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Exam 17: Queueing Theory

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Question 1

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Customers arrive at a fast food restaurant with one server according to a Poisson process at a mean rate of 30 per hour.The server has just resigned,and the two candidates for the replacement are X (fast but expensive)and Y (slow but inexpensive).Both candidates would have an exponential distribution for service times with X having a mean of 1.2 minutes and Y having a mean of 1.5 minutes.Restaurant revenue per month is given by $6,000/W where W is the expected waiting time (in minutes)of a customer in the system.Determine the upper bound on the difference in their monthly compensations that would justify hiring X rather than Y.

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For both alternatives,we have an M/M/1 queueing system.Using one hour as the unit of time,we also have \(\lambda\) = 30 in both cases.With candidate X: µ = 50,so W = 11ea84c6_c8b9_9ce1_83dc_effa0b689ca6_TB2462_11 = 3 minutes,Monthly revenue = 11ea84c6_c8b9_c3f2_83dc_03afee34c4d3_TB2462_11 = $2000.With candidate Y: µ = 40,so W = 11ea84c6_c8b9_eb03_83dc_113edbab8722_TB2462_11 = 6 minutes,Monthly revenue = 11ea84c6_c8b9_eb04_83dc_7b5d5be7c078_TB2462_11 = $1000.Since the difference in monthly revenues is $1000,the upper bound on the difference in their monthly compensations that would justify hiring candidate X rather than candidate Y is $1000.

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The Copy Shop is open 5 days per week for copying materials that are brought to the shop.It has three identical copying machines that are run by employees of the shop.Only two operators are kept on duty to run the machines,so the third machine is a spare that is used only when one of the other machines breaks down.When a machine is being used,the time until it breaks down has an exponential distribution with a mean of 2 weeks.If one machine breaks down while the other two are operational,a service representative is called in to repair it,in which case the total time from the breakdown until the repair is completed has an exponential distribution with a mean of 0.2 week.However,if a second machine breaks down before the first one has been repaired,the third machine is shut off while the two operators work together to repair this second machine quickly,in which case its repair time has an exponential distribution with a mean of only 1/15 week.If the service representative finishes repairing the first machine before the two operators complete the repair of the second,the operators go back to running the two operational machines while the representative finishes the second repair,in which case the remaining repair time has an exponential distribution with a mean of 0.2 week.(a)Letting the state of the system be the number of machines not working,construct the rate diagram for this queueing system.(b)Use the balance equations to find the steady-state distribution of the number of machines not working.(c)What is the expected number of operators available for copying?

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(a)Let state n = number of machines brok...

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Consider a single-server queueing system with a Poisson input,where the server must perform two distinguishable tasks in sequence for each customer,so the total service time is the sum of the two task times (which are statistically independent),where times are being expressed in units of a minute.(a)Suppose that the first task time has an exponential distribution with a mean of 3 minutes and that the second task time has an Erlang distribution with a mean of 9 minutes and with the shape parameter k = 3.Which queueing theory model should be used to represent this system? Also identify the parameters of the model.(b)Suppose that part (a)is modified so that the first task time also has an Erlang distribution with the shape parameter k = 3 (but with the mean still equal to 3 minutes).Which queueing theory model should be used to represent this system? Also identify the parameters of the model.

Airplanes arrive for take-off at the runway of an airport according to a Poisson process at a mean rate of 20 per hour.The time required for an airplane to take off has an exponential distribution with a mean of 2 minutes,and this process must be completed before the next airplane can begin to take off.Because a brief thunderstorm has just begun,all airplanes which have not commenced take-off have just been grounded temporarily.However,airplanes continue to arrive at the runway during the thunderstorm to await its end.Assuming steady-state operation before the thunderstorm,determine the expected number of airplanes that will be waiting to take off at the end of the thunderstorm if it lasts 30 minutes.

You are given an M/M/1 queueing system in which the expected waiting time and expected number in the system are 120 minutes and 8 customers,respectively.Determine the probability that a customer's service time exceeds 20 minutes.

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 $\lambda$ ,and service times have an exponential distribution with parameter $\mu$ . (a)Find L_q and W_q. (b)Construct the rate diagram for this queueing system. (c)Use the balance equations to find the expression for P_n in terms of P₀. (d)Find P₀. (e)Find L and W.

Sally Gordon has just completed her MBA degree and is proud to have earned a promotion to Vice President for Customer Services at the First Bank of Seattle.One of her responsibilities is to manage how tellers provide services to customers,so she is taking a hard look at this area of the bank's operations.Customers needing teller service arrive randomly at a mean rate of 30 per hour.Customers wait in a single line and are served by the next available teller when they reach the front of the line.Each service takes a variable amount of time (assume an exponential distribution),but on average can be completed in 3 minutes.The tellers earn an average wage of $18 per hour.(a)If two tellers are used,what will be the average waiting time for a customer before reaching a teller? On average,how many customers will be in the bank,including those currently being served? (b)Company policy is to have no more than a 10% chance that a customer will need to wait more than 5 minutes before reaching a teller.How many tellers need to be used in order to meet this standard? (c)Sally feels that a significant cost is incurred by making a customer wait because of potential lost future business.Sally estimates the cost to be $0.50 for each minute a customer spends in the bank,counting both waiting time and service time.Given this cost,how many tellers should Sally employ? d)First Bank has two types of customers: merchant customers and regular customers.The mean arrival rate for each type of customer is 15 per hour.Both types of customers currently wait in the same line and are served by the same tellers with the same average service time.However,Sally is considering changing this.The new system she is considering would have two lines-one for merchant customers and one for regular customers.There would be a single teller serving each line.What would be the average waiting time for each type of customer before reaching a teller? On average,how many total customers would be in the bank,including those currently being served? How do these results compare to those from part a.(e)Sally feels that if the tellers are specialized into merchant tellers and regular tellers,they would be more efficient and could serve customers in an average of 2.5 minutes instead of 3 minutes.Answer the questions for part d again with this new average service time.