Packets arrive at a certain node on the university's...

Packets arrive at a certain node on the university's intranet at 10 packets per minute, on average. Assume packet arrivals meet the assumptions of a Poisson process. (a) Calculate the probability that exactly 15 packets arrive in the next 2 min. (b) Find an expression for the probability that more than 75 packets arrive in the next 5 min. (c) Calculate the probability that the next packet will arrive in less than 15 s. (d) What is the average time between successive packet arrivals? (e) Calculate the probability that the fifth packet arrives in less than 45 s.

Packets arrive at a certain node on the university's intranet at 10 packets per minute, on average.
Assume packet arrivals meet the assumptions of a Poisson process.
(a) Calculate the probability that exactly 15 packets arrive in the next 2 min.
(b) Find an expression for the probability that more than 75 packets arrive in the next 5 min.
(c) Calculate the probability that the next packet will arrive in less than 15 s.
(d) What is the average time between successive packet arrivals?
(e) Calculate the probability that the fifth packet arrives in less than 45 s.

Key Concepts

Memoryless Property

The memoryless property states that the probability of an event occurring in the future is independent of how much time has already elapsed. This property is inherent in the exponential distribution and is crucial when modeling and analyzing the time intervals between independent events in a Poisson process, simplifying the analysis of waiting times.

Gamma (Erlang) Distribution

The Gamma distribution, particularly in its Erlang form when the shape parameter is an integer, is used to model the waiting time until a specified number of events occur in a Poisson process. This distribution extends the concept of the exponential distribution to the time until the kth event, thereby allowing the calculation of probabilities related to the timing of multiple events.

Poisson Distribution

The Poisson distribution represents the probability of a given number of events occurring within a fixed interval of time or space when the events happen independently and with a constant average rate. It is derived from the Poisson process and is characterized by the parameter ? (lambda), which is the average rate of occurrence, making it essential for calculating the probability of a specific number of arrivals in a set time period.

Poisson Process

A Poisson process is a stochastic process used to model the occurrence of events that happen independently and randomly at a constant average rate. In contexts such as network traffic or customer arrivals, the process accurately describes the number of events within a given time interval, assuming each event occurs with a fixed mean rate and independently of the time since the last event.

Exponential Distribution

The exponential distribution is the probability distribution of the time between successive events in a Poisson process. It is memoryless, meaning that the probability of an event occurring in the future is independent of the past history. This property simplifies calculations for waiting times, such as the interval until the next event, using a parameter that is the inverse of the average event rate.

Transcript

00:01 In this question we are given two pieces of information.

00:05 The first is we are told that packets arrive at an average rate of 10 per minute, and the second piece of information is that the packet arrivals meet the assumptions of a posseum process.

00:19 So if this is the case, then we can also say that the number of arrivals over a given period of time is distributed according to a posseum distribution.

00:29 So if we define x as the number of arrivals in a given amount of time, we can say the probability of x equaling a certain number is given by e to the negative lambda t times lambda t to the exponent x divided by x factorial.

01:01 So just to clarify, a poisson distribution is a discrete probability distribution for the number of events over again.

01:09 Given range and usually this range is time, but it can be any other parameter.

01:16 So here we've said x is the number of arrivals.

01:22 More generally, you can say that's the number of events.

01:28 Lambda is the average rate of the arrivals.

01:33 So in this case, it is equal to 10 packets per minute.

01:39 And also we have t, which is the time over which we are going to count the events that occur or the arrival of the packets.

01:46 So we have x is distributed according to a poisson distribution.

01:55 And the poisson distribution has only a single parameter, and that is the average number of events in a given time, which you can also say is the average rate of the events.

02:11 Now for part a, we are asked to calculate the probability that exactly 15 packets arrive in the next two minutes.

02:20 So we want the probability that the number of packets that arrive is equal to exactly 15.

02:32 So that is equal to the negative pen.

02:37 So that rate is 10 per minute.

02:40 And now we're talking about a time interval of two minutes.

02:43 So we multiply by 2.

02:48 We have lambda times time in brackets to the exponent x.

02:54 In this case, x is 15.

02:55 And we divide by x factorial, which is 15 factorial.

03:07 And if you calculate this, you should get 0 .056.

03:19 And moving on to part b, we're asked to find an expression for the probability that more than 75 packets arrive in the next five minutes.

03:33 So here we have t is equal to 5, and we are looking for the probability that more than 75 packets arrive in that period of time.

03:52 This is also equal to 1 minus the probability that the number that arrives is less than or equal to 74.

04:11 In fact, that should be less than or equal to 75...

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