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Queuing Theory and Telecommunications: Networks and Applications
2nd Edition Solution Manual
© 2014 Queuing Theory and Telecommunications: Networks and Applications -
All rights reserved Queuing Theory and Telecommunications: Networks and Applications
|BY |
|GIOVANNI GIAMBENE |
|DIPARTIMENTO DI INGEGNERIA DELL'INFORMAZIONE E |
|SCIENZE MATEMATICHE, UNIVERSITÀ DEGLI STUDI DI |
|SIENA, VIA ROMA, 56 - 53100 SIENA, ITALY, EMAIL: |
|GIAMBENE@UNISI.IT |
CONTENTS
Contents v Preface vii 1. Exercises on Part I of the book 1 2. Exercises on Chapter 4: Survey on Probability Theory 23 3. Exercises on Chapter 5: Markov Chains and Queuing Theory 39 4. Exercises on Chapter 6: M/G/1 Queuing Theory and Appl. 83 5. Exercises on Chapter 7: Local Area Networks Analysis 117 6. Exercises on Chapter 8: Networks of Queues 145
Preface This is the solution manual of the book "Queuing Theory and
Telecommunications: Networks and Applications", second edition. Exercises
are given here for the various Chapters with a detailed solution, allowing
the reader to understand the models and the approaches adopted. The purpose
of this solution manual is to provide tools and examples for solving
typical teletraffic engineering problems. This solution manual is an
essential complement to the book and contains more than 80 solved exercises
on the different topics.
Please note that the numbering of the exercises is the same as in the
book, but Section numbering does not match Chapter numbering in the book.
Figures are numbered independently of the book, but a mapping of the
numbering is provided for those figures that are also in the book. |SECTION 1 | |
|Exercises on Part I of the Book | |
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|DIGITAL NETWORKS AND The INTERNET | |
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|Exercises on Part I of the book | |
This Section contains some exercises on the first part of the book. The
main interest is on traffic regulators, Dijkstra routing, deterministic
queuing, and cwnd behavior of TCP.
Ex I.1 We have a Frame Relay network, which applies a policer to control
the access of traffic sources. Let us consider a traffic source, which has
a periodic ON-OFF bit-rate as a function of time as shown in Figure 1.1
(Figure 3.57 in the book), with parameters b (= burst bit-rate), T (= time
length of the source cycle), and l (= xT, burst duration). The policer uses
the following parameters: measurement interval, Tc, committed burst size,
Bc, and excess burst size, Be. We make the following assumptions: . Bc/Tc = Rc, a constant value, . Be/Tc = Re, a constant value, . bl/T = bx = Rs, mean source bit-rate, . A rectangular pulse (burst) represents a single packet in Figure 1.1; . The measurement interval Tc is applied to the periodic source according
to the "phase" shown in Figure 1.1, so that the source cycle T contains
an integer number of measurement intervals Tc (Tc = yT, with y = 1, 1/2,
1/3, etc.); . Constraints: bx = Rs ? Rc (so that there is enough capacity to service
the traffic source) and Tc ( xT ( y ( x (the measurement interval is
larger than the burst duration).
It is requested to determine the conditions to have marking or dropping
of all generated packets. [pic]
Figure 1-1. Periodic traffic source (source cycle T) and measurement
interval Tc (in this graph Tc ( T, y = 1).
Solution
The men bit-rate Rs = bx and the peak bit-rate is equal to b. Then, the
burstiness ? of the traffic source can be obtained as: [pic] Hence, parameter x directly controls the burstiness of the traffic
source. For the sake of simplicity, we could consider x ( y, even if we
have maintained the use of both symbols in the following derivations.
Note that bxT represents the burst size in bits and one burst,
corresponding to one packet, is generated per cycle. The condition to have
marking or dropping of all generated packets is: [pic] We can combine this result with the constraints Rs ? Rc and y ( x: [pic] The maximum of the burst duration l = xT that can be allowed without
marking is constrained as follows: [pic] The condition to drop all generated packets is: [pic] The combined condition for packet dropping is: [pic] Summarizing the above considerations, we can conclude that under the
condition x ? y (i.e., for a sufficiently high burstiness value ? = 1/x),
all packets are marked if yRc < Rs ? y(Rc + Re); moreover, all the packets
are dropped if Rs > y(Rc + Re). Hence, marking or dropping depend on the Rs
value: if Rs increases, we can have first packet marking and then packet
dropping.
An equivalent approach could be carried out on the basis of the graph of
the arrival curve of the traffic source, as shown in Figure 1.2.
[pic]
Figure 1-2. Arrival curve of the traffic source and measurement interval
Tc; example with packet marking.
Ex I.2 Let us consider an ATM switch with a switching table, which
manages virtual paths and virtual channels, as shown in Figure 1.3 (Figure
3.58 in the book). It is requested to determine the VPI and VCI fields to
be used for an input cell if we like that this cell leaves the switch from
output line A; in this case, we are also asked to provide the VPI and VCI
fields of the corresponding output cell. [pic]
Figure 1-3. ATM switch and its switching table Solution
On the basis of the switching table provided, an input cell must have VPI
= 1 and VCI =1 to have that this cell leaves the switch from output line A.
The corresponding output cell has VPI = 2 and VCI = 3. Ex I.3 Let us consider the network depicted in Figure 1.4 (Figure 3.59 in
the book): it is requested to determine the sink tree for node A by
applying the Dijkstra shortest path routing algorithm. [pic] Figure 1-4. Network with bidirectional links, labeled by (a, c), where
''a'' denotes the link number and ''c'' represents the link cost.
Solution
We start the Dijkstra algorithm by labeling all nodes with an infinite
cost. Then, we take node A as a reference and we label the nodes connected
to A:
|[pic] | |
| |Between nodes B |
| |and F we select |
| |node B, which has|
| |the lowest cost. |
| |Then, we need to |
| |re-label the |
| |nodes connected |
| |to node B (i.e., |
| |node C). |
|[pic] |Between nodes C |
| |and F we select |
| |node C, which has|
| |the lowest cost. |
| |Then, we need to |
| |re-label the |
| |nodes connected |
| |to node C (i.e., |
| |nodes D and E). | |[pic] |Among nodes D, E,|
| |and F we select |
| |node F, which has|
| |the lowest cost. |
| |Then, we need to |
| |re-label the |
| |nodes connected |
| |to node F. |
| |However, the |
| |label of node E |
| |does not change |
| |because the cost |
| |would increase. |
| |Moreover, the |
| |label of node D |
| |does not change |
| |because the cost |
| |of the path |
| |through F would |
| |not change. | |[pic] |Between nodes D |
|