Exercise 2.2.
Draw the traffic profile of a periodic ON-OFF source characterized by the following parameters:
• average rate,
,
• capacity of a data block (data issued in the active source state),
,
• inactivity period,
.
, 
, 
are known, neither of the two Equations 2.1 and 2.2 can be solved
individually. They can therefore be combined to allow one of the unknown parameters to be obtained, for
example the duration of the inactivity period, knowing that 
through the relation
![\[P=\frac{AL}{l-AT_{OFF}}=200 \mathrm{kbit/s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-df58457239f59cb24dfc3e51df9c2c7e_l3.png "Rendered by QuickLaTeX.com")
at the average rate 
, that is 
, from which we obtain ![\[T_{ON}=\frac{L-AT_{OFF}}{A}=20 \mathrm{ms}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-74e13e918ba829605f253d0297eb2f20_l3.png "Rendered by QuickLaTeX.com")
Exercise 2.4.
Draw the traffic profile of a periodic ON-OFF source characterized by the following parameters:
• ratio between duration of activity and inactivity periods, 
,
• time interval between two consecutive source transitions from the ON state to the OFF state, 
,
• total amount of data issued in an interval of 
, 
.
![\[B=\frac{T_{ON}}{T_{ON}+T_{OFF}}=\frac{R}{R+1}=0,8\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-ebd2d390fdc64f1924f4e94255178a63_l3.png "Rendered by QuickLaTeX.com")
![\[T_{ON}=B(T_{ON}+T_{OFF})=BT=40 \mathrm{ms},\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-06a2ee834be025e0fd9f633e602f9854_l3.png "Rendered by QuickLaTeX.com")
![\[T_{OFF}=T-T_{ON}=10 \mathrm{ms}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-de8d8ce0153d490ff945c2618f9ebbcb_l3.png "Rendered by QuickLaTeX.com")
, which gives the peak rate 
. The resulting traffic profile is therefore shown in Figure E2.2.
Exercise 2.7.
Consider a one-way ring network consisting of 4 nodes X, Y, Z, W, connected in series. The
X-Y, Y-Z, Z-W and W-X connections between adjacent nodes have a capacity C bit/s. Assuming that each
node receives traffic from a CBR source and that all sources offer traffic with the same peak rate P, determine
the maximum throughput 
that characterizes this network and the maximum rate Pmax that does not
cause traffic losses.
. In the second case the traffic relations are X-W, Y-X, Z-Y, W-Z and therefore each link supports three of these flows, that is, all those that do not originate and terminate on the two ends of the link itself. It follows that the intensity of each flow cannot exceed one third of the connection capacity and therefore 
.
Exercise 2.11.
A source A generates a flow of IUs obtained by encoding a voice signal CBR at a peak rate 
by silence suppression. For simplicity, it is assumed that silence pauses (of constant duration 
) alternate with speech intervals (of constant duration 
) and each silence pause involves the removal of 
from the data stream. Source A therefore behaves as a periodic VBR ON-OFF source
with burstiness factor BA = 0.375. The source is connected by means of a link of negligible length to a packet
transmission node X. The IUs received by A, after a processing delay equal to 2/3 of , are retransmitted
by X as packets to destination B without adding any overhead and without any queueing delay. Node X transmits
at peak rate 
. B is a space module in orbit around the Moon and is located at an approximate distance from X 
. Calculate the total transfer time 
from A to B of the IUs issued by the source in the interval [0,40] s, also graphically representing with the correct scale ratios: (i) the flow of IUs issued by A (
), (ii) the flow of IUs issued by X (
), (iii) the flow of IUs received by B (
).
, as ![\[T_{OFF-A} =\frac{L_S}{P_A}=\frac{80 \cdot 10^3 \cdot 8}{64 \cdot 10^3}=10 \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-ad7133f0645081f808a8e74533f1c954_l3.png "Rendered by QuickLaTeX.com")
, obtaining ![\[T_{ON-A}=\frac{B_A}{1-B_A}T_{OFF-A}=6 \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-490dee8b76d7d461a8c8be5c6f0d0afe_l3.png "Rendered by QuickLaTeX.com")
bit) allows us to derive the burst transmission time by node X ![\[T_X=\frac{L_{ON-A}}{P_X}=\frac{384000}{128\cdot 10^3}=3 \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-982a0e79c6857bf25d98e299fca4ca04_l3.png "Rendered by QuickLaTeX.com")
in node X and the propagation time 
along the link X-B are obtained ![\[T_P=\frac{2}{3}T_{ON-A}=4 \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-98bc8fa32ba4deb7e3307a90448a8a9a_l3.png "Rendered by QuickLaTeX.com")
![\[\tau_X=\frac{d_X}{c}=\frac{400000}{300000}=1,3 \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-1558de396d3e24dde24d41bca109b13b_l3.png "Rendered by QuickLaTeX.com")
![\[T_{tot}=3T_{ON-A}+2T_{OFF-A}+T_P+T_X+\tau_X=46,3 \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-32dbf57dac439d88fe2fc861788f7687_l3.png "Rendered by QuickLaTeX.com")
Exercise 2.15.
Two terminal stations A and B are connected by a link crossing two packet-switched nodes X
and Z with the following numerical values:
• capacity of the two access links A-X and Z-B, 
,
• capacity of the internode link X-Z, 
,
• length of the two access links A-X and Z-B, 
,
• length of the internode link X-Z, 
,
• signal propagation speed on all connections, 
,
• processing times in nodes X and Z, 
and 
.
