Please answer the attached Queuing Model problem solving. Thanks 3. Five administrative assistants use an office copier. The average time between arrivals foreach assistant is 40 minutes, which is equivalent to an arrival rate of 1/40 = 0.025 arrivalper minute. The mean time each assistant spends at the copier is 5 minutes, which is equivalent to aservice rate of ½ = 0.20 per minute. Use the M/M/1 model with a finite callingpopulation to determine the following:a. The probability that the copier is idleb. The average number of administrative assistants in the waiting linec. The average number of administrative assistants at the copierd. The average time an assistant spends waiting for the copiere. The average time an assistant spends at the copier

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Answered: 3. Five administrative assistants use…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter12: Queueing Models

Section12.5: Analytic Steady-state Queueing Models

Problem 18P

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Chapter 5. (M/M/s/K Model). Leia is responsible for 12 computers in the IT offices in a small business called DataWish. Each computer runs on an average for 4 weeks and requires an average of ½ or 0.5 weeks to get reset by Leia before it can start up again. If we assign the Poisson distribution to the “arrival” of each computer for service and an exponential distribution to the actual service provided, answer the following questions: a. The arrival rate would be units per week and the service rate would be units per week. b. Calculate the following operating characteristics for the M/M/s/K system: Utilization Factor (ρ) % Performance Measures P0 (probability that sytem is empty) % L (average units in system) units Lq (average units waiting in queue) units W (average time in the system) weeks Wq (average time in the queue) weeks c. If we assign a cost of waiting as $750 per week per computer and $800 per week of…
An average of 12 jobs per hour arrive at our departmentalprinter.a Use two different computations (one involving thePoisson and another the exponential random variable) todetermine the probability that no job will arrive duringthe next 15 minutes. b What is the probability that 5 or fewer jobs will ar-rive during the next 30 minutes?
Consider the Dynamic Program formulation of the n-class optimal booking problem with sequential arrivals. In thisproblem, show some monotonicities of the value function.• Jt(x) − Jt(x − 1) is non-decreasing in t, ∀x.• Jt(x) − Jt(x − 1) is non-increasing in x, ∀t.Consider that the second property implies the first property. To prove the second property, given Dt, consider the discrete concavity of Ht(Dt, x) = max0≤u≤min{Dt,x} (pt)u + Jt−1(x − u) in x, i.e., Ht(Dt, x + 1) − Ht(Dt, x) ≤ Ht(Dt, x) − Ht(Dt, x − 1) for any x ∈ Z+ for any t. Image attatched for clarity.
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A petrol station in the capital Kingstown has a single pump manned by one attendant.Vehicles arrive at the rate of 20 customers per hour and petrol filling takes 2 minutes on anaverage. Assume the arrival rate is Poisson probability distribution and service rate isexponentially distributed. Arrivals tend to follow a Poisson distribution, and service timestend to be exponential. The attendant is paid $10 per hour, but because of lost goodwill andsales, station loses about $15 per hour of customer time spent waiting for the attendant toservice and order. Part BThe Petrol station is considering adding a second pump with an attendant to servicecustomers. The station would pay that person the same $10 per hour.Using appropriate formula for the multiple channel model, answer the following questions:a. What is the probability that no customers are in the system (Po)?b. What is the average number of customers waiting for service (Lq)? c. What is the average number of customers in the system…
A petrol station in the capital Kingstown has a single pump manned by one attendant.Vehicles arrive at the rate of 20 customers per hour and petrol filling takes 2 minutes on anaverage. Assume the arrival rate is Poisson probability distribution and service rate isexponentially distributed. Arrivals tend to follow a Poisson distribution, and service timestend to be exponential. The attendant is paid $10 per hour, but because of lost goodwill andsales, station loses about $15 per hour of customer time spent waiting for the attendant toservice and order. a. What is the probability that no customers are in the system (Po)?b. What is the average number of customers waiting for service ( Lq)? c. What is the average number of customers in the system (L)?d. What is the average time a customer waits for service(Wq)? e. What is the average time in the system (W)?f. What is the probability that a customer will have to wait for service (Pw)?g What is the probability that there is exactly 2…
A shipping company operates 10 container ships that can each carry 5000 containers oneach journey between ports. They want to be able to load and unload 20,000 containerseach week. Assume their ships always travel fully loaded. What is the longest averagetravel time between ports that allows them to meet their goal of 20,000 containers perweek?
The VBO grocery store has a checkout counter with a full-time cashier. Customers arrive randomly with an average rate of 20 per hour. The distribution of service time is exponential with an average of 2 minutes. Occasionally, this situation causes a long queue and customer complaints. Since there is no room for another cashier, the manager plans to hire another person to help pack the groceries and reduce the expected service time to 1.5 minutes, still with exponential distribution. The manager wants the percentage of time when there are more than 2 customers at the cash register to be less than 25%. He also wants no more than 5% of customers to have to wait 5 minutes or more before starting service or 7 minutes or more before finishing service. a) Use the formulas from the M/M/1 model to calculate L, Lq, W and Wq, π0, π1 and π2 of the current transaction. What is the probability that there are more than 2 customers at the cash register? (b) Repeat (a) for the helper alternative.
On average, 100 customers arrive per hour at GothamCity Bank. It takes a teller an average of two minutesto serve a customer. Interarrival and service times areexponentially distributed. The bank currently has fourtellers working. The bank manager wants to compare thefollowing two systems with regard to the average numberof customers present in the bank and the probability that acustomer will spend more than eight minutes in line.■ System 1: Each teller has his or her own line (andno moving between lines is permitted). Arrivingcustomers are equally likely to choose any teller.■ System 2: All customers wait in a single line forthe first available teller.If you were the bank manager, which system wouldyou prefer?
The Seaboard Shipping Company has a warehouse terminal in Spartanburg, South Carolina. The capacity of each termi-nal dock is three trucks. As trucks enter the terminal, the dri-vers receive numbers, and when one of the three dock spaces becomes available, the truck with the lowest number entersthe vacant dock. Truck arrivals are Poisson distributed, and the unloading and loading times (service times) are expo-nentially distributed. The average arrival rate at the terminal is five trucks per hour, and the average service rate per dockis two trucks per hour (30 minutes per truck).a. Compute L, Lq, W, and Wq. b. The management of the shipping company is consider-ing adding extra employees and equipment to improve the average service time per terminal dock to 25 min-utes per truck. It would cost the company $18,000 per year to achieve this improved service. Managementestimates that it will increase its profit by $750 per yearfor each minute it is able to reduce a truck’s waitingtime.…

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