$5\mathrm{.}15$ For a high$-$performance system such as a B$-$tree index for a database, the page size is determined mainly by the data size and disk performance. Assume that on average a B$-$tree index page is $70\%$ full with fix$-$sized entries. The utility of a page is its B$-$tree depth, calculated as log$2($entries$).$ The following table shows that for $16-$byte entries, and a $10-$year$-$old disk with a $10$ ms latency and $10$ MB$/$s transfer rate, the optimal page size is $16$ K$.$

Page Size $($KB$)$ Page Utility or B$-$Tree Depth $($Number of Disk Accesses Saved$)$ Index Page Access Cost $($ms$)$ Utility$/$Cost

$26\mathrm{.}49($or log$2(2048/16\backslash$times $0\mathrm{.}7))10\mathrm{.}20\mathrm{.}6447\mathrm{.}4910\mathrm{.}40\mathrm{.}7288\mathrm{.}4910\mathrm{.}80\mathrm{.}79169\mathrm{.}4911\mathrm{.}60\mathrm{.}823210\mathrm{.}4913\mathrm{.}20\mathrm{.}796411\mathrm{.}4916\mathrm{.}40\mathrm{.}7012812\mathrm{.}4922\mathrm{.}80\mathrm{.}5525613\mathrm{.}4935\mathrm{.}60\mathrm{.}38$

$5\mathrm{.}15\mathrm{.}1[10]<5\mathrm{.}7>$ What is the best page size if entries now become $128$ bytes?

$5\mathrm{.}15\mathrm{.}2[10]<5\mathrm{.}7>$ Based on $5\mathrm{.}15\mathrm{.}1,$ what is the best page size if pages are half full?

$5\mathrm{.}15\mathrm{.}3[20]<5\mathrm{.}7>$ Based on $5\mathrm{.}15\mathrm{.}2,$ what is the best page size if using a modern disk with a $3$ ms latency and $100$ MB$/$s transfer rate? Explain why future servers are likely to have larger pages. Keeping $$frequently used$($or $$hot$)$ pages in DRAM can save disk accesses, but how do we determine the exact meaning of $$frequently used$$ for a given system? Data engineers use the cost ratio between DRAM and disk access to quantify the reuse time threshold for hot pages. The cost of a disk access is $Disk$/$accesses$\_$per$\_$sec$,$ while the cost to keep a page in DRAM is $DRAM$\_$MB$/$page$\_$size. The typical DRAM and disk costs and typical database page sizes at several time points are listed below:

Year DRAM Cost $($$$/$MB$)$ Page Size $($KB$)$ Disk Cost $($$$/$disk$)$ Disk Access Rate $($access$/$sec$)$

$19875000115,00015199715820006420070\mathrm{.}05648083$

$5\mathrm{.}15\mathrm{.}4[20]<5\mathrm{.}7>$ What other factors can be changed to keep using the same page size $($thus avoiding software rewrite$)?$ Discuss their likeliness with current technology and cost trends.