EE132B-HW Set #3
UCLA 2014 Fall
Prof. Izhak Rubin
Problem 1 Consider a broadcasting bus system as shown in Figure 1. Attaching to the bus are N stations, and they share the channel under pure ALOHA scheme. Suppose that the transmission rate is R (bits/sec), and the message size is M bits. Neglect the effect of propagation delay. Statin i 1
2
i
i+1
N
Broadcasting Bus
Figure 1: A broadcasting bus system with N statoins (a) How long does it take to transmit a single message? (b) Let Ti denote the idle time for station i. Suppose that Ti is exponentially distributed with parameter λi , and {Ti | i = 1, 2, . . . , N } is a set of independent random variables. Let T denote the idle time for the broadcast channel. Note that the channel is idle when every station is idle. What is the distribution function for T (i.e., P (T ≤ t))? (c) Suppose that the channel is idle and station i initiates a message transmission at time 0. What is the collision probability for this message transmission? Ans: (a) If no collision occurs, it will take
M R
seconds to transmit a single message.
(b) The distribution for the idle time of the broadcast channel is given by P (T > t) = P (min{Ti | i = 1, 2, . . . , N } > t) = P (T1 > t, T2 > t, . . . , TN > t) = P (T1 > t)P (T2 > t) . . . P (TN > t) = e−λ1 t e−λ2 t . . . e−λN t = e−(λ1 +λ2 +···+λN )t . Therefore, P (T ≤ t) = 1 − e−(λ1 +λ2 +···+λN )t , which is an exponential distribution with parameter λ1 + λ2 + · · · + λN . 1
EE132B-HW Set #3
UCLA 2014 Fall
Prof. Izhak Rubin
(c) The probability that this message transmission will not be unsuccessful. P (no collision) = P (T1 > =e
M M M M , . . . , Ti−1 > , Ti+1 > , . . . , TN > ) R R R R
−(λ1 +···+λi−1 +λi+1 +···+λN )× M R
−
PN
=e
k=1,k6=i
P
P (collision) = 1 − P (no collision) = 1 − e
−
N k=1,k6=i
λk × M R
λk × M R
.
Problem 2 Consider a slotted token ring system with N stations. The token travels in the counter clockwise direction as shown in Figure 2. In this polling system, a busy station will capture the token and hold it for the duration of its frame transmission. It then releases the token and makes it available for capture by the neighboring station along the ring network. Assume for our analysis that each station holds the token for a random period of time. Let Hi denote the token holding time of station i (measured in units of slots). Assume that {Hi | i = 1, 2, . . . , N } is a set of independent and identically distributed (i.i.d.) random variables. We also assume that Hi is governed by a geometric distribution with parameter 1 − p such that P (Hi = n) = p(1 − p)n , ∀n = 0, 1, . . . . Note that p represents the probability that station i releases the token at the end of a time slot. Station 1
Station 1 2 3
Random holding time
Token ring direction
H1 H2 H3
Station N
Station 2
Station j
N
Station 3
HN
Figure 2: The slotted token ring network (a) Neglect the propagation delay in the system. Suppose that station 1 receives the token at the start of time slot 0. Calculate the probability that station 3 receives the token at the start of slot k. Note that k can be equal to 0, since we neglect the propagation delay as well as assume (as an approximation) that a non-busy station will hold the token for negligible period of time that is set here equal to 0. (b) The probability that station 1 will receive the token again at the start of slot k. (Hint: The sum of r i.i.d. Geometric random variables with parameter p follows a negative binomial distribution with parameter (r, p).) 2
EE132B-HW Set #3
UCLA 2014 Fall
Prof. Izhak Rubin
(c) Suppose that the propagation delay from any station to its adjacent station is τ slots. What is the throughput rate of this system? (HINT: Consider a cycle period (occurring between successive visits of the token to the same station). Calculate the throughput rate as the ratio between the average amount of "work" performed by stations during a cycle to the average duration of the cycle. A station is said to perform "work" when it holds the token; its amount of "work" is set equal to the time that it holds the token.) Ans: (a) The probability that station 3 receives the token at the start of slot k is given by P (H1 + H2 = k) = fH1 ⊗ fH2 (k) = = = =
k X
P (H1 + H2 = k | H1 = n)P (H1 = n)
n=0 k X n=0 k X n=0 k X
P (H2 = k − n)P (H1 = n) p(1 − p)k−n p(1 − p)n p2 (1 − p)k = (k + 1)p2 (1 − p)k .
n=0
(b) Based on the hint, the probability that station 1 receives the token ring again at the start of slot k is given by P (H1 + H2 + · · · + HN = k) = fH1 ⊗ fH2 ⊗ · · · ⊗ fHN (k) !
