(a)
(b) (c) and (d)
Since the mean arrival rate is 20 requests per second. The mean inter-arrival time is 1 20
= 50ms.
The mean number of requests arriving in 1 minute = 20 requests per seconds × 60 seconds / minute = 1200 requests per minute.
Recalling that for Poisson arrivals with mean arrival rate λ and time interval t, the probability of n arrivals is
(λt)n exp(−λt). (1) n!
For this question, λ = 20 and t = 60, so λt = 1200.
In order to calculate the probability of no arrivals in a minute, we put n = 0 to obtain
exp(−λt) = exp(−1200) (2) In order to calculate the probability of 10 arrivals in a minute, we put n = 10 to obtain
Solution to COMP9334 Revision Questions Week02A — Part 2
Question 1
(1200)10 exp(−1200) 10!
(3)
Question 2
In order to refer to the two Poisson processes in a convenient way, I call them P1 and P2. The Poisson processes P1 and P2, have rates r1 and r2, respectively.
Consider a time interval T. Since P1 is a Poisson process with rate r1, we know that the probability that there are k arrivals in time interval T is
e−r1T(r1T)k k!
Similarly, the probability that there are j arrivals in time interval T from P2 is e−r2T(r2T)j
j!
(4)
(5)
Let us consider the aggregation of the two Poisson processes P1 and P2 over the time interval T. The arrivals can come from P1 or P2. Let us find the probability that there are n arrivals in T. If there are n arrivals from P1 and P2 together, this can be resulted from
• 0 arrivals from P1 and n arrivals from P2
• 1 arrivals from P1 and (n − 1) arrivals from P2
1
=
=
Probability that there are n arrivals over time T from P1 and P2 together n
Probability of i arrivals over time T from P1 × Probability of (n − i) arrivals over time T from P2 i=0
n e−r1T(r1T)i e−r2T(r2T)n−i i=0 i! (n−i)!
• 2 arrivals from P1 and (n − 2) arrivals from P2 …
• (n − 1) arrivals from P1 and 1 arrivals from P2 • n arrivals from P1 and 0 arrivals from P2 Therefore
1 e−(r1+r2)T n n! (r1T)i(r2T)(n−i) n! i=0 i!(n−i)!
=
= 1e−(r1+r2)T((r1+r2)T)n
n!
This shows that the aggregation of P1 and P2 is a Poisson process with rate r1 + r2.
2