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Topic: A "simple" problem
Replies: 7   Last Post: Jun 20, 1996 6:12 PM

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L.N. Chung

Posts: 5
Registered: 12/18/04
A "simple" problem
Posted: Jun 13, 1996 7:06 PM
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Need some help in solving the following problem:

The system is a slotted-aloha system with multiple receivers, and
transmissions in a slot do not always collide. The arrival process is
assumed to be Poisson. Each transmission (arrival) is then aligned in
time slot of length T sec. The frequency of each transmission is
selected in random (uniformly distributed) over frequency range F.

A collision is only declared, when two (or more) transmissions in the
same timeslot are within Fc Hz (i.e. F1- F2 <= Fc) of each other.

The system is also limited to N receivers per timeslot, i.e. if more
the N transmissions are not collided in one timeslot, they will be

The object is find the throughtput of the system in terms of average
no. of success transmission per time slot.

My attempt at a solution go like this:

throughput S = SUM ( Ps(k) * k )

Ps(k) = prob. of k success tx in a timeslot
= SUM ( P(j) * Ps(k|j) )
j = k

P(j) = prob. of j transmission in T = Pisson distribution

Ps(k|j) = prob. of k success given j transmission in T.

my main problem is with getting the generalized equation for the last
expression. Of course, there is also the possibility of not able to
sum everything to inf.

However, I got a feeling that the last expression is fairly standard
problem. So, please send in any comment, solution or references to
published work etc..

Many thanks in advance.

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