Consider a Jackson network with three service facilities having the parameter values shown below.(a) Find the total arrival rate at each of the facilities.(b) Find the steady-state distribution of the number of customers at facility 1, facility 2, and facility 3. Then show the pr...
06 Aug 2020Consider a system of two infinite queues in series, where each of the two service facilities has a single server. All service times are independent and have an exponential distribution, with a mean of 3 minutes at facility 1 and 4 minutes at facility 2. Facility 1 has a Poisson i...
06 Aug 2020Consider a queueing system with two servers, where the customers arrive from two different sources. From source 1, the customers always arrive 2 at a time, where the time between consecutive arrivals of pairs of customers has an exponential distribution with a mean of 20 minutes....
06 Aug 2020A particular work center in a job shop can be represented as a single-server queueing system, where jobs arrive according to a Poisson process, with a mean rate of 8 per day. Although the arriving jobs are of three distinct types, the time required to perform any of these jobs ha...
06 Aug 2020Consider the model with nonpreemptive priorities presented in Sec. 17.8. Suppose there are two priority classes, with λ1 = 4 and λ2 = 4 In designing this queueing system, you are offered the choice between the following alternatives: (1) one fast server (μ = 10) and (2) two slow ...
06 Aug 2020Southeast Airlines is a small commuter airline serving primarily the state of Florida. Their ticket counter at the Orlando airport is staffed by a single ticket agent. There are two separate lines—one for first-class passengers and one for coach-class passengers. When the ticket ...
06 Aug 2020Reconsider Prob. 17.7-6. Management has adopted the proposal but now wants further analysis conducted of this new queueing system.(a) How should the state of the system be defined in order to formulate the queueing model as a continuous time Markov chain?(b) Construct the corresp...
06 Aug 2020The maintenance base for Friendly Skies Airline has facilities for overhauling only one airplane engine at a time. Therefore, to return the airplanes to use as soon as possible, the policy has been to stagger the overhauling of the four engines of each airplane. In other words, o...
06 Aug 2020Antonio runs a shoe repair store by himself. Customers arrive to bring a pair of shoes to be repaired according to a Poisson process at a mean rate of 1 per hour. The time Antonio requires to repair each individual shoe has an exponential distribution with a mean of 15 minutes.(a...
06 Aug 2020Marsha operates an expresso stand. Customers arrive according to a Poisson process at a mean rate of 30 per hour. The time needed by Marsha to serve a customer has an exponential distribution with a mean of 75 seconds(a) Use the M/G/1 model to find L, Lq, W, and Wq(b) Suppose Mar...
06 Aug 2020Consider the M/G/1 model with λ = 0.2 and μ 0.25.(a) Use the Excel template for this model (or hand calculations) to find the main measures of performance—L, Lq, W, Wq— for each of the following values of σ : 4, 3, 2, 1, 0.(b) What is the ratio of Lq with σ = 4 to Lq with σ = 0? ...
06 Aug 2020Consider the M/G/1 model.(a) Compare the expected waiting time in the queue if the servicetime distribution is (i) exponential, (ii) constant, (iii) Erlang with the amount of variation (i.e., the standard deviation) halfway between the constant and exponential cases.(b) What is t...
06 Aug 2020The Dolomite Corporation is making plans for a new factory. One department has been allocated 12 semiautomatic machines. A small number (yet to be determined) of operators will be hired to provide the machines the needed occasional servicing (loading, unloading, adjusting, setup,...
06 Aug 2020At the Forrester Manufacturing Company, one repair technician has been assigned the responsibility of maintaining three machines. For each machine, the probability distribution of the running time before a breakdown is exponential, with a mean of 9 hours. The repair time also has...
06 Aug 2020Janet is planning to open a small car-wash operation, and she must decide how much space to provide for waiting cars. Janet estimates that customers would arrive randomly (i.e., a Poisson input process) with a mean rate of 1 every 4 minutes, unless the waiting area is full, in wh...
