Conference rusure::math

2004.0. "complex probability question" by SNOFS1::JONESCHRIS (Chris Jones) Mon Oct 16 1995 22:05

I am looking for a probability model for the following system.  I have tried
simple probability theory and come unstuck, as when the system was trialled
the results didn't match the generated theory, by a significant margin.

The system has two basic elements.  The first is a transmitting device, that
transmits a uniquely identifable message, and has a choice of 'n' channels on
which it can transmit.  The channel selection is controlled by a Pusedo Random
Binary Seqeunce generator and can be considered effectively a random channel
selection source.  The second element of the system is the reader which can
read 'c' simultaneous channels.

The conditions of the system are as follows:
1.      All messages from the transmitting devices are of a fixed length.
2.      In the first case, it can be assumed all messages are synchronous,
        including when the transmitting device changes channels.  Later on the
        case becomes more complex (real system) where the tags are all
        effectively asynchronous (both cases can exist).
3.      If any two transmitters occupy the same channel, the message is
        garbled and can not be received and decoded.
4.      The number of transmitters can exceed the number of available channels.
5.      The channel selection is based on PRBS generation and are considered
        to be unique.
6.      The maximum number of transmitters is presently unbounded (<1000).
7.      The number of available channels is between 1 and 32.

A typical system, can have between 6 and 300 transmitters and the receiver has
12 receiving channels.

The questions become:
1.      What is the probability of a discrete transmitter being received?
2.      For a given population of transmitters and a given number of available
        channels, how many iterations (transmissions) are used before the
        probability of a percentage of the transmitters (99% for instance) are
        seen?
3.      If some of the channels became unavailable, how does that change the
        result (ie 10 transmission channels enabled but only 7 are working).

If there is any comment or questions on this, I would be pleased to answer if
I can.  Many thanks.

Chris,

	Here are some answers to your questions. I am not sure that I understand
the problem correctly, because the set-up seems so wierd. Let's handle the
synchronous case to keep things simple.

	You can plug in the numbers for yourself.


>1.      What is the probability of a discrete transmitter being received?

	The probability that a discrete transmitter is received is the
probability that no other transmitter picks the same channel, multiplied by
the probability that the receiver happens to be watching that channel.

	So if we have T transmitters (including the one we are watching), and
C channels, of which c are being monitored by the receiver, then the above
probability is

	              (T-1)
	P1 = ((C-1)/C)      * (c/C)

>2.      For a given population of transmitters and a given number of available
>        channels, how many iterations (transmissions) are used before the
>        probability of a percentage of the transmitters (99% for instance) are
>        seen?

	I cannot parse the above sentence! The expected number of iterations
before a given transmitter is detected is 1/P1. The expected number of
transmitters detected per iteration is T*P1. 

>3.      If some of the channels became unavailable, how does that change the
>        result (ie 10 transmission channels enabled but only 7 are working).

	I presume that transmitters may still send messages to the duff
channels, and the receiver may still attempt to read from them. In that case
the probability for receiving a certain signal is:

	P1' = P1 * (C'/C)

where C' is the number of working channels.

Comment: One of the things which seems odd in the model that you propose is
that you can increase the probability of messages getting through by adding
extra channels which do not work. These will cause the transmitted messages
to be spread more thinly, and the chance of any of them being alone is
therefore increased. Is this really the case?

	If I can help you further, please get in touch with me. I am located
in South Africa (6 hours behind Sydney?) and my telephone number is:
[27]-11-320-4370.

Cheers,
Andy.