintaining speech quality. 7. A basic rate ISDN transmission system uses TDM. Frames are transmitted at a rate of 4000 frames/ second. Sketch a possible frame structure. Recall that basic rate ISDN provides two 64 kbps channels and one 16 kbps channel. Assume that one-fourth of the frame consists of overhead bits.

4 Assuming that the 16 kbps D channel is followed by the two 64 kbps B channels and then the overhead (which is not necessarily the case), in the time domain, the frame could look as follows: D channel B channel B channel Overhead bits t [ms] The T-1 carrier system uses a framing bit to identify the beginning of each frame. This is done by alternating the value of the framing bit at each frame, assuming that no other bits can sustain an alternating pattern indefinitely. Framing is done by examining each of 193 possible bit positions successively until an alternating pattern of sufficient duration is detected. Assume that each information bit takes a value of 0 or 1 independently and with equal probability. a. Consider an information bit position in the frame. Calculate the average number of times this bit position needs to be observed before the alternating pattern is found to be violated. Define the random variable X as the number of times the bit position needs to be observed before the alternating pattern is violated. Define P a,b as the probability that a is observed the first try and b on the second. Similarly, define P a,b,c as the probability that a,b and c are observed in succession. P[X 1] = 0 P[X = ] = P 0,0 + P 1,1 = 1/4 + 1/4 = 1/ P[X = 3] = P 1,0,0 + P 0,1,1 = 1/8 + 1/8 = 1/4 P[X = 4] = P 1,0,1,1 + P 0,1,0,0 = 1/16 + 1/16 = 1/8 P[X = n] = 1/ n + 1/ n (n -1) = 1/ Define p = P[a one bit is observed] = P[a zero bit is observed] = ½ E[X] = i= ip[ X = i 1 d i d i d i] = ip = p = p = [ p dp dp dp i= i= i= i= 0 i 1 p] d = [ 1 p] = 1 = 1 dp 1 p (1 p) (1 1 ) = 3 b. Now suppose that the frame synchronizer begins at a random bit position in the frame. Suppose the synchronizer observes the given bit position until it observes a violation of the alternating pattern. Calculate the average number of bits that elapse until the frame synchronizer locks onto the framing bit.

5 On average, if the frame synchronizer starts at a random location, it will have to examine 18/ = 91 incorrect bit locations before locking onto the framing bit. For each such incorrect bit location, on average, a non-alternating pattern will be discovered on the third observation, at which time the synchronizer will move one bit forward to the next bit. Thus, the average number of bits elapsed until the synchronizer observes the framing bit is: Average number of bits elapsed until frame bit found = 91 x [( x 193) + 1] = bits At this point, the synchronizer will begin observing the framing bit. It can t be sure that it has located the framing bit until an alternating pattern has been observed for a sufficient number of bits, say n. The synchronizer can never be sure that it has locked onto the frame bit, so n must be chosen sufficiently large so that the synchronizer is reasonably confident that it has indeed found the frame bit. Thus, the total number of bits elapsed before the synchronizer has locked onto the framing bit is: Average number of bits elapsed until locked onto frame bit = (n 1) bits. 9. The CEPT-1 carrier system uses a framing byte at the beginning of a frame. A CEPT-1 frame is 3 bytes = 56 bits long. Let p be the probability that a specific byte pattern is observed. Thus, p = (½) 8 Also, let us assume that the framing byte is chosen in such a way that any other occurrence of the byte pattern cannot include any of the bits in the frame byte. a. Suppose that all frames begin with the same byte pattern. What is the probability that this pattern occurs elsewhere in the frame? Assume that each information bit takes a value of 0 or 1 independently and with equal probability. In addition to its correct position, the an occurrence of the frame byte pattern can begin at any of Thus, = 40 locations P[pattern occurs elsewhere in the frame] = P[pattern occurrence begins in bit 9] + + P[pattern occurrence begins in bit 10] + + P[pattern occurrence begins in bit 49] = 40(1/) 8 = b. Consider an arbitrary information bit position in the frame. Calculate the average number of times that the byte beginning in this bit position needs to be observed before it is found to not be the framing byte. Define the random variable X as the number of times the bit position needs to be observed to discover that it is not the location of the framing byte.

