2. In a different warehouse, book orders are handled by two members of staff, A and B, who taketurns to pack each orders as it arrives (this is a non-random alternation). Assume orders arriveas a Poisson process with rate A per hour, that A packs the first order, and that orders arepacked instantaneously.Let NA(t) be the number of orders packed by A by time t, and let X₁, X2.... be the inter-arrivaltimes for the process (NA()).(a) What is the probability distribution for X.?Hint: relate the X, to the times between orders.(b) Is (NA(t)) a Poisson process? Explain your reasoning.

Given that, In a warehouse, book orders are handled by two members of staff, staff A and staff B.…

Answered: 2. In a different warehouse, book…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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A club recruit member either through active recruitment by its current members or through advertising in neighborhood billboards. Suppose that each member of the club recruit new members at a rate of 3 members per week and that advertising in neighborhood billboards generate new members at a rate of 2 members per week. Suppose further that each member is expected to stay a club member for 1 week, on average. Assume that the number of members of the club at any given time follows a birth and death process. Given that the club started with 5 members, what is the expected number of club members after the first two weeks? (Hint: refer to the theorem below) Theorem For a linear growth process with immigration and Xo = i, we have 0 · (e(x-μ)t - 1) + ie(x-μ)t, x‡μ E (X₂) = x-fl Ot + i, λ = μ.
Consider a Poisson process with rate X= 2 and let T be the time of the first arrival. 1. Find the conditional PDF of T given that the second arrivat came before time t = 1. Enter an expression in terms of A and t. 2. Find the conditional PDF of T given that the third arrival comes exactly at time t = 1.
A grasshopper hops between three flowers, labelled 1, 2, 3. It starts from flower 1. Once it arrives at a flower, it sits there for a period of time distributed exponentially with parameter 2. Then it makes a jump, as follows: if it is located on flower 1, it jumps to 2 or 3 with equal probability; if it is located at 2, it jumps to 3, and if it is at 3, it jumps to 2. (a) Model the grasshopper using a continuous-time Markov chain. Find the states, the initial distribution and the jump rates! (b) What is the probability that at time t the grasshopper will sit on the third flower?
3. 3 jobs, with i.i.d. runtimes each U[0, 1], are to be run in parallel on 2 processors. By how much time does the expected flowtime decrease if the schedule is changed from a best fixed schedule to a best schedule with recourse?
3. Jobs arrive in a computer queue in the manner of a Poisson process with intensity λ. The central processor handles them one by one in the order of their arrival, and each has an exponentially distributed runtime with parameter , the runtimes of different jobs being independent of each other and of the arrival process. Let X (t) be the number of jobs in the system (either running or waiting) at time t, where X (0) = 0. Explain why X is a Markov chain, and write down its generator. Show that a stationary distribution exists if and only if λ < μ, and find it in this case.
3.) Johns Hopkins reports two numbers daily for all the US states: P(t): the total number of cases confirmed on day t or before N(t): number of new cases reported on day t a.) What is P(t+1)-P(t) in terms of N? b.) Compare (P(t+1))-(P(t))/1 with (P(t-Δ))-(P(t))/Δt What is the value of Δt and explain why P ≈ N in words.
Thank you
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?
A certain market has both an express checkout line and a super-express checkout line. Let X, denote the number of customers in line at the express checkout at a particular time of day, and let X, denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X, and X, is as given in the accompanying table. X2 1 3 0.08 0.08 0.04 0.00 1 0.07 0.11 0.05 0.06 X1 0.05 0.04 0.10 0.06 3 0.00 0.03 0.04 0.07 4 0.00 0.01 0.05 0.06 The difference between the number of customers in line at the express checkout and the number in line at the superexpress checkout is X, – X,. Calculate the expected difference.
Find out whether each of the following processes {Xt} is weakly stationary or nonstationary. For the weakly stationary processes, find the mean and autocovariance functions. Consider in each question that {et} ~ WN(0, 0²). and X₁ = Xt-1 for t ≥ 2. X₁ = { 1 with probability p otherwise 0
A certain market has both an express checkout line and a superexpress checkout line. Let X, denote the number of customers in line at the express checkout at a particular time of day, and let X, denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X, and X, is as given in the accompanying table. X2 1 3 0.09 0.07 0.04 0.00 1 0.05 0.15 0.05 0.04 X1 0.05 0.04 0.10 0.06 0.01 0.02 0.04 0.07 4 0.00 0.02 0.05 0.05 (a) What is P(X, = 1, X, = 1), that is, the probability that there is exactly one customer in each line? P(X, = 1, X, = 1) = | (b) What is P(X, = x,), that is, the probability that the numbers of customers in the two lines are identical? P(X, = X,) = (c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X, and X,. O A = {X, 2 2 + X2U X, 2 2 + Xq} O A = {X, s 2 + X2 U Xq 2 2 + Xq} O A = {X, s 2 + X2 U Xq S2+ Xq} O A = {X, 2 2 + X2 U Xq S2+ Xq} Calculate…

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