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Assignment 4

# Assignment 4 - STATS 320/725 Applied Stochastic Modelling...

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STATS 320/725 Applied Stochastic Modelling Assignment 4 Due 4 pm, Thursday, 5 June, 2008 Note: This is the compulsory computing assignment. 1 In this question we will again be considering the single server queue with Poisson arrivals at rate 3 per unit time and service rate 4 per unit time. This time we will do our simulations using the method of batched means. There is a sample program for a simulation of an M/M/ 1 queue using the method of batched means attached to this assignment (see also Cecil). a Modify the script for batched means simulation to produce a batched- means estimate of the average number in the system, L , in steady state in an M | G | 1 queue with arrival rate 3 per unit time. The service time distribution should be changed so that with probability 2/3 a service time is equal to 1/8, and with probability 1/3 it is equal to 1/2. Set up the script so the simulation will have an initial transient period of 100 time units and then carry out 100 subruns of 200 time units each. Attach a copy of your R or Matlab script to the assignment. b Let X i denote the average queue length in the i th subrun. i Give the output that you observed from your simulation run, i.e. print X 1 , X 2 , . . . , X 100 . ii Give the point estimate, ¯ X , of L . Assuming the subruns are indepen- dent, give a 95% confidence interval for L . iii Use your estimates for L to give a point estimate and 95% confidence interval for the time that a customer spends in the queue waiting for service (not including the service time). iv Give estimates, ˆ γ ( k ) of Cov ( X i , X i + k ) for k = 1 , 2 , 3 , 4 , 5 , 6, and use acf to plot the autocorrelations. Is there any evidence that the sub- runs are not independent?

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