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Cork Institute of Technology
Bachelor of Science (Honours) in Software Development and
Computer Networking – Award
(NFQ– Level 8)
January 2006
Network Design
(Time: 2 Hours) Answer any four questions for full marks Examiners: Dr D. Pesch Dr. J. Buckley Mr. A. Kinsella
Q1.
(a) Consider a LAN with a large number of PCs and a single database server. The server receives 100 requests per minute during normal operation according to a Poisson process. The server requires 0.4 seconds on average to process a request. The processing time is exponentially distributed. (i) Calculate the total time each user has to wait until she receives a reply to her query using a suitable queuing model. (ii) By what factor can the request rate be increased if during peak times users are willing to wait twice as long for a reply compared to the normal case of part i) of the question? Assume that communication delay on the LAN is negligible. [10 marks]
(b) A transmission system, comprising of a single wireless transmission link and a multiplexer, is shared among 10 computers. Each computer generates fixed size data packets of size 1024bits according to a Poisson process with rate λ = 50/sec. Transmissions are protected by an ARQ scheme which results in 4 effective data rates Ri , i = 1, …, 4, depending on the error rate on the link. The rates are R 1 = 256kbit/s for 10% of the time, R 2 = 512kbit/s, 20% of the time, R 3 = 1024kbit/s, 40% of the time, and R 4 = 2048kbit/s, 30% of the time. (i) Calculate the throughput of the system. (ii) Calculate the average delay data packets experience in the multiplexer before transmission. (iii) What single data rate would be required for the wireless link in order to maintain the same average delay if the packet size was exponentially distributed with mean 1024bits? [15 marks]
Q2.
(a) Consider a telecommunications switch with a total of 40 output circuits without a call attempt queue.
- Find the probability that a telephone user of the switch finds all output lines busy when the mean call duration is 240sec and the arrival rate of calls to the switch is λ = 7.5/min.
- Determine how many Erlang the telephone switch can carry when it has 25 output lines and the call blocking probability is restricted to P (^) B = 0.05.
- Determine the blocking probability when the switch with 25 output lines has now also a buffer to allow 10 calls to wait until a free line becomes available. [15 marks] NOTE : You might find Table 1 useful for some of your calculations.
(b) The input to a router is a data packet stream modelled by a Poisson process with arrival rate λ = 50 packets/sec. The length of the data packets is exponentially distributed with mean L = 512bytes.
- Assume that the router has m = 4 output lines of data rate R = 64kbit/s, which transmit packets from the head of the queue. Calculate the probability that arriving packets have to queue before transmission.
- Assume that the 4 output lines are aggregated into a single output line of data rate R = 256kb/s. Calculate the probability that an arriving packet has to queue for this case. What do you observe and discuss your observations in the context of efficient design of multiplexers and routers. [10 marks] [Total: 25 marks] Q3.
(a) The queuing network shown in Figure 1 is given. Calculate the average queue length and average queuing delay for each of the three queues. The parameter values shown are as follows: r 1 = 2/sec, r 2 = 3/sec, μ 1 = 3/sec, μ 2 = 5/sec, μ 3 = 5/sec, p 13 = 0.7, p 23 = 1.0, p 32 = 0.4, all other pij = 0.
μ 1
μ 2
μ 3
r 1
r 2
p (^13)
p (^23) p 32
Figure 1. Queuing Network
Q5.
(a) Describe a method for computer generated pseudo random numbers that are uniformly distributed over the interval [0, 1] and show with the help of mathematical expressions how exponentially distributed random variates can be generated based on those pseudo random numbers. [10 marks]
(b) Briefly describe the structure of a stochastic discrete-event simulation program for an M/G/n queuing system. You may use the functionality of a particular simulation system or programming language to help you in your description. [15 marks]
You might find the following formulae useful in answering questions 1, 2, and 3.
Little’s theorem: A = λ T and AQ = λ W
M/M/1/∞ Queuing System: state probability of Markov chain pi = ( 1 −ρ) ρ i
A ,
M/M/m/∞ Queuing system:
Q Q 1
A P and
Q 1
A m P
Probability of queuing in M/M/m: (^ )
ρ −
(^0) m! 1 P p m m Q , where
−
1 (^0 0)!! 1
m i
i m m
m i
p m
ρ ρ and
m
Pollazcek-Khinchine formula for an M/G/1 queuing system:
( X )
W X
2 and (^) T = X + W , X is service time random variable
with ∑ ( )
∞
k 0 k^ k
X X PX and ( k)
k 0
X 2 ∑∞ X^2 kPX
=
= , for discrete probability distributions
∞
−∞
X = x⋅p(x)dx, ∫
∞
−∞
X 2 = x^2 p(x)dx; for continuous probability distributions
The solutions of the quadratic equation ax^2^ + bx + c = 0 are a
x b b ac 2
1 , 2
=− ±^ −
Jackson’s Theorem:
r p j K
K
i
j j i ij ;^1 ,^2 ,Κ
1
=
λ λ
j
j j μ
λ
P ( ) n P ( n ) P ( n ) Λ PK ( nK )
= 1122 with n ( n 1 , n 2 ,Λ, nK )
and Pj ( n^ j )^ = ρ njj ( 1 − ρ j )^ ; nj ≥ 0
