The McDougal Sandwich Shop has two windows available

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.

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.

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?
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.

5. 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
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