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CS Homework 241 Winter 1993 Craig A. Rich
Recall the variables used in analyzing page replacement strategies: K is the number of virtual pages referenced (or number of time steps). r is a reference string r(0)r(1)r(2) . . . r(K - 1) consisting of virtual page numbers. m(t) is the number of physical pages available in memory (RAM) at time 0 t < K; it may be called m if constant over time. S(m, t) is the set of virtual pages whose contents appear in memory (RAM) consisting of m physical pages at time 0 t < K. S(t) is the set of virtual pages whose contents appear in memory at time 0 t < K. S(t) is the number of such pages. Z(t) is the set of virtual pages whose contents are replaced between time t and time t + 1. A page replacement strategy has the stack property if S(m, t) S(m + 1, t) for all physical memory sizes m and times t when using that replacement strategy. LRU, OPT, LFU, LIFO, and MRU are page replacement strategies which the have stack property; FIFO does not have the stack property (e.g., it exhibits Belady's anomaly). If a page replacement strategy has the stack property, then S(t) can be "stacked by priority" S1 (t), S2 (t), S3 (t), . . ., defined by the following: If r(t) S(t), then let p be such that r(t) = Sp (t); otherwise let p be S(t) + 1. if i = 1; r(t), Si (t + 1) = min({S1 (t), . . . , Sp-1 (t)} - {S1 (t + 1), . . . , Si-1 (t + 1)}), if 1 < i p; Si (t), if i > p. 1 Consider the reference string r = 1 2 1 (3 1 2)5 3 4 5 4 (6 4 2 7 5)3. For each page replace...