k+N −1 = (1 − p)N pk . k (c) The throughput rate measures the average amount of work per unit time. The amount of performed work is measured as the total time spent by stations in holding the token. Define a cycle to be the time elapsed from the instant the token is received by any station to the earliest instant it is received by the same station. Consider the amount of work during a cycle denoted by W . Then, the average amount of work during a cycle is given by E[W ] = E
"N X i=1
#
Hi =
N X i=1
3
E [Hi ] =
N (1 − p) , p
EE132B-HW Set #3
UCLA 2014 Fall
Prof. Izhak Rubin
and the average length of a cycle is E[W ] + N τ . Thus, we have the throughput rate (denoted as η), which is given by E[W ] η= = E[W ] + N τ
N (1−p) p N (1−p) + Nτ p
=
1−p p (1−p) + p
τ
=
1−p . 1 − p + τp
(1)
Problem 3 (a) In a few words, describe the most significant difference between circuit-switching and packet-switching systems. Is email system a circuit-switching or a packetswitching system? (b) Outline the key differences between datagram and virtual-circuit switching systems. (c) Consider the following situations in conjunction with multiple access schemes. Which multiple access scheme is typically employed in each situation? (i) You are waiting at an intersection with 4-way stop sign. (ii) You present your research work at a small conference where you raise your hand before you speak. (iii) You let your child use your car every Saturday night. (iv) You are trying to take the elevator in ENGINEERING IV Bldg. Ans: (a) The most critical difference between circuit switching and packet switching systems is that the circuit switching system provides dedicated space, time, and/or frequency resource to a user, regardless of the frequency and the efficiency of the resource utilization by the user. The packet switching network system does not dedicate link capacity resources to a user flow. It allocates resources to a user when the user needs such a resource. Email applications are typically carried over a packet switching network system. (b) In the virtual circuit approach to packet switching, the relationship between all packets belonging to the message or a session is preserved. A single route is chosen between the sender and the receiver at beginning of the session. When the data are sent, all packets of transmission travel one after another along that route. In the other approach of packet switching that is the datagram approach, each packet is treated independently of all others. Even if one packet is just a piece of a muti-packet transmission, the network treats it as though it is existed alone. 4
EE132B-HW Set #3
UCLA 2014 Fall
Prof. Izhak Rubin
(c) There is no absolutely correct answer. Possible answers are: (i) Statistical multiplexing: ATDM (ii) Reservation (demand assigned TDMA) (iii) TDMA (iv) Reservation (demand assigned TDMA)
Problem 4 A
I
B
J
C D E
1 2 3 4
4 by 4 switch
1 2 3 4
K L M
F
N
G
O
H
Time-division multiplexer
P
Figure 3: A switching network Consider a switching network illustrated in the Figure 3. Suppose that this network is a circuit switching system. Also assume that this network is a one-directional system (i.e., the users on the left hand can only send and the users on the right can only receive). The users are identified by indices A, B, . . . , P , and the lines are number as 1, 2, 3, 4. The switch is a 4 by 4 switch. Each input port of the switch receives the output of a time-division multiplexer which produces two slots per frame. For example, at Line 1 on the transmitting terminals side, User A is assigned slot 1 and user B is assigned slot 2. Assume that the user on the top is always assigned slot 1. During busy hours, the routing table for the switch is set in Table 1. (a) Let (X, Y ) denote that user X is connected to user Y . List all the connections in the system based on the routing Table 1. (b) If Lines 2 and 4 on the transmitting side fail, and Line 1 on the receiving side fails, which connection(s) survive(s)? (c) Suppose that we only have connections (A, O), (D, P ), (G, N ), (H, I), and (E, M ). Make a new routing table.
5
EE132B-HW Set #3
UCLA 2014 Fall
Input Port Line NO. 1 3 4 3 2 4 1 2
Slot NO. 1 1 1 2 2 2 2 1
Output Port Line NO. 3 2 3 4 4 2 1 1
Prof. Izhak Rubin
Slot NO. 1 1 2 1 2 2 1 2
Table 1: Routing table (d) Suppose that the service provided is the transmission of stereo digital audio signals which is sampled at 22.4 kHz. These samples are quantized to 256 levels. What is the minimum transmission rate (bits/sec) for the time-division multiplexer? Ans: (a) (A, M ), (E, K), (G, N ), (F, O), (D, P ), (H, L), (B, I), (C, J). (b) (A, M ), (E, K), (F, O). (c) Input Port Line NO. 1 2 4 4 3
Slot NO. 1 2 1 2 1
Output Port Line NO. 4 4 3 1 3
Slot NO. 1 2 2 1 1
Table 2: The new Routing table
(d) The frame duration time must be equal to the length of sampling interval. Thus, 1 kHz = 44.64 µsec. Since each frame has two slots, the frame duration time is 22.4 the slot duration time is 22.32 µsec. The number of quantization levels is 256 so that 6
EE132B-HW Set #3
UCLA 2014 Fall
Prof. Izhak Rubin
each sample is represented as a binary code of 8 bits. Note that a slot per frame is assigned for a station. Therefore, a station must transmit at least 8 bits during a slot 8 bits ) = 358 kbps. and the minimum transmission rate is therefore equal to ( 22.32 µsec
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