06 Aug 2020You are given an M/M/1 queueing system in which the expected waiting time and expected number in the system are 120 minutes and 8 customers, respectively. Determine the probability that a customer’s service time exceeds 20 minutes.
06 Aug 2020In the Blue Chip Life Insurance Company, the deposit and withdrawal functions associated with a certain investment product are separated between two clerks, Clara and Clarence. Deposit slips arrive randomly (a Poisson process) at Clara’s desk at a mean rate of 16 per hour. Withdr...
06 Aug 2020Section 17.6 gives the following equations for the M/M/1 model:Show that Eq. (1) reduces algebraically to Eq. (2).(a) The M/M/1 model (b) The M/M/s model
06 Aug 2020A gas station with only one gas pump employs the following policy: If a customer has to wait, the price is $1 per gallon; if she does not have to wait, the price is $1.20 per gallon. Customers arrive according to a Poisson process with a mean rate of 15 per hour. Service times at...
06 Aug 2020Airplanes arrive for takeoff at the runway of an airport according to a Poisson process at a mean rate of 20 per hour. The time required for an airplane to take off has an exponential distribution with a mean of 2 minutes, and this process must be completed before the next airpla...
06 Aug 2020Consider the M/M/s model with a mean arrival rate of 10 customers per hour and an expected service time of 5 minutes. Use the Excel template for this model to obtain and print out the various measures of performance (with t = 10 and t = 0, respectively, for the two waiting time p...
06 Aug 2020The Friendly Neighbor Grocery Store has a single checkout stand with a full-time cashier. Customers arrive randomly at the stand at a mean rate of 30 per hour. The service-time distribution is exponential, with a mean of 1.5 minutes. This situation has resulted in occasional long...
06 Aug 2020Customers arrive at a single-server queueing system in accordance with a Poisson process with an expected interarrival time of 25 minutes. Service times have an exponential distribution with a mean of 30 minutes. Label each of the following statements about this system as true or...
06 Aug 2020Consider the following statements about an M/M/1 queueing system and its utilization factor ρ. Label each of the statements as true or false, and then justify your answer(a) The probability that a customer has to wait before service begins is proportional to ρ.(b) The expected nu...
06 Aug 2020It is necessary to determine how much in-process storage space to allocate to a particular work center in a new factory. Jobs arrive at this work center according to a Poisson process with a mean rate of 3 per hour, and the time required to perform the necessary work has an expon...
06 Aug 2020Show that the second strong Wolfe condition (3.7b) implies the curvature condition (6.7)
06 Aug 2020Show that (5.24d) is equivalent to (5.14d).
06 Aug 2020Show that if f (x) is a strictly convex quadratic, then the function f (x0 σ0 p0 ··· σk−1 pk−1) also is a strictly convex quadratic in the variable σ (σ0, σ1,...,σk−1T
06 Aug 2020Implement Algorithm 5.2 and use to it solve linear systems in which A is the Hilbert matrix, whose elements are Ai,j =1/(i j − 1). Set the right-hand-side to b = (1, 1,..., 1)T and the initial point to x0 =0. Try dimensions n =5, 8, 12, 20 and report the number of iterations req...
06 Aug 2020Show that if B is any symmetric matrix, then there exists λ ≥ 0 such that B λI is positive definite.
06 Aug 2020Derive the solution of the two-dimensional subspace minimization problem in the case where B is positive definite.
06 Aug 2020Show that (4.43) and (4.44) are equivalent. Hints: Note that(from (4.39)), and
06 Aug 2020Theorem 4.5 shows that the sequence {∥g∥} has an accumulation point at zero. Show that if the iterates x stay in a bounded set B, then there is a limit point x of the sequence {xk } such that g(x) 0.Theorem 4.5:Let η 0 in Algorithm 4.1. Suppose that ∥BK∥≤ β for some constant β, t...
