Probability of transmitting during the same time slot

[kevins963]

Have a question regarding probability. We have a device that communicates wirelessly to a main router and no devices can transmit with in the same time period as any other device. Can anyone provide an equation that would solve what the probability is of any device transmitting at the same time slot when there is X number of devices trying to communicate?

Details:
Devices take 5 msec to transmit all data
Devices will transmit new data every 5000 msec
Data with be corrupt if two or more devices transmit during any part of each others active transmission.

For example first device transmits at time 0 msec and device two transmits at time 4.5msec there will be a collision of data at the last 500 msec and cause a fault.

What the probability when you have X number of devices running?

* This is a actual problem we are trying to solve at our office, not a homework question, have a few answers but each solution is different by a decent amount.

 

[Stephen Tashi]

You haven't stated a problem that involves probability, except perhaps for the time that each device makes its first transmission, which you didn't address.

 

[kevins963]

Sorry I meant What the probability of a single transmit collision when you have X number of devices running?

 

[Stephen Tashi]

Sorry I meant What the probability of a single transmit collision when you have X number of devices running?

That doesn't specify whether you mean the probability of a single collision in the first 20 seconds after they start running or the probability of a single collision in the next 10 years after they start running.

The point of my last post is that you have stated the problem in a deterministic fashion. According to you statement, if device A starts transmitting at time t = t1 and device B starts transmitting at time t = t2 then the question of whether and where the subsequent transmissions overlap doesn't involve probability because you have stated an exact interval of 5000 msecs between their subsequent transmission and an exact transmission length of 5 msecs.

 

[mXSCNT]

Suppose you have X devices starting at T1 = 0, T2, T3, ..., TX, where 0 <= Tj < 5000 ms. Assume a uniform distribution over the start times of each device. We can simplify/discretize by saying that the start times are divisible by 5 ms.

So the number of ways to have NO collisions is Choose(1000,X) * X! (choosing X distinct time slots out of 1000, then permuting it by X), And the total number of ways to choose times allowing collisions is 1000^X. So the probability of at least one collision with X devices over 5000 ms is 1 - Choose(1000,X) * X! / 1000^X.

 

[moonman239]

You didn't say how long the transmissions take. I will use the Poisson distribution (though a normal distribution could be a better fit if your data says so), assume that the probability of any of those devices starting a transmission = 1/86400000, the time each transmission takes = 500 ms, and the router and devices are on for 1 day.

The average time at which a transmission from any given device takes place, then, is 43200000.5 ms.
Multiply that number by 3 and you get 129600001.5 the mean of the Poisson distribution.

Plug that number (either as the mean, average, or average rate of success) and 500 ms (as the value of the Poisson variable) into a calculator and it should give you the cumulative Poisson probability (the probability that x <=500) (or should it be x < 500?). Unfortunately I tried multiple calculators which could not give me the answer.