answer theses question
Problem # 1 (10 points)
Parts arrive for packaging at a Poisson rate of 30/hour. The time required for
packaging is normally distributed with a mean of 3 minutes and a standard
deviation of 1 minute. If the system operates as an Erlang loss system, how
many stations should be provided in order to have no greater than 5%
Problem # 2 (10 points)
An accumulation conveyor is to be provided at a workstation. When the
conveyor is full, parts are diverted to another area for processing. Parts arrive at
a Poisson rate of 2.0 /minute. The time required to process a part at the
workstation is exponentially distributed with a mean of 18 seconds per part. It is
desired to provide an accumulation line sufficiently long such that less than 1% of
the arriving parts will be diverted to another area of processing. What is the
minimum number of waiting spaces that will satisfy the objective?
Consider a distribution center that uses a fleet of six battery-powered automated
guided vehicles for material delivery. Due to the wide variety of demands placed
on the AGV, the life of a battery, before recharging is required, is a random
variable. Suppose the time from recharging a battery for an AGV until it requires
recharging again is exponentially distributed with a mean value of 10 hours. Also,
suppose the AGV is idle during battery recharging. Finally, suppose the time
required to recharge a battery is also exponentially distributed with mean of 2
hours. There are two battery-recharging stations, and the distribution center
operated 24 hours/ day and seven days per week.
a) Determine the probability of at least one battery recharging station
b) Determine the average number of AGVs waiting to be recharged.
Problem # 4 (5 points}
Discuss the different assumptions of open and closed queuing network models.
Confronted with a real system, what factors would determine which modeling
approach to use?
Problem # 5 (10 points)
Three prospective designs are under consideration for a manufacturing cell. The
first uses 3 parallel machines each with service rate p, a common queue, and
total arrival rate A. The second option uses the same machines but has a
separate queue for each machine. Arriving part batches will be randomly
assigned to a machine. The third configuration consists of one fast machine,
which serves at the rate of 3 p. Ignoring the capital and operating costs, which
system has the lowest throughput time and WIP levels? You may assume
inter-arrival and service times are exponential and let p: 5 and A: 10.
Problem # 6 (10 points)
A job shop has three types of machines; two mills, one drill press, and one
surface grinder. Orders arrive to the shop at a rate of 2 per day. About 60% of
these go to milling first. The other 40% start at the drill. One half of the drilling
jobs go next to milling, whereas the other one half leave the system. Thirty
percent of jobs being milled are sent for grinding, and the others leave the shop.
Jobs always leave the system after grinding. Operation times are exponentially
distributed, averaging one day per job for milling, drilling, and grinding. Find the
average number ofjobs in the system.
Problem # 7 (10 points)
Four doctors work in a hospital emergency room that handles three types of
patients. The time a doctor spends with each type of patient is exponentially
distributed, with a mean of 15 minutes. Interarrival times for each customer type
are exponential, with the average number of arrivals per hour for each patient
type being as follows: type 1, 3 patients; type 2, 5 patients; type 3, 3 patients.
Assume that type 1 patients have the highest priority, and type 3 patients have
the lowest priority (no preemption is allowed). What is the average length of
time that each type of patient must wait before seeing a doctor?