STUDY SET
23.05.2013
1) Parts-finished components from a particular production process are input into one of two identical machines in order to complete their manufacture. The components arrive at random at an average rate of 100/hr and final processing time on the second machine has an exponential distribution with mean 1 minute.
a)How many components an average are waiting on the line to receive their final processing?
b)Storage problems will arise if the number of components in the system exceeds 10. What is the probability that storage problems will arise?
c)Find the expected length of time each component spends in the system.
d)Find the fraction of time that a particular machine is idle.
2) A semiconductor company uses 5 robots in the manufacture of its circuit boards. The robots breakdown periodically and the company have 2 repair people to do service when robots fail. When one is fixed, the time until the next breakdown is thought to be exponentially distributed with a mean of 30 hr. The shop always has enough of a work backlog to ensure that all robots in operating condition will be working. The repair time for each service is thought to be exponentially distributed with mean of 3 hr. The shop manager wishes to know
a)The average number of robots operational at a given time.
b)The expected downtime of a robot that requires repair.
c)The expected percentage of idle time of each repairer.
3)Customers arrive at a two-server system at a Poisson rate 3. An arrival finding the system empty is equally likely to enter service with either server. An arrival finding one customer in the system will enter service with the idle server. An arrival finding two customers in the system will wait in line for the first free server. An arrival finding three in the system will not enter. All service times are exponential with rate 5, and once a customer is served (by either server), he departs the system.
a)Define states.
b)Find the long-run probabilities.
c)Suppose a customer arrives and finds two customers in the system. What is the expected time he spends in the system?
d)What proportion of customers enters the system?
e)What is the average time an entering customer spends in the system?
4)Potential customers arrive at a single server station in accordance with Poisson process with rate λ. However if the arrival finds n customers already in the system, he or she joins the system with probability αn. Assuming an exponential service rate µ,
a)Set up this problem as a birth and death (B&D) process; determine the B&D rates.
b)Let α0 = 1 and αn = Compute the stationary distribution of the number of customers in the system.
c)Let αn = 1/2. Compute the stationary distribution of the number of customers in the system.
5) Consider a self-service model in which the customer is also the server. Note that this corresponds to having an infinite number of servers available. Customers arrive according to a Poisson process with parameter λ, and service times have an exponential distribution with parameter μ.
a)Construct the rate diagram.
b)Use the balance equations to find the expression for Pnin terms of P0.
c)P0.
d)Find L and Lq.
e)Find W and Wq.
6)Metalco is in the process of hiring a repair person for a 5-machine shop. Two candidates are under consideration. The first candidate can carry out repair at the rate of 5 machines per hour and earns $15 an hour. The second candidate, being more skillful, receives $20 an hour and can repair 8 machines per hour. Metalco estimates that each broken machine will incur $50 an hour because of lost production. Assuming that machines break down according to a Possion distribution with a mean of 3 per hour and the repair time is exponential, which repair person should be hired?
7)In a multiclerk tool crib facility, requests for tool exchange occur according to a Poisson distribution at the rate of 17.5 requests per hour. Each clerk can handle an average of 10 requests per hour. The cost of hiring a new clerk in the facility is $12 an hour. The cost of lost production per waiting machine per hour is approximately $50 an hour. Determine the optimal number of clerks for the facility.
8) A factory has N machines working independently and subject to breakdown. There is one repairman who works on one machine at a time. The repair times are exponentially distributed with rate µ. Machines that breakdown when the repairman is busy wait their turn. Breakdown time for each machine is exponentially distributed with parameter λ.
a)What would be the state of the queuing system?
b)Obtain the arrival and service rate for this queuing system.
c)Determine the steady-state distribution for the states.
d)Show that the expected number of working machines is .
e)Apply your results in parts (b)-(d) when there are 10 machines, the mean time to breakdown is 6 hours, and the mean repair time is 1.5 hours.