Reference no: EM13576875

Q1. The McDougal Sandwich Shop has two windows available for serving customers, who arrive at a Poisson rate of 40/hr. Service time is exponentially distributed with a mean of 2 min. Only one window is open as long as there are three or less customers in the shop. An identical server opens a second window when there are four or more customers in the shop. The manager of the shop helps the two attendants when there are six or more customers in the shop. When the manager is helping, the mean service time for each of the two servers is reduced to 1.5 min. The shop cannot legally hold more than 20 customers. Use the general birth-death process to determine:

A) the expected number of customers in the shop

B) the probability that the manager will be helping serve customers and,

C) the probability that the store is full.

Q2. The manager of the Hippon Drug Store is remodeling his facility and is considering three ways of organizing his branded-items business. The first way is to employ a fast clerk to wait on all customers. Assume such a clerk serves customers exponentially at an average rate of two minutes per customer. The second way is to employ two moderately fast clerks to wait on all customers. Assume each serves customers exponentially at an average rate of four minutes per customer. The third way is to use a self-service system in which each customer waits on himself. Self-service, at an average rate of six minutes, is slower than clerk service. The manager wants to calculate the average number of customers in the store, the average time each spends in the store, and the average time each spends waiting for service, under each way of organizing. Assume that customers arrive completely randomly, one at a time, at the rate of 15 per hour. Calculate the operating characteristics of interest to the manager.

Q3. An urn contains nine white balls and 11 black balls. A ball is drawn and replaced. If the ball is white your opponent pays you 25 cents. If it is black you pay him 25 cents. You have one dollar and your opponent has 50 cents. Play continues until one of you is broke. What is the probability that you lose all your money? What is the expected number of times that you play prior to the game ending? What is the answer to these two questions if both you and your opponent started with 75 cents?

Q4. Messages are transmitted from low speed terminals and arrive at a message concentrator at a Poisson rate of 600/hr. They are held in a buffer until a hi-speed trunk line is free to transmit them. The trunk line transmission time is exponential with a mean of 30 secs. Determine the smallest integer number of trunk lines needed so that tq the waiting time in the queue satisfies the relation t P[tq < 60 sec.]> 0.95, i.e., the probability exceeds 95% that the time the message spends in the buffer is less than 60 sec. Compute L and Lq for the number of trunk lines you determined.

Q5. Rivets used to secure the stainless steel sheet metal of aircraft wings are designed to withstand certain shearing forces. It is known that because of random variations in location the load on a rivet is Weibull distributed with a scale parameter of 750 pounds and a shape parameter of 3.0. The strength of the rivets in pounds is also random because of variations in the material characteristics and the dimensional tolerances. It has been found that the strength is also Weibull distributed with the same shape parameter but with a scale parameter of 1000 pounds.

Let x denote the load on a given rivet and let y denote the strength of that rivet. The rivet will not fail if x is less than y. The Reliability may therefore be expressed as P[X<Y]. Use @Risk to estimate the reliability. One approach is to set up a cell to yield random values of X and another cell to yield random values of Y. Then compute X-Y in a third cell and make it an output cell. From the simulated distribution of X-Y determine the probability that X-Y is less than 0.

A second approach is to compute a binary variable having the value 1 if X<Y and 0 otherwise. The average value of the binary variable will be an estimate of the reliability. Whichever method you choose use 1000 iterations. This problem may be solved analytically but for other distributions this may not be possible. The simulation will be valid for any choices of the distributions of load and strength. The theoretical value for the present case is 70.3%. If you do it correctly your answer should be close to this.