Reference no: EM13563640

Question 1. 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

1) the expected number of customers in the shop

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

3) the probability that the store is full.

Question 2. 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.

Question 3. 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?

Question  4. 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.