Download Collaboration - Suraj @ LUMS

Winter Quarter 2001
CS 472 Advanced Networks
Assignment Number 1:
To be done individually
Due on 11:45pm,Friday 4th Januray ,2002
Collaboration
I strongly encourage you to discuss the topics with anyone you can. That's the way good science
happens. As a professional, you should acknowledge significant contributions or collaborations in your
homework/projects. Keep in mind that there is a lot of difference between helping and cheating.
Note: Use a spreadsheet and the iterative version of the loss formula, this will ease your
calculations.
1.
A loss-system full availability group consists of five devices. If the mean call holding
time is 180 seconds, and the call intensity is 80 calls/hour, what is the mean load per
device?
A=lambda*S=80*180/(1*60*60)= 4 erlangs.
Per device=4/5 =0.8 Erlangs
2.
In a particular, system, it is found that during the busy hour, the average number of calls
in progress simultaneously in a certain full availability group of circuits was 15. All
circuits were busy for a total of 30 seconds during the busy hour. Calculate the traffic
offered to the group.
Average calls in progress
Time when the system is busy=30 secs.
Carried traffic=15 erlangs
Offered traffic=carried traffic + lost traffic
OT=15+ OT*traffic loss ratio
Traffic loss ratio=30/3600=1/120
Therefore OT=15.126
3.
A group of eight circuits is offered 6 erlangs of traffic. Find the time congestion of the
group, and calculate how much traffic is lost.
A=6,N=8
E[n]=0.121876 (use the formula)
Lost Trafic= A* E[n]=0.731255 erlangs
4.
The overflow traffic from the eight circuits in the problem 3 is fed to a ninth circuit.
What traffic will it carry?
Overflow traffic=0.731255
Use recursive formula for N=1
Answer=0.422384*0.731255=0.421
5.
A system of six telephones has full availability access to six devices. Find the probability
that 1,2, …, 6 devices are busy. What is the (a) the call, and (b) the time congestion of
the system if the carried traffic is 2.4 erlangs?
S=N
Bernoulli case
Carried traffic=2.4 Erlangs
Traffic per device=2.4/6=0.4 Erlangs
A=(alpha)(1+alpha)=carried traffic per device=0.4
[I]=(S choose I) ai(1-a)s-1
for 1=BINOMDIST(1,6, 0.4,FALSE)= 0.18662
for 2=BINOMDIST(2,6, 0.4,FALSE)= 0.31104
for 3=BINOMDIST(3,6, 0.4,FALSE)= 0.27648
for 4=BINOMDIST(4,6, 0.4,FALSE)= 0.13824
for 5=BINOMDIST(5,6, 0.4,FALSE)= 0.03686
for 6=BINOMDIST(6,6, 0.4,FALSE)= 0.0041
call congestion=0 (no more calls since S==N)
time Congestion=[N]=A6= 0.0041
6.
(a) Two erlangs of traffic are fed to three devices. What is the congestion, and how much
traffic is lost?
(b) Two erlangs of traffic are fed to one device. The overflow is fed to a second device,
and the overflow from that to third device. Is the value of traffic lost the same as in
(a)?
e[3]=0.210536 =answer part (A)
Lost traffic =0.421053
Same as Q4
Traffic offered to first device=2 erlangs
Overflow from first device=e[1]*2=1.333
Traffic offered to 2nd device=traffic overflow from first device=1.3333
Overflow from 2nd device=e[1]*1.3333=0.76951
Traffic offered to 3rd device=traffic overflow from 2nd device=0.76951
Overflow from 3nd device=e[1]*0.76951 = 0.33
Differnet from the first case.
7.
The state transition diagram below represents a system with an infinite number of devices
subjected to calls arriving at random with fixed mean arrival rate, .

0


2
1

2

3
3
4
If / = A, the mean offered traffic. Assuming statistical equilibrium, show that:
[i] 
Ai e  A
i!
8. Find the probability of the network being in each of its possible states, i, and check that
 [i]  1 , if an infinite number of sources are feeding traffic into 8 devices for the
i
following three cases:
(a) traffic fed = 2 erlangs
(b) traffic fed = 5 erlangs
(c) traffic fed = 7 erlangs
Plot [i] against i, for each case. What do you infer from these distributions.
9. Suppose that in a local exchange office serving 10000 subscribers, the average subscriber
calling rate (originating traffic load) is 0.04 erl, out of which 10% is directed to a
particular transit exchange.
(a) What is the minimum number of trunk circuits that you would require to the transit
exchange with a blocking probability no greater than 1%.
(b) Calculate the blocking probability when the traffic is doubled under overload
conditions.
0.04 erl per subscriber
total offered traffic=0.04*10000=400 erlangs
traffic directed to the exchange = 10%= 40 erlangs
recurse E(n) formula till e(n)=0.01
answer part (a)=53 circuits
blocking probablity under overload condition =35%
10. A key telephone system with 4 telephone sets is connected to a local exchange through a
PBX via 2 subscriber lines. Suppose that each telephone set originates 2 calls per hour at
random, and is used for three minutes on average for each call. Then calculate the
blocking probability when the system operates on a non-delay (loss) basis.
2 calls per hour
S= 3 mins
Therefore alpha=traffic offered per free source=2*3/60=0.1 erlangs
Since s>N
Therefore ENGSET distribution
Use [B] formula for N=2
Answer=0.225
11. Derive the iterative version of the Erlang B loss formula:
Ei ( A) 
AEi 1 ( A)
i  AEi 1 ( A)