06 Aug 2020Write a program that implements the dogleg method. Choose Bk to be the exact Hessian. Apply it to solve Rosenbrock’s function (2.22). Experiment with the update rule for the trust region by changing the constants in Algorithm 4.1, or by designing your own rules.
06 Aug 2020Let Q be a positive definite symmetric matrix. Prove that for any vector x, we havewhere λn and λ1 are, respectively, the largest and smallest eigenvalues of Q. (This relation, which is known as the Kantorovich inequality, can be used to deduce (3.29) from (3.28).)
06 Aug 2020Show that the one-dimensional minimizer of a strongly convex quadratic function always satisfies the Goldstein conditions (3.11).
06 Aug 2020Show that the one-dimensional minimizer of a strongly convex quadratic function is given by (3.55).
06 Aug 2020Program the steepest descent and Newton algorithms using the backtracking line search, Algorithm 3.1. Use them to minimize the Rosenbrock function (2.22). Set the initial step length α0= 1 and print the step length used by each method at each iteration. First try the initial poin...
06 Aug 2020Show that the symmetric rank-one update (2.18) and the BFGS update (2.19) are scale-invariant if the initial Hessian approximations B0 are chosen appropriately. That is, using the notation of the previous exercise, show that if these methods are applied to f (x) starting from x0 ...
06 Aug 2020Show that the symmetric rank-one update (2.18) and the BFGS update (2.19) are scale-invariant if the initial Hessian approximations B0 are chosen appropriately. That is, using the notation of the previous exercise, show that if these methods are applied to f (x) starting from x0 ...
06 Aug 2020Suppose that f (x) xT Qx, where Q is an n×n symmetric positive semidefinite matrix. Show using the definition (1.4) that f (x) is convex on the domain Rn. Hint: It may be convenient to prove the following equivalent inequality:for all α ∈ [0, 1] and all x, y ∈ Rn
06 Aug 2020Prove that all isolated local minimizers are strict. (Hint: Take an isolated local minimizer x∗ and a neighborhood N . Show that for any x ∈ N , x≠ x∗ we must have f (x) > f (x∗).)
06 Aug 2020Consider the function f : R2 → R defined by f (x)= ∥x∥2. Show that the sequence of iterates {xk } defined bysatisfies f (xk1) k ) for k 0, 1, 2,.... Show that every point on the unit circle {x | ∥x∥2 1} is a limit point for {xk }. Hint: Every value θ ∈ [0, 2π] is a limit point of...
06 Aug 2020Show that the functionhas only one stationary point, and that it is neither a maximum or minimum, but a saddle point. Sketch the contour lines of f .
06 Aug 2020Compute the gradient ∇ f (x) and Hessian ∇2 f (x) of the Rosenbrock functionShow that x∗ (1, 1)T is the only local minimizer of this function, and that the Hessian matrix at that point is positive definite.
06 Aug 2020Write the integer programming formulation of F4 | prmu | Cmax with the set of jobs in Exercise 6.1.Exercise1:Consider F4 | prmu | Cmax with the following 5 jobs under the given sequence j1,...,j.Find the critical path and compute the makespan under the given sequence
06 Aug 2020Consider F4 | prmu | Cmax with the following 5 jobs under the given sequence j1,...,j.Find the critical path and compute the makespan under the given sequence
06 Aug 2020Consider Pm | rj , prmp | Lmax. Show through a counterexample that the preemptive EDD rule does not necessarily yield an optimal schedule.
06 Aug 2020ConsiderDevelop a heuristic for minimizing the makespan subject to total completion time optimality. (Hint: Say a job is of Rank j if j − 1 jobs follow the job on its machine. With two machines in parallel there are two jobs in each rank. Consider the difference in the processing...
06 Aug 2020Consider Q2 | prmp | Cmax with the jobsand machine speeds v1 = 2 and v2 = 1.(a) Find the makespan under LRPT when preemptions can only be made at the time points 0, 4, 8, 12, and so on.(b) Find the makespan under LRPT when preemptions can only be made at the time points 0, 2, 4, ...
06 Aug 2020
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