6 P[X = 1] = 1 p P[X = ] = p(1 p) P[X = i] = p i - 1 (1 p) We see that this sequence rapidly decreases (geometrically), so we expect the average of X to be very close to 1. E[X] = i= 1 ip [ X = i] = i= 1 ip i 1 (1 p) = (1 p) i= 1 d dp p i = (1 p) d dp i= 1 p i = = d i d p p ( 1 ) [ p 1] = (1 p) [ 1] = = dp dp p p p = 1 (1 ) 1 i 0 Since p = (½) 8, we have: E[X] = c. Now suppose that the frame synchronizer begins at a random bit position in the frame. Suppose the synchronizer observes the byte beginning in the given bit position until it observes a violation of the alternating pattern. Calculate the average number of bits that elapse until the frame synchronizer locks onto the framing byte. On average, if the frame synchronizer starts at a random location, it will have to examine 55/ = 17.5 incorrect bit locations before locking onto the framing bit. For each such incorrect bit location, on average, an incorrect pattern will be discovered after observations, at which time the synchronizer will move one bit forward to the next bit. At this point, the synchronizer will begin observing the framing byte. It can t be sure that it has located the framing byte until the framing byte pattern has been observed a sufficient number of times, say n. The synchronizer can never be sure that it has locked onto the framing byte, so n must be chosen sufficiently large so that the synchronizer is reasonably confident that it has indeed found the framing byte. This will entail n 1 full frames elapsing plus the 8 bits to check the framing byte the n th time. Thus, the total number of bits elapsed before the synchronizer has locked onto the framing byte is: Average number of bits elapsed until locked onto frame bit = [(E[X] 1) x ] x (n 1) + 8 = (n 1) 10. Suppose a multiplexer has two input streams, each at a nominal rate of 1 Mbps. To accommodate deviations from the nominal rate, the multiplexer transmits at a rate of. Mbps as follows. Each group of bits in the output of the multiplexer contains 18 positions that always carry information bits, nine from each input. The remaining four positions consist of two flag bits and two data bits. Each flag bit indicates whether the corresponding data bit carries user information or a stuff bit because user information was not available at the input. a. Suppose that the two input lines operate at exactly 1 Mbps. How frequently are the stuff bits used? In this case, the stuff bits are always used because the information bits alone only provide an aggregate bit rate of 1.8 Mbps.

7 b. How much does this multiplexer allow the input lines to deviate from their nominal rate? This multiplexer provides either 9 or 10 bits for each stream per -bit frame. Thus, it allows either of the two input streams to transmit as low as 0.9 Mbps and as high as 1.0 Mbps. 11. Calculate the number of voice channels that can be carried by an STS-1, STS-3, STS-1, STS-48, and STS-19. Calculate the number of MPEG video channels that can be carried by these systems. A conventional voice channel is 64 kbps. An MPEG video signal can vary in bit rate. Assuming a bit rate of 5 Mbps, the results are summarized in the table below: Bit rate of signal (Mbps) Voice Channels MPEG Video STS STS , STS ,70 14 STS , STS ,50 1, Consider a SONET ring with four stations. Suppose that tributaries are established between each pair of stations to produce a fully connected logical topology. Find the capacity required in each hop of the SONET ring in the following three cases, assuming first that the ring is unidirectional and then that the ring is bidirectional. a. The traffic between each pair of stations is one STS-1. Unidirectional: STS-6 Bidirectional: STS-3 b. Each station produces three STS-1 s worth of traffic to the next station in the ring and no traffic to other stations. Unidirectional: STS-3 Bidirectional: STS-3 c. Each station produces three STS-1 s worth of traffic to the furthest station along the ring and no traffic to other stations. Unidirectional: STS-6 Bidirectional: STS Consider a set of 16 sites organized into a two-tier hierarchy of rings. At the lower tier a bidirectional SONET ring connects four sites. At the higher tier a bidirectional SONET ring connects the four lower-level SONET rings. Assume that each site generates traffic that requires an STS-3. The high-tier ring must carry all inter-ring traffic form the lower tiers. The capacity required on the high-tier ring is thus statistically dependent on the amount of inter-ring traffic generated from the lower rigs and the distribution of such traffic.

