During the daily lunch rush, arrivals at the drive-thru at a nearby fast food restaurant follow

During the daily lunch rush, arrivals at the drive-thru at a...

Key Concepts

Poisson Process

A Poisson process is a stochastic process that models the occurrence of events randomly over a fixed period of time or space with a given average rate. It has the key properties of independent increments and stationary increments, meaning that the number of events in non-overlapping intervals are independent and that the process’s behavior is uniform over time. This concept is essential for analyzing situations where events occur randomly, such as customer arrivals at a service facility.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, grounded on the average rate of occurrence. In contexts like customer arrivals, it allows the computation of probabilities for different counts over a specified time period. The mean and variance of the Poisson distribution are equal, which simplifies calculating measures such as the standard deviation.

Exponential Distribution

The exponential distribution describes the time between successive events in a Poisson process. It is a continuous probability distribution characterized by its memoryless property, which implies that the probability of an event occurring in the future is independent of how much time has already elapsed. This concept is particularly useful for calculating waiting times between arrivals or events in various systems.

Gamma Distribution

The gamma distribution generalizes the exponential distribution and is used to model the waiting time until the occurrence of the k-th event in a Poisson process. When summing multiple exponentially distributed waiting times (for example, the time until the 100th arrival), the resulting distribution has mean and variance that can be derived from the parameters of the underlying exponential distributions. This concept is crucial when analyzing cumulative arrival times in processes.

Memoryless Property

The memoryless property is a characteristic of the exponential distribution, stating that the probability of an event occurring in the future is independent of the elapsed time. This property, whereby the process 'restarts' after each event, greatly simplifies the analysis of waiting times in stochastic processes like the Poisson process, facilitating the calculation of conditional probabilities associated with the occurrence of one event given the occurrence of another.

Transcript

00:01 In this question, the scenario is that it is lunchtime at a given restaurant, and customers are arriving at the drive -thru in a random manner, and we were told that they follow a poisson process with a rate of 0 .8 customers per minute.

00:18 So we can say the rate is 0 .8 customers per minute.

00:25 Now, when something follows a poisson process, there's two things that we can say about it.

00:31 The first is that the number of arrivals that occur, and non -overlapping time intervals are independent.

00:40 And the second is that the number of arrivals in a given time interval follows a poisson distribution with mean lambda t.

00:51 So if we're given a certain time interval, the mean of the poisson distribution is equal to lambda t.

01:06 Now, for part a, we are asked, what is the expected number of customers in one hour, and what is the corresponding standard deviation.

01:17 So if we define x as the number of customers that arrive in one hour, we know that x follows a poisson distribution with mean equal to lambda times t.

01:39 And so the mean for a poisson distribution is simply lambda times t, which is 0 .8 times 60 minutes in one hour because this rate is 0 .8 per minute and so that we expect that there will be 48 arrivals on average in one hour.

02:10 Now another property of the plus zone distribution is that the variance on the number of arrivals is equal to the expected number of arrivals, which tells us that the standard deviation on the number of arrivals is equal to the square root of 48 and this is equal to 6 .93.

02:43 Now for part b we are given information that the drive -through workers cannot handle more than 10 customers in any five -minute span and we're asked to find the probability that too many customers arrive for the workers to handle between the time of 1215 p .m.

02:59 And 1220 p .m.

03:05 So we're talking about a time of five minutes.

03:11 Because it's a poisson process, it doesn't matter if we're talking about the time interval from 1205 to 1210 or 1210 to 1215 or 1215 to 1220.

03:23 All that matters is the length of time that passes.

03:28 So if we want to find the probability that the drive -thru cannot handle the number of customers that occur in that five minutes, we're looking for the probability that the number of customers who arrive in that five minutes is greater than 10 because we know that they can handle up to 10 customers.

03:45 Five minutes.

03:48 And this is equal to one minus the probability that the number of customer arrivals is at most 10.

04:01 So this is equal to one minus the summation from x is equal to 0 to 10 of e to the minus lambda times t, which is equal to minus 4.

04:23 I got that from t times lambda, which is 0 .8 times lambda times t to the exponent x over x factorial.

04:45 Just to explain this, remember the probability of getting x arrivals is equal to e to the negative lambda t times lambda t to the exponent x over x factorial.

05:06 So here we're finding the sum of the probabilities of getting 0 through 10 arrivals in the next five minutes and then subtracting that from one.

05:23 And so if you calculate this, it comes out to about 0 .0028.

05:32 So that is the probability that too many customers come between the time of 12 .15 p .m.

05:40 And 1220 p .m.

05:41 For the drive -through staff to handle.

05:45 And moving on to part c, we are told that a customer has just arrived and we're asked to calculate the probability that another customer will arrive in the next 30 seconds.

05:58 So one way to think of this question is that for a poisson process, the inter -arrival times are distributed according to an exponential distribution with rate lambda...

Please give Ace some feedback