YOUTOOLCOMPUTE has 10 portable personal computers available...

YOUTOOLCOMPUTE has 10 portable personal computers available for rent. The average rental time is 2.5 days and is exponentially distributed. Customers arrive randomly at an average rate of five per day. If a computer is not available, a customer will go to HELL (Hewlett, Ernest, Leland, and Lial) for a computer. (a) What fraction of arriving customers will be lost? (b) What is the average number of computers on rent? (c) What is the probability a personal computer on rent will be on rent for more than five days? (d) Management estimates that the profit on a computer being rented is $$\$ 20$$ per day (they rent by the day, only). One of the computers is destroyed in an accident and cannot be replaced for 30 days. What is the maximum YOUTOOLCOMPUTE can afford to pay per day for a 30 day replacement?

YOUTOOLCOMPUTE has 10 portable personal computers available for rent. The average rental time is 2.5 days and is exponentially distributed. Customers arrive randomly at an average rate of five per day. If a computer is not available, a customer will go to HELL (Hewlett, Ernest, Leland, and Lial) for a computer.
(a) What fraction of arriving customers will be lost?
(b) What is the average number of computers on rent?
(c) What is the probability a personal computer on rent will be on rent for more than five days?
(d) Management estimates that the profit on a computer being rented is $$\$ 20$$ per day (they rent by the day, only). One of the computers is destroyed in an accident and cannot be replaced for 30 days. What is the maximum YOUTOOLCOMPUTE can afford to pay per day for a 30 day replacement?

Key Concepts

Steady-State Probabilities

Steady-state probabilities refer to the long-term probabilities of the various states in a stochastic system, such as the number of busy servers in a queueing model. In an M/M/c/c system, these probabilities allow for the computation of key performance measures like the loss probability, average number of customers served, and system utilization. These probabilities are derived from balance equations that equate the flow in and out of each state, ensuring the system's stability over time.

Erlang B Formula

The Erlang B formula calculates the probability that all servers in an M/M/c/c queue are busy, leading to an arriving customer being lost. This formula is crucial for determining loss systems' performance, such as the fraction of customers who are denied service. It incorporates the arrival rate, service rate, and the number of servers to provide a single measure of system congestion.

Poisson Process

The Poisson process is a model used to describe random events occurring independently over time at a constant average rate. It is often applied in queueing theory to represent customer arrivals, where the number of arrivals in a given time period follows the Poisson distribution. This property is essential for modeling systems where events such as customer arrivals occur unpredictably yet with a known average frequency.

Exponential Distribution

The exponential distribution is widely used to model the time between events in a Poisson process, such as the service time in a queueing system. Its key property is memorylessness, which simplifies the analysis and calculations in systems involving random waiting or service times. This distribution is fundamental in many queueing models where the duration of an activity is assumed to have no memory of past events.

M/M/c/c Queueing Model

The M/M/c/c queueing model is a common framework used to analyze systems with 'c' identical servers and no waiting space. In such models, arriving customers who find all servers occupied are blocked or lost. This model is characterized by Poisson arrivals and exponential service times, and it helps measure performance metrics like the probability of losing a customer and the average number of servers in use.

Transcript

00:01 In question 78, we have information about a video rental store, dvd rental store, called video to go.

00:08 And we're given a probability model in which we have x to be 0 through 5.

00:14 They're representing the number of dvd rentals per day for a customer, and then the probability of each of those things happening.

00:21 Now, the 4 has 0 .70 beside it.

00:25 And looking at that, that exceeds 100%, and that would not be a legitimate probability model.

00:30 And then we would not be able to answer questions following this.

00:34 So i think that's supposed to be .07.

00:37 I think that might be a textbook error, so i'm going to use this as .07 for the rest of the problem.

00:43 Also notice that because a probability model should add up to one, there's a gap beside three, so we're meant to, we're supposed to find that gap, that gap is .12, and now each of those probabilities when you add them together, that adds up to one.

00:58 Let's answer part a.

01:00 A says describe the random variable x in words.

01:02 So x is going to be the number of dvd rentals from video to go.

01:06 That is per day per customer.

01:09 B, find the probability that a customer rents three dvds.

01:13 Well, that was the gap in the probability model.

01:15 We've already figured that out.

01:17 That is 0 .12.

01:21 C, find the probability that a customer rents at least four dvds, which that means four or five.

01:29 So or means add.

01:30 So 4 is 0 .07.

01:32 5 is 0 .04.

01:33 Add those together, and we get 0 .11.

01:37 D, we want to find their probability that customer rents at most two dvds.

01:42 So at most 2 means 0 or 1 or 2.

01:47 We can find each of those probabilities in the distribution above.

01:50 0 .03 plus 0 .5 plus 0 .24.

01:53 That gives us 0 .77 as our total.

01:59 And then we're given information about a new store, entertainment headquarters.

02:03 We're given their probability distribution, 0 through 5, for 3 .5, for the number of, dvd rentals and all of their probabilities.

02:08 The probabilities do add up to one or 100%.

02:12 Now, to make things simpler, i'm going to change the x and the p of x and this one to y and p of y.

02:18 Just when i calculate them, there's a difference between video to go and entertainment headquarters.

02:24 In e, it says at which store is the expected number of dvd rentals higher? well, we're going to find to the expected value of x, which is videos to go from the top, and then the video of y, which is our new distribution...

Please give Ace some feedback