8 For this question, let us assume that the traffic generated is distributed evenly throughout the high-tier network that is, each low-tier ring sends an equal amount of traffic to each of the other low-tier rings. The lower-tier ring must accommodate 100% of the traffic sent from each of its sites. Since each site will generate as much traffic as it will receive, each site will send STS-3 of bandwidth. a. Discuss the bandwidth requirements that are possible if 80% of the traffic generate by each site is destined to other sites in the same tier ring. If 80% of the traffic is going to stay within the same ring, then the required bandwidth on the hightier ring will be reduced as most traffic is confined locally to the lower-tier rings. The number of links to be traversed by the average message is between one and two. Assuming that the traffic is distributed evenly on all of the links in the low-tier ring, the bandwidth required on each link will be around 1.5 STS-3. On the high-tier ring, 0% of the traffic from each sub-ring will be routed equally to each other sub-ring. Thus, each message will travel an average of ( )/3 = 1.33 hops. Since each ring produces 4 0. STS-3 amount of traffic. The bandwidth required on each link of the hightier ring is on the order of STS-3 STS-3. STS STS-3 b. Discuss the bandwidth requirements that are possible if 80% of the traffic generate by each site is destined to sites in other rings. If 0% of the traffic is going to stay within the same ring, then the required bandwidth on the hightier ring will be increased. The required bandwidth on the lower-tier rings will not change significantly, since each source still produces and receives STS-3 of traffic. The higher-tier ring will now support 4 times more traffic. Thus, the bandwidth requirement is around STS Consider a four-station SONET ring operated using the four-fiber bidirectional line-switched approach. Adjacent stations are connected by two pairs of optical fibers. One fiber pair carries working traffic and one fiber pair provides protection against faults. Traffic can be routed between any two stations in either direction, and no limit is placed on the occupancy in each working fiber. a. Sketch a break in the ring and explain how the system can recover from this fault.

9 a a d Protection loops b d b c a Working loops c d b Fiber not used for traffic between a and b Fiber carrying traffic between a and b c If a break in the ring occurs and all four fibers are affected, the system recovers by rerouting the traffic between the two affected nodes around the ring over the protection fibers. This protection is commonly referred to as line protection. b. Suppose a failure affects only one of the fiber pairs connecting two adjacent stations. Explain how the system can recover from this fault. In the case of such a failure, the system can simply route the information between the two adjacent stations over the protection fiber pair. This type of protection is commonly referred to as link protection. c. How does this ring operation differ from the ring operation in Figure 4.1 in terms of available bandwidth? In terms of available bandwidth, the working two fibers of the 4-fiber SONET ring can carry traffic up to capacity. In -fiber SONET rings, no explicit protection fibers exist, so the protection fibers must be embedded within the working fibers. Thus, half of the available bandwidth on each of the two fibers must be reserved to provide line protection. Note, however, that in both cases, half of the total system bandwidth on each link is reserved for protection because, in the 4-fiber case, the entire bandwidth of the protection fibers is reserved for protection. 15. Consider the operation of the dual gateways for interconnecting two bidirectional SONET rings shown in Figure The primary gateway transmits the desired signal to the other ring and simultaneously transmits the signal to the secondary gateway which also routes the signal across the ring and then to the primary gateway. A service selector switch at the primary gateway selects between the primary and secondary signals. Explain how this setup recovers from failures in the link between the primary gateways.

