1. Customers entering a shop are served in the order of their arrival by the single server. Theyarrive in the manner of a Poisson process with intensity, and their service times are independentexponentially distributed random variables with parameter u. By considering the jump chain, showthat the expected duration of a busy period B of the server is (μ-2)-1 when λ < . (The busyperiod runs from the moment a customer arrives to find the server free until the earliest subsequenttime when the server is again free.)

Answered: Image /qna-images/answer/09c6463e-6ae5-4e8e-9a66-5e8271d1c22d.jpg

Answered: 1. Customers entering a shop are served…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose that X and Y are independent random variables each with a x²(4) distribution. Find the moment generating function of U = X +5Y and evaluate the expectation value E(U). State clearly and justify all steps taken. Hint: You may use without proof the fact that the moment generating function of a x²(v) random variable W is Mur-(t) = (-)". 2t
1a) Derive a maximum-liklihood estimator for the unknown parameter Y 1b) An experienced sales person completes sales following a Poisson distribution, with a mean rate of Y = 1.8 sales/month. A junior sales person completes sales at a mean rate of Y = 0.85 sales per month Find the probability of the joint sales team (experienced and junior) completing exactly 2 sales in any given months, assuming the two sales people act independently of each other. 1c) Find the probability of the joint sales team (experienced and junior) completing more than 2 sales in any given months
9. Consider a bank office where customers arrive according to a Poisson process with an average arrival rate of customers per minute. The bank has only one teller servicing the arriving customers. The service time is exponentially distributed and the mean service rate is μ customers per minute. It turns out that the customers are impatient and are only willing to wait in line for an exponential distributed time with a mean of 1/μ minutes. Assume that there is no limitation on the number of customers that can be in the bank at the same time. a. Construct a rate diagram for the process and determine what type of queuing system this correspond to on the form A1/A2/A3. b. Determine the expected number of customers in the system when λ = 1 and μ = 2. c. Determine the average number of customers per time unit that leave the bank without being served by the teller when λ = 1 and µ = 2.
Pedestrians approach a crossing from the left and right sides following independent Poisson processes withaverage arrival rates of λL = 5 and λR = 1 arrivals per minute. Each pedestrian then waits until a lightis flashed, at which time all waiting pedestrians must cross to the opposite side (either from left to right orfrom right to left). Assume that the left and right arrival processes are independent, that the light flashesevery T = 2 minutes, and that crossing takes zero time – it is instantaneous.1. What is the probability that in a particular crossing, there are total 10 pedestrian and they are allcrossing from left to right?
EXERCISE 7. Let X be a random variable, for each real number t, define P(t) = E(e*). The function @(t) is called the moment generating function of X. Please try to give the moment generating function of the Poisson distribution with mean 2.
Let X(1), X(2), ..., X(n) be the order statistics of a set of n independent uniform (0, 1) random variables.Find the conditional distribution of X(n) given that X(1) = s(1), X(2) = s(2), ..., X(n-1)=s(n-1).
Given two independent Poisson processes of rates λ1 and λ2, what is the probability that process 1 has more arrivals than process 2 in 1 second?
Calls arrive at a switchboard according to a Poisson process with parameter λ = 5 per hour. (a) In a switchboard, what is the probability that it will be at least 15 minutes till the next call? (b) Find the probability that at least seven of the next ten waits between calls will be longer than 6 minutes. (c) Compute the probability that the first wait shorter than 6 minutes is the next wait.
b) Let X₁, X₂,..., X and Y₁, Y₂, ..., Ym be random samples from populations with moment generating 25 functions Mx₁(t) = ³t+t² and My(t) = (₁-¹)²5, respectively. i) Find the sampling distribution of the statistic W = X₁ + 2X₂ − X3 + X4 + X5. ii) What is the value of the sample size n, if P[Σ1(X¡ − X)² > 68.3392] = 0.025? iii) What is the value of the sample size m, if P(|Ỹ - µy| ≥10) < 0.04?
The lifetime of a bulb is modeled as a Poisson variable. You have two bulbs types A andB with expected lifetime 0.25 years and 0.5 years, respectively. When a bulb’s life ends, it stops working. You start with new bulb of type A at the start of the year. When it stopsworking, you replace it with a bulb of type B. When it breaks, you replace with a type Abulb, then a type B bulb, and so on.1. Find the expected total illumination time (in years), given you do exactly 3 bulb re-placements2. Your replacements are now probabilistic. If your current bulb breaks, you replaceit with a bulb of type A with probability p, and with type B with probability (1 –p). Find the expected total illumination time (in years), given you do exactly n bulbreplacements, and start with bulb of type A
A service company receives on average 4 service requests per day. The requests are received randomly according to Poisson process. The company has 2 service engineers and sends one engineer to attend each request. 1 An engineer needs an exponentially distributed service time with the mean of day(s). 2 The company's policy is to have maximum of 2 requests waiting in the queue If this number is reached, all incoming requests are rejected (sent to a competitor). Answer the following questions based on the information provide above: (a) Using the Kendall's notation, indicate what type of queueing system it is: (b) Compute the system state probabilities (provide at least 3 decimals): Po = P1= P2 = P3 = Pa = (c) Compute the expected total number of customer requests (waiting and served) in the system. ELL] = (d) Compute the expected number of accepted requests. Aaccepted = (e) Compute the expected total processing time (waiting + being served) for the accepted requests. E[Time] =
Exercise 2. A r.v. X is Poisson distributed with parameter λ = 5.5. (a) Calculate the probability that X = 3. (b) Calculate the probability that X < 2. (c) What is the most likely (i.e. highest probability) value for X ? (You can make a graph of f(x) to find the answer).

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