Assuming an infinite storage capacity in the nodes, calculate the total end-to-end delay required for the transmission from A to B of a 10,000-byte data unit which is broken into two packets of equal length with a packet header 
(no other data units cross the nodes X and Z).
and a header 
and 
the transmission times of each packet on the wired access line and internode link (given the value of the propagation speed), with 
and 
the respective propagation times, then the numerical values
of the delays that arise are: ![\[\tau_a=d_a/v= 5\mu s\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-4d8f6d437b4ca28248f088ebb21c0000_l3.png "Rendered by QuickLaTeX.com")
![\[T_{ta}=(L_p+L_h)/f_a=40,8 \mathrm{ms}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-de76bfd1d07baa430e736c6b3a79bd63_l3.png "Rendered by QuickLaTeX.com")
![\[\tau_i=d_i/v=0,5 \mathrm{ms}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-1120dd31e5f21e6f806572287185b8fa_l3.png "Rendered by QuickLaTeX.com")
![\[T_{ti}=(L_p+L_h)/f_i=8,16 \mathrm{ms}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-1c89be0a83b399e656e4de305f6fb1b5_l3.png "Rendered by QuickLaTeX.com")
, requiring a transmission time 
. As in any transmission of multiple data units in packet-switched networks from a source to a destination, the transmission, propagation and processing times required by the different units are partially overlapped as they affect different network elements (nodes and/or links). Note that, since the processing time 
. Furthermore 
is constant, there is no queuing even in node Z since the first packet requires a transmission time ![\[T_{tot}=\tau_a+T_{ta}+T_{pX}+\tau_i+T_{ti}+T_{pZ}+\tau_a+T_{ta}+T_{ta}=3T_{ta}+T_{ti}+T_{pX}+T_{pZ}+2\tau_a+\tau_i\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-d57ecc050fd4f9cfe21c14dd193f8d7f_l3.png "Rendered by QuickLaTeX.com")
![\[T_{tot}=\tau_a+2T_{ta}+T_{pX}+\tau_i+T_{ti}+T_{pZ}+\tau_a+T_{ta}=3T_{ta}+T_{ti}+T_{pX}+T_{pZ}+2\tau_a+\tau_i\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-0404bf00e2be1f608572bea8d2937038_l3.png "Rendered by QuickLaTeX.com")
.
Exercise 2.17.
Determine the symbolic expression analogous to Equation 2.6 which provides the total time required to transmit and receive (completely) a sequence of N packets of equal length from a source S to a destination D along a path that crosses H nodes, 
. All nodes are characterized by the same parameters 
(transmission time of a packet), 
(processing time of a packet), and all the links are characterized by the same delay 
. The nodes are assumed to have infinite storage capacity and no other data units pass through the nodes.
![\[T=(N-1)T_t+T_t+\tau+H(T_p+T_t+\tau)=(N-H)t_t+HT_p+(H+1)\tau\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-3e38913dc2b556f7447eb410709f7d1c_l3.png "Rendered by QuickLaTeX.com")
Exercise 2.20.
Consider a cascade of 3 packet switched nodes, A, B, C, connected by cable and characterized
by the following parameters on the respective outgoing connections: 
, 
, 
, 
. Node A issues packets of constant length at rate 
. Assuming that node B starts to retransmit a packet immediately after receiving it completely and that its processing time is zero, calculate the overall delay for the transmission of 3 packets from A to C.
![\[\tau_A=d_{A-B}/v= 500 \mu \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-8052924e137f47ef29555d6baa6042f9_l3.png "Rendered by QuickLaTeX.com")
![\[T_{A}=L/f_A=400 \mu \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-9d38e1b132288314186edb80021be254_l3.png "Rendered by QuickLaTeX.com")
![\[\tau_B=d_{B-C}/v=1 \mathrm{ms}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-ca03dc1749c15ba8628782316173beca_l3.png "Rendered by QuickLaTeX.com")
![\[T_{B}=L/f_B=800 \mu \mathrm{s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-255f238380bf917493ef74cd18b64fde_l3.png "Rendered by QuickLaTeX.com")
, which corresponds to a time between the transmission start events of consecutive packets given by 
. Figure E2.6 shows the sequence of reception, transmission and propagation times seen by the different nodes A, B and C. So the overall delay for the transmission from A to C of three packets is given by ![\[T=\frac{2}{P}+\tau_A+T_A+\tau_B + T_B=4,7 \mathrm{ms}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-314bcf4cc38998d319b35b4536d6efbc_l3.png "Rendered by QuickLaTeX.com")
Exercise 2.26.
Consider a packet switched network topology with 6 nodes (A, B, C, D, E, F) and 7 edges (AB,
B-C, C-D, D-E, F-A, B-E, C-F) with virtual circuit service where the following virtual connections are
active:
1 H-A-B-C-H
2 H-A-B-C-D-H
3 H-F-C-D-H
4 H-E-B-C-D-H
5 H-D-C-B-A-H
6 H-E-B-A-H
Fill in the forwarding tables of the nodes assuming that:
• the connections have been established in the order of the list,
• the logical channel identifier has 2 bits available,
• the logical channel identifier assigned by each node/host is the smallest available.
Exercise 2.30.
An S-TDM synchronous multiplexer generates an output flow with rate 
, frames of period 
which include 36 traffic time slots each, each with capacity of n bits, and an alignment time slot whose capacity is half of that available in a traffic slot. Determine the frame length 
and the capacity 
available in each traffic time slot.
![\[L_m=f_m\cdot T_m=657 \mathrm{bit}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-200a1cb970efb580b843b5cfa2c6b74a_l3.png "Rendered by QuickLaTeX.com")
. So the capacity of each tributary is ![\[f_c=n/T_m=36 \mathrm{kbit/s}\]](http://internetandnetworks.com/wp-content/ql-cache/quicklatex.com-ad884f9f37bc034dd2e648531f25f9f9_l3.png "Rendered by QuickLaTeX.com")