A certain city the Transportation Authority (CTA) has trollies leaving regularly throughout the day beginning at 5:00 a.m. Although the length of a trip and the number of stops are constant, the time taken for each trip varies due to weather, traffiffiffic, and time of the day. According to the CTA, the mean time taken for the trip is 50 minutes. Suppose the travel time is Normally distributed with standard deviation σ = 4 minutes. A random sample of 25 trips is obtained, and the time for each is recorded. Find the probability that the sample mean time will be less than 48 minutes.

WhatYouNeed is an online retailer with operations in Northern Indiana. Their network includes five big warehouses. All of the warehouses are fully automated: packaging is performed by robots on top of electric conveyors. Although the company has saved a lot in labor costs by the use of this packaging line, it has resulted in some excessive waste of material. After some analysis of this waste issue, the company has discovered that the number of product types that are assigned to the packaging line is strongly correlated with the amount of wasted material. Based on this insight, their operations manager has built a mathematical expression that estimates the expected amount of waste from the packaging line as a function of the number of product types in it. The expression is as follows: W = (n-93)2 – (n-88) + 227 In this expression, n is the number of product types you are packing on the line and W is the hourly waste of packaging material (in lbs). Based on this expression, how much will…

At a one man barber shop, customers arrive according to poison distribution with a mean arrival rate of 5 per hour and hair cutting time was exponentially distributed with an average hair cutting time was exponentially distributed with an average hair cut taking 19 minutes. It is assumed that because of excellent reputation, customers were always willing to wait. Calculate the following a. Average number of customers in the shop and average numbers waiting for a haircut b .Percentage of time arrival can walk in right without having to wait c. The percentage of customers who have to wait before getting into the barber’s chair

The Fast Shop Drive-In Market has one checkout counter where one employee operates the cash register. The combination of the cash register and the operator is the checkout counter in this waiting line system.  Customers arrive at a rate of one every 3 minutes according to a Poisson distribution, and service times are exponentially distributed, with a mean rate of 30 customers per hour. a) The market manager wants to determine the following for this waiting line system: Average number of customers in the total system Average time spent in the system per customer Probability that there are more than 2 customers waiting in line

For each of the following queuing systems, indicate whether it is a single- or multiple-servermodel, the queue discipline, and whether its calling population is infinite or finite. Bank

A small grocery store has a single cashier lane where groceries line up to pay for their purchases. Interarrival time of customers is exponentially distributed with mean of 3 mins. Due to the variety in type and number of items purchased, transaction time is approximately exponentially distributed with mean 2.5 mins. a. Compute for the average number of customers in the cashier lane. b. Find the probability that there are at most 4 customers in the lane. c. What is the proportion of time that the cashier is idle? d. Find the average time a customer spends in the cashier, including queuing time and transaction time. e. Buying a POS machine will help reduce average transaction time to only 2 mins. (20% reduction) with the aid of a barcode reader - eliminating the need to type item codes manually. If implemented, find the percent reduction in the average time a customer spends in the cashier.

A grocery store checkout counter operates with a single server to process customertransactions. Customers arrive at the counter with an average rate of 20 customers perhour, following a Poisson distribution. The average service time per customer is 2.5minutes, following an exponential distribution. Calculate the following performancemeasures for the checkout counter: a. Average time a customer spends in the system (waiting time + service time).b. Probability that the server is idle (not serving any customer).c. Determine the probability that there will be exactly two customers in the system(both in the queue and being served) at any given moment. What does thisprobability indicate about the checkout counter's efficiency in handling customerflow?d. Discuss how the probability in part f above might change if the average servicetime per customer increases or decreases.

The Peachtree Airport in Atlanta serves light aircraft. It has a single runway and one air trafficcontroller to land planes. It takes an airplane 12 minutes to land and clear the runway. Planes arriveat the airport at the rate of four per hour. The FAA has a rule that an air traffic controller can, on average, land planes a maximum of 45 minutes out of every hour. There must be 15 minutes of idle time available to relieve the tension. Will this airport have to hire an extra air traffic controller?

TRUE OR FALSE?  In the Poisson distribution we assume that inter-arrival times are random.

Juhn and Sons Wholesale Fruit Distributors employ one worker whose job is to load fruit on outgoing company trucks. Trucks arrive at the loading gate at an average of 24 per day, or 3 per hour, according to a Poisson distribution. The worker loads them at a rate of 4 per hour, following approximately the exponential distribution in service times. Determine the operating characteristics of this loading gate problem. What is the probability that there will be more than three trucks either being loaded or waiting? Discuss the results of your queuing model computation.