January 2023
Intermediate to advanced
272 pages
5h 3m
English
Content preview from Queues Applied to Telecoms
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11Chapter 3 Exercises
EXERCISE 3.1.– Length of a queue
According to Little’s law from Proposition 3.2:
Now, E(Τs) = E(Τf) + 1/μ since we assume that the duration of service for a customer (on average 1/μ) is independent of the duration of time spent waiting in the queue.
Therefore, E(Ns) = λe(E(Τf) + 1/μ ) = λeE(Τf) + λe/μ = E(Nf) + λe/μ. We deduce that E(Nf) = E(Ns) − λe/μ.
EXERCISE 3.2.– M/M/1 queue
We have an M/M/1 queuing system. A customer arrives on average every 10 minutes: λ = 1/10, and the average duration of service is 7 minutes: μ = 1/7. The offered traffic is 
. We use πk to denote the coordinate of the stationary distribution of the system: πk = (1 − ρ)ρk.
The probability р1 that at least two customers will wait to be served is equal to the probability that the system will contain at least three customers (one served and at least two waiting):
The probability р2 that an arriving customer will have to wait before being served is equal to the probability that the system is busy serving a customer:
The probability р3 that an arriving customer will find a queue in front ...
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