Help with a technical telecomm question?

[Stir]

Can anyone out there point me in the right direction on this?

I need a method (and the math behind it) to estimate the the total messages lost due to collision (or the %) on a channel with no collision detection, and no acks, where each sender randomizes its delay before sending over time T, with message volume m (or alternatively message length L).

I've derived a solution (which I'd like to confirm or correct) but am certain this must be readily available out there ... but my Internet search hasn't succeeded.

Help ??!!

[Yalius]

Ok, if I'm getting it right, over a time T, where average length of message is L and number of messages is M, probability of messages getting lost in transmission ought to be ((L*M)/T)^2. Method behind it is, one transmitter is using the channel (L*M)/T fraction of time T; probability of a second transmission occurring during that fraction is also (L*M)/T, so product of the two is the probability of both occurring at the same time.

[Stir]

Yalius, thanks for the response. I see that I didn't state the problem very well. I should have made it clear that there are not just two contenders for the channel, but rather m (each sender sends exactly one message during the time period, at a random time). Sorry to have been unclear ... any help?

[slyjoe]

I don't have a lot of time right now, but look up ALOHAnet - it was an early collision scheme, and sounds a little like what you are describing (random backoff is something to look up also).

[Yalius]

For that case, I'm getting probability of any particular message making it through without interference as ((T-L)/T)^(M-1); number of messages that would come through without interference would be M*(((T-L)/T)^(M-1)).

[DickK]

Sorry, I don't know the answer, but shouldn't any answer account for the bandwidth on the channel?

[slyjoe]

Yalius: Your equation assumes that the collision occurs if the messages EXACTLY overlap. If sender 1 sends a single message at t0, with L=1 sec, and we assume T=10 sec, a collision can occur if sender 2 tries to send at t=0, 0.1,....1.0.

The problem is not that simple. When is a sender allowed to send? i.e., what length slots are available?

ETA: Is this a synchronous or asynchronous channel?

[Yalius]

Yalius: Your equation assumes that the collision occurs if the messages EXACTLY overlap. If sender 1 sends a single message at t0, with L=1 sec, and we assume T=10 sec, a collision can occur if sender 2 tries to send at t=0, 0.1,....1.0.

Would it work properly to use ((T-2L)/T)^(M-1)? That would account for any overlap in the messages, but would only hold true assuming L<<T and M large.

[slyjoe]

Would it work properly to use ((T-2L)/T)^(M-1)? That would account for any overlap in the messages, but would only hold true assuming L<<T and M large.

No - the probability should be calculable no matter the size. It should, in the limit, approach 0 as M (number users) decreases. There is no closed expression based on the original post. The slotted ALOHA example is a pretty good starting point. The original post did not say if a message occupies an entire slot or what the slot spacing is. Also, what is meant by "random" in this case? ALOHA isn't really random - there are fixed retry offsets, as with Ethernet as the channel.

The other problem, for the real world anyways, is that attempts to access a channel are generally not a gaussian distribution. Poisson or binomial works pretty well. Again, based on the OP, I don't know what Stir is really after.

Hope this helps :)

[slyjoe]

Sorry, I don't know the answer, but shouldn't any answer account for the bandwidth on the channel?

Sorry I missed this post, DickK.

With the parameters in the OP, the bandwidth is not defined, and doesn't need to be. The message length is given in seconds. One doesn't care how many bits are in that period of time. You can think of this as "percent occupancy", since the period of integration is given as T.

[DickK]

Sorry I missed this post, DickK.

With the parameters in the OP, the bandwidth is not defined, and doesn't need to be. The message length is given in seconds. One doesn't care how many bits are in that period of time. You can think of this as "percent occupancy", since the period of integration is given as T. Thanks slyjoe.

[Stir]

slyjoe, Yalius:

Thanks much. It seems difficult to state this type of question precisely enough, so let me clarify : there are no slots, and there are no retries (so the ALOHA analyses I have found aren't directly applicable); the probability distribution is uniform (that is an attempted transmission from any sender is equally possible at any time over T), each sender sends exactly one message; and in fact T>>m, and m is large.

Since I posted this originally I built a simple simulation program to test my proposed model ... which is the same as Yalius' last attempt, and the simulation results match modeled expectations very closely.

Thanks again! Stir