- Solve Question Number 3 Highlighted In Red Color 1 (143.58 KiB) Viewed 40 times
Solve question number 3 highlighted in red color.
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Solve question number 3 highlighted in red color.
Solve question number 3 highlighted in red color.
2. Suppose that the output process from a queueing simulation is L(t), 0 <t<T, the total number in queue at time t. A continuous-time output process can be converted into the sort of discrete- time process Y1, Y2, ... described in this chapter by first forming k = T/m batch means of size m time units: 1 Y; = Som = L(t) dt m (-1) for j = 1, 2, ..., k. Ensemble averages of these batch means can be plotted to check for initial- condition bias. (a) Show algebraically that the batch mean over 2m time units can be obtained by averaging two adjacent batch means over m time units. [Hint: This implies that we can start with batch means over rather small time intervals m and build up batch means over longer intervals without reanalyzing all of the data.] (b) Simulate an M/M/1 queue with a = 1 and u = 1.25 for a run length of 4000 time units, computing batch means of the total number in queue with batch size m = 4 time units. Make replications and use a mean plot to determine an appropriate number of batches to delete. Convert this number of batches to delete into a deletion time. 3. Using the results of Exercise 2 above, design and execute an experiment to estimate the steady- state expected number in this M/M/1 queue, L, to within +0.5 customers with 95% confidence. Check your estimate against the true value of L=1/(u - 2).
2. Suppose that the output process from a queueing simulation is L(t), 0 <t<T, the total number in queue at time t. A continuous-time output process can be converted into the sort of discrete- time process Y1, Y2, ... described in this chapter by first forming k = T/m batch means of size m time units: 1 Y; = Som = L(t) dt m (-1) for j = 1, 2, ..., k. Ensemble averages of these batch means can be plotted to check for initial- condition bias. (a) Show algebraically that the batch mean over 2m time units can be obtained by averaging two adjacent batch means over m time units. [Hint: This implies that we can start with batch means over rather small time intervals m and build up batch means over longer intervals without reanalyzing all of the data.] (b) Simulate an M/M/1 queue with a = 1 and u = 1.25 for a run length of 4000 time units, computing batch means of the total number in queue with batch size m = 4 time units. Make replications and use a mean plot to determine an appropriate number of batches to delete. Convert this number of batches to delete into a deletion time. 3. Using the results of Exercise 2 above, design and execute an experiment to estimate the steady- state expected number in this M/M/1 queue, L, to within +0.5 customers with 95% confidence. Check your estimate against the true value of L=1/(u - 2).
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