Please only answer the question 5. 1 Problem Consider a memory system which only
ID: 3313575 • Letter: P
Question
Please only answer the question 5.
1 Problem Consider a memory system which only allows you to do sequential search. For example a read/write tape drive. If you want to look for a file you have to search sequentially looking at the first file, then the second file and so on until you find the file. A reasonable strategy would be place the most recently retrieved file at the front (imagine that the tape system can magically do this). This way the files that are accessed more often will be "at the front" and require less searching time in the long run. Consider the case with only 3 files A, B, and C. ·Let Xn denote the sequence of the memory system after the nth search. For example, if the files were ordered A and then B followed by C, then ABC. Enumerate the state space. 2. If Xo -ABC, list all possible states of Xi 3. If PA,PB, and PC = 1-PA-PA are the probabilities with which files A, B, and C are accessed, respectively, determine the one-step state transition matrix. 4. If PA 0.6,pB 0.10, pc 0.3, determine the steady state probability for the file order ABC 5. In general show that the steady state probability of the state ABC is given byExplanation / Answer
As the files tha are kept in a system is squential the possible number of states at any point in time are
{ABC,ACB,BAC,BCA,CAB,CBA}
the future states for these states after any of the three searches A,B or C are:-
ABC A-[ABC],B-[BAC],C-[CAB]
ACB,A-[ACB],B-[BAC],C-[CAB]
BAC A-[ABC] B-[BAC] C-[CBA]
BCA A-[ABC] B-[BCA] C-[CBA]
CAB A-[ACB] B-[BCA] C-[CAB]
CBA A-[ACB] B-[BCA] C-[CBA]
Thus the final probabilities of the states {ABC,ACB,BAC,BCA,CAB,CBA} is equal at 3/18
we can see that ABC comes from either ABC or BAC or BCA thus
Pabc=Pa*(Pabc+Pbac+Pbca) or Pabc=Pa*(Pbac+Pbca)/(1-Pa) or Pa*(Pbac+Pbca)/(Pb+Pc)
Pbac+Pbca=Pb in the steady state as in the inifinite case Pbac+Pbca=Pb*(Pac+Pca)=Pb.
Thus Pabc=Pa*Pb/(Pb+Pc)