1. A model is classified as a (M/G/4):(FCFS/8/30) queueing system. Fill in the b

Question

1. A model is classified as a (M/G/4):(FCFS/8/30) queueing system. Fill in the blanks below

The “M” tells us __________________

The “G” tells us __________________

The “4” tells us __________________

The “FCFS” tells us ________________

Could this system have balks? Why or why not?

2. Select the MOP and/or parameter in the table below that would be most helpful in answering each question. The situation is a single-server finite-capacity Birth-Death model with reneging. Customers will only balk if the system is full. In some cases, the table item may lead to the answer, rather than provide a direct answer. In that case, write an expression that will produce the necessary result. Rates and time averages are in hours. Not all MOP/parameters will be used. Some questions require more than one MOP/parameter. Finally, some questions have more than one correct answer.

a. Average number of customers in the system.

b. Average number of customers who enter the system and have to wait for service.

c. Average time a customer who enters the system will spend waiting for service.

d. Average number of customers entering the system per hour.

e. Average number of customers served per hour.

3. Cars arrive at a toll plaza following a Poisson process at a rate of 135 per hour and trucks arrive following an independent Poisson process at a rate of 40 per hour.

a. What is the expectation of the number of vehicles (cars + trucks) to arrive between 8:00 am and noon?

b. If 60 trucks arrive between 9am and 10am how may cars are expected between 9am and 10am?

c. If car tolls are $5 and truck tolls are $12, how much revenue is expected per hour?

d. If the last five vehicles were trucks, what is the probability of the next vehicle being a truck?

e. If 10 trucks arrived in the last 20 minutes, what is the probability that at least one truck will arrive in the next two minutes?

4. Two mechanics Bill and Sally, have been working on cars. At this time, Bill has been working on his car for one hour and Sally has just started working on her car. The time Bill takes to do this sort of repair is an exponential random variable with a mean of three hours. Sally’s time is an independent random variable with a mean of two hours.

a. What is the expectation of the remaining time until the first one of the two cars will be done?

b. What is the probability that at least one of the two cars will be done one hour from now?

c. Why?

5. A small travel agency has three telephone lines and two clerks. Arrivals follow a Poisson distribution with a rate of 8 per hour and both the time to complete a call and the time until a customer on hold balks are exponentially-distributed. The mean time to complete a call is 15 minutes if nobody is on hold and 12 minutes if the clerks see the hold light blinking. The mean time a customer waits on hold is 5 minutes. Sketch a state diagram using the number of busy telephone lines as the state. Be sure to label all rates and state the units you are using. Do not attempt to solve this problem.

6. A work cell consists of six machines and one operator. At any time, however, no more than four machines are running. The fifth and sixth machines are spares. If one of the four running machines stops, a spare is immediately switched into service. If four machines are running when the operator fixes the problem, that machine becomes a spare. If three or fewer machines are running, the newly-repaired machine becomes a running machine. The time to failure for each machine is an exponential random variable with MTTF equal to 60 minutes, if the machine is running. Assume that machines that are spares do not fail. The time to repair a machine is an exponential random variable with a mean of 12 minutes. Sketch a state diagram using the number of busy telephone lines as the state. Be sure to label all rates and state the units you are using. Do not attempt to solve this problem.

7. You watched a auto parts store from 10am to 11am. During that time, you observed the following service times. (There is more than one server). A dash (-) in the Start column means the service had started prior to you observing the system and a dash in the End column means the service was still in progress when you stopped observing.

a. What is the maximum-likelihood estimate of the mean service time , assuming the service times have an exponential distribution?

b. Calculate a 90% confidence interval for using the appropriate confidence interval.

Explanation / Answer

1)

(M/G/4):(FCFS/8/30) is a type of queueing system,

In this,

M stands for Markovian i.e. Poisson arrivals or exponential service time. It is arrival process.

G stands for General.It is service Process. General probability for arrivals or service time.

4 tells us how many number of servers are in the system.

FCFS means First Come First Serve system, In this the request is exicuted in the sequence it enters in the system. Their is no priority in this.

Yes, this system have balks. It is a FCFS system in which the work content also represents the queuing time, and hence the model represents customers balking in face of long waits. Customer have to wait in the queue for their turn as the first standing customer or the input will enter in the system first and others have to wait.

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