- 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 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 st...
- Consider 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 Consider 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 exponentia...
- Consider 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 Consider 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...
- A 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 job A 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 ...
- Consider 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 be Consider 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,...
- Southeast 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 Southeast 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. Th...
- Reconsider 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 Reconsider 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 b...
- The 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 The 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 po...
- Antonio 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 Antonio 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 tim...
- Marsha 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 Marsha 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 exp...
- Consider 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 Consider 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— f...
- Consider 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 Consider 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 amoun...
- The 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 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 operator...
- At 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 At the Forrester Manufacturing Company, one repair technician has been assigned the responsibility of maintaining three machines. For each machine, the probability distri...
- Janet 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 Janet 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...
- You 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 You 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 p...
- In 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 In the Blue Chip Life Insurance Company, the deposit and withdrawal functions associated with a certain investment product are separated between two clerks, Clara and Cla...
- Section 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 Section 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
- A 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 A 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...
- Airplanes 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 Airplanes 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 a...
- Consider 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 Consider 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 ...
- The 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 The 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 servic...
- Customers 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 Customers 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 exponenti...
- Consider the following statements about an M M 1 queueing system and its utilization factor Label each of the statements as true or false and then justify your answer a The probability that Consider 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...
- It 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 It 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 Po...
- Show that the second strong Wolfe condition 3.7b implies the curvature condition 6.7 Show that the second strong Wolfe condition (3.7b) implies the curvature condition (6.7)
- Show that 5.24d is equivalent to 5.14d Show that (5.24d) is equivalent to (5.14d).
- Show that if f x is a strictly convex quadratic, then the function f x0 σ0 p0 σk1 pk1 also is a strictly convex quadratic in the variable Show 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
- Implement 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 =T and the initial point Implement 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,....
- Show that if B is any symmetric matrix then there exists λ 0 such that B λI is positive definite Show that if B is any symmetric matrix, then there exists λ ≥ 0 such that B λI is positive definite.
- Derive the solution of the two-dimensional subspace minimization problem in the case where B is positive definite Derive the solution of the two-dimensional subspace minimization problem in the case where B is positive definite.
- Show that 4 43 and 4 44 are equivalent Hints Note that from 4 39 and Show that (4.43) and (4.44) are equivalent. Hints: Note that(from (4.39)), and
- Theorem 4.5 shows that the sequence 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 Theorem 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 seq...
- Write 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 Write 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 ...
- Let Q be a positive definite symmetric matrix Prove that for any vector x, we have where λn and λ1 are respectively the largest and smallest eigenvalues of Q This relation, which is know Let 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 rel...
- Show that the one-dimensional minimizer of a strongly convex quadratic function always satisfies the Goldstein conditions Show that the one-dimensional minimizer of a strongly convex quadratic function always satisfies the Goldstein conditions (3.11).
- Show that the one-dimensional minimizer of a strongly convex quadratic function is given by Show that the one-dimensional minimizer of a strongly convex quadratic function is given by (3.55).
- Program 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 Program 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...
- Show 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 Show 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, ...
- Show 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 Show 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, ...
- Suppose 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 Suppose 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...
- Prove that all isolated local minimizers are strict Hint Take an isolated local minimizer x and a neighborhood N. Show that for any Prove 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∗).)
- Consider the function f R2 R defined by f x 2 Show that the sequence of iterates xk defined by satisfies f xkb1 k for k 0 1 2 Show that every point on the unit circle Consider 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...
- Show that the function has only one stationary point and that it is neither a maximum or minimum but a saddle point Sketch the contour lines of Show 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 .
- Compute the gradient f x and Hessian ∇2 f x of the Rosenbrock function Show that x∗ (1, 1)T is the only local minimizer of this function and that the Hessian matrix at that point is positive Compute 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...
- Write the integer programming formulation of F4 prmu Cmax with the set of jobs in Exercise 6.1. Exercise Consider F4 prmu Cmax with the following 5 jobs under the given sequence Write 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...
- Consider F4 prmu Cmax with the following 5 jobs under the given sequence j1 Find the critical path and compute the makespan under the given sequence 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
- Consider Pm rj prmp Lmax Show through a counterexample that the preemptive EDD rule does not necessarily yield an optimal schedule Consider Pm | rj , prmp | Lmax. Show through a counterexample that the preemptive EDD rule does not necessarily yield an optimal schedule.
- Consider Develop 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 ConsiderDevelop 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 mac...
- Consider Q2 prmp Cmax with the jobs and 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 Consider 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,...