10 Two logical connections are maintained between the primary and secondary gateway. The first is a direct physical connection via the primary gateway link. The second is connected indirect via the secondary gateway as described in the question. This redundancy provides a backup in case of a link failure. If the primary gateway link fails, the service selector switch can select signals for the alternate indirect path until the primary gateway link is restored. 16. Consider the synchronous multiplexing in Figure Explain how the pointers in the outgoing STS-1 signals are determined. The outgoing pointers are all synchronized to a common clock such that all the outgoing STS-1 signals are synchronized with respect to each other. 17. Draw a sketch to explain the relationship between a virtual tributary and the synchronous payload envelope. Show how 8 T-1 signals can be carried in an STS-1. Tributaries are multiplexed in stream. Each tributary carries four T-1 channels and occupies twelve columns of the SPE, as shown below: 87 Columns T 1 T T 3 T 4 T 5 T 6 T 7 9 rows t The first column is path overhead. The next 84 columns are the seven tributaries and the last two columns are unused. 18. Show that the 100 nm window in the 1300 nm optical band corresponds to 18 terahertz of bandwidth. Calculate the bandwidth for the 100 nm window in the 1550 band. For a channel of bandwidth W, wavelength window λ, and center wavelength λ: Frequency Bandwidth = c λ λ Thus, in the 1300 nm optical band, we have:

11 8 (3 10 )( Frequency Bandwidth = 9 ( ) 9 ) = 17.75THz Similarly, in the 1550 band, we have: Frequency Bandwidth = 8 (3 10 )( ( ) 9 ) = 1.49THz 19. Consider WDM systems with 100, 00, and 400 wavelengths operating at the 1550 nm region and each carrying an STS-48 signal. a. How close do these systems come to using the available bandwidth in the 1550 nm range? One STS-48 signal carries Mbps of traffic. Thus, the aggregate bit rate of each of these systems is: System Bit rate 100 wavelengths Gbps 00 wavelengths Gbps 400 wavelengths Gbps The available bandwidth of this window is 1.49 THz. Thus, none of these systems even comes close to using all of the available bandwidth. b. How many telephone calls can be carried by each of these systems? How many MPEG television signals? The number of telephone calls and MPEG video signals that can be carried each system is shown below: System Telephone Calls MPEG Video Signals 100 wavelengths wavelengths wavelengths Although, none the systems comes close to using all of the available bandwidth, they can still carry huge amounts of traffic. This shows how WDM provides the potential for huge amounts of bandwidth and radically expands the bandwidth of existing optical links. 0. Consider the SONET fault protection schemes described in problem 14 and also earlier in the chapter. Explain whether these schemes can be used with WDM rings. The fault protection schemes described in problem 14 are applicable to any ring network structure. To use these schemes in WDM, the exact same procedures can be carried out except that instead of allocating half of the physical fibers for protection, half of the wavelengths in the WDM link would be allocated for protection and remain unused unless a fault occurred. 1. Calculate the spacing between the WDM component signals in Figure What is the spacing in hertz and how does it compare to the bandwidth of each component signal?

