DSC4821_2022_Assignment_04
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School
University of South Africa *
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4821
Subject
Industrial Engineering
Date
Nov 24, 2024
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3
Uploaded by ChancellorFang8501
DSC4821: Questions for Assignment 04 Question 1 Market orders arrive at a certain small stock exchange market according to a Poisson process with a rate of 10 per hour. The stock exchange opens at 9:00. (a) What is the probability that no market order has arrived by 09:207 (b) What is the probability of the event: “Exactly three market orders arrive between 10:00 and 10:30 and exactly one market order arrives between 10:30 and 11:00”7 (¢) What is the expected time that the 15th market order arrives at the exchange? (d) Suppose that no market order has arrived by 09:30, what is the expected time until the first market order arrives? Question 2 A drive-in banking service is modeled as an M /M /1 queueing system with customer arrival rate of 5 per minute. It is desired that 95% of the time no more than 6 customers would be lining up for service (including the customer being served). How fast should the service rate be? That is, what should be the appropriate service rate? Question 3 A small supermarket has two self-service scanners where customers can pay for their pur- chases. Customers arrive at these randomly at an average rate of 20 per hour and have independent exponentially distributed service times taking, on average, 5 minutes each to complete their purchases. Currently there is a single queue for both scanners but a new manager is considering two further options: (i) relocating one of the scanners to the opposite end of the shop so that there will be separate queues for each machine with customers assumed equally likely to enter either queue; (i) selling one scanner leaving only one to serve all the customers. For the three possible options (including the existing setup) calculate, where possible, the proportion of customers who will be served immediately on arrival (that is without joining the queue) and advise the manager appropriately. Question 4 Suppose that X is a discrete random variable with the following probability mass function (where 0 < 6 < 1 is a parameter). X 0 1 2 3 P(X) | 26/5 | 36/5 | 2(1—6)/5 | 3(1 — 6)/5 The following 10 independent observations were sampled from such a distribution: (2;0;3;2;2;2;1;0; 15 1). What is the maximum likelihood estimate of 67 Question 5 A telephone switch has 10 output lines and a large number of incoming lines. Upon arrival a call on the input line is assigned an output line if such line is available — otherwise the call is blocked and lost. The output line remains assigned to the call for its entire duration which is of exponentially distributed length. Assume that 180 calls per hour arrive in Poisson fashion whereas the mean call duration is 110 seconds.
Question 6 Question 7 Question 8 Question 9 Question 10 (1) Determine the blocking probability, that is the probability that a random call will be blocked. (2) On average, how many calls are rejected per hour? Substantiate your answer. (3) What is the proportion of time that all the 10 output lines are available. A company has a single production machine as a key work centre on its factory floor. Jobs arrive at this work centre according to a Poisson process at a rate of 2 per day. The processing time to perform each job has an exponential distribution with a mean of i day. Because the jobs are bulky, those not being worked on are currently being stored in a room some distance from the machine. However, to save time in fetching the jobs, the production manager is proposing to add enough in-process storage space next to the production machine to accommodate 3 jobs in addition to the one being processed. (Excess jobs will continue to be stored temporarily in the distant room.) Under this proposal, what proportion of the time will this storage next to the turret lathe be adequate to accommodate all waiting jobs? A small bank has two tellers, one for deposits and one for withdrawals. The service time for each teller is exponentially distributed with a mean of 1 minute. Customers arriving at the bank according to a Poisson process, with mean rate 40 per hour, it is assumed that depositors and withdrawers constitute separate Poisson processes, each with mean rate 20 per hour, and that no customer is both a depositor and a withdrawer. The bank is thinking of changing the current arrangement to allow each teller to handle both deposits and withdrawals. The bank would expect that each teller’s mean service time would increase to 1,2 minutes, but it hopes that the new arrangement would prevent long lines in front of one teller while the other teller is idle, a situation that occurs from time to time under the current setup. Analyse the two arrangements with respect to the average idle time of a teller and the expected number of customers in the bank in any given time. A service station has one petrol pump. Cars arrive at the station according to a Poisson process at a mean rate of 15 per hour. However if the pump already is being used, these potential customers may balk (drive on to another service station). In particular if there are n cars already in the service station, the probability that an arriving potential customer will balk is n/3 for n = 1;2;3. The time required to service a car has an exponential distribution with a mean of 4 minutes. (a) Assume that this system is modelled by a birth-and-death process. Give the birth and death parameters. (b) Develop the steady-state equilibrium equations. (¢) Solve these equations to find the steady state probabilities po; p1; p2;- - - (d) Find the expected time spent in the system for those cars that stay. Consider a sample X1; Xo;...;X,, of independent and identically distributed random vari- ables where for each ¢, X; has a geometric distribution with probability mass function f(z, 0) = Prob{X; =2} =0(1 —0)*"!, Vee{l1;2;3;...} where the success probability 6 satisfies 0 < # < 1 and is unknown. (a) Give the likelihood function of the sample {x1;z2;...;x,}. (b) Determine the maximum likelihood estimator of the success probability 6. A bank employs 4 tellers to serve its customers. Customers arrive according to a Poisson process at a mean rate of 2 per minute. However, business is growing and management projects that the mean arrival rate will be 3 per minute a year from now. The transaction
time between a teller and a customer has an exponential distribution with a mean of 1 minute. Management has established the following guidelines for a satisfactory level of service to customers. ) Use the M/M/s model to determine how well these guidelines are currently being satisfied. (b) Evaluate how well the guidelines will be satisfied a year from now if no change is made in the number of tellers. Hint: Use the fact that in an M/M/s system, if #;, denotes the time that a customer spends waiting in the queue, then s—1 P{W, >t} = (1 = Pfl) —sn(—p)t n=0 1 The time T (in seconds) for a chemical reaction to take place at a certain pressure has the following distribution 28te=t%* ift >0 t) = fr(®) { 0 otherwise where the constant 5 > 0 is unknown. Independent observations ty, t2,..., t, of the time T are made. Find the maximum likelihood estimator of the parameter [.
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