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Is there an error in the following analysis:

Objective: find lower and upper bounds on the probability that two or more events are initiated at the same time in a 30 days period. (edited)

System:
Events of type A, B, and C are initiated randomly with the following frequency:

  • Event A is initiated 48 times every 24 hours
  • Event B is initiated 128 times every 24 hours
  • Event C is initiated 84 times every 24 hours

When any event is initiated it precludes other events from occurring for 5 minutes. For example if an Event A is initiated, then another Event A or an Event B or an Event C cannot be initiated for 5 minutes. This is the time it takes an event to complete.

For our purposes time resolution is 1 second, so that two events are considered simultaneous if they are initiated at the same second.

Additional information about the Events A, B, and C. For event A, the A system controls two sets of 3 devices. Every 3 hours the system will initiate the devices in a sequential manner, so that every 24 hours each device is initiated 8 times. Similar for event B: 16 devices every 3 hours; and event C: 28 devices every 8 hours. The exact timing of the initiating is also effected by external events, so the events are not initiated at precisely regular intervals.

Analysis: a 24 hour time interval has 86400 seconds = 24 hrs x 60 min/hr x 60 sec/min

Events cannot be initiated for 78000 seconds = total number of events occurring in a 24 hour period multiplied by the time an even takes to complete = (48 + 128 + 84 events) * 5 min/event = 60 sec/min

events may be initiated in a time window of 8400 seconds = total seconds in a 24 hour period - time that events are running = (86400 - 78000 seconds)

For a lower bound, consider the probability of each type of event being initiated at the available times for a 24 hour period are:

  • P(A) = probability of event A = 48 / 8400 = 1/175
  • P(B) = probability of event B = 128 / 8400 = 8/525
  • P(C) = probability of event C = 84 / 8400 = 1/100

The probability of two events being initiated at the same time over a 24 hour period is the product of their probabilities. And the probability of all three event being initiated at the same time is the product of all three probabilities.

For the probability over a 30 day period, multiply by 30.

Result

Lower bound on the probability of simultaneous initiation of events over a 30 day period

Psimultanous30 = 30 ( P(A) P(B) + P(A) P(C) + P(B) P(C) + P(A) P(B) P(C) ) = 0.48

Notes; Similar to this question: Probability of two events occurring in the same time period

user6546
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  • Are the numbers $48,128,84$ expected or exact values? – user Jun 10 '24 at 08:13
  • the numbers 48, 128, and 84 are exact values. – user6546 Jun 10 '24 at 12:42
  • Then you cannot use constant propabilities because they now depend on the previous history and can take on any value from $0$ (all allowed events already happened) to $1$ (no event has happened and there remain too few time slots). – user Jun 10 '24 at 12:47
  • some more information about the Events A, B, and C. For event A, the A system controls two sets of 3 devices. every 3 hours the system will initiate the devices in a sequential manner, so that every 24 hours each device is initiated 8 times. Similar for event B: 16 devices every 3 hours; and event C: 28 devices every 8 hours. – user6546 Jun 10 '24 at 13:04
  • Rather than an exact value, am really interested in a lower and upper bound. Could we say that the result provided is a lower bound, The probability of simultaneous events is greater than the provided result? (will change the question) – user6546 Jun 10 '24 at 13:06
  • Every time you have simultaneous events, it reduces the number of seconds during which events cannot be initiated, so your calculation of the number 78000 represents an extreme case (no simultaneous events). – David K Jun 10 '24 at 13:19
  • @David K -- I agree with that. This will add some inaccuracy to the result, which is acceptable. Would be good if we can estimate the amount of the inaccuracy, if it is signficant – user6546 Jun 10 '24 at 13:31

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