12 In Figure 4.19, there are 16 wavelengths distributed equally over the spectrum from 1545 nm to 1557 nm. Thus the spacing between each wavelength is: = 0.75nm 16 The spacing in Hertz is: Frequency Bandwidth = 8 (3 10 )( ( ) 9 ) = 93.65GHz Thus, the maximum achievable bit rate that each wavelength can achieve using on-off modulation is = 187 Gbps. Each signal carries.5 Gbps of traffic, so we see that because of the relatively large guard bands needed, WDM wavelengths have a very large spacing compared to their width.. How does WDM technology affect the hierarchical SONET ring topology in Figure 4.13? In other words, what are the consequences of a single fiber providing large number of high-bandwidth channels? WDM allows for much more aggregate bandwidth on each fiber. Thus, in the context of the hierarchical SONET ring topology, the high-tier ring can use WDM in order to increase its bandwidth and the number of channels it can support and, in turn, the number of lower-tier rings that it can service. WDM allows it to achieve this increased service ability without the need to lay down more physical fibers over its links. 3. WDM and SONET can be used to create various logical topologies over a given physical topology. Discuss how WDM and SONET differ and explain what impact these differences have in the way logical topologies can be defined. Both WDM and SONET define logical topologies by setting up semi-permanent paths between nodes in the network. WDM creates these paths by assigning lightpaths between nodes. WDM without wavelength conversion has an inherent restriction imposed on the logical topologies that it can create, because no to lightpaths that share the same link can use the same wavelength. In SONET, since tributaries are electrically defined, the only restriction imposed on the topologies is the total bandwidth utilized on each link. 4. Compare the operation of a multiplexer, an add-drop multiplexer, a switch, and a digital crossconnect. A multiplexer is a 1:N device. It takes N separate signals on N different inputs and combines them into one higher rate signal on one output port.

13 An add-drop multiplexer takes in N signals on one input port and replaces one of them with a new signal from a separate input port. The new aggregate signal is routed to an output port and the signal that was replaced is dropped to a separate output port. A switch takes in N inputs and routes them to N different outputs. A switch can provide full nonblocking interconnections between the input and output ports, but this is always the case. Generally, switches are configured using signaling that establishes paths across the network. A digital cross connect is similar to a switch except that it is semi-permanent, usually configured by network operators rather than signaling processes. Their configuration is also done on a larger time scale (days or weeks). Digital cross-connects provide a basic network topology on which routing can be applied. 5. Consider a crossbar switch with n inputs and k outputs. a. Explain why the switch is called a concentrator when n > k? Under what traffic conditions is this switch appropriate? This switch is a concentrator because traffic comes in on n lines and is concentrated onto k lines. Traffic on the output lines is higher than that of the input lines by a factor of n/k on average. The switch has inherent multiplexing functionality. b. Explain why the switch is called an expander when n > k? Under what traffic conditions is this switch appropriate? This switch is an expander because the number of lines is expanded from n to k. This switch is appropriate when the outgoing lines carry less traffic than the incoming lines. One such example is at the egress of a backbone network to an access network. The switch has inherent demultiplexing functionality. c. Suppose an N x N switch consists of three stages: an N x k concentration stage; a k x k crossbar stage; and a k x N expansion stage. Under what conditions is this arrangement appropriate? The number of crosspoints in an N x N crossbar switch is N. If the traffic load on the inputs is relatively low, this configuration may save on hardware costs. The number of crosspoints needed in the three-stage switch described is:

14 k + kn Depending on the values of n and k, k + kn may be less than n, especially if n is much greater than k. d. When does the three-stage switch in part (c) fail to provide a connection between an idle input and an idle output line? Assuming unicast traffic, if more than k inputs require a path to one of the third stage s n outputs at the same time, blocking will occur in the first stage, since it only has k outputs. 6. Consider the multistage switch in Figure 4.3 with N = 16, n = 4, k =. a. What is the maximum number of connections that can be supported at any given time? Repeat for k = 4 and k = 10. For N = 16, n = 4 and k =, we have the following switch architecture: 4 x x 4 4 x 4 4 x x 4 4 x 4 x 4 x 4 x 4 x 4 Thus, the second stage is the bottleneck, and blocking can occur in the first stage. Thus, eight connections can be supported at a time. If k = 4, then no blocking will occur at any stage in the switch, so 16 connections can be supported at a time. If k = 10, then there are ten 4 x 4 switches in the second stage. However, since there are only 16 inputs and 16 outputs, clearly only up to 16 connections can be supported at one time. b. For a given set of input-output pairs, is there more than one way to arrange the connections over the multistage switch? As shown in the picture in part (a), it is clear that each input-output pair can be connected through any one of the k second-stage switches. Thus, there are numerous ways to arrange the connections over a multi-stage switch. 7. In the multistage switch in Figure 4.3, an input line is busy 10% of the time. a. Estimate the percent of time p that a line between the first and second stage is busy.

15 Although, the busyness of each output line of the first stage depends on the switch control, on average, we can estimate that each of the output lines will have n/k the amount of traffic as the input lines. Thus p = (n/k) 10% b. How is p affected by n and k? p is directly proportional to the ratio of n to k. c. How does this p affect the blocking performance of the intermediate crossbar switch? Blocking occurs at the second switches when two of its inputs are to be routed to the same output. Although the control and routing algorithm of the switch can minimize the chance of this occurring, the greater the value of p, the greater the chance that such blocking will occur. d. Supposing that the blocking probability of the intermediate crossbar is small, what is the proportion of time p that a line between the second and third stage is busy? Assuming that the blocking probability of the second stage switches is small, the percentage of time that that the output lines on a switch in the second stage are busy is equal to the percentage of time that its input lines are busy. This is the case because the second stage switches are kxk components. Thus, p = p = (n/k) 10% For a given input and output line, what is the probability that none of the N/n paths between the input and output lines are available? In this question N/n should be k, since there are k possible paths from each input to each output (one different path possible through each second-stage switch). Assuming independence of all lines, which is not the case generally, for any one of the k paths between a given input and the output to be unavailable, one of the jumps must be unavailable. P[either one of two jumps busy] p For none of the paths to be available, all k must be busy. Since we assumed that all are independent of each other, P[none of the paths available] = (p) k 8. Consider the multistage switch in Figure 4.3 with N = 3. Compare the number of crosspoints required by a nonblocking switch with n = 16, n = 8, n = 4, and n =. For any switch to be non-blocking, we require k nb = n 1. The total number of crosspoints is Nk + k(n/n). The resulting number of crosspoints necessary for different values of n is shown below. N n k Number of Crosspoints needed

16 For a one-stage N x N switch with n = 3, we would require 103 crosspoints. Thus we see that, just as was noted in question 5c, for k much less than n, multistage switches can provide good hardware economy while remaining non-blocking. 9. A multicast connection involves transmitting information from one source user to several destination users. a. Explain how a multicast connection may be implemented in a crossbar switch. In a crossbar switch, multicasting can be implemented by copying the multicast input to multiple outputs. b. How does the presence of multicast connections affect blocking performance of a crossbar switch? Are unicast calls adversely affected? Unlike unicast switches, flow in multicast switches is not conserved. Generally, if multicasting is implemented in a switch, the total traffic flow out of the switch is more than the flow into the switch. Thus the probability of blocking is increased because the traffic inside the switch is higher. Unicast traffic is also affected by this increased flow because fewer resources in the switch are available. c. Explain how multicast connections should be implemented in a multistage switch. Should the multicast branching be done as soon as possible or as late as possible? In a multistage switch with a given multicast traffic profile, each branching procedure represents an increase in traffic flow beyond the branch point. Thus, to keep traffic load in the switch to a minimum, branching should be done as late as possible. 30. Show that the minimum number of crosspoints in a three-stage nonblocking Clos switch is given by n equal to approximately (N/) 1/ Given the number of inputs N, the number of crosspoints in a three-stage non-blocking Clos switch can be calculated as: N(n 1) + (n 1)(N/n) This number can be minimized with the correct choice on n by taking the derivative of the above expression with respect to n and setting the result to zero: d dn N 4nN N + n N n [ N N 4N + 3 n n ] = 0 = 0

17 N (1 n) = 3 n Assuming n>>1, this reduces to: n = N/ n = 31. What is the delay incurred in traversing a TSI switch? N The arriving frame is written onto the switch register and after the frame is completely written in, it needs to be read out in permuted order. 3. Explain how the TSI method can be used to build a time-division multiplexer that takes four T-1 lines and combines them into a single time-division multiplexed signal. Be specific in terms of the number of registers required and the speeds of the lines involved. Each input line carries T-1 traffic. The input frame size is n = 4 slots. Each frame that is output is of length 4n = 96 slots. After a full set of 96 slots is written into the 4 registers, the slots are read out in the outgoing line Mbps AA Mbps AB Mbps AC Mbps AD 4 slots 4 slots 4 slots 4 slots AA B A AC AD Mbps 33. Suppose that the frame structure is changed so that each frame carries two PCM samples. Does this affect the maximum number of channels that can be supported using TSI switching? Yes. Now each slot requires two memory reads and two memory writes. Thus, the maximum number of channels that can be supported is: Max = 15µ s 4 mem In other words, one half as many slots per frame can be supported as compared to when each slot requires one PCM sample.

18 34. Examine the time-space-time circuit-switch architecture and explain the elements that lead to greater compactness, that is, smaller physical size in the resulting switching system. In a space switch, n inputs are switched simultaneously in a switching time of T seconds. The space-time circuit switch switches each of the n inputs in succession with a switching time of T/n each, for a total switching time of T. By switching the inputs one at a time instead of simultaneously, the switching matrix needed at any given time is reduced in size accordingly and greater hardware compactness is achieved. The implicit tradeoff, however, is that higher line and switch speeds are needed. 35. Consider the three-stage switch in problem 5c. Explain why a space-time-space implementation of this switch makes sense. Identify the factors that limit the size of the switches that can be built using this approach. A space-time-space implementation would have the middle stage replaced by a TSI switch stage, and thus reduce hardware costs. The maximum number of slots in a TSI switch, say M, is limited by the length of the memory cycle. This limitation means that N/n M, which in turn places a limitation on N, the number of inputs to the switch (since n has a maximum that is determined by technological constraints). The limitation on the number of slots M suggests that the approach is suitable when concentration can be used in the input stage because of light loading in the input lines. 36. Consider n digital telephones interconnected by a unidirectional ring. Suppose that transmissions in the ring are organized into frames with slots that can hold one PCM sample. a. Suppose each telephone has a designated slot number into which it inserts its PCM sample in the outgoing direction and from which it extracts its received PCM sample from the incoming direction. Explain how a TSI system can be used to provide the required connections in this system. At some point in the ring, TSI is used to swap the information in the time slots of all pairs of connected users b. Explain how the TSI system can be eliminated if the pairs of users are allowed to share a time slot? If all user pairs share a time slot, for any connected pair of users we can assign the same timeslot to both. To communicate, each of the users inserts outgoing PCM samples on the slot and extracts incoming PCM samples from the same slot. c. How would connections be established in parts (a) and (b)? In the system described in part (a) the TSI switch monitors all of the slots continuously for connection request made by the users. If user A requests a connection to user B, the central switch controller makes the connection if the slot corresponding to user B is available. In the system described in part (b) a mechanism is needed to coordinate the sharing of slots between all the pairs of users. The most efficient mechanism would have half as many timeslots as users. The central switching mechanism would assign each pair of users a timeslot to

19 communicate over. Call setup would require all users to communicate with the central switch on a setup slot that would have to be accessed via a MAC protocol. An alternative simple distributed mechanism would involve each user monitoring their own timeslot. Assuming that the users have time slot tunability, a call could be initiated by sending a sample on the intended receivers time slot (provided it is free). Communication would occur on the receiver s time slot in this scenario. The drawback of this mechanism is that no more than half of the available bandwidth would ever be used. 37. Consider the application of a crossbar structure for switching optical signals. a. Which functions are the crosspoints required to implement? The crosspoints must be able to open, close, hold open, and hold closed. b. Consider a x crossbar switch and suppose that the switch connection pattern is (1 1, ) for T seconds and (1, 1) for T seconds. Suppose it takes τ seconds to change between connection patterns, so the incoming optical signals must have guardbands to allow for this. Calculate the relationship between the bit rate R of the information in the optical signals, the number of bits in each frame, and the values T and τ? For R in the range from 1 gigabit per second to 1 terabit per second and τ in the range of 1 microsecond to 1 millisecond, find values of T that yield 50% efficiency in the use of the transmission capacity. The transition time is τ seconds, and the hold time is T seconds. The optical transmission rate is R bps and the actual information bit rate is R effective, and letting n be the number of bits per frame (including overhead), we have: R effective = (total bit rate of connection)(proportion of useful bits) T n = R ( ) = T + τ T + τ The value of T that yields 50% efficiency is T = τ. The actual bit rate between each input/output port is R effective /. For T = τ = 1 microsecond, we have n = RT which gives frame size of n = 1000 bits at 1 Gbps and n = bits at 1 Tbps. For T = τ = 1 millisecond, the frame sizes are 1000 times longer. 38. Suppose an optical signal contains n wavelength-division multiplexed signals at wavelengths λ 1, λ,, λ n. Consider the multistage switch structure in Figure 4.3 and suppose that the first stage switches consist of an element that splits the incoming optical signal into n separate optical signals each at one of the incoming wavelengths. The jth such signal is routed to jth crossbar switch in the middle stage. Explain how the resulting switch structure can be used to provide a rearrangeable optical switch. In the resulting architecture, the second-stage inputs to any given switching element have the same wavelength. Thus, each single wavelength signal inputted to the second stage can be directed to any one of the third stage elements. Each third stage element corresponds to a single output port. It is assumed that the third stage consists of elements that recombine the n incoming signals, each with a different wavelength, into a WDM signal. The operation of this switch is very

20 similar to that of the switch in figure 4.3. However, it differs in that there is only one possible route for each input-output-wavelength combination. 39. Consider the equipment involved in providing a call between two telephone sets. a. Sketch a diagram showing the various equipment and facilities between the originating telephone through a single telephone switch and on to the destination telephone. Suppose first that the local loop carries analog voice; then suppose it carries digital voice. In an analog connection, the telephone set is connected by a pair of wires to the telephone office. The wires are terminated in a line card that digitizes the voice signal and transfers the digital signal to a digital switching system. The digital signal travels along a fixed path from the telephone switch in the originating office to the telephone switch in the destination office. The two or more offices are interconnected by digital transmission lines. If the telephone set were to generate digital voice then a digital signal would be transmitted along the local loop that connects the telephone to the central office. The operation of the line card would differ, as the arriving signal would already be in digital form. b. Repeat part (a) in the case where the two telephone calls are in different LATAs. If the two telephones are in different LATAs then the digital voice signal must traverse tandem switches that connect the two LATAs. c. In parts (a) and (b) identify the points at which a call request during setup can be blocked because resources are unavailable. Call requests can be blocked when there are insufficient resources to complete the call. These resources can include time slots and crosspoints in the switches and time slots in the digital lines. 40. Suppose that an Internet service provider has a pool of modems located in a telephone office and that a T-1 digital leased line is used to connect to the ISP s office. Explain how the 56K modem (that was discussed in Chapter 3) can be used to provide 56 kbps transfer rate from the ISP to the user. Sketch a diagram showing the various equipment and facilities involved. The ISP office is connected to the user s local central office, which is in turn connected to the user. The ISP can use the T-1 line to send digital information at a rate of 56 kbps into and across the telephone network to the line card that connects to the user. A modem can then deliver information at 56 kbps to the user. Recall that the user cannot send at 56 kbps because of limitations on the available SNR. Modem User s comp Twisted Pair Telephone Network T-1 line ISP